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The CFHT open star cluster survey Kalirai, Jasonjot Singh 2001

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THE CFHT OPEN STAR CLUSTER SURVEY By Jasonjot Singh Kalirai B. Sc. University of British Columbia, 2000 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S A N D A S T R O N O M Y We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A August 2001 © Jasonjot Singh Kalirai, 2001 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 and Astronomy The University of British Columbia 6224 Agricultural Road Vancouver, B .C. , Canada V6T 1Z1 Date: ABSTRACT We have used the Canada-France-Hawaii Telescope to obtain deep and ac-curate multi-colour photometry of 19 open star clusters in our Galaxy. The quality and size of the data set are unprecedented when compared with previ-ous studies of open star clusters. The clusters in the survey differ in richness, age, metal content and location in the Galaxy and therefore provide a very diverse database for furthering the research in several areas of astrophysics. Some of the key issues which we address include the comparisons between the observational results and up-to-date theoretical stellar evolutionary models, star cluster dynamics, distance and age determinations, chemical evolution scenarios, and star formation histories. Of particular interest is the study of the properties of white dwarf stars, such as the initial-final mass relationship and the upper mass limit to production. The depth gained in the present sur-vey (limiting V magnitude ~ 25) has allowed us to establish a large catalogue of white dwarf candidates, and our current spectroscopic investigations will provide the much needed observational constraints to white dwarf theoretical models. The colour-magnitude diagrams for the rich, young clusters in the survey possess very tightly constrained, long main-sequences and show a much richer cluster population than previous studies. This thesis summarizes how this was done as well as key results for the two richest clusters in the survey, NGC 6819 and NGC 2099. For NGC 6819 we find: distance = 2500 pc, age = 2.5 Gyrs, size = 9.5', and mass = 2600 M 0 . The cluster is found to be dynamically evolved and exhibits mass-segregation effects. A strong cooling trail of white dwarfs is found to be in excellent agreement with a 0.70 M Q theoretical white dwarf cooling model. For NGC 2099 we find: distance = 1500 pc, age = 0.5 Gyrs, size = 13.9', and mass = 2500 M 0 . The cluster exhibits some signs of dynamical evolution, although not as severe as for NGC 6819. The white dwarf cooling age of NGC 2099 is found to be in excellent agreement with the turn-off age. ii Table of Contents Abstract ii Table of Contents iii List of Tables vi List of Figures vii Preface ix Acknowledgements x 1 Introduction 1 1.1 The Study of Open Star Clusters 1 1.2 The Survey 2 1.3 Thesis Goals 4 1.4 Present Status 6 2 Observations ^ 3 Data Reduction 1 0 3.1 FLIPS 1 0 3.1.1 Pre-processing 10 3.1.2 Combining Multiple Exposures 11 4 Photometry and Calibration 13 4.1 PSFex 1 3 4.2 Calibration of Instrumental Data 16 4.2.1 Landolt Standard Stars 16 4.2.2 On-Field Standards 18 i i i 5 The Clusters NGC 6819 and NGC 2099 21 5.1 N G C 6819 21 5.1.1 Previous Efforts 21 5.1.2 Summary of Current Data Set 21 5.2 N G C 2099 22 5.2.1 Previous Efforts 22 5.2.2 Summary of Current Data Set 23 6 Star Counts and Cluster Extent 26 6.1 N G C 6819 Extent 26 6.1.1 Comparison to King Model 27 6.2 N G C 2099 Radial Density Distribution 30 7 Colour-Magnitude Diagrams and Cluster Parameters 32 7.1 The N G C 6819 Colour-Magnitude Diagram 32 7.1.1. Reddening 32 7.1.2 Cluster Distance by Main-Sequence Fitting 34 7.1.3 Stellarity 34 7.2 The N G C 2099 Colour-Magnitude Diagram 35 7.2.1 Cluster Metallicity 38 7.2.2 Cluster Reddening and Distance - Main-Sequence Fitting with the Hyades 39 8 Theoretical Isochrones 42 8.1 Model Description 42 8.2 N G C 6819 Fit 4 3 8.3 N G C 2099 Fit 4 3 9 Dynamical Studies 4 8 9.1 Incompleteness Corrections and Counting Uncertainties 49 9.1.1 Incompleteness Errors 4 9 9.2 Final Star Counts 5 1 9.2.1 Luminosity Functions 52 9.3 Dynamical Relaxation 55 iv 9.4 Mass Segregation 57 9.5 Mass Functions 61 9.6 Star Counts Down to the H-burning Limit 64 10 White Dwarf Stars and the Initial-Final Mass Relationship 65 10.1 White Dwarfs in Open Clusters 66 10.1.1 Methods 68 10.2 White Dwarfs in N G C 6819 69 10.2.1 Continuity Arguments and Field Object Subtraction 69 10.2.2 White Dwarf Analysis 71 10.2.3 Interpretation of Cooling Sequence 73 10.3 White Dwarfs in N G C 2099 75 10.3.1 Removing Non-cluster Objects 75 10.3.2 White Dwarfs in Binaries 77 10.3.3 White Dwarf Luminosity Function and Cooling Age 78 10.3.4 Evaluation of Errors 81 10.3.5 Summary of N G C 2099 White Dwarf Studies 82 11 Conclusions 84. Bibliography 86 v List of Tables 1.1 The C F H T Open Star Clusters 3 2.1 Observational Data for the C F H T Open Star Clusters 9 4.1 Calibration Stars For the 12 CCDs (00-12) 19 5.1 Observational Data for N G C 6819 and N G C 2099 23 6.1 Geometry of Annuli 27 9.1 Completeness Corrections for N G C 2099 Data 51 9.2 Cluster Star Counts (Raw / Corrected) for N G C 6819 52 9.3 Cluster Star Counts (Raw / Corrected) for N G C 2099 53 10.1 White Dwarf Continuity Analysis - Predicted Number vs Observed Number 71 10.2 White Dwarf Luminosity Function 81 vi List of Figures 2.1 CFH12K high resolution wide field imaging camera 8 4.1 Source Extractor star-galaxy classification 14 4.2 PSFex vs A L L S T A R 15 4.3 Histograms of PSFex - A L L S T A R plane 16 4.4 O-C diagram for calibration stars 20 5.1 Colour image of N G C 6819 22 5.2 Colour image of N G C 2099 24 5.3 PSFex photometric error plot 25 6.1 Star counts and cluster extent of N G C 6819 28 6.2 Single mass King model fit to the radial density distribution of N G C 6819 29 6.3 Single mass King model fit to the radial density distribution of N G C 2099 31 7.1 N G C 6819 colour-magnitude diagram 33 7.2 N G C 2099 colour magnitude diagram 36 7.3 Comparison of current data to previous efforts for N G C 2099 37 7.4 Similarities between the colour-magnitude diagrams of N G C 2099 and the Hyades 41 8.1 Theoretical isochrone fit for N G C 6819 • • 44 8.2 Theoretical isochrone fits for N G C 2099 47 9.1 Input vs output stars in the incompleteness tests 50 9.2 Global luminosity function for N G C 6819 54 9.3 Global luminosity function for N G C 2099 55 9.4 Comparison of luminosity functions 56 9.5 Mass segregation in N G C 6819 58 9.6 Radial colour-magnitude diagrams for N G C 6819 59 vii 9.7 Mass-segregation in N G C 2099 60 9.8 Radial mass functions in N G C 6819 62 9.9 The mass function of N G C 2099 63 10.1 0.5 M 0 white dwarf cooling sequence 66 10.2 Statistical subtraction and source classification for white dwarfs in N G C 6819 72 10.3 The N G C 6819 white dwarf cooling sequence 73 10.4 The N G C 6819 white dwarf luminosity function 74 10.5 The termination of the N G C 2099 white dwarf luminosity function . . . . 78 10.6 The N G C 2099 white dwarf cooling sequence 79 10.7 White dwarf cooling ages in N G C 2099 80 viii Preface In accordance with the University of British Columbia thesis regulations, a summary of publications which have resulted from this work is presented to the reader. The work has all been completed under the supervision of Harvey Richer and Gregory Fahlman and the guidance of the co-authors. The thesis author's role in this work involved the reduc-tion of the data and the writing of the papers. The co-authors helped in interpreting the results and making suggestions/corrections to the work. Co-authors were also involved in data acquisition. This statement is understood and agreed upon by the co-authors' who contributed to this help. The photometry of the clusters, goals of the survey, reduction of the data and cali-bration is all discussed in Kalirai et al. 2001, Astronomical Journal, 122, 257. Specific results on the richest cluster in the survey N G C 6819, such as observational vs theoretical comparisons of the C M D , cluster dynamics and detailed white dwarf star studies, are pre-sented in Kalirai et al. 2001, Astronomical Journal, 122, 266. A detailed paper describing similar studies for the rich cluster N G C 2099 (M37), including the white dwarf cooling age of the cluster, has also been accepted for publication in the Astronomical Journal (in press). Additionally, I have written two short review articles related to this work for the Canada-France-Hawaii Telescope Bulletin Series (CFHT Bulletin # 42, (2000B) and C F H T Bulletin # 43, (2001A)). I presented preliminary results for this work (in the form of a poster) at the May 2000 C A S C A Meeting (Vancouver, BC) . I also gave a summary talk of the key results in the thesis during a conference in Coimbra, Portugal this sum-mer (2001). The proceedings of the conference will be published in an upcoming ASP Conference Series. I will also discuss the third paper in our series, "The White Dwarf Cooling Age of N G C 2099", at the upcoming January 2002 A A S Meeting in Washington D.C. Harvey Richer presented a description of the survey during the 196 t h A A S Meeting in Rochester, N Y (June 20nnV ix Acknowledgements I owe thanks to many people who have contributed to this thesis. First, my parents for encouraging me to pursue my interests which are outside the conventional stream. Your support is responsible for my success. This thesis would not be possible without the encouragement and support that my supervisors (and friends) Prof. Harvey B . Richer and Prof. Gregory G . Fahlman have provided me with. I'm glad that you started me on this project. Thanks guys. A significant portion of the work in this thesis was completed at the Canada-France-Hawaii Telescope (summer 2000) and the Osservatorio Astronomico di Roma (summer 2001). I thank these institutions for providing me with computing facilities and ac-commodations. Thanks to Greg Fahlman and Roberto Buonanno for taking me on. Many collaborators have contributed to the success of this project and deserve credit: Francesca D'Antona, Gianni Marconi, Patrick Durrell, Jean-Charles Cuillandre and Em-manuel Bertin. I would especially like to thank Paolo Ventura for including this work in his Ph.D thesis. Thanks to Harvey, Greg, Pat and Jaymie Matthews for reviewing drafts and helping me interpret the results. I am completely indebted to Harvey Richer who has made this project very enjoyable. Thanks for allowing me to travel and do research in beautiful places around the world. The financial support for these trips/expenses is gratefully acknowledged. Many thanks also to Harvey, Greg and Gianni for providing funding for publication costs through their grants. I would also like to acknowledge the financial support that I've received through an NSERC PGS-A Research Grant. Although they'll never admit it, the motivation I've received from the Q B C boys encourages me to succeed (EIP 4 Life). You know who you are. Thanks to R. Penhale for believing in me, you're two steps ahead of everyone and got this all started back in High School. Finally, I would like to extend my gratitude to Mandeep, my best friend. Thanks for everything. Chapter 1 Introduction 1.1 The Study of Open Star Clusters The Milky Way Galaxy is known to contain over a thousand open star clusters (see e.g., Binney & Merrifield 1998 [14]). The majority of these are often referred to as stellar associations, consisting of simple gravitationally interacting systems of only a few tens of stars. However, some of the rare rich open star clusters can contain several thousands of stars and it is these systems which contain the most useful astrophysical informa-tion. Open star clusters are particularly interesting because of their diverse nature. The youngest clusters are only several 10's of millions of years old whereas the oldest, N G C 6791, is believed to be almost 10 billion years old. This age range, coupled with different metal abundances, sizes and locations in our Galaxy have motivated many different stud-ies in both the observational and theoretical planes for these clusters. When combined, the agreements and discrepancies between the two planes further our knowledge of the properties of stellar systems. There are many motivations for studying clusters of stars as opposed to individual stars. First and most importantly, independent distance determinations to individual stars are not possible outside of a few 100's of parsecs, the limit set by the precision of parallax measurements. As I will later discuss, although not trivial, getting the distance of a coeval system of stars such as a star cluster is relatively easy. This distance deter-mination is crucial in transforming the observables, such as the apparent brightness of the stars, into physical quantities such as their luminosities. Open star clusters are also very interesting objects to study because they spatially sample the disk of our Galaxy. Therefore, chemical and stellar evolutionary studies of these systems can yield direct information on the properties of the Galactic Disk. The oldest systems can even set a lower limit to the age of the Disk. Because stars in a cluster are formed at relatively the same time with the same composition and stellar evolution is driven primarily by the initial masses of stars, the examination of different populations of stars in a cluster 1 Chapter 1. Introduction 2 can give us a snapshot of the life stages of a single star (see Renzini & Fusi-Pecci 1988 [92]). The younger clusters contribute to this study mostly by providing a population of higher mass (3 M 0 - 6 M 0 ) , unevolved stars. The older clusters contribute to stellar evolutionary study mostly through the population of stars in post main-sequence phases, such as white dwarfs. There are also advantages for studying open star clusters as opposed to the globular clusters. The major effort in star cluster photometry has always been directed towards the study of globular clusters as they are some of the oldest structures in the Universe and can set a lower bound for the age of the Universe. These clusters are generally distributed in various orbits and inclinations in the halo of our Galaxy and contain redder, more metal poor Population II stars when compared to the Population I open star clusters. Globular clusters are also much richer than any open cluster and can contain up to a million stars. This often complicates the photometry as the central regions of these systems tend to be very crowded. Since the globular clusters in the Milky Way all formed ~12—14 billion years ago, all stars more massive than ~0.8 M 0 have evolved off the main-sequence. Although very interesting for post main-sequence evolutionary studies, globular clusters do not allow for tests of stellar evolution theory for intermediate-mass hydrogen-burning stars. 1.2 The Survey The present survey is a large BVR imaging data set of 19 open star clusters in our Galaxy (see Table 1.1). This data set was taken with the high-resolution wide-field CFH12K mosaic C C D (42' x 28') and the majority of the clusters were imaged in ex-cellent photometric, sub-arcsecond seeing conditions. There are very few other programs currently underway to address the detailed properties of open star clusters partly because a large field of view and a large telescope are required. Many papers have been published on individual open star clusters (some of which are in this survey), but the only major current survey similar to ours is the W I Y N Open Star Cluster Survey (WOCS). There are a few overlapping clusters in these two surveys however some of the science goals are different (see §1.3). The C F H T data for these clusters is unprecedented with regard to the diversity of the sample, the size of the data set, and the precision of the measurements. The sample of Chapter 1. Introduction 3 Table 1.1: The C F H T Open Star Clusters Cluster a (1950) S (1950) Night Log(Age)° (m-M)° E(B -V)° Ang. Size (')° Ref b NGC 1039 (M34) .... 2 38.8 42 34 1 8.26 8.82 0 10 35 1,2 NGC 2323 (M50) .... 7 00.8 -8 16 1 8.05 10.73 0 23 16 3,4 NGC 6633 18 25.3 6 32 1 8.66 8.43 0 16 27 5,6,7 NGC 1342 3 28.4 37 10 1 8.71 9.46 0 26 14 — NGC 2343 7 05.9 -10 34 1 7.36 10.19 0 15 7 8,9 NGC 6819 19 39.6 40 04 1 9.3 12.3 0 10 19 10-14 IC 4665 17 43.8 5 44 2 7.58 8.29 0 19 > 30 15-20 NGC 1778 5 04.7 36 59 2 8.05 11.93 0 34 7 21,22 NGC 225 0 40.5 61 31 2 8.23 9.74 0 24 12 23 STOCK 2 2 11.4 59 02 2 8.23 8.62 0 38 > 30 24 NGC 1647 4 43.1 18 59 2 8.22 9.85 0 41 45 25 NGC 1960 (M36) .... 5 32.8 34 06 2 7.5 11.25 0 22 12 26 NGC 2301 6 49.2 0 31 2 8.19 9.79 0 04 12 27,28 NGC 1750 : 5 00.9 23 35 2 8.30 10.08 0 34 > 20 29,30 NGC 2099 (M37) .... 5 49.1 32 32 2 8.7 11.55 0 21 28 31-34 NGC 7243 22 13.3 49 38 2 7.9 10.11 0 24 21 35 NGC 2169 6 05.6 13 58 3 7.18 10.67 0 19 7 36-40 NGC 2251 6 32.0 8 24 3 8.31 11.26 0 22 10 — NGC 2168 (M35) .... 6 05.8 24 21 3 8.05 10.44 0 25 28 41,42 ° These parameters are not constrained well. 6 See current papers (Preface) and this study for constraints. TABLE REFERENCES : (1) Jones & Prosser 1996; (2) Ianna & Schlemmer 1993; (3) Claria, Piatti, & Lapasset 1998; (4) Schneider 1987; (5) Jeffries 1997; (6) Reimers & Koester 1994; (7) Sanders 1973; (8) Maitzen 1993; (9) Claria 1972; (10) Rosvick & VandenBerg 1998; (11) Auner 1974; (12) Lindoff 1972; (13) Sanders 1972; (14) Burkhead 1971; (15) Menzies & Marang 1996; (16) Prosser & Giarapapa 1994; (17) Prosser 1993; (18) Sanders & van Altena 1972; (19) Abt, Bolton & Levy 1972; (20) Hogg & Kron 1955; (21) Garcia-Pelayo & Alfaro 1984; (22) Barbon & Hassan 1974; (23) Lattanzi, Massone & Munari 1991; (24) Foster et al. 2000; (25) Turner 1992; (26) Sanner et al. 2000; (27) Marie 1992; (28) Harrington 1992; (29) Tian et al. 1998; (30) Galadi-Enriquez, Jordi, Trullols 1998; (31) Mermilliod et al. 1996; (32) Becker & Svolopoulos 1976; (33) West 1967; (34) Upgren 1966; (35) Hill & Barnes 1971; (36) Pena & Peniche 1994; (37) Peria & Peniche 1994; (38) Perry, Lee & Barnes 1978; (39) Sagar 1976; (40) Cuffey & McCuskey 1956; (41) Sung & Bessell 1999, (42) Cudworth 1972. clusters, which span a large range of angular sizes, age and metallicity values, were chosen on the basis of stellar density, age and distance. Previous photometry of these clusters has been mostly limited to photoelectric and photographic observations, and usually concen-trated on bright to intermediate magnitude ranges. The colour-magnitude diagrams from these observations generally show a large amount of scatter both in the main-sequence and turn-off stars. The faintest stars recorded in most studies are ~17—18th visual mag-nitude. While, the present photometry includes the bright end of the main-sequence, the program is primarily intended to find white dwarf stars down to V = 25, and therefore provides some of the deepest images ever taken for open star clusters. A very well de-fined main-sequence can be seen over a long magnitude range for the rich, young clusters, Chapter 1. Introduction 4 which will allow for a wide range of investigations (see §1.3). A small number of the clus-ters are very sparsely populated, but also among the youngest in the survey. These are especially important for white dwarf searches because any white dwarfs found in these clusters would have had to form from very massive progenitors, and therefore they will establish a constraint on the upper mass limit to white dwarf production. 1.3 Thesis Goals The ultimate goal of the C F H T Open Star Cluster Survey is to catalogue the white dwarf stars for each cluster and constrain both the initial-final mass relationship for these stars and the upper mass limit to white dwarf production, both of which are currently rather poorly constrained by observational data (see Weidemann 2000 [119] for a review). A n accurate measure of these quantities will require supplemental spectroscopic observations using larger telescopes, such as Gemini and Keck. In addition to studying white dwarfs we will also undertake several other studies as described in this thesis for the rich clusters N G C 6819 and N G C 2099. Measurements of key properties for the clusters (such as age and distance) will refine previous values significantly by using higher quality data and more updated physics in the models. By converting the theoretical plane of the models into the observational plane, we will test stellar evolutionary theory over a wide parameter space. In the past, this testing has been mostly limited to the upper main-sequence and turn-off due to large scatter and errors from photometric uncertainty for low signal to noise stars in the lower main-sequence. The depth in the current data set (V ~ 25) will provide an excellent test to theoretical isochrones on the lower main-sequence (10 < M y < 12). Of particular interest is the change in slope of the main-sequence caused by the various stages of stellar evolution and structure, such as the onset of H 2 dissociation-recombination in the stellar envelope at ( B - V ) 0 = 1.0-1.1, or the 'kink' in the main-sequence at ( B - V ) 0 = 0.3-0.4 caused by changes from convective envelope models to radiative models. These slope changes will be used as guidelines for the fit of the observations with the theoretical models. The models being used are up-to-date in the input physics and calculated especially for the survey (P. Ventura and F. D'Antona 2000, private communication). In addition to main-sequence and turn-off model testing, there are many other sci-entific goals which we hope to complete with this data set. For N G C 6819 and N G C Chapter 1. Introduction 5 2099, these goals have been completed and are described in this thesis. For example, a main-sequence turn-on could potentially provide an independent age measurement for some of the very young clusters for which the low mass stars have not yet reached the main-sequence. As mentioned earlier, the faint magnitudes reached by such a study will also allow the study of white dwarf stars. Fitting models to the end of the white dwarf cooling sequence, if bright enough, will allow a third determination of the age measurement of some of the youngest clusters. For example, we use white dwarf cooling models (Richer et al. 2000 [96]) to determine that the white dwarf cooling sequence in an intermediate age cluster (~250 Myrs) will terminate at M y ~ 11.2, depending on the mass of the stars, whereas for a much older cluster (~2 Gyrs), the end of the cooling sequence occurs at a significantly fainter magnitude of M y ~ 13.5. Therefore, even for a moderately close cluster, the depth of the current photometry should allow us to es-tablish the white dwarf cooling age for a large number of the clusters in the survey (see Table 1.1). For the older clusters for which the termination of the white dwarf cooling curve is fainter than our limiting magnitude, we should nevertheless be able to identify a significant number of white dwarf candidates above our mean cut off of V ~ 25. Other studies that will be looked at involve producing luminosity and mass functions for these clusters. In particular, these will be explored for evidence of dynamical evolution in the older clusters. A n updated mass-luminosity relation from new stellar models will also be used to test mass segregation in the clusters by comparing the mass functions for various annuli at different distances from the centers of the clusters. This test will be possible for a wide range of masses (~0.5 — 5 M Q ) for most of the intermediate aged clusters. Improved distance estimates of the clusters will be found by establishing a fiducial of the main-sequence and comparing to the Hyades main-sequence which has a very well established distance from Hipparcos measurements (de Bruijne, Hoogerwerf & de Zeeuw 2001 [17]). Chemical evolution theories for old stars in the clusters will be tested by comparing the masses of white dwarfs to their progenitors on the main-sequence. Num-ber counts involving the red giant/white dwarf ratio will be carried out by assuming conservation of star number through various stages during stellar evolution, to establish the number of expected white dwarfs. Binary star tests will be undertaken to determine the population of binaries (relative to main-sequence stars) in each cluster. These have been shown to have important dynamical evolutionary effects, especially for the younger, less dense clusters (de la Fuente Marcos 1996 [33]). Chapter 1. Introduction 6 1.4 Present Status The science goals mentioned above have all been completed for the richest two clusters in our survey, N G C 6819 and N G C 2099. Additionally, we have produced a photometric catalogue in order to encourage and expedite studies outside the scope of our goals. The data set for each cluster will be made available as soon as the results have been published. For example, the data for the first published cluster, N G C 6819, has been shared with six groups at the time of writing this thesis, for studies involving astrometry, proper motions, variable stars, blue-straggler populations and synthetic colour-magnitude tests. At the present time, three papers have been published on the survey (see Preface). Additionally, further science goals of the two rich, young clusters N G C 2168 and N G C 2323, such as main-sequence fitting and theoretical isochrone comparisons have been completed. This program has lead to several supplemental studies which are currently underway. For example, the authors have been successfully granted telescope time (in collaboration with M . Rich) with the LRIS instrument on Keck to obtain spectral classification for potential white dwarf objects in N G C 2099. A similar study (in collaboration with T. von Hippel) is underway to investigate the white dwarf cooling sequence in M67 with GMOS on Gemini. A proposal to obtain spectra of candidate white dwarf stars in N G C 6819 has also been submitted to Gemini. Additionally, the original survey data set has been supplemented with deep C F H T photometry of two very rich, old clusters (NGC 7789 and N G C 6791). Chapter 2 Observations The data for the 19 clusters (see Table 2.1) was taken during an excellent three-night observing run with the Canada-France-Hawaii Telescope (CFHT) during October 15-18 1999, using the CFH12K camera. The optical detector is a 12,288 x 8,192 pixel C C D mosaic camera for high-resolution wide-field imaging at the C F H T prime focus. The C C D camera was ideal for our purposes as it covers a large area on the sky (42 by 28 arc-minutes, or about 1.5 times the area of the full moon), and also contains a large number of pixels (> 108) to ensure a high angular sampling. The camera is equipped with twelve 2048x4096 pixel CCDs with an angular size of 0.206 arc-seconds per pixel at the f/4 prime focus. The orientation of the CCDs within the mosaic is such that chips 00-05 form a sequence on the bottom row from left —> right, and chips 06-11 are directly on top of the bottom row (see Figure 2.1). Therefore the inner 4 CCDs are chips 02, 03, 08 and 09, and the outer 4 CCDs are chips 00, 05, 06 and 11. Data for each cluster was taken in three filters: B , V and R. The R images were usually rather shallow and are used to provide some leverage for the reddening of the clusters based on colour-colour diagrams. The optimal exposure time to be used was determined so as to achieve a limiting magnitude 1 magnitude fainter than the oldest expected white dwarfs in most of the clusters. This was found by equating the cluster age to the white dwarf cooling time (Richer et al. 2000 [96]). For a few of the older clusters (such as N G C 6819), it was not feasible to try and observe the end of the white dwarf cooling sequence from ground based imaging as it was expected to be too faint, so we hope to detect as many white dwarfs as possible above our limiting magnitude. To achieve a higher signal to noise ratio, multiple 300-second exposures were taken for some of the clusters in both the V and B filters. Additionally, single 50 and 10 second exposures were obtained in all three of the B , V , R filters. The long exposures in each band were dithered from one another to prevent a star from landing on a bad pixel in more than one frame. The multiple fields were averaged and combined together 7 Chapter 2. Observations 8 Figure 2.1: CFH12K high resolution wide field imaging camera. using the FITS Large Images Processing Software (FLIPS) (see §3.1). In many cases the CFH12K images for these clusters are the deepest yet, and indicate a much richer cluster population than previously thought. The images also provide a more complete aerial coverage for each of the clusters than most previous studies. In addition to these frames, several flat-field images and bias frames were also taken (see Chapter 3). Five of the clusters in our sample have sizes that exceed that of the C C D mosaic field of view, and thus additional images of neighboring blank fields were also taken. These will be used for background subtraction by obtaining an estimate of the number of field stars in the region around the cluster. Table 2.1 summarizes the observational data that was taken for each cluster in the survey, as well as other relevant information pertaining to the exposures. Since most of these clusters have not been extensively studied in the past it is difficult to compare results with 'good' previous photometry. Furthermore, none of the clusters contain any well established standard stars in the field. These factors lead to the critical calibration stage of the reduction which we address by taking multiple images of the SA-92 and SA-95 standard star fields (Landolt 1992 [66]). Calibration is further discussed in §4.2 (see Tables 2.1 & 4.1 for particulars). Chapter 2. Observations 9 Table 2.1: Observational Data for the C F H T Open Star Clusters Cluster V (300/50/10)" NGC 1039c (M34) 1/1/1 NGC 2323d (M50) 1/1/1 NGC 6633 1/1/1 NGC 1342 9/1/1 NGC 2343 1/1/1 NGC 6819d'e 9/1/1 IC 4665c 1/1/1 NGC 1778 1/1/1 NGC 225 1/1/1 STOCK 2C 1/1/1 NGC 1647c 1/1/1 NGC 1960 (M36) 1/1/1 NGC 2301 1/1/1 NGC 1750 1/1/1 NGC 2099d-> (M37) .... 3/1/1 NGC 7243c 1/1/1 NGC 21696 2/1/1 NGC 22516 2/1/1 NGC 2168d (M35) 2/1/2 Field V (5/10)^g~ Calibration SA-92 1/2 Calibration SA-92 2/2 Calibration SA-92 0/2 Calibration SA-95 0/2 Frame V Flat Field Pre-Processing _33 B (300/50/10)° R (50/10)" Seeing (")b (V/B/R) Air-mass X 1/1/1 1/1/1 1/1/1 9/1/1 1/1/1 9/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/1/1 1/1/1 3/1/1 1/1/1 2/1/2 2/1/1 2/1/2 1/1 ~(0.8/0.75/0.65) 1.18 1/1 ~(0.85/0.95/0.76) 1.15 1/1 ~(0.6/0.75/0.60) 1.2 1/1 ~(0.81/0.72/0.55) 1.09 1/1 ~(0.87/1.05/0.76) 1.23 1/1 ~(0.70/0.89/0.66) 1.1 - 1.65 1/1 ~(0.88/l.1/0.96) 1.3 - 1.8 1/1 ~(0.76/0.87/0.74) 1.15 1/1 ~(0.82/0.92/0.77) 1.37 1/1 ~(0.78/0.82/0.75) 1.36 1/1 ~(0.65/0.77/0.63) 1.23 1/1 ~(0.80/0.82/0.67) 1.05 1/1 ~(0.85/0.88/0.75) 1.37 1/1 ~(0.77/0.93/0.85) 1.20 1/1 ~(0.90/0.82/0.81) 1.03 1/1 ~(0.78/0.79/0.80) 1.16 2/1 ~(1.8/1.8/1.6) 1.06 1/1 ~(1.35/1.30/1.75) 1.37 1/1 ~(1.35/1.20/0.90) 1.60 B (5/10/15/20)"'" R (5/10)"'3 2/0/2/0 2/2/0/1 0/0/1/0 0/2/0/0 1/2 2/2 0/2 0/2 B Flat Field R Flat Field Bias (Dark Filter) 20 13 a The number of exposures taken in this filter, for each exposure time (sec) listed. 4 The data for these clusters suffer from one or more of the following : high humidity, poor focus, or cirrus. c An equivalent number of blank field images were also obtained for these clusters (offset by 1 degree). d Further reductions have been completed and published for these clusters (or to be published shortly). e Additional 0.2 second and 1.0 second images were obtained at a later date. ' Additional 0.5 second images were obtained at a later date. 9 These images are intentionally taken over a range of air-masses to obtain air-mass terms. Chapter 3 D a t a Reduc t ion The data for the survey were reduced and organized locally at C F H T by the author. The first stages of the reduction involve pre-processing the raw data as shown in equation (3.1): raw image — bias Preprocessea Frame — — — . (31) flat — bias First several zero exposure bias frames are taken and subtracted from the images (de-biasing) to account for counts read out even if no light falls on the C C D . A dark current is also subtracted from the longer exposures. After the images are de-biased, the data is flat-fielded to account for pixel-to-pixel variations. For the flat-field frames, we combined twilight flats taken from all three nights of the observing run. 3.1 F L I P S 3.1.1 Pre-processing FITS Large Images Processing Software (FLIPS) is a highly automated software package developed at C F H T (Cuillandre 2001 [29]). FLIPS was originally developed for early C C D mosaic wide field imagers at C F H T , such as M O C A M (4k x 4k pixels, 1994) and UH8k (8k x 8k pixels, 1995), but has now been updated and upgraded with new functions to handle CFH12K images. This software is ideal for our survey as it is optimized for both speed and requires limited memory resources. FLIPS is not a photometric reduction package and should be thought of as a package which performs similar functions to the commonly used I R A F task mscred. FLIPS is designed to operate on individual chips within the CFH12K mosaic. The first steps involve averaging the 'good' exposures for each of the bias, dark and flat exposures. For each filter, these exposures can be accepted or rejected based on the on-screen statistics (ie. if image level is too high (saturation) or too low (no flux)). The 10 Chapter 3. Data Reduction 11 flats used in the survey are twilight exposures averaged and sigma clipped (iteratively eliminating if ±sigma cut is not satisfied) from all three nights of the run, whereas the darks were combined by taking a median from a sigma clipped sample. FLIPS will then search through all the images for each cluster and apply the above corrections (bias, dark, flat) based on input parameters specific to the C C D and data. These detailed parameters for each of the operations have been optimized to produce the best processing. One of the important features of FLIPS is that it normalizes the background sky value to the chip with the highest sky value (lowest gain), C C D 04. This provides for a scaled data set with a smooth background on all chips. Therefore, the instrumental zero points for the data set will all be almost equal. The final processed data show very small variations from a completely smooth background. By measuring the statistics in small boxes of size comparable to the mean stellar aperture size at various positions in the mosaic, we find the flat-fielding to be good to ~0.5% in V and ~0.7% in B , averaged over 11" sq. patches. 3.1.2 Combining Multiple Exposures In order to achieve a higher signal to noise ratio and a deeper colour-magnitude diagram, we obtained several deep exposures for some clusters. These exposures were dithered from one another and then aligned and combined into one image. We use a program called align, within FLIPS, which searches through a specified area on each C C D and finds patterns of common stars based on both the positions and fluxes of the stars. This pattern recognization algorithm works very quickly for moderately rich clusters, however it is quite slow for sparse fields. FLIPS align uses SExtractor (Bertin & Arnouts 1996 [11]) to create an X Y catalogue of aperture magnitudes and positions which align searches through to find the geometric patterns. This catalogue does not not contain the final photometry as these magnitudes are only used for this one purpose. A n important aspect of align, as we will shortly see, is that it defines the true sky background and atmospheric transmission across the data set. Next we invoke the imcombred command within FLIPS to register the frames with respect to one common image, and average the data. The parameters for the variable transmission (if present) and sky background are taken and applied directly from the align output. Imcombred re-centers the individual images based on the X k Y offsets derived for each C C D . For a given pixel coordinate, a Chapter 3. Data Reduction 12 column of pixels from each of the individual images is created with the proper background and transmission corrections. Then a C C D sigma clipping of ± 5 is applied on that set of pixels to determine the exact shifts. This requires a determination of the readout noise and gain to evaluate properly the expected noise in the signal and reject outliers. We found that the point-spread-function fit to the stars was being skewed in the average because the centers of stellar positions were not being located accurately enough. In order to correct the problem, we use a FLIPS imcombred option called sub-pixelling. This sub-pixelling parameter can be given a value of n = 1, 3 or 5, and will split each CFH12K pixel (0.206") into n 2 boxes, and then locate the centre within each of these sub-divisions. This procedure is similar to the drizzle technique in I R A F and works to better align the stellar profiles directly on top of one another in the above combining procedures. The sub-pixelling feature within FLIPS also maintains the original resolution of the image and does not lose any spatial information in the combining process. The resulting colour-magnitude diagram (for n = 5) was also found to go about 0.3 magnitudes fainter, thereby indicating that the additional step allowed for the measurement of lower signal to noise objects. This division led to a very slow computing process of averaging the frames (as expected for 2.5 billion pixels on each mosaic), however the results were excellent. The resulting data images are next used in the T E R A P I X PSF-extraction and mod-elling tool, PSFex, to find magnitudes, colours, errors, and stellarity. Chapter 4 Photometry and Calibration 4.1 PSFex Point-Spread-Function Extractor (PSFex) is a highly automated program which will be integrated in the code of the next version of SExtractor (E. Bertin 2000, private com-munication). SExtractor detects sources through a segmentation process consisting of 6 essential steps: estimation of the sky background, thresholding, deblending of overlap-ping images, filtering of the detections, photometry, and star/galaxy separation. Further details and simulation results are given in the program manual (Bertin & Arnouts 1996 [11]). SExtractor is most commonly used to distinguish between stars and galaxies at faint magnitudes by assigning a stellarity index to the objects. This process also elimi-nates bad pixels caused by cosmic ray hits. The stellarity is key to our project as white dwarf stars and faint star forming galaxies both appear as faint, blue objects. SEx-tractor determines the stellarity of objects by computing a neural network (a group of connected units) which learns based on other (high signal-to-noise) stars in the field. The classification is highly dependent on the seeing of the image and generally will be less accurate for faint stars than faint galaxies because of crowding: faint stars have a higher probability of catching wings added to the profile by a background galaxy which would result in misclassification. The shape of all diagrams (stellarity vs magnitude) that we produced using SExtractor agree with that of the Monte Carlo experiments executed during the tests of the package by Emmanuel Bertin. Furthermore, our results seem to indicate that a simple constant magnitude stellarity cut (0.95 is commonly used - see von Hippel & Sarajedini (1998) [117]) may not be a real indication of the separation of stars from galaxies. We find a common pattern in the stellarity vs magnitude diagram for all rich clusters, and this may be a more accurate indication of the separation between the galactic and stellar sequences. The above points are illustrated in Figure 4.1, for the stars of the young, rich open cluster N G C 2099. This type of cut would have to be used with caution however as there are some objects that fall under the line, yet are brighter 13 Chapter 4. Photometry and Calibration 14 and have a slightly higher stellarity than sources that are over the line. More data will eventually allow us to decide on an optimal stellarity cut. 0.5 S E x t r a c t o r C l a s s i f i c a t i o n . for *GC- 20&9 10 15 20 B m a g n i t u d e 25 Figure 4.1: Star-galaxy classification from SExtractor seems to indicate two sequences (split by the line), where the top sequence may be indicative of stars and the bottom of galaxies. PSFex automatically creates a PSF based on a set of bright, unsaturated and isolated stars. Included in the PSF are polynomial basis functions which can map the variations of the PSF across the field. This is done immediately after the SExtractor find catalogue is created. Next, it uses this PSF to derive PSF magnitudes and colours for the objects in the original catalogue. This entire process is not very computer intensive and is easily executed with very little user interaction. For the C F H T Open Star Cluster Survey, we have chosen to use a variable PSF for all fits to account for small changes in the profiles of stars over the large range of each C C D . Additionally, there are differences between each of the 12 CCDs on the mosaic so the analysis is done on a single chip basis for all exposures. Preliminary results, based on the differences in measured magnitudes of PSFex and A L L S T A R (Stetson 1992 [108]), the shape of the main-sequence in PSFex compared to A L L S T A R , and the number Chapter 4. Photometry and Calibration 15 20 I B 16 , 1 . . 1 . . . . - Photometr ic Error Bar/?&3 —i—i—i—i—j—i—i—i—i— i f •' -— —:—: '—VN^ jg ^Limiting Magnitude (B~24.6) I . - • • i • >t - ' M • : i • i • '"'?>•••' J NGC 6819 - - c h i p 0 8 (300 sec) • • • • 1 • i i i 1 i i—i—i 1—i—i—i—i— AB ( P S F e x - ALLSTAR) Figure 4.2: Comparison of PSFex magnitudes with ALLSTAR magnitudes indicates good agreement between the two packages within the errors. A very small bias is seen at faint magnitudes (B > 25), which is fainter than our limiting magnitude cutoff. The bias effect on other CCDs and in other filters was less than that shown here. of stars that were measured indicate consistency between the two packages. To better illustrate the small scatter measured for common stars in these two packages, we show a comparison of the difference in magnitudes vs magnitude for the two packages in Figure 4.2. Results on other CCDs and other frames were all much better than this, some by several factors. A photometric error bar (combining errors from both programs) is also shown in order to judge the agreement between the two systems. The spread (2/3cr) is within the errors along the tight vertical sequence of stars, except at the very bright and faint ends. The scatter at the bright end is in part due to saturation and also due to the handling of blended objects in the frames. Although the low end scatter may also appear to be larger than the errors this may be simply due to different detection parameters between the two packages. Our expected limiting magnitude is indicated on Figure 4.2. The results for different magnitude cuts, in the form of a histogram, are shown in Figure Chapter 4. Photometry and Calibration 16 4.3. The distribution peaks strongly at A B = 0 and quickly falls off indicating a very small amount of scatter in all but the very faint case (see lower right diagram). 300 0.5 -0.5 22.5 < B < 24.0 -0.5 0 0.5 -0.5 0 0.5 AB (PSFex - ALLSTAR) AB (PSFex - ALLSTAR) Figure 4.3: Histograms for different magnitude cuts show a large pile up of stars with A B = 0 in the PSFex - ALLSTAR plane. The spread in the data is small for all but very faint objects. 4.2 Calibration of Instrumental Data 4.2.1 Landolt Standard Stars Observations of Landolt standard stars in SA-92 and SA-95 (Landolt 1992 [66]) were obtained in order to convert the instrumental magnitudes to real magnitudes as shown in equations (4.1) and (4.2): vinstr = V + cxX + f3(B-V) + Zv, (4.1) Chapter 4. Photometry and Calibration 17 knstr = B + a'X + p'(B -V) + Zb. (4.2) In these equations vinstr and binatr are instrumental magnitudes (discussed more later). For equation (4.1), a is the coefficient of the air-mass term X , ft is the coefficient of the colour correction term ( B - V ) , and Zv is the zero point shift for the V-band. Similarly, in equation (4.2), binatr, a', 3', and Zb are the corresponding parameters for the B-band images. We use a total of 23 calibration frames of different exposure times, air-masses, and wavelength filters to transform the data from instrumental magnitudes to calibrated magnitudes (see Table 2.1). There are systematic differences between chips on the C C D so separate calibrations are required for each chip. The colour term does not change with different exposures or nights of observations. The zero points are the most critical part of this first stage of the calibration equations. This shift tells us the relationship between the instrumental magnitudes and the real magnitudes, taking the exposure time into account. The magnitudes of the standard stars used by Landolt are based on non-linear trans-formations of the magnitudes and colours of the stars in several different filter bands. The final magnitudes derived for the stars are based on the total amount of flux emitted in a filter and within a large enough aperture so that no light is lost. However, the fitting parameter used in our PSF photometry is one that maximizes the signal-to-noise ratio. In order to collect the entire light from the star without an increase in the background level, we use aperture photometry for calibration. For those stars which are not saturated and appear in both Landolt's paper (Landolt 1992 [66]) and our images of SA-92 and SA-95, we use D A O P H O T to produce a curve-of-growth of flux for aperture sizes ranging from 8 to 20 pixels. At an aperture size of 16 pixels it is found that the magnitude begins to level off (less than 0.01 magnitude difference in all cases). Therefore, an aperture size of 16 pixels collects almost all of the light from the star (> 99%). For these stars we use the flux from this aperture in our calibration equations. To prevent Landolt objects from being saturated on our images, we chose extremely short exposure times depending on the filter. The number of exposures that were used in the calibration for each night is summarized in Table 2.1. The calibration parameters are found by using a least squares method to determine the coefficients of the air-mass, colour and zero point terms. The specific algorithm is Chapter 4. Photometry and Calibration 18 based on a method designed by Harris, Fitzgerald & Reed (1981) [43]. These parameters are solved for in those chips which contained enough calibration stars for the program to produce accurate (low sigma) results. The chips with only a few or no calibration stars do not help this analysis. Some sections of Landolt's frames were highly concentrated with calibration stars; however, the C F H T chips did not align with these regions when the calibration frames were taken. It would be strongly preferred to have enough cali-bration stars in each chip to get accurate parameter values, which could then be used individually. Additionally, dithering the calibration frames could have allowed for the measurement of common standards on different chips, which would also help in better constraining individual C C D calibration parameters. We are forced to average the 'good' chips (02,04,07,08) together and use the resulting parameters on other chips. There is minimal scatter in the parameters for different chips so this method is reliable. For night 3 there are too few observed standards to establish low sigma calibration parameters. Therefore, we use the transformation equations from night 1 and 2 to create more stan-dards and then apply these to the night 3 data. These results were also well constrained. The number of calibration stars on each C C D are provided in Table 4.1. To illustrate the small scatter between the actual magnitude of each standard star, and the calculated value from the least squares solution, we present a composite O-C diagram in Figure 4.4. There is clearly no obvious bias in this diagram, which includes all calibration stars on all nights, in all bands. O-C diagrams for data in each of the separate V and B filters, individual nights of the observing run, or red vs blue segregation also shows no trends or biasses. The small vertical spread is caused by measurements of the same star at various air-masses and/or exposure times. 4.2.2 On-Field Standards As mentioned earlier, we have determined all of the best fit calibration parameters in both the V and B filters. One can now simply use these in equations (4.1) and (4.2) to determine a preliminary zero point term. Next, we find Z'v and Z'b, which are denoted in equations (4.3) and (4.4) as the final shifts of the instrumental magnitudes to the calibrated magnitudes, by creating on-field standards on the science frames: V = vPSF-Z'v-p(B-V), (4.3) Chapter 4. Photometry and Calibration Table 4.1: Calibration Stars For the 12 CCDs (00-11) 19 Exposure" Air-mass X 00 01 02 03 04 05 06 07 08 09 10 11 N I G H T 1 - SA-92 V/5 1.085 0 0 3 1 5 0 1 6 2 1 0 0 B/5 1.101 0 0 3 1 5 0 1 6 3 2 0 0 B/5 1.099 0 0 3 1 5 0 1 6 2 1 0 0 V/10 1.088 0 0 3 1 4 0 1 5 2 0 0 0 V/10 2.105 0 0 3 1 5 0 1 6 2 1 0 0 B/15 1.092 0 0 3 1 5 0 1 6 2 0 0 0 B/15 2.038 0 0 3 1 5 0 1 6 2 0 0 0 N I G H T 2 - SA-92 B/5 1.519 0 0 3 1 5 0 1 6 3 1 0 0 B/5 1.060 0 0 3 1 5 0 1 6 0 1 0 0 B/10 1.508 0 0 3 1 5 0 1 6 2 1 0 0 B/10 1.060 0 0 3 1 5 0 1 6 2 1 0 0 B/20 1.496 0 0 3 1 5 0 1 6 2 1 0 0 V/5 1.549 0 0 3 1 5 0 1 6 2 1 0 0 V/5 , , .. 1.059 0 0 3 1 4 0 1 5 2 1 0 0 V/10 1.535 0 0 3 1 5 0 1 5 2 1 0 0 V/10 1.059 0 0 3 1 4 0 0 4 2 1 0 0 N I G H T 2 - SA-95 B/10 1.389 1 0 0 0 0 0 3 2 4 0 2 0 B/10 1.398 0 0 0 0 0 0 3 2 4 0 2 0 V/10 ,,, 1.413 1 0 0 0 0 0 3 2 4 0 0 0 V/10 1.422 1 0 0 0 0 0 3 3 4 0 1 0 N I G H T 3 6 -- SA-92 V/10 1.076 0 0 8 1 5 0 5 8 2 1 0 0 V/10 1.078 0 0 8 1 5 0 5 8 2 1 0 0 B/15 1.081 0 0 8 1 8 0 6 9 2 1 0 0 ° Column organized as filter/exposure time (seconds). 6 We used night 1 calibration results to create additional stars for this night. B = bPSF-Zb-B'(B-V). (4.4) These new zero points will include the correction needed in going from aperture to PSF magnitudes. The stars that are used for this are isolated and bright, yet non-saturated. Finally, equations (4.1) and (4.2) are solved for V + B(B - V) and B + 6'{B - V) which are then used in equations (4.3) and (4.4), along with the P S F magnitude values to derive Z'v and Z'b for each chip. These new zero points can be used to calibrate the entire data set to calibrated magnitudes as they are particular to the science frames only. This shift is employed by first solving the calibration equations for a colour term, as shown in equation (4.5), Chapter 4. Photometry and Calibration 20 o.i 0.05 3 O « u i 9) > J 5 o - 0 . 0 5 - 0 . 1 13 14 15 H a g _ l L_ 16 17 Figure 4 . 4 : O-C diagram for calibration stars on all three nights, and in both the V and B filters, shows no biases or trends. The small vertical spread is caused by slightly different measurements of common stars at different air-masses and/or exposure times. B-V = (l + {p- - V ) P S F - (Z'b - Z'v)], (4 .5 ) and then applying the colour term to equations ( 4 . 3 ) and ( 4 . 4 ) to get the V and B mag-nitudes. This method can be used for all clusters in the survey by using the determined calibration parameters for the night in which the respective cluster was imaged. Chapter 5 The Clusters NGC 6819 and NGC 2099 5.1 NGC 6819 5.1.1 Previous Efforts N G C 6819 was identified as an old stellar system almost 30 years ago (Lindoff 1972 [73]; Auner 1974 [3]). These early studies used photographic plates or photoelectric detectors and concentrated on bright stars. Cluster ages were determined by calibrating relative positions of turn-off and red giant branch stars on the colour-magnitude diagram, and by comparing the bright stars in the cluster to those of the evolved system M67. These methods produced a large range of ages for the cluster: 2 Gyrs (Lindoff 1972 [73]) - 4 Gyrs (Kaluzny & Shara 1988 [58]). More recently, a detailed isochrone fit to the cluster photometry has been carried out by Rosvick and VandenBerg (1998) [97], and a new age estimate of 2.4 Gyrs has been determined using models with a significant amount of convective core overshooting (the age dependent on the amount of overshooting). The limiting magnitude of their C C D photometry (V ~ 18.5), the deepest for this cluster at the time, is too bright to include the low mass main-sequence stars and far too bright to detect any cluster white dwarfs. 5.1.2 Summary of Current Data Set The observational data for N G C 6819 comes from the first night of our three night observing run. We obtained nine 300 second images in both B and V , as well as single 50 and 10 second images in each of the B, V and R filters. At a later time, we also acquired short 1 second frames so as to obtain unsaturated images of the bright stars in the cluster. Blank field images are not necessary as the outer four chips of the mosaic can be used to correct for field star and galaxy contamination (cluster radius ~ 9.5'). The seeing for the images deviated slightly from mean values of 0.70" in V and 0.90" in B (see Table 5.1). Photometric skies combined with this good seeing produced very sharp 21 Chapter 5. The Clusters NGC 6819 and NGC 2099 22 Figure 5.1: Colour image of NGC 6819 created from individual V, B and R 50 s frames. The image size is 42' x 28'. images of the cluster. A colour image created from the processed V , B and R images is shown in Figure 5.1. 5.2 N G C 2099 5.2.1 Previous Efforts Similar to N G C 6819, N G C 2099 was established as a relatively rich and large star cluster very early on and thus has been the focus of many studies, for example: photo-metrically (von Zeipel & Lindgren 1921 [118]; Becker 1948 [7]; Hoag et al. 1961 [47]); spectroscopically (Zug 1933 [126]; Lindblad 1954 [72]); and astrometrically (Nordlund 1909 [78]; Giebeler 1914 [39]; Lindblad 1954 [72]; Upgren 1966 [113]; Jeffreys 1962 [53]). The first significant photometric study that shed light on cluster parameters was that Chapter 5. The Clusters NGC 6819 and NGC 2099 Table 5.1: Observational Data for N G C 6819 and N G C 2099 23 Filter Exposure Time (s) No. of Images (6819/2099) Seeing (") (6819/2099) Air-mass (6819/2099) V 300 9/3 0.70/0.85 1.30/1.03 50 1/1 0.70/0.97 1.16/1.04 10 1/1 0.68/0.99 1.15/1.04 1 1/0 0.78/— 1.27/— 0.5 0/1 - / l . l —/1.19 B 300 9/3 0.90/0.79 1.40-1.76/1.03 50 1/1 0.82/0.85 1.38/1.03 10 1/1 0.84/0.85 1.37/1.03 1 1/0 1.1/- 1.25/— 0.5 0/1 - / l . l —/1.25 R 50 1/1 0.64/0.81 1.14/1.04 10 1/1 0.66/0.81 1.13/1.04 of West (1967) [121]. West used photographic plates to establish a colour-magnitude diagram for 930 member stars brighter than V = 17.7. The photometric spread in the cluster main-sequence from this study is quite large and consequently, there are large uncertainties in the results. That U B V study concluded with an estimated cluster age of 220 Myrs, distance modulus of (m—M)v = 11.6, and a constant reddening value of E(B—V) = 0.27. The first significant dynamical study of the stellar density distribution near the location of N G C 2099 was based on three-colour Schmidt plate photometry in 1975 (Becker &: Svolopoulos 1976 [8]). The latest study of the cluster used radial ve-locity and U B V photoelectric observations to establish a much cleaner main-sequence based on cluster membership (Mermilliod et al. 1996 [77]). The results of this study concluded with ( m - M ) v = 11-5, E ( B - V ) = 0.29, age = 450 Myrs (comparison to theo-retical isochrones from Schaller et al. (1992) [106] and Bertelli et al. (1994) [10]) and Z = 0.020. The colour-magnitude diagram of this study exhibits less than 100 objects and has a limiting magnitude brighter than V = 17. The quality of the data in the present study is unprecedented when compared with these previous efforts for both clusters (see Figure 7.3 for a comparison of the present N G C 2099 data to previous efforts). 5.2.2 Summary of Current Data Set For N G C 2099, the data set (three 300 second images in both B and V , as well as single 50 and 10 second images in each of the B, V and R filters) was obtained during the second night of the observing run. Shorter 0.5 second frames were obtained later to complete Chapter 5. The Clusters NGC 6819 and NGC 2099 24 the photometry for the brightest cluster stars. As with N G C 6819, blank field images are not necessary due to the cluster's apparent size (~13.9') relative to the CFH12K mosaic. A l l images were obtained in excellent conditions (see Table 5.1); a colour image from the deep V and B exposures is shown in Figure 5.2. Figure 5.2: Colour image of NGC 2099 created from the V and B 300 s frames. The image size is 42' x 28'. The data set was flat-fielded, and bias and dark corrected using FLIPS as described in Chapter 3. The photometric calibration was obtained from numerous exposures of the Landolt standard fields SA-92 and SA-95 (Landolt 1992 [66]) as described in §4.2. The photomet-ric uncertainty in the zero points for the standard stars during night 1 was measured to be ~0.015 in V and ~0.014 in B. The air-mass coefficients were determined to be 0.088 ± 0.01 in V and 0.165 ± 0.005 in B, both in good agreement with C F H T estimations of 0.10 and 0.17 respectively. During night 2, the photometric uncertainty in the zero points for the standard stars was measured to be ~0.021 in V and ~0.025 in B. The extinction in magnitudes per air-mass during night 2 was determined to be 0.084 ± 0.012 in V and Chapter 5. The Clusters NGC 6819 and NGC 2099 25 0.184 ± 0.008 in B, both close to the standard C F H T values of 0.10 and 0.17 respectively and similar to those on the first night of the run. The colour terms were averaged over the three night observing run and are in agreement with C F H T estimations in the V filter and slightly lower than estimations for the B filter. 0.6 1 1 1 1 1 1 1 i , | i t i i | 0.4 -•i" - •'.V • 0.2 J 0 I l i I _ l I I I I 1 1 1 1 1 L _ 15 20 25 V Figure 5.3: Statistical errors in the PSFex photometry. These are negligible up to V = 22, at which point they start rising rapidly. The mean error at V = 23 is 0.032 and at V = 24 is 0.075. Very few outliers are present in the photometry. Two small 'glitches' at 13 < V < 15 and V ~ 17.5 indicate the magnitudes at which the different exposure time data sets were merged. The data was reduced using a preliminary version of the new T E R A P I X photometry routine PSFex (Point Spread Function Extractor) (E. Bertin 2000, private communica-tion) as described in Chapter 4. We used a separate, variable PSF for each C C D in the mosaic. The mean errors in the photometry were 0.015 mag at V = 22, 0.032 mag at V = 23 and 0.075 at V = 24. A statistical error plot (photon statistics for each star) for the cluster N G C 2099 is shown in Figure 5.3. Chapter 6 Star Counts and Cluster Extent 6.1 NGC 6819 Extent The existing measures of the apparent diameter of N G C 6819 makes it one of the small-est, rich open star clusters known. No clear definition of the methods used to determine the previously adopted cluster size have been published, however these results were de-rived using much smaller CCDs or photographic/photoelectric instruments which do not include a significant estimation of the background stars directly around the cluster. Most early estimates favored a larger cluster radius of 7' (Barkhatova et al. 1963 [5]) or 6.5' (King 1964 [60]; Burkhead 1971 [18]), although some are as small as 4' (Lindoff 1972 [73]). Each of the CFH12K CCDs are 7' in their shortest direction, and therefore the entire cluster would be concentrated within the inner 4 of the 12 CCDs (assuming the cluster is roughly centered in the mosaic). To determine the cluster centre from our data we first count stars in thin vertical and horizontal strips across the mosaic. This method is valid since there is no obvious trend in the background distribution, therefore suggesting no biases from differential extinction across the CCD's . The resulting profile was less than gaussian in shape, and produced limitations on the accuracy with which the centre could be determined. A n alternative approach which we implemented consisted of estimating the centre by eye, and then counting the number of stars in four equal quadrants around this centre. Next we moved the centre location around a small area and re-counted the number of stars until the value agreed in all four quadrants. Both approaches lead to similar values which estimate the centre of the cluster to be at (x,y) = (6250,4020) on a global coordinate system (aJ2ooo = 19 / ,41m17.73,5 J 2ooo = +40°11'17"). The error in each of the x and y directions is ~40 pixels (8.2"). This system combines the 6 chips (horizontal) and 2 chips (vertical) on the mosaic (2048 pixels horizontal/CCD 12288 total and 4096 pixels vertical/CCD 8192 total) into one coordinate system taking into account the gaps between the CCDs. The extent of N G C 6819 can now be found by counting the number of stars in different 26 Chapter 6. Star Counts and Cluster Extent 27 Table 6.1: Geometry of Annuli Annulus Radius (') Radius (pixels) N G C 6819 A l 0 < R < 0.5 0 < R < 145 A2 0 < R < 1.0 0 < R < 291 A3 0.5 < R < 1.5 145 < R < 437 A1+A3 0 < R < 1.5 0 < R < 437 A4 1.5 < R < 2.5 437 < R < 728 A5 2.5 < R < 3.5 728 < R < 1019 A6 3.5 < R < 4.5 1019 < R < 1311 A7 4.5 < R < 5.5 1311 < R < 1602 A8 5.5 < R < 6.5 1602 < R < 1893 A9 6.5 < R < 7.5 1893 < R < 2184 A10 7.5 < R < 8.5 2184 < R < 2475 A l l 8.5 < R < 9.5 2475 < R < 2767 Global 0 < R < 9.5 0 < R < 2767 N G C 2099 Al 0 < R < 3.5 0 < R < 1019 A2 3.5 < R < 7.0 1019 < R < 2039 A3 7.0 < R < 10.5 2039 < R < 3058 A4 10.5 < R < 13.9 3058 < R < 4049 Global 0 < R < 13.9 0 < R < 4049 annuli around the centre of the cluster. To avoid significant biasses from selection and incompleteness effects, we only use the stars with 15 < V < 20. Each successive annulus that we use is 1' in width, with the exception of those near the centre of the cluster (see Table 6.1 for annulus geometry). The value for the number of stars in each annulus is then normalized by the area of the respective annulus. We expect a drop-off and stabilization in the resulting distribution of number of stars vs radius from the centre as soon as we clear the cluster and are simply counting the constant background. This approach is the first ever for this cluster and is only made possible due to the large size of CFH12K. Figure 6.1 shows the results for the extent of N G C 6819, and indicates the cluster to be larger than all previous estimates. We find a drop off caused by the boundary of the cluster and the background between R = 8.0'-9.5' (1' = 290 pixels). There are most likely still a very small number of cluster member stars outside of R = 9.5', however it is difficult to resolve these from background. 6.1.1 Comparison to King Model The mass range of the stars on the main-sequence in the magnitude range 15 < V < 20 is not very large (0.70 - 1.5 M 0 ) . In a classic series of papers King (1962; 1966) [59, 61] Chapter 6. Star Counts and Cluster Extent 28 R ( a r c - m i n u t e s ) 1 .0 ' 3 . 0 ' 5 . 0 ' 7 . 0 ' 9 . 0 ' 1 1 . 0 ' 1 3 . 0 ' - 3 . 2 . 1 1 1 1 1 1 1 . - 3 . 4 -- 3 . 6 i - [ i : i i -- 3 . 8 - { i - 4 Cluster Extent R~9.5 '« --- 4 . 2 i i i—i—1—i i i—i—1— i— i— i— . i . . . . i . 0 1000 2 0 0 0 3 0 0 0 4 0 0 0 R ( p i x e l s ) Figure 6.1: Star counts in NGC 6819. The radius of the cluster is found to be ~9.5'. The flat distribution of stars after this point is just background. Error bars reflect both Poisson errors in the cluster and blank field star counts, and the error in locating the centre of the cluster. described how the density distributions of stars in globular clusters, open clusters, and some elliptical galaxies can all be represented by the same empirical law. This single-mass density law is very simple to use and involves only three parameters: a numerical factor, a core radius, and a limiting (tidal) radius. For systems with a large range in mass (more than a factor of 10), there are also more complicated multi-mass King models that can be used to represent density profiles with more relaxed constraints on mass ratios and distributions (e.g., DaCosta & Freeman 1976 [31]). In order to fit a King model to the density profile shown in Figure 6.1, we first subtract off the background star counts from each annulus (described in §9.2). Next we determine a value for the tidal radius of N G C 6819. The tidal radius of a star cluster could potentially cause the cluster extent to be truncated at a finite value. This is determined by the tidal influence of massive objects in the Milky Way (e.g., GMCs), which will remove the highest velocity stars from the cluster as they venture out to large distances from the centre. The tidal radius for N G C Chapter 6. Star Counts and Cluster Extent 29 -3.5 *—V a 0) u < -4.5 0 0.5 1 L o g ( R ) Figure 6.2: Single-mass King model shown to agree well with the cluster density distribution of NGC 6819. The arrows correspond to the core and tidal radii of the cluster. Error bars reflect both Poisson errors in the cluster and blank field star counts, and the error in locating the centre of the cluster. 6819 can be estimated using equation (6.1) (taken from Binney & Tremaine 1987 [13]), r.~0"3A (61) where m is the mass of the cluster (~26OOM 0) (see §9.5), M is the mass of the Galaxy within the cluster orbit, and D is the Galactocentric distance (Clemens 1985 [25]) of N G C 6819 (8.17 kpc - the distance is determined in §7.1.2). Therefore, the tidal radius for N G C 6819 is rt ~17 pc, much larger than the cluster radius (~6.9 pc). Additionally, Wielen (1991) [122] showed that it is not expected that a cluster of the size and mass of N G C 6819 would be dissolved before a minimum age of ~7 Gyrs (much older than the cluster age). This analysis takes into account both the effects of the Galactic tidal field for internal and external processes of cluster dissolution, and the evaporation of a star Chapter 6. Star Counts and Cluster Extent 30 cluster due to its internal relaxation (see §9.3). We fit the model to the data by varying the concentration parameter (Log(r t/r c)) until the best fit is obtained. This allows us to determine the core radius, rc ~ 1.75 pc, which strictly speaking has no physical meaning, although for many clusters it corresponds to the radius at which the surface brightness drops to half the central value. The model then gives us a tidal radius of rt ~ 17.5 pc. This model value compares well to the dynamical value found above, and the resulting King model profile is shown in Figure 6.2 to agree very well with the data. 6.2 NGC 2099 Radial Density Distribution Using similar techniques to those used for N G C 6819, we first determine the centre of N G C 2099 to be at coordinates aJ2ooo = 05 / l52m17.6*, 5J2ooo = +32°32'08" with an accuracy of about 30". The annuli used to build the radial density distribution are each 2' in width and all stars with V < 22 are used in the counting (see Table 6.1). The background field star population is taken from the outer four CCDs of CFH12K and then scaled to match the area in each annulus. The resulting distribution is shown in Figure 6.3 with a superimposed single-mass King Model. For a core radius of 2 pc, the model tidal radius is found to be ~ 18 pc. Finally, we compare the tidal radius from the model to the expected tidal radius of the cluster given the cluster's mass and it's position in the Galaxy as shown in equation 6.1. The mass of the cluster in this calculation (m = 2515 M©) has been corrected by including those stars between our faintest magnitude bins and the hydrogen burning limit (0.08 M 0 ) as described in §9.6. M is the mass of the Galaxy within the cluster's orbit (1.02 x 10 1 1 - Clemens 1985 [25]) and D is the Galactocentric distance of N G C 2099 (~10 kpc, almost directly towards the anti-centre of the Milky Way - see §7.2.2). Solving equation 6.1 for the tidal radius of N G C 2099 gives rt = 20.2 pc, in excellent agreement with the model value. The apparent cluster radius of N G C 2099 is only ~6 pc and therefore some stars may have evaporated past this radius towards the outer parts of the cluster (this may be an explanation for the faint, red feature in the background C M D - see Figure 7.2). Such an effect would be important as it would bias the faint end luminosity function. We use the King model to determine the total mass of N G C 2099 out to the tidal radius of the cluster. This involves calculating the projected central density of the cluster as well as Chapter 6. Star Counts and Cluster Extent 31 Log(R) Figure 6.3: Single mass King model found to be in good agreement with the NGC 2099 radial density distribution. Only stars with V < 22 have been counted to avoid incompleteness effects. The arrows mark the core, half-mass and tidal radii of NGC 2099. several model dependent parameters (see Gunn & Griffin 1979 [40]). This value is found to be 2550 M© and is in excellent agreement with the value found by integrating the mass function to our limiting radius (2500 M 0 - see §§9.5 and 9.6). Further discussion of the dynamical state of the clusters can be found in §9.4 Chapter 7 Colour-Magnitude Diagrams and Cluster Parameters 7.1 The NGC 6819 Colour-Magnitude Diagram Figure 7.1 shows colour-magnitude diagrams for both the entire cluster (R < 9.5') and a blank field taken from an equal area of the outer chips on the mosaic. The cluster colour-magnitude diagram (left) shows a very tight main-sequence and turn-off region. For this diagram we adopt a 0.50 stellarity confidence limit (see §7.1.3). The short 1-second exposures have allowed for the measurement of the red giant clump and giant branch at the bright red end of the colour-magnitude diagram for N G C 6819. A significant contribution of these red giant stars is rare in most open clusters due to their young age and poor population, but are crucial for testing stellar models (see Chapter 8). Some potential white dwarf candidates can also be seen in the faint blue end of the colour-magnitude diagram. Further analysis of these stars is presented in Chapter 10 where we fit cooling curves to the white dwarf sequence after eliminating field objects and applying more stringent confidence limits. We also note the presence of a significant contribution from approximately equal mass binary stars in the cluster. 7.1.1 Reddening Prior knowledge of key cluster parameters, such as reddening, distance and metallicity, is very important in order to transform the theoretical plane of isochrones into the obser-vational data plane. The uncertainty in these values can lead to an ambiguous isochrone fit. For N G C 6819, the reddening value causes the largest concern as E ( B - V ) has been quoted as large as 0.30 (Lindoff 1972 [73]; Auner 1974 [3]) and as low as 0.12 (Burkhead 1971 [18]). The earlier high estimates have been acknowledged as being derived from poor U filter photographic photometry. More recently, a reddening value of 0.15 (Canterna et al. 1996 [19]) from photoelectric observations was found to agree with that obtained by Burkhead (1971) [18]. A much more recent C C D study has found E ( B - V ) = 0.16 for 32 Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 33 Figure 7.1: Rich, tight main-sequence and turn-off of NGC 6819, clearly seen (left). Also shown is a blank field of equal area taken from the outer four CCDs of CFH12K (right). A 0.50 stellarity cut has been applied to the data. the reddening, and ( m - M ) v = 12.35 for the distance modulus (Rosvick k VandenBerg 1998 [97]). This reddening value has been determined by first comparing the red giant clump of N G C 6819 with that of M67, and then adjusting the value slightly based on the theoretical isochrone fits. Additionally, a recent spectroscopic study of the red clump stars in N G C 6819 has yielded a reddening value of E ( B - V ) = 0.14 ± 0.04 (Bragaglia et al. 2001 [16]), for a cluster metallicity of [Fe/H] = +0.09 ± 0.03. Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 34 7.1.2 Cluster Distance by Main-Sequence Fitting N G C 6819 is four times the age of the Hyades (Perryman et al. 1998 [83]) and therefore a much smaller region of the colour-magnitude diagram is available to fit to the Hyades main-sequence fiducial for the cluster distance because the high-mass stars have evolved off the main-sequence. Fortunately, our deep C C D photometry provides a longer un evolved main sequence for the fitting. For a reddening of E(B—V) = 0.10 (which has been model-massaged, but is consistent within the uncertainty in the recent Bragaglia et al. 2001 [16] study), we find ( m - M ) v = 12.30 ± 0.12. Correcting for extinction ( A v = 3 .1E(B-V)) , the absolute distance to N G C 6819 is therefore 2500 pc. Our value is slightly larger than most previous studies due to the lower adopted reddening value (which allows a slightly better isochrone fit): Rosvick and VandenBerg (1998) [97] (d = 2350 pc), Auner (1973) [3] (d = 2170 pc), and Lindoff (1972) [73] (d = 2200 pc). 7.1.3 Stellarity Source classification is very important for studies of faint objects in the cluster so that an accurate distinction between star or galaxy can be made. The star/galaxy cut will affect both evolutionary tests of stars that become white dwarfs and the luminosity function for these stars. Our data set extends to V ~ 25 implying that most of the objects measured here are faint and have low signal-to-noise ratios. We use SExtractor (Bertin & Arnouts 1996 [11]) to assign a stellarity index to all objects on all CCDs in our data. This stellarity index is determined through a robust procedure that uses a neural network approach. The coefficients of this neural network were found by Emmanuel Bertin by training the routine in simulations of artificial data. The classification scheme attempts to determine the best hypersurface for an object which can be described by either of two different sets of parameter vectors; one for stars and the other for galaxies. Figure 4.1 showed the variation of this stellarity index with magnitude. The difficulty arises in choosing a confidence limit to separate out stars from galaxies. Objects with an index of 0 are most likely galaxies and objects with an index of 1 appear stellar. Additionally, some of the objects in the 'clump' seen in Figure 4.1 at 23 < B < 24.5 and 0.75 < stellarity B < 0.95 are most likely stars. We find a clear separation between a white dwarf cooling sequence and background objects at approximately a 0.50 stellarity limit, and an even better separation at 0.75, however at this more strict cut we lose some Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 35 faint objects (see §10.2.1). Later, we will use a statistical method to eliminate possible galaxies and background objects. We note however that it is difficult to determine this cut accurately without a spectroscopic classification for these objects. We can estimate the number of galaxies that we expect in our cluster field (NGC 6819 - inner 9.5'), by considering galaxy counts (see Woods & Fahlman 1997 [124]) at high latitude and correcting for extinction in our field. Based on these statistics and prior to any stellarity cut, there are far more objects in our faint magnitude bins than the number of expected galaxies. For example, we expect less than 7% of all objects to be galaxies for 21 < V < 22, less than 22% for 22 < V < 23, whereas the number of expected galaxies rises to ~70% for 23 < V < 24. Clearly, this last bin is of interest as the number of expected galaxies is comparable to the total number of all objects. However, after applying a 0.75 stellarity cut, we eliminate 1673 of 1701 objects from this bin, a number far greater than the number of expected galaxies, 1189. Therefore we can conclude that a significant number of faint objects in the cluster field are most likely stellar, however we may need to relax our 0.75 stellarity cut to include some of these fainter stars. A cut at 0.50 provides a better agreement between the number of objects thrown out and the number of expected galaxies. Further details of the stellarity index of faint blue objects in N G C 6819 is given in §10.2.1. 7.2 The NGC 2099 Colour-Magnitude Diagram In Figure 7.2 we present colour-magnitude diagrams for both the cluster and a background field after applying a 0.50 stellarity cut from SExtractor (Bertin & Arnouts 1996 [11]) to remove obvious faint galaxies (see §7.1.3). The cluster field in this Figure has been scaled down in area to be consistent with the area represented in the background field (ie. each object in the background field represents 1.37 cluster field objects). This has been done by eliminating a ring of objects centered close to the half-mass radius (4.5 pc) of the cluster (a full colour-magnitude diagram consisting of the entire cluster population is presented later, Figure 7.3). The cluster colour-magnitude diagram shows a magnificent tightly constrained main-sequence extending over 12 magnitudes from the cluster turn-off V ~ 12 ( M v = 0.45) to V ~ 23.5 ( M v = 11.95). A tight red giant clump is seen consisting of 20 stars (see Figure 7.3 for all stars). There is also a significant background disk star population below the main-sequence in the cluster colour-magnitude diagram Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 36 and in the background field. This distribution arises due to the low Galactic latitude of the cluster and clearly overlaps significantly with the faintest cluster members on the colour-magnitude diagram. In fact, the feature in the background field at 20 < V < 23.5 and B—V ~ 1.7 (which can also be seen in the N G C 6819 data set) may be faint cluster red stars located between our apparent cluster radius and the tidal radius (see §6.2). Figure 7.2: Rich, tight and long main-sequence of NGC 2099 clearly separated from the back-ground/foreground distribution. The cluster field has been scaled down in area by a factor of 1.37 (as described in text) so that the relative populations on the two diagrams can be compared (see Figure 7.3 for all stars). A 0.50 stellarity cut has been applied to both diagrams. The cluster colour-magnitude diagram in Figure 7.2 also shows several faint, blue objects that will be discussed in detail in Chapter 10. Although there is field star and background galaxy contamination, the density of objects is clearly greater in the cluster colour-magnitude diagram indicating that a significant number of these objects are cluster Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 37 T i j i i i i p—i i i i | i r n i | i i i r - | i i [ i | i rp i | i i i i | i r—i i | i r . . I i i i . I i i i i I • T I T , I • . . . I . . . . I • Tli • I i i i i I i i i i I i i 0 1 2 0 1 2 0 1 2 B - V B - V B - V Figure 7.3: Present CCD photometry compared to NGC 2099 studies of West (photographic) and Mermilliod (photoelectric). Unlike Figure 7.2, this CMD (right) contains all cluster stars out to R = 13.9'. white dwarf members (see §10.3.1 for statistical arguments involving the subtraction of field stars and removal of galaxies). These stars have quickly evolved (~104—105 years) from the red giant phase to the white dwarf phase and are now slowly piling up on the colour-magnitude diagram at a magnitude that can be used to independently infer the cluster age. The N G C 2099 colour-magnitude diagram clearly shows the faintest objects in this 'clump' to be at V = 23.5, after which point there is a significant gap signifying that we have detected the termination point in the white dwarf cooling sequence (the complete colour-magnitude diagram in Figure 7.3 shows this better). We also note that the photometry for these bluer objects is fainter than for the main-sequence by over 1 magnitude. The cluster colour-magnitude diagram (from Figure 7.3) also shows several Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 38 (5) very blue and bright objects, some of which may in fact be stars that have shed their outer layers during the planetary nebula stage are now cooling to become white dwarfs. We do not expect to detect many, if any, objects at this stellar evolutionary stage because the stars spend a very small amount of time during this phase. A crude estimate of the expected number of such objects can be determined by comparing the evolutionary time (<105 years) of these stars to the numbers (20) and timescales (100 Myrs) of red giants in the clump of N G C 2099. Although the theoretically expected number is less than one, three of the objects in the colour-magnitude diagram are centrally concentrated in the cluster, not image defects, and have a high stellarity index. Spectroscopic confirmation is needed to determine whether any of these objects are cluster members and potentially among the most interesting stars in N G C 2099. If confirmed, these objects would provide crucial observational constraints on the phases of evolution from the planetary nebulae stage to the white dwarf stage (Iben & Renzini 1983 [50]) and on neutrino cooling of pre-white dwarf stars (O'Brien et al. 1998 [79]). 7.2.1 Cluster Metallicity A n ideal testing of theoretical stellar isochrones by using observational colour-magnitude diagrams requires prior knowledge of the cluster metallicity, reddening and distance. Unfortunately, prior N G C 2099 data have not set tight constraints on any of these. Although the published literature does not show any detailed spectroscopic abundance studies of the cluster, Lynga's Fifth Catalogue of Cluster Parameters (see e.g., Janes, Tilley & Lynga 1998 [51]) lists the cluster metallicity to be [Fe/H] = 0.09 or Z ~ 0.020. Additionally, the study of Mermilliod et al. (1996) [77] experimented with different models (Schaller et al. 1992 [106] and Bertelli et al. 1994 [10]) and examined the colours of the red giant clump in N G C 2099 also concluding with a best fit Z = 0.020 isochrone. Since there have been no studies contradicting this metallicity, we adopt the Z = 0.020 value as the most likely cluster metallicity, but will also consider a Z = 0.025 (Hyades-like) value (see §8.3). Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 39 7.2.2 Cluster Reddening and Distance - Main-Sequence Fitting with the Hyades The only previous model-independent reddening estimate of N G C 2099 comes from the photographic work of West (West 1967 [121]), E ( B - V ) = 0.27. The main sequence of these data contains a large scatter. The colour-colour plots also suffer from large uncertainties (photometric spread ~0.5 magnitudes in B - V ) . The study of Mermilliod et al. (1996) [77] compared their photoelectric data to stellar isochrones and estimated the reddening to be E ( B - V ) = 0.29. The advantages of determining a reddening value from the present data over the previous work include the higher stellar density, deeper colour-magnitude diagram and lower uncertainties in the photometry. See Figure 7.3 for a comparison of the present colour-magnitude diagram with these studies. Unfortunately, the R filter photometry for N G C 2099 was unable to provide reddening constraints as the slope of the reddening vector (E(V—R)/E(B—V) = 0.78) is too similar to the observed main-sequence in the colour-colour plot. An accurate U B V or B V I analysis is desirable for the cluster. A n alternative method of determining the reddening of a star cluster was demon-strated in the Richer et al. analysis of M4 (Richer et al. 1997 [94]), where a set of subdwarf stars were fit to the observed main-sequence to simultaneously solve for both the reddening and apparent distance modulus. Due to the degeneracy between these two quantities, the results established more stringent ranges on the reddening than for the apparent distance modulus. We adopt a similar technique by using the well established Hyades main sequence to fit N G C 2099. This method is ideal for the present comparisons due to the long main sequences of the two clusters (stars with mass below 2.6 M Q in N G C 2099 have not yet evolved off the main-sequence), accurate photometry (several observ-able features are present in both main-sequences), similar metallicity of the clusters (Z = 0.024 for the Hyades) and similar age for the clusters (see §8.3). The present fiducial main-sequence of the Hyades open star cluster is based on a new analysis of Hipparcos individual parallaxes by de Bruijne, Hoogerwerf & de Zeeuw [17]. The fiducial was calculated by binning known cluster members at 0.5 magnitude intervals from My = 1.5 to M y = 9 with a small colour cut to remove outliers. Binaries were removed, however we include the brightest binary component which defines the turn-off of the cluster (star 92TA). We first adjust the N G C 2099 colour-magnitude diagram by +0.03 Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 40 in B - V to account for the A Z ~ 0.005 metallicity difference in the clusters. This was determined by comparing two isochrones based on the same input physics, but of different metallicity. We then compare observable features in our tight main-sequence, such as the 'kink' in the main-sequence at ( B - V ) 0 ~ 0.4 caused by changes from convective envelope models to radiative envelopes, as well as the general shape and slope of the main-sequence to the Hyades fiducial to solve for the best combination of reddening and apparent distance modulus. The results point towards a reddening of E ( B - V ) = 0.23 ± 0.03 and an apparent distance modulus of (m—M) v = 11.65 ± 0.13, where the uncertainty in the latter value reflects the range of main-sequence fitting distance moduli obtained by using reddening values at the extremes of the ±0.03 uncertainty. Figure 7.4 illustrates the remarkable agreement between the detailed features of the two main-sequences for these values (the bright main-sequence is more strongly affected by evolution in the Hyades due to its increased age). Additional reddening constraints are also provided by comparing the colours of the N G C 2099 red giant clump to the Hyades clump (which is reproduced well by our current theoretical models). The small age difference between the two clusters (see §8.3) would affect the luminosity of the clump stars significantly, but only slightly affect the colour. On this metallicity corrected plane, the clump of N G C 2099 can not be redder than the Hyades (would imply Z > 0.024), however for a reddening value E(B—V) < 0.18, this condition is violated and therefore such a low reddening can be ruled out. Given the uncertainties in the reddening, distance modulus and model colours (dis-cussed in §8.3) we incrementally massage the reddening by —0.02 to allow a slightly better isochrone fit of the model to the un-evolved main-sequence of N G C 2099. There-fore, our best estimate of the reddening of N G C 2099 is E ( B - V ) = 0.21 ± 0.03, with a corresponding distance modulus of ( m - M ) v = 11.55 ± 0.13. The uncertainties in these values are not known well enough to establish different upper and lower limits based on the adjustment from the main-sequence fitting values. The higher limits established ear-lier are ruled out by the models so to remain consistent we will use the model-massaged values throughout. The corresponding distance to N G C 2099, using A v = 3 .1E(B-V) , is 1513 ± 133 pc. The error in the true distance modulus (10.90 ± 0.16) combines four indi-vidual errors, and accounts for correlations between them: (1) a scale factor translation ( A ( B - V) to AV) from the B - V axis to the V axis to account for the reddening un-certainty, (2) an estimated main-sequence fitting uncertainty at a fixed reddening value, (3) the error in the extinction, A,4V = Z.1AE(B-V) and (4) the colour uncertainty due Chapter 7. Colour-Magnitude Diagrams and Cluster Parameters 4 1 Figure 7.4: NGC 2099 main-sequence brought to the Hyades (red points) plane by adjusting the metallicity, reddening and distance as described in §§7.2.1 and 7.2.2. The resulting detailed shape of the NGC 2099 main-sequence is found to be in excellent agreement with the Hyades cluster, thus simplifying the main-sequence fitting. The turn-off and clump stars in the Hyades clearly indicate a slightly older cluster, as confirmed by the isochrones (Aage ~ 100 Myrs). to an estimated metallicity uncertainty, Az. We evaluate and provide a more detailed description of these terms in §10.3.4. Chapter 8 Theoretical Isochrones Testing theoretical stellar evolutionary models requires knowledge of the cluster metal-licity, reddening and distance. This testing is very important in order to further refine the models. For example, theoretical models involving the amount of convective core overshooting that should be used or the temperature at which the slope changes in the faint-end of the main-sequence (caused by opacity effects), lack observational constraints. Additionally, the testing allows for an age determination for the cluster. The present pho-tometry of these clusters is ideal for these comparisons as there are a large number of stars on a tightly constrained main-sequence. The turn-off of the cluster, as well as the red giants and giant branch stars are also useful as they allow for modeling of stars in the helium burning phases of evolution. 8.1 Model Description The theoretical models that we will use for this project have been calculated especially for the C F H T Open Star Cluster Survey at the Rome Observatory and are up-to-date in the input physics (R Ventura & F. D'Antona). The tracks were built by adopting the ATON2.0 code for stellar evolution, a detailed description of which can be found in Ventura et al. (1998) [115]. Convection has been addressed within the Full Spectrum of Turbulence (FST) framework (Canuto k Mazzitelli 1992 [20]). Chemical mixing and nuclear burning have been treated simultaneously with a diffusive scheme: convective velocities have been computed according to eqs. (88), (89) and (90) in Canuto, Gold-man & Mazzitelli (1996) [21]. We assume that the helium abundance scales with the metallicity content according to the relation A Y / A Z = 2. The models adopt convective core overshooting by means of an exponential decay of turbulent velocity out of the for-mal convective borders as fixed by Schwarzschild's criterion; this behavior of velocity is consistent with approximate solutions of the Navier-Stokes equations (Xiong 1985 [125]), and with the results of numerical simulations (Freytag, Ludwig & Steffen 1996 [36]). A 42 ) Chapter 8. Theoretical Isochrones 43 value of C = 0.03 for the free parameter giving the e-folding distance of the exponential decay has been adopted (Ventura et al. 1998 [115]). The theoretical isochrones were transformed into the observational plane by making use of the Bessell, Castelli & Plez (1998) [12] conversions. The lower main-sequence (M < 0.7 M 0 ) has been calculated by adopting NextGen atmosphere models (Hauschildt, Allard & Baron 1999 [44]). Models based on a grey atmosphere approximation give almost identical results for larger masses. For M < 0.47 M 0 (or T e f f < 3500 K) the transformations of Hauschildt, Allard & Baron (1999) [44] in B—V are not very reliable and so the faint end of the isochrones terminate at this mass. However, the bolometric correction for cooler stars in the models can still be used to establish a mass-luminosity relationship extending to M ~ 0.25 M 0 (this is used in §9.5). 8.2 N G C 6819 Fit Figure 8.1 shows the fit of the N G C 6819 data to a 2.5 Gyr old stellar isochrone. We use a reddening value of E(B—V) = 0.10 and a derived distance modulus of (m—M) v = 12.30 ± 0.12 for this and further theoretical comparisons. It is clearly seen that the slope of the bright main-sequence as well as the turn-off and red giant clump are all in excellent agreement with this model, which uses Z = 0.02. Although still quite good, the fit to the lower main-sequence is slightly blue of the data, which is most likely a result of incorrect colour transformations in the models for cool stars (see next section for a more detailed discussion). 8.3 N G C 2099 Fit Figure 8.2 shows three panels with two isochrones assuming extra mixing beyond the formal convective borders, but for different ages, and one isochrone calculated assuming no extra mixing. We argue the best fit in Figure 8.2 is in the left panel (520 Myrs -assuming core-overshooting) and for this combination of reddening, distance, metallicity and age, the isochrone reproduces several features of the colour-magnitude diagram nicely. Several comparisons can be drawn from the fit of the theory to the observations. First the lower main-sequence in N G C 2099 is clearly redder than the isochrone. This slope change in the main-sequence, caused by the onset of H 2 dissociation-recombination in Chapter 8. Theoretical Isochrones 44 0 1 2 B-V Figure 8.1: Theoretical isochrone of age 2.5 Gyrs, found to fit the turn-off well. The slope of the main-sequence and the location of the red giant clump also agrees well with the isochrone. Some potential white dwarfs are also evident in the faint blue end of the colour-magnitude diagram (we have applied a 0.50 stellarity cut). the stellar envelope (Copeland, Jensen & Jorgensen 1970 [26]) is not reproduced well by stellar models employing grey atmosphere boundary conditions (see e.g., Castellani, DeglTnnocenti & Prada Moroni 2001 [22] for the lower main-sequence of the Hyades). The non-grey boundary conditions employed in our models for the lower main-sequence (J. Montalban 2001, private communication) are in better agreement with the general shape of the main-sequence, but the models are still too blue by ~0.05 mag in B—V. This disagreement can be explained in part due to the use of the Bessell, Castelli & Plez (1998) [12] colours at such low temperatures. Photometric spread in the data for faint stars may also partly contribute to such an effect. Also, the turn-off of N G C 2099 falls Chapter 8. Theoretical Isochrones 45 slightly redder than the model, although the shape is consistent with the data. Despite considerable effort, we find it difficult to reproduce the details of this feature in the N G C 2099 colour-magnitude diagram with current models (while maintaining a good fit to the luminosity of the red giant clump). The problem here is very similar to that for the Hyades cluster, for which several groups are unable to provide a good fit to the turn-off. A n extensive discussion concerning the difficulty of fitting both the turn-off and the lower main-sequence region of the Hyades by adopting the same set of colour-temperature calibrations can be found in de Bruijne, Hoogerwerf & de Zeeuz (2001) [17]. The fact that the N G C 2099 data is in excellent agreement with the Hyades data (see Figure 7.4), and models fail to reproduce the turn-off feature in both clusters, will motivate a deeper understanding of this feature of the colour-magnitude diagram for intermediated aged clusters. Conversely, both the colour and luminosity of the red giant clump are in excellent agreement with the data; the latter provides the age. The shape and slope of the main-sequence (from M v = 2.5 — 8.0) is also in excellent agreement with the Z = 0.020 model. For example, the 'kink' in the main-sequence at ~1.3 M Q described earlier is modelled perfectly. The age estimate of 520 Myrs for N G C 2099 is also consistent with Figure 7.4 which shows both the peel-off of the main-sequence stars in the Hyades (age ~ 600 Myrs with our models) and the luminosity of the clump stars to be slightly fainter than that for the corresponding stars in N G C 2099. Fitting exactly the location of the turn-off region of the cluster requires a slightly higher age. The middle panel of Figure 8.2 shows the fit for an age of 580 Myrs. The shape of the turn-off of the cluster, as well as the hook at the top of the main-sequence caused by the contraction of stars that have exhausted their hydrogen supply (very difficult to see), are both modelled well. However, a close examination of the luminosity of the red giant clump shows a discrepency of ~0.20 magnitudes. Given the importance and relative simplicity of fitting the clump luminosity rather than the turn-off, we believe this to be a poorer fit than that presented in the left panel. Due to the uncertainty in the reddening value established in §7.2.2, we also note that we can not completely rule out ages as low as 450 Myrs for N G C 2099 (the value found by Mermilliod et a l ) . Using a reddening value near our upper limit ( E ( B - V ) ~ 0.25), we obtain a decent fit to the luminosity of the clump stars, however, the subsequent fit to the main-sequence for this reddening is poor. We also examined the possibility of fitting the observed N G C 2099 stars with an Chapter 8. Theoretical Isochrones 46 isochrone of metallicity Z = 0.025. For this, we derived the reddening and distance modulus by comparing the N G C 2099 main-sequence with the Hyades fiducial with no metallicity shift. The corresponding best fit to the luminosity of the clump is obtained for an age of ~550 Myrs. The N G C 2099 clump is brighter and slightly redder than the Hyades clump, indicating a younger age for N G C 2099 with respect to the Hyades. The fit to the slope of the main-sequence for this metallicity is clearly worse than for a Z = 0.020 model. As we will discuss later in Chapter 10, one of the possible uses of white dwarfs as chronometers is to check the consistency of the turn-off and white dwarf ages. If we fit the cluster colour-magnitude diagram with theoretical isochrones based on stellar models of the same metallicity (Z = 0.020), but calculated assuming no extra mixing beyond the formal convective borders, the best agreement between the theory and observations is obtained for an age of ~300—350 Myrs, using the same distance modulus and reddening as in the overshooting case. In this case, which is based on reproducing the luminosity of the clump, we get a very poor fit to the turn-off region, which is noticeably bluer than the observed stars (see right panel of Figure 8.2). A n age of ~400—450 Myrs would lead to an acceptable fit of the turn-off of the cluster, but in this case the theoretical luminosity of the clump would be ~0.4 magnitudes fainter than the observed clump stars, a gross disagreement. The above comparisons suggest that a certain amount of overshooting from the border of the convective core is required and that a solar metallicity (Z = 0.020) model of age 520 Myrs provides the best fit to the observed N G C 2099 colour-magnitude diagram (based on the luminosity of the clump). However, a slightly larger age provides a better fit to the turn-off of the cluster. Models without convective core overshooting do not reproduce the observed data well. Ultimately it is the white dwarf cooling age of the cluster that will be the decisive discriminant (see §10.3.3). Chapter 8. Theoretical Isochrones 47 0 1 2 0 1 2 0 1 2 (B-V)„ (B-V) 0 (B-V) e Figure 8.2: Left panel shows the best fit isochrone, based on the luminosity of the clump stars, to the observed NGC 2099 data. A better fit to the turn-off of the main-sequence is shown in the middle panel, however, a close examination of the clump stars also shows a ~0.20 magnitude discrepency when compared to this model. A non core-overshooting model (right panel) provides a poor fit to the data. See §8.3 for a discussion of these results. Chapter 9 Dynamical Studies The rich stellar populations of these clusters motivate star count studies to determine the cluster populations, their dynamical state, and investigate the luminosity and mass functions. The shapes of the luminosity functions for these clusters are very important to establish observational constraints on the timescales and levels at which we expect dynamical evolutionary effects in clusters to take place (see §9.3). The colour-magnitude diagram of N G C 6819 indicates some similarities to other old open star clusters such as M67, N G C 6633, N G C 752 (Prancic 1989 [35]), and N G C 188 (von Hippel & Sarajedini 1998 [117]) with regards to the distribution of stars. For example, by simply looking at the density of stars along the main-sequence as a function of magnitude, it is clear that the majority of the cluster members are bright, not faint. After accounting for incompleteness at the faint end, such an effect can be investigated by accurately counting the number of cluster stars as a function of magnitude. This resulting luminosity function for such a cluster will clearly be either flat {jj^ ~ 0) or slightly negatively sloped (jj^ < 0). N G C 2099 is a much younger cluster and should possess a rising luminosity function at faint magnitudes as is seen for other young clusters, such as the Pleiades (Lee & Sung 1995 [69]). We define the cluster stars by first creating a main-sequence fiducial (clipping ob-jects with (B—V) > 3.5a from the mean) after isolating the main-sequence from the background distribution. We then use a clipping routine to create an envelope around this fiducial based on the errors in the photometry (envelope broadens out towards faint magnitudes). The counting of the stars is done within this envelope, for both the cluster colour-magnitude diagram and the background colour-magnitude diagram, with the raw cluster luminosity function coming from the difference between the counts in the two fields (after accounting for aerial differences - see next section). However, in order to accurately count stars, we must first determine incompleteness corrections. 48 Chapter 9. Dynamical Studies 49 9.1 Incompleteness Corrections and Counting Uncertainties Before we can interpret results from the star counts in our data we must correct the data for incompleteness in the number of detected objects. This effect is typically negligible for brighter objects and increases for fainter sources. We produce an artificial catalogue of input stars for which we know the magnitudes and colours. A small number of these stars is added uniformly in several trials (5—7) in proportion to their numbers in the raw cluster luminosity function so as not to affect the crowding statistics of the field. These stars are chosen along the same slope of the main-sequence, and a separate sequence for the location of the white dwarfs (see Figure 9.1). After adding these stars to our data frames, we re-reduce the new data in an identical manner to the original data in the cluster. This involves running PSFex on the data and merging the output V and B files. We then count the number of stars per magnitude bin that were recovered (in both V and B - see Figure 9.1). This analysis is carried out for 2 cluster CCDs and 2 background field CCDs in order to make the analysis reliable. The cluster CCDs represent 2 of the central chips of CFH12K, and provide statistics on a 14' x 14' region near and including the central core of the clusters. Each of the background field CCDs represent a 14' x 7' area taken on opposite sides of the cluster. To give an example of the typical incompleteness, we find the N G C 2099 data set to be 100% complete down to V = 19 in the background field and 100% complete to V = 17 in the cluster field. For main-sequence stars the correction at V = 23 in the cluster field is 1.219 (ie. Nadded/Nrecovered = 1.219). The white dwarfs are more complete than the main-sequence stars in almost all bins. This is expected as these stars are brighter in the B band for a given V band magnitude than the main-sequence stars. A summary of the incompleteness factors for the 2 cluster and background field CCDs in N G C 2099 is given in Table 9.1. 9.1.1 Incompleteness Errors With these star counts, it is also important to have an estimation of the errors. Bolte (1989) [15] gives a careful account in determining incompleteness errors in the analysis of M30. This reasoning assumes that the counting uncertainties are derived from a Poisson distribution, and that the artificial star count uncertainties are derived from a binomial distribution. Furthermore the errors in the incompleteness and the raw star counts are assumed to be uncorrelated. If we write the corrected number of stars in any magnitude Chapter 9. Dynamical Studies 50 Figure 9.1: Both input stars (left) and recovered stars (right) in the incompleteness tests for both the main-sequence and potential white dwarf cooling sequence. bin as n = n^JL, where nobs are the raw counts in the bin, and / is the ratio of the number of recovered artificial stars to the number added for each magnitude bin (/ = n™c-™e"A), '^added then the variance in n is 2 - - 2 ( ^ ) 2 + * / Q 2 - ( 9 - 1 ) The variance in nobs is simply aloia = n o b s , and the variance in / is o) = Performing the partial differentiations in equation (9.1) gives the desired variance in n, ^1 _ nobs _|_ (1 — f)nob$ £g 2) n P naddedP Chapter 9. Dynamical Studies Table 9.1: Completeness Corrections for N G C 2099 Data 51 V mag No. Stars Input No. Stars Recovered (Cluster/B'kgn'd) Completeness Correction (Cluster/B'kgn'd) Main-Sequence 11.0-12.0 34 34/34 1/1 12.0-13.0 24 24/24 1/1 13.0-14.0 26 26/26 1/1 14.0-15.0 46 46/46 1/1 15.0-16.0 34 34/34 1/1 16.0-17.0 54 54/54 1/1 17.0-18.0 50 48/50 1.042/1 18.0-19.0 58 56/58 1.036/1 19.0-20.0 70 66/68 1.061/1.029 20.0-21.0 56 54/54 1.037/1.037 21.0-22.0 100 94/96 1.064/1.042 22.0-23.0 78 64/66 1.219/1.182 23.0-24.0 100 72/84 1.389/1.190 24.0-25.0 62 36/50 1.722/1.240 White Dwarfs 21.0-22.0 14 13/14 1.077/1 22.0-23.0 30 26/28 1.154/1.071 23.0-24.0 38 30/32 1.267/1.188 24.0-25.0 34 20/26 1.700/1.308 We use this method of determining errors for both the field and background stars, and then add the errors for the difference in quadrature. 9.2 Final Star Counts The completeness-corrected number of stars in the cluster can now be determined in three steps. First, we multiply the cluster field incompleteness correction by the number of stars in the cluster in that magnitude range. Second, we multiply the background field incompleteness correction by the number of background field stars, and multiply the result by 1.37 to account for the difference in aerial coverage of the background (for the case of N G C 2099). Finally, we subtract the two and obtain the corrected star counts. For the global star counts, we use the inner 13.9' for the N G C 2099 cluster field (inner 9.5' for N G C 6819) and establish a background field area from the outer portions of the outer CCDs. The cluster density at this radius has dropped off significantly from the central regions, however, in the absence of a strong field star population, cluster members Chapter 9. Dynamical Studies Table 9.2: Cluster Star Counts (Raw / Corrected) for N G C 6819 52 V mag A1+A3 A4 A6 A8 A10 V mag GLOBAL 15.5-16.5 (Raw) 63 91 68 23 29 15.0-16.0 441 Corrected 64.3 (8.4) 96.8 (11.3) 72.6 (10.6) 25.6 (9.8) 32.1 (10.6) 459.5 (30.4) 16.5-17.5 33 51 52 21 15 16.0-17.0 351 Corrected 35.4 (7.7) 57.1 (9.9) 58.5 (10.8) 25.9 (12.6) 20.2 (14.2) 378.5 (32.5) 17.5-18.5 20 33 61 36 13 17.0-18.0 315 Corrected 22.4 (6.7) 38.0 (9.1) 71.3 (14.6) 44.4 (15.5) 19.5 (16.0) 333.9 (40.5) 18.5-19.5 14 23 35 37 18 18.0-19.0 315 Corrected 15.1 (4.9) 25.5 (7.1) 39.2 (9.6) 42.5 (11.7) 22.8 (12.2) 385.6 (46.3) 19.5-20.5 3 28 19 12 25 19.0-20.0 209 Corrected 3.2 (4.7) 33.5 (7.3) 25.0 (10.4) 19.2 (13.4) 36.4 (16.5) 263.1 (40.3) 20.5-21.5 6 23 22 14 34 20.0-21.0 201 Corrected 8.6 (4.7) 31.4 (9.7) 32.1 (13.1) 25.1 (17.4) 52.7 (21.0) 293.4 (59.7) 21.5-22.5 18 35 29 13 21.0-22.0 193 Corrected - ( - ) 26.8 (11.2) 53.1 (19.3) 49.9 (31.3) 29.8 (15.4) 252.8 (87.9) 22.5-23.5 24 17 — 22.0-23.0 169 Corrected - ( - ) - ( - ) 47.1 (20.1) 42.0 (18.1) - ( - ) 286.6 (181.8) could have been detected at greater radii. The final corrected star counts for N G C 6819 are presented in Table 9.2 and for N G C 2099 in Table 9.3. In these tables, the first row of each magnitude bin consists of raw counts (cluster field — blank field x aerial correction) whereas the row underneath this one contains the incompleteness corrected numbers. Also shown in parentheses are the errors in these counts, as calculated from the analysis given above. 9.2.1 Luminosity Functions The global luminosity function for N G C 6819 is plotted in Figure 9.2, where the dashed line represents the raw counts and the solid line the incompleteness corrected counts. As expected, the luminosity function is almost flat (slightly negatively sloped), most likely due to dynamical evolution in this relaxed cluster (see §9.3). We do not see a drop off at the faint end of the luminosity function. Although large errors make it difficult to determine, this evidence suggests that the lowest mass main-sequence stars may not have been detected in this deep photometry. Integrating the luminosity function and Chapter 9. Dynamical Studies Table 9.3: Cluster Star Counts (Raw / Corrected) for N G C 2099 53 V mag A l A2 A3 A4 GLOBAL 11.0-12.0 (Raw) — — — — 40.1 Corrected - ( - ) - ( - ) - ( - ) - ( - ) 40.1 (7.3) 12.0-13.0 (Raw) 32.6 37.8 15.0 11.6 97.0 Corrected 32.6 (5.9) 37.8 (6.8) 15.0 (5.4) 11.6 (5.5) 97.0 (11.9) 13.0-14.0 (Raw) 66.4 50.2 36.9 8.3 161.8 Corrected 66.4 (8.5) 50.2 (8.1) 36.9 (7.9) 8.3 (6.6) 161.8 (15.6) 14.0-15.0 (Raw) 42.8 61.4 40.1 17.1 159.4 Corrected 42.8 (7.4) 61.4 (9.9) 40.1 (10.1) 17.1 (10.0) 159.4 (18.8) 15.0-16.0 (Raw) 46.8 63.5 33.2 20.9 158.5 Corrected 46.8 (8.5) 63.5 (11.9) 33.2 (12.8) 20.9 (14.1) 158.5 (24.0) 16.0:17.0 (Raw) 28.7 70.2 50.7 37.6 183.3 Corrected 28.7 (8.2) 70.2 (13.6) 50.7 (15.6) 37.6 (17.3) 183.3 (28.1) 17.0-18.0 (Raw) 41.9 52.7 36.6 34.8 155.0 Corrected 44.5 (9.4) 57.4 (13.7) 42.1 (16.0) 41.7 (18.2) 174.2 (31.4) 18.0-19.0 (Raw) 32.8 51.3 67.8 31.3 171.1 Corrected 34.4 (8.0) 54.5 (12.0) 72.6 (15.0) 35.7 (15.2) 184.8 (27.1) 19.0-20.0 (Raw) 22.6 59.9 71.1 34.7 178.3 Corrected 23.3 (7.4) 61.7 (12.7) 73.3 (15.5) 35.8 (16.0) 183.8 (27.7) 20.0-21.0 (Raw) 24.0 69.9 84.8 54.3 208.9 Corrected 24.9 (8.4) 72.5 (14.8) 88.0 (18.4) 56.3 (19.7) 216.7 (34.2) 21.0-22.0 (Raw) 16.0 65.8 96.8 119 289.5 Corrected 18.0 (11.0) 73.0 (20.3) 107.9 (26.6) 133.3 (31.4) 323.7 (53.5) 22.0-23.0 (Raw) — 34.6 54.8 115.9 181.7 Corrected - ( - ) 52.9 (39.9) 84.6 (58.6) 165.4 (76.9) 277.8 (155.9) 23.0-24.0 (Raw) — — — 145.9 Corrected - ( - ) - ( - ) - ( - ) 391.2 (125.6) accounting for the evolved stars brighter than V = 15, gives a lower limit to the total cluster population of ~2900 stars. This number makes N G C 6819 one of the richest open star clusters known. The global luminosity function for N G C 2099 is plotted in Figure 9.3, where again the dashed line represents the raw counts and the solid line the incompleteness corrected counts. The global luminosity function is almost flat from V = 13.5 ( M v ~ 2) to V = 19.5 (My ~ 8) and slowly rises beyond this point. This rise is due to the change of slope in the mass-luminosity relation at M y ~ 8 for solar chemistry (see e.g., D'Antona 1998 [30]), which can also be seen as a change in slope in the cluster main-sequence (see §8.3 or colour-magnitude figures). Integrating the luminosity function and accounting for red giants and white dwarfs provides a lower limit to the total cluster population of ~2600 stars. This observed N G C 2099 stellar content makes it very similar to that of N G C 6819, and therefore one of the richest open star clusters known. In Figure 9.4 we compare the N G C 2099 luminosity function to that for the Solar Chapter 9. Dynamical Studies 54 Figure 9.2: Global (R < 9.5') luminosity function for NGC 6819 shown before (dashed) and after (solid) incompleteness corrections. The almost flat luminosity function is most likely due to dynamical evolution which has caused the high mass stars to sink to the inner regions of the cluster. The error bars reflect a combination of Poisson errors and incompleteness errors as discussed in §9.1.1. Neighbourhood (Binney & Merrifield 1998 [14]), the Pleiades (Lee & Sung 1995 [69]) and N G C 6819 (Paper II). We have normalized the luminosity functions to the number of stars in N G C 2099 at the M v = 5 bin. The masses of the stars in this bin are slightly less than solar and no stellar evolutionary effects have yet taken place for any of the clusters. The luminosity function for N G C 2099 exhibits a similar slope to both the Pleiades and the Solar Neighbourhood from M v = 6 - 10. The cluster N G C 6819 is 10 times older than its relaxation time and therefore has lost some of its low mass, faint stars (see Paper II). It is clear from the figures that even deeper colour-magnitude diagrams are required in both N G C 2099 and N G C 6819 to map out the shapes of the luminosity functions at fainter limits. Chapter 9. Dynamical Studies 55 Figure 9.3: The global (R < 13.9') luminosity function for NGC 2099 is shown before (dashed) and after (solid) incompleteness corrections. The rising portion of the luminosity function beyond V ~ 19.5 (My ~ 8) corresponds to a slope change in the lower main-sequence of the cluster colour-magnitude diagram, and is caused by an inflection in the mass-luminosity relationship. The error bars reflect a combination of Poisson errors and incompleteness errors as discussed in §9.1.1. 9.3 Dynamical Relaxation The cluster luminosity function is determined by the initial mass function as well as the subsequent effects of dynamical evolution to the present epoch. These dynamical effects are caused by equipartition of energy between stars of different masses. The initial distribution of stars of any mass will roughly follow a density profile given by a King model - ie. an isothermal sphere (Binney & Tremaine 1987 [13]). Therefore the initial density of stars for any mass will always be highest in the centre of the cluster and decrease as a function of increasing radii from the centre. However, as the cluster ages we can expect the density distribution to expand from the centre. The low mass stars will gain energy in the interactions and have higher velocity dispersions than the high mass stars, which sink to the centre of the cluster. The relaxation timescale is Chapter 9. Dynamical Studies 56 Figure 9.4: The NGC 2099 luminosity function is compared with the Solar Neighbourhood distribution of stars, the Pleiades luminosity function and the NGC 6819 luminosity function. All luminosity functions have been normalized to the number of stars in NGC 2099 at My = 5 to avoid evolutionary effects. NGC 6819 exhibits an inverted slope with respect to the other functions due to significant dynamical effects. proportional to the number of crossings of a star across the cluster that are required for its velocity to change by order of itself (Binney & Tremaine 1987 [13]). If the cluster is older than its relaxation time, then we can expect that the stellar encounters within the cluster have caused the stars to relax rapidly toward equipartition, with low-mass stars travelling faster than the high-mass stars. To estimate the relaxation time for N G C 6819, we must first determine the crossing time of a star across the cluster, tcross = ^. Since we know both the distance to the cluster (see §7.1.2) and its angular extent (see §6.1), it is trivial to solve for the linear radius (~6.9 pc). Additionally, the velocity of a star across the cluster can be calculated using v2 ~ where Nm is the mass of the cluster (estimated in §9.5). The relaxation time is then given by equation (9.3), N Pretax ~ tcross 0 ; „ j\r' (^-3) Chapter 9. Dynamical Studies 57 where is the number of crossings of a star which are required for its velocity to change by an order of itself. N G C 6819 is a very rich cluster which gives a relatively large relaxation time compared to average open clusters with several hundreds of stars. However, this relaxation time (220 Myrs) is still a factor of 10 smaller than the cluster age (2.5 Gyrs), so we expect the cluster to be relaxed: the lower mass stars gain energy through gravitational encounters with higher mass cluster stars (which sink to the centre of the cluster) and slowly diffuse out of the cluster if their escape velocity is great enough (Hawley, Tourtellot & Reid 1999 [45]). Therefore, for N G C 6819, the lower mass stars are more likely to be observed at a larger radii than the higher mass stars. The evolution does however depend on parameters such as binary fraction and cluster richness (de la Fuente Marcos 1997 [32]). If old enough the cluster may even lose some stars due to these and other external processes such as tidal interactions with the disk of the Galaxy (Wielen 1991 [122]). For N G C 2099, we use the distance (1.5 kpc), population size (see §9.6), linear size (6.2 pc) and mass (see §9.6) to determine a relaxation time of ~300 Myrs (see equation 9.3), about 0.6 x the age of the cluster. Therefore we can expect some dynamical effects to have taken place in the cluster. 9.4 Mass Segregation Mass segregation is a consequence of dynamical evolution where evaporation and redis-tribution of low mass stars may have occurred in the cluster. Although this process has been long known to occur in open clusters (van den Bergh & Sher (1960) [114]), one of the first efforts to catalogue dynamical effects in many open clusters occurred when Francic investigated mass functions of eight clusters (Francic 1989 [35]). Francic's study clearly showed that the effects of mass segregation are more prominent in older clusters. Other recent studies have also confirmed that the mass functions for some open clusters have likely changed over time due to dynamical evolution (for example the Hyades (Reid 1992 [87]); N G C 188 (von Hippel & Sarajedini 1998 [117]); Praesepe and N G C 6231 (Raboud & Mermilliod 1998 [86]); N G C 2420 (Lee, Kang & Ann 1999 [68]); N G C 2516 (Hawley, Tourtellot & Reid 1999 [45]); M i l (Sung et al. 1999 [110]); M35 (Barrado y Navascues et al. 2001 [6]) and the Pleiades (Adams et al. 2001 [2]). Some of these Chapter 9. Dynamical Studies 58 Figure 9.5: Luminosity function shown for three regions in NGC 6819. The counts in the central and outer annuli have been normalized to the number in the inner annulus at V = 17. This demonstrates clear evidence for mass segregation in NGC 6819. clusters are quite young, such as the Pleiades, but others such as N G C 2420 are several billions of years old. We should also mention that Sagar & Griffiths (1998) [98] looked at mass segregation effects for five, distant open clusters and found that the effects are not correlated with cluster age. N G C 6819 and N G C 2099 are ideal clusters for these types of dynamical studies because they are far richer in stellar content than most of these clusters. This affects the evolutionary scenario as the escape velocity increases with the cluster mass. Additionally, N G C 2099 is neither very old nor very young but rather splits the above sample in age and therefore represents an important cluster for setting timescales of evolutionary effects. In order to look for evidence of mass segregation in N G C 6819, we produce three lumi-nosity functions for different annuli from the centre of the cluster. To keep the statistics reasonable we split the cluster into three components, a central portion (0'—2.5'), mid-dle portion (2.5'-5.5'), and outer portion (5.5'-8.5'). Figure 9.5 shows the luminosity Chapter 9. Dynamical Studies 59 Figure 9.6: Colour-magnitude diagrams for NGC 6819 in each alternate annuli, illustrating the main-sequence density as a function of increasing radius from the centre. The figure shows that the lower mass (faint) stars are located in the outskirts of the cluster. A prominent binary sequence is also evident in the intermediate regions of the cluster ( top right). function for each of these portions. We have normalized the luminosity function for each of the annuli with respect to the first annulus at V = 17. The shapes of the luminos-ity functions provide clear evidence for mass segregation in N G C 6819. The high mass stars on the main-sequence are clearly concentrated in the central regions of the cluster, whereas the outer annuli show a greater relative concentration of low mass stars. In Figure 9.6 we plot six colour-magnitude diagrams, each for an increasing radial annulus to show the richness of the main-sequence as a function of radial position. Clearly, the colour-magnitude diagram for annulus A l l (8.5' < R < 9.5') contains very few main-sequence stars, which is consistent with Figure 6.1. The general trend from inner to Chapter 9. Dynamical Studies 60 outer annuli in this figure confirms that the faint stars are not centrally concentrated. Surprisingly, Figure 9.6 also shows a prominent binary sequence in the intermediate-outer regions of the cluster (see Annulus A6 —>• 3.5' < R < 4.5'). Figure 9.7: The luminosity function is shown for four increasing concentric annuli around NGC 2099. The counts in all annuli have been normalized to the number at My = 5. There is some evidence for mass segregation in the faintest bins. For N G C 2099, we split the cluster into four annuli, each 3.5' in radial extent, with geometry summarized in Table 6.1. Figure 9.7 displays luminosity functions in each annulus, scaled to the number of stars at M v = 5. Although not as prominent as in N G C 6819, there is some evidence for mass segregation of the faintest stars in N G C 2099 (cluster age ~1.7xdynamical age). The outer annuli of the cluster show a greater relative concentration of lower mass stars (0.40 M Q — 0.60 M 0 ) . Chapter 9. Dynamical Studies 61 9.5 Mass Functions The mass function of a stellar population is typically expressed as the number of stars / unit mass. Since it is the luminosity that is measured, not the mass, this mass function is usually expressed as in equation (9.4), N(m) = N(MV)[^]. (9.4) In this equation, m is the mass, N(My) is the luminosity function, and is the mass-luminosity relation. Therefore, the observed luminosity function must be multiplied by the slope of the My-mass relation to obtain the mass function. Typically, the mass function is assumed to be a power law so that * ( m ) o c m - ( 1 + j ) , (9.5) where x takes on a value of 1.35 in the work of Salpeter (1955) [100]. A discussion of the observational constraints and differing values of this slope in young clusters, young field stars, old open clusters, low mass disk stars, globular clusters, and the Galactic spheroid and halo is given by Richer & Fahlman (1996) [93]. Additionally, Francic (1989) [35] has shown that the mass functions for some old Galactic clusters (NGC 6633, N G C 752, and M67) are weighted towards the higher mass stars. This analysis also showed the slope of the mass function for younger open clusters to be about x = 1. The inverted mass function for the older clusters may be due to dynamical processes in the cluster which work to better retain the higher mass stars. We have shown that it is likely that these processes may have already occurred in N G C 6819 and N G C 2099 (see §9.3). In order to better quantify the dynamical evolution, we can observe the change in the mass function for different annuli from the centre of the cluster. We use the Rome theoretical isochrones (see §8.1) to create a mass-luminosity rela-tionship for N G C 6819. The slope of this mass-luminosity relationship is used to convert the number of stars in each magnitude bin to the number of stars per unit mass. We derive the slope using the end points of each magnitude bin that is used in the analysis, and from this slope, derive the mass function. As expected, the global (R < 9.5') mass function (bottom-right of Figure 9.8) is almost flat ( ^ = constant). Fitting a power law to this global function (as in equation (9.5)) gives a value of x = -0.15. For comparison, Chapter 9. Dynamical Studies 62 S 3 .5 d * 3 I 2.5 £ 2 I 1 5 1 -I—I—|—I—I—r 0 , <B<1.6' 3.5 | -3 E" 2.5 2 ^ 1.5 1 | I I I I—|—i—i—i—r i.ef<n<z.V _ l I I I • 1 • -]—i—i—i—i—I—I—i—i—r-i 2.&<R<3.& : -0.5 3.5 3 ^ a 2.5 £ 2 0 0.5 -0.5 — i — i — j — i — i i i ' 3.6'<R<4.6' I . . . . I . . . . 3.5 | -3 p-2.5 2 I 1-5 : 4.6 ,<R<6.6* ' . F_J I L _ l I I—I—L_l—I--0.5 S 3.5 a * 3 | 2.5 £ 2 JJ 1.5 1 6.6'<R<7.6' 0 .5 - 0 .5 3 .5 3 2 . 5 2 i 1.5 | -. F I i i i i 7.6'<R<8.6' -0.5 0 0.5 -0.5 0 0.5 -0.5 0 0.5 L o g ( M / M . ) L o g ( M / i l . ) L o g ( M / M . ) Figure 9.8: Series of mass functions for eight different annuli, illustrating a general trend of positive to negative slope as a function of increasing distance from the cluster centre. The global mass function (x = —0.15) of the cluster is clearly flatter than a Salpeter value (x = 1.35, dashed line, bottom right). The error bars are taken from the errors in the luminosity functions (Poisson and incompleteness) and then multiplied by the slope of the mass-luminosity relation. we also plot a Salpeter value (x = 1.35) in the global plot which is much steeper than the N G C 6819 mass function. Figure 9.8 also shows a series of mass functions for annuli at increasing radial distances from the cluster centre. There is a systematic change in the slope of the mass function with increasing radius (positive slope —¥ negative slope) which is consistent with the expectations of dynamical evolution in N G C 6819. Integrat-ing the global mass function provides a total cluster mass of ~2600M Q (to our limiting magnitude). N G C 2099 is only about 1.7 times older than its dynamical relaxation time, and therefore should exhibit a much steeper global mass function than what was observed in N G C 6819 if all clusters form initially with a mass function slope near that of a Salpeter Chapter 9. Dynamical Studies 63 i 1 i 1 1 1 i 1 1 1 i 1 1 1 i 1 1 1 i 1 1 • i -| Log(Mass) Figure 9.9: The global mass function of NGC 2099 (solid, x = 0.60) is found to be flatter than a Salpeter IMF (dashed, x = 1.35) from ~0.5 M Q to ~1.8 M Q . The slope of the high mass end of the mass function is found to be very steep, x = 2.5. value. Figure 9.9 shows the mass function for the cluster (solid) and a Salpeter slope (dashed). This mass function extends from ~0.3 M© (the limit of our photometry) to ~2.6 M 0 (the main-sequence turn-off) and has a best fit slope of x = 0.60 between ~0.5 M© and ~1.8 M©, somewhat flatter than the Salpeter value. The high mass end of the mass function (between ~1.8 M© and ~2.6 M©) is much steeper with a best fit slope of x = 2.5. The two lowest mass bins (~0.3 M© to ~0.5 M©) of the mass function show a very flat distribution. For such low masses, colour transformation equations are not available for the non-grey boundary conditions used in our models, however, we have extrapolated a mass-luminosity relation (as discussed in §8.1). Integrating the global mass function, and accounting for the evolved stars, provides a total cluster mass of ~2300 M©. Chapter 9. Dynamical Studies 64 9.6 Star Counts Down to the H-burning Limit Both the estimated total cluster populations from §9.2.1 and the total cluster masses from §9.5 are lower limits due to our photometric limit (V ~ 23.5 on the main-sequence). To obtain a more realistic measure of the total cluster population we can extrapolate our observed luminosity function to the hydrogen burning limit. For example, for N G C 2099 the counts between our faintest bin (~0.30 M 0 ) and the hydrogen burning limit (0.08 M 0 ) are obtained by normalizing the Pleiades luminosity function (Lee & Sung 1995 [69]) to the N G C 2099 function (see Figure 9.4). The results provide an additional 1350 stars with a total mass of 210 M 0 . This raises the N G C 2099 cluster population to just under 4000 stars and a total cluster mass of just over 2500 M 0 . We also note that the total mass of the clusters may be larger due to the presence of binary stars. The total cluster populations may also be higher due to the missed stars between our cluster extent and the tidal radius. Chapter 10 White Dwarf Stars and the Initial-Final Mass Relationship White dwarfs represent the end point in the life of low-mass stars (M < 7 M 0 ) . These burnt-out stellar cinders have extinguished their nuclear fuel and are now radiating away any remaining stored thermal energy. Younger than an age of about 8 Gyrs, white dwarfs will simply cool and fade, becoming dimmer and redder as time passes. Bolometrically they will continue to do so older than 8 Gyrs however in certain colours the spectral energy distribution shifts to the blue at this point (see Figure 10.1) due to molecular hydrogen opacity in the atmospheres of the DA stars (Hansen 1999 [41]). White dwarfs typically have 10 < M v < 17.5 (depending on the age of the star - the upper limit set by the oldest white dwarfs in our Galaxy), making them difficult to detect at moderate distances. This has made it nearly impossible to set observational constraints on certain theoretical models of white dwarfs. For example, the initial-final mass relationship of the progenitor-white dwarf star, or the upper mass limit to white dwarf production, have never been tested with the detail that is now possible (Weidemann 2000 [119]; Weidemann 1987 [120]; Reimers &; Koester 1988b [90]). A progenitor-white dwarf initial-final mass relationship is particularly important for Galactic chemical evolutionary studies as it would provide important information on the amount of stellar mass loss in post main-sequence evolutionary stages. The primary goal of the C F H T Open Star Cluster Survey is to catalogue a large number of white dwarf stars by analyzing deep photometry (V ~ 25) of rich stellar environments (as presented in this thesis for N G C 6819 and N G C 2099). By supple-menting the photometry with spectroscopic observations, we hope to vastly increase the sample size of objects that is available for such studies as outlined above. The only major large-scale observational program to attack the issues outlined above has been the almost two-decade long study of Dieter Reimers and Detlev Koester (see Reimers & Koester 1988b [90] for an early summary). The quality and size of the present data set are unprecedented when compared with previous efforts. 65 Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 66 Figure 10.1: 0.5 M© white dwarf cooling sequence shows a strong blue hook for ages upward of ~8 Gyrs due to molecular hydrogen opacity (Hansen 1999 [41]). 10.1 White Dwarfs in Open Clusters For many reasons star clusters represent excellent environments in which we can learn more about the properties of white dwarf stars. The assumed single-burst star formation event that creates a star cluster produces stars with a spectrum of masses yet similar metallicity, age and (most importantly) distance. Since the evolution of stars is driven primarily by their initial mass, examining different populations of stars in a cluster can give us a snapshot of the life stages of a single star: turn-off, red giant, planetary nebula, white dwarf, etc. (see Renzini & Fusi-Pecci 1988 [92]). By applying a statistical argument to objects in different locations of a cluster colour-magnitude diagram, we can infer information about the properties of objects in different stellar evolutionary stages, such as white dwarfs. Studying white dwarfs in practice, however, is not this simple. The richest star clusters, the globular clusters, contain large numbers of white dwarfs but these clusters are also very old. Therefore the limiting magnitude of the coolest white dwarfs Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 67 in these clusters occurs at magnitudes that are too faint for ground based observations. For example, the limiting white dwarf magnitude for a 12 Gyr cluster occurs at M v = 17.5 (Richer et al. 2000 [96]). The only program currently underway to establish the white dwarf cooling age of a globular star cluster is a deep HST study of M4, the nearest globular cluster (see Richer et al. 1997 [94]). Success in this study may be possible because it takes advantage of new models that predict radically different behavior in the emergent spectra of cool white dwarfs with ages > 8 Gyrs or TeS < 4000 K (Hansen 1999 [41]; Saumon & Jacobson 1999 [105]). A n alternative method of studying white dwarfs is to utilize the much younger and more metal-rich open star clusters. There are four major difficulties in using open star clusters to study white dwarfs: (1) the majority of these clusters are not old enough to have produced a sizeable white dwarf population, (2) most are not rich enough to contain many white dwarfs, (3) these clusters lie in the plane of the Galaxy so that foreground and background contamination is high, and (4) the photometric depth of most studies has not been deep enough to clearly see white dwarfs. The first two factors result in very few white dwarfs, often quite scattered somewhere in the faint, blue end of the colour-magnitude diagram, and the third results in a large amount of contamination from both field stars and background star forming galaxies, which also appear as faint, blue objects. The fourth factor has always been a deterrent to the serious study of cluster white dwarfs. White dwarf cooling models (Wood 1994 [123]; Bergeron 1995 [9]), indicate that a significant number of white dwarfs are not expected brighter than an absolute magnitude of My ~ 10. At the opposite end, it is known that very old, cool white dwarfs may reach M v > 17.5. Even the bright end of the white dwarf cooling sequence is often too faint for the limiting magnitude that could possibly be reached by many telescopes, even for moderately close clusters. The new large field detectors and imagers on 4 m class telescopes are ideal instruments to study these faint magnitudes in open star clusters. This was demonstrated in the white dwarf analysis of M67 (Richer et al. 1998 [95]). What we require to address the properties of white dwarf stars are very rich, yet intermediate aged open star clusters with little reddening. The clusters must be old enough so that a significant number of white dwarfs have formed, yet young enough so that the end of the white dwarf cooling sequence does not occur at an unobservable magnitude. N G C 6819 and N G C 2099 are ideal for these studies. Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 68 10.1.1 Methods We will attack these issues by first producing rich colour-magnitude diagrams which show a population of white dwarfs. White dwarf cooling models (Wood 1994 [123]) will allow us to establish a most likely statistical location for white dwarfs in our colour-magnitude diagram. The cooling models will yield a preliminary value for the final masses of the stars and, when combined with the luminosity of the star, will provide a white dwarf cooling age. We can then use the cluster age to determine the main-sequence lifetime for each of the progenitor stars to the white dwarfs. Finally, main-sequence stellar evolutionary models will be used to provide initial masses for these stars. The largest uncertainty in the above analysis is determining accurately the final masses of the white dwarfs. First, we do not know exactly which objects are bona-fide cluster members, and secondly, the objects do not closely follow any particular mass cooling sequence. The current project will therefore identify possible candidates that will then be spectroscopically observed using multi-object spectrographs on 8-metre class telescopes to isolate cluster members and measure surface gravities and effective temperatures (which gives the radius and therefore a more accurate value of the final mass). This is a similar approach to that of Reimers and Koester who used spectroscopy to confirm and identify bright white dwarfs (individually) in sparsely populated open clusters that had been discovered in previous published imaging projects. Only a few objects were found in each study and the results for all clusters were combined to establish constraints on the relationships outlined above. Among the clusters studied were N G C 2516 (Koester & Reimers 1996 [62]; Reimers & Koester 1982 [91]), N G C 6633 (Reimers & Koester 1994 [88]), N G C 3532 (Koester & Reimers 1993 [63]), N G C 2168 (Reimers & Koester 1988 [89]), IC 2391 and N G C 2451 (Koester & Reimers 1985 [64]), and N G C 2287 and N G C 2422 (Koester & Reimers 1981 [65]). The clusters studied by Reimers and Koester do not generally contain enough white dwarfs to accurately establish a cooling sequence and therefore, an independent age measurement. However, white dwarf cooling ages have been established for older open star clusters such as M67 (Richer et al. 1998 [95]) and N G C 2420 (von Hippel & Gilmore 2000 [116]). The difficulties in these studies are that the photometry does not extend much fainter than the end of the cooling sequence and that the termination of the white dwarf cooling sequence is buried in field stars or unresolved galaxies making it very Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 69 difficult to actually isolate it on the colour-magnitude diagram. The statistics in these clusters do show that the end of the cooling sequence has been detected, but these are affected by large errors. Clusters such as N G C 2099 and N G C 6819 possess advantages over both the young clusters studied by Reimers and Koester and these older clusters. The clusters are several times richer than any of the younger clusters and the photometry is both more accurate and deeper. Therefore, spectroscopic observations of the objects in these clusters will not only increase the sample size by many factors, but also establish constraints from fainter, older white dwarfs. The clusters are also approximately three times richer than the richest of the older clusters for which a white dwarf cooling age has been established. The white dwarf population is well separated from the bulk of the field stars thereby making the statistics simpler (see §10.2.3). Additionally, for N G C 2099 our photometry extends almost 1.5 magnitudes fainter than the termination of the white dwarf cooling sequence and the cluster colour-magnitude diagram clearly shows that we have detected the end of the cooling sequence. For N G C 6819, a larger telescope is needed to detect the end of the cooling sequence, however a nice trail of potential white dwarfs is clearly seen. 10.2 White Dwarfs in NGC 6819 10.2.1 Continuity Arguments and Field Object Subtraction N G C 6819 is very rich in stellar content, thus a significant number of both hydrogen (DA) and helium (DB) white dwarfs are expected. We can predict the number of expected white dwarfs above a limiting magnitude cut-off in the cluster by counting the number of stars in the red giant phase and applying a continuity argument for these stars. From the masses of the red giant stars and the Rome stellar evolutionary sequence, the lifetime of the red giants in the 'clump' (V = 13, B - V = 1.2) is determined to be 5 x 107 years for models with no convective overshooting and 9 x 107 years for an overshooting model. We favour the latter model as it is in better agreement with our data. Additional evidence for the justification of core-overshooting models for N G C 6819 was given in detail in the analysis of Rosvick and VandenBerg (1998) [97]. The continuity argument which we apply to determine the number of expected white dwarfs (NWD) follows from the hypothesis that all stars of mass less than ~7 M 0 will evolve into white dwarfs. First we Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 70 determine the number of objects in the red giant 'clump' (NRQ) after correcting for field star contamination; 13. Next we can use the white dwarf cooling models to determine the white dwarf cooling ages {tcooung) at a certain magnitude (V = 23, 23.5, and 24) to which the stars have cooled, for varying white dwarf masses (M = 0.5, 0.6, and 0.7 M 0 ) . For post main-sequence evolution the number of stars in a given evolutionary phase is proportional to the time spent in that phase. Therefore, we can estimate the number of white dwarfs that we expect to see in the cluster and compare this with the number observed after both field star subtraction and incompleteness corrections (see §9.1), by using equation (10.1): NWD{< Mv) = ^ - t c o o l i n g { < Mv). (10.1) The field star subtraction is addressed statistically by comparing the location of ob-jects in the lower-left (faint-blue) section of the cluster colour-magnitude diagram to the background field colour-magnitude diagram. We take each object within this location on the background colour-magnitude diagram and eliminate the corresponding closest object in the cluster colour-magnitude diagram. We can estimate the uncertainties in the expected number of white dwarfs in a similar manner to that used for the main-sequence stars: use Poisson errors for both the observed number of white dwarfs and red giants, and a binomial distribution for the incompleteness errors (see §9.1.1). Additionally, there is an uncertainty in the white dwarf cooling age which is found by multiplying the slope of the cooling curve by the error in the magnitude as determined by PSFex. We find that for a model with a large amount of core-overshooting (producing a larger convective core so time in R G phase increases) the predicted number of white dwarfs far exceeds the number observed (to V = 24) at a strict 0.90 stellarity cut. However, the predicted number agrees very well if we impose a less stringent confidence limit of 0.80, especially for 0.7 M© objects. Similar results are also seen for intermediate core-overshooting, where the number of predicted white dwarfs agrees well with the number observed up to V = 23.5. For a brighter magnitude cut of V = 23, we find too many white dwarfs in all but the low core-overshooting cases with M = 0.6 and 0.7 M 0 . It is difficult to make predictions from this analysis because the uncertainties remain large. Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 71 Table 10.1: White Dwarf Continuity Analysis - Predicted Number vs Observed Number V mag cut Observed (Raw) Observed (Corr) Pred (0.6 M 0 ) Pred (0.7 M 0 ) Pred (0.8 M 0 ) For 0.80 Stellarity Cut 13 .0±3 .6 Red Giants tRG = » x 107 48 (±14) <24 27 (±5) 53 (±7) 61 (±17) 55 (±16) <23.5 21 (5) 35 (6) 29 (8) 25 (7) 21 (6) <23 17 (4) 28 (6) 11 (3) 9(3) 8(2) For tRG = 7 x 107 61 (17) <24 27 (5) 53 (7) 78 (22) 71 (20) <23.5 21 (5) 35 (6) 37 (10) 32 (9) 27 (8) <23 17 (4) 28 (6) 15 (4) 12 (3) 10 (3) For tRG = 5 x 107 86 (24) <24 27 (5) 53 (7) 109 (31) 99 (28) <23.5 21 (5) 35 (6) 52 (15) 44 (13) 38 (11) <23 17 (4) 28 (6) 20 (6) 17 (5) 14 (4) For 0.90 Stellarity Cut tRG = 9 x 107 48 (14) <24 17 (4) 34 (6) 61 (17) 55 (16) <23.5 13 (4) 22 (5) 29 (8) 25 (7) 21 (6) <23 12 (3) 19 (5) 11 (3) 9(3) 8 (2) For tRG = 7 x 107 61 (17) <24 17 (4) 34 (6) 78 (22) 71 (20) <23.5 13 (4) 22 (5) 37 (10) 32 (9) 27 (8) <23 12 (3) 19 (5) 15 (4) 12 (3) 10 (3) For tRG = 5 x 107 86 (24) <24 17 (4) 34 (6) 109 (31) 99 (28) <23.5 13 (4) 22 (5) 52 (15) 44 (13) 38 (11) <23 12 (3) 19 (5) 20 (6) 17 (5) 14 (4) The analysis however indicates that a core-overshooting model is preferred for the cluster if we are to consider all potential white dwarfs to our limiting magnitude. This assumes that a substantial number fraction of the white dwarfs are not tied up in binaries. These results are summarized in detail in Table 10.1, with errors in parentheses. 10.2.2 White Dwarf Analysis Figure 10.2 shows the colour-magnitude diagram for N G C 6819 before any source rejec-tion (left) and after we have imposed some constraints (middle and right). The constraints are (1) only accept objects with a stellarity confidence index above 0.50 (middle), and (2) only accept those objects which also survive a statistical subtraction to remove field objects (right). Criterion (1) is arbitrary as explained in §7.1.3. Even at a 0.50 stellarity cut, some of these objects could still be faint unresolved galaxies, A G N , or some other Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 72 Figure 10.2: Uncorrected colour-magnitude diagram for NGC 6819 (left). There is a general spread of stars in the lower left corner. After correcting for extended sources (center) and field star subtraction (blue of dashed line, middle), a potential white dwarf cooling sequence is evident (right). We also show a 0.7 M© white dwarf cooling sequence which agrees with the bluest potential white dwarfs. This analysis is a purely statistical method of determining the most likely location on the colour-magnitude diagram of the cluster white dwarfs. non-cluster object. We note however that a significant portion of the remaining objects are in fact stellar to within a 0.90 stellarity confidence. This is shown in more detail in Figure 10.3, where we have zoomed into the hot faint end of the colour-magnitude dia-gram. Criterion (2) has been addressed by eliminating possible cluster field objects that are in the same vicinity of the colour-magnitude diagram as background field objects as described in §10.2.1. We invoke this approach for a small region in the colour-magnitude diagram surrounding and including all possible cluster white dwarfs (objects below the dashed line in Figure 10.2 (middle)). The statistical subtraction shows that there is an over-density of objects in the cluster field, however, we can not say for sure whether the Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 73 objects that we have removed are in fact not cluster objects. In Figure 10.3 we also indicate the stellarity for each of the white dwarf candidates, which is found to be > 0.90 for the majority of the objects brighter than V = 24. > 20 ' 1 1 1 1 .1 | 1 « 0.60 < Stellarity < 0.70" . Photometric Error Bkr 0 0.70 < Stellarity < 0.80-21 - \ • 0.80 < Stellarity < 0.90J • \ + Stellarity > 0.00 22 > :\ V * \ \ 8 23 M — t—i W \ • \ 8 24 1 1 \ • , \ «« 25 " . , 1 t.O M 8 \ \0 .50 lle -0.5 0 0.5 B - V Figure 10.3: All potential white dwarf candidates within 0.3 magnitudes (colour) of a 0.7 M© cooling sequence and after a statistical subtraction. A photometric error bar is also shown as a function of magnitude. A large number of the objects are determined to be very high confidence stars (diamonds). 10.2.3 Interpretation of Cooling Sequence The statistically subtracted and star/galaxy corrected colour-magnitude diagram (Figure 10.2 (right)) indicates a clear separation between the white dwarfs and the field stars. This potential white dwarf cooling sequence is separated from these field stars by an average of ~0.6 magnitudes in colour on the colour-magnitude diagram. There are very few objects between the two populations. The putative white dwarf cooling sequence revealed however is not particularly tight as there is some evidence for a gap between this (the bluer objects) and a redder potential white dwarf sequence (apparent in Figure Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 74 2 1.5 1 W 3 0.5 0 "~20 21 22 23 24 25 V Figure 10.4: Luminosity function of the left-most trail of white dwarfs in the colour-magnitude dia-gram; agrees well with the slope of the theoretical luminosity function (dashed line - Liebert, Dahn, & Monet 1988 [71]). 10.3). For the adopted reddening value of the cluster, no reasonable mass white dwarf cooling sequence fits the reddest objects in this location of the colour-magnitude diagram. In the Figure, we show cooling models for the two extreme masses 0.5 and 1.0 M Q . It is unlikely that photometric spread is causing some of these objects (the redder objects) to deviate so much in colour from those that agree with the 0.7 M© cooling sequence (Wood 1994 [123]). To better judge this, we have plotted a photometric error bar as a function of magnitude at 0.5 magnitude intervals in Figure 10.3. These errors, as determined by PSFex, are consistent with those found for the recovered artificially added stars in the incompleteness tests. It is clear that the spread in data points is larger than the error bars. We also attempted to reconcile the split by considering a D A / D B segregation. This dichotomy can however be ruled out as the observations would require a very large surface gravity in the D A objects. A more likely scenario to explain the positions of these objects in the colour-magnitude diagram is that they are just excess background or Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 75 foreground objects which were not removed in the statistical subtraction. Alternatively, some could also be highly reddened background white dwarfs. For those objects which closely follow the 0.7 M Q white dwarf cooling sequence we determine a luminosity function after correcting for both a 0.75 stellarity cut and an incompleteness correction. The slope of this luminosity function agrees very well with theoretical expectations (see Liebert, Dahn & Monet 1988 [71]) (shown in Figure 10.4). The bright end of this theoretical function has been extrapolated to include our brightest objects (overlap region is 22.5 < V < 24). There is no obvious agreement with the theoretical function if all objects in Figure 10.3 are considered. Although we believe the bluest objects in this Figure to be bona-fide white dwarfs, spectroscopic confirmation is required for these as well as those that were eliminated in the statistical subtraction. Fortunately, multi-object spectroscopy will allow measurement of multiple objects in the faint-blue end of the colour-magnitude diagram. 10.3 White Dwarfs in NGC 2099 10.3.1 Removing Non-cluster Objects The background field colour-magnitude diagram for N G C 2099 (Figure 7.2) shows that there is a significant population of field stars and/or faint background galaxies in the region where cluster white dwarfs are expected (see 22 < V < 25, 0 < B—V < 0.6). However, this population is also clearly more dense in the cluster field. Although we can not delineate those objects that are cluster members from the colour-magnitude diagram alone, we can use the background field to statistically determine the number of expected cluster members and their most likely location in the colour-magnitude diagram as done for N G C 6819. We count all objects in the background field within each 0.50 magnitude interval in V and remove 1.37 times the number of field objects in the same magnitude bin. This approach is only invoked for possible white dwarfs in the faint, blue end of the colour-magnitude diagram. It is important to note that we may in fact remove all white dwarf candidates in this process if there were no cluster white dwarfs. This statistical technique no longer allows us to consider individual objects as potential white dwarfs: multi-object spectroscopy of all potential candidates, before any cuts, with instruments such as GMOS on Gemini North or LRIS on Keck, will provide the definitive answer as to which objects really are white dwarfs. If we assume that the clustering properties Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 76 of background galaxies is almost uniform across the 42' x 28' field, then this statistical subtraction should also eliminate the correct number of galaxies from the cluster field. As mentioned before, it is difficult to estimate which stellarity cut is an optimal separation of galaxies from stars, so we must consider the expected number of galaxies in our cluster field (R < 13.9'). As in §7.1.3 this is done by considering galaxy counts (see Woods & Fahlman 1997 [124]) at high latitude and correcting for extinction in our field. Based on these statistics and prior to any stellarity cut, there are far more objects in the brightest of our faint magnitude bins than the number of expected galaxies. For example, we expect less than 15% of all objects to be galaxies for 21 < V < 22. However, for 22 < V < 23, we expect 37% of our objects to be galaxies, and for 23 < V < 24 fully 98% of our objects are expected to be galaxies. Remarkably, invoking a 0.50 stellarity cut removes approximately the correct number of galaxies from two of these three bins. For 21 < V < 22, we expect 336 galaxies in our cluster field and we remove 430 objects with a 0.50 stellarity cut. For the 22 < V < 23 bin, the number of objects removed at a 0.50 stellarity cut is 1015 and the number of galaxies is 1016. For 23 < V < 24, the number of objects removed at a 0.50 stellarity cut is 1774 and the number of expected galaxies is 2552. A 0.75 stellarity cut in this last bin is almost perfect in eliminating the correct number of objects. It eliminates 2556 of the expected 2552 galaxies. Perhaps after the study of many clusters in the survey we can establish with certainty the correct stellarity cut for separating galaxies from stars, at a given signal to noise ratio. For this work, we will use a 0.50 stellarity cut which is consistent with that used for the N G C 6819 data. After statistical subtraction and galaxy removal as described above, most of the re-maining objects in the faint-blue end of the colour-magnitude diagram should be white dwarfs. There may also be some unresolved galaxies or field blue objects. We can esti-mate the number of expected field white dwarfs and compare this value to the number of faint-blue objects in our field. We use the hot white dwarf luminosity function (Figure 2 in Leggett, Ruiz & Bergeron 1998 [70], taken from Liebert, Dahn & Monet 1988 [71]) to count the number of expected white dwarfs per pc 3 above our limiting magnitude. First, a white dwarf cooling model for 0.70 M 0 (Wood 1994 [123]) is used to convert our ab-solute magnitude bins to bolometric magnitude bins. This then provides a value for the bolometric luminosity of the white dwarfs at each bin centre. The number of expected white dwarfs, above that luminosity and in that bin, simply follows from the Figure. Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 77 Adding up the numbers in the bins from M v = 10.5 to 13 and multiplying by the volume created by the cone that represents our cluster field, yields a population of 63 expected field white dwarfs. The number of objects in our background field, after correcting for incompleteness in our data and removing background galaxies, is 35, which when scaled up to match the size of the cluster field gives 48. The difference between the expected and the observational values can be most likely attributed to errors in the galaxy counts and therefore in the stellarity cut used and also to white dwarfs that are in binaries and therefore missed in our study (see next section). 10.3.2 White Dwarfs in Binaries Evidence supporting a population of white dwarfs in binaries in N G C 2099 is provided from a very crude estimate of the expected number of white dwarfs in the cluster based on an extrapolation of the observed mass function at the high mass end. As mentioned in §9.5, the high mass end slope of the mass function is x = 2.5. Using the number counts in the m = 2.22 M 0 mass bin to solve for the proportionality constant (A = 3684), and integrating equation 9.5 from the turn-off (M ~ 2.6 M 0 ) to the upper limit to white dwarf production (M ~ 7 M 0 ) provides a number for the expected number of white dwarfs in the cluster; 113. This value is ~2 times greater than the number of detected white dwarfs (see next section). A significant number of these missing white dwarfs are most likely in main-sequence binaries, however, a few additional arguments can also help explain the discrepancy. First, the cluster will lose stars through tidal stripping and evaporation. The white dwarfs may be more prone to these effects because they have gone through the planetary nebulae stage. Any asymmetric mass-ejection would induce a recoil velocity on the white dwarf that could eject it from the cluster. Another effect that could lead to fewer detected white dwarfs is relaxation (see §§9.3 and 9.4). Since the white dwarfs have a small mass compared to their progenitors, relaxation will increase their velocity dispersion and therefore make them less frequent in the cluster centre (compared to the progenitors) and more susceptible to evaporation. This effect should not, however, be very dramatic as the white dwarf cooling ages for most of the candidates are comparable to the relaxation age of the cluster. Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 78 40 30 20 s 3 z 10 0 -10 22 23 24 25 V Figure 10.5: White dwarf luminosity function rises to a peak at V = 23.5 ( M v = 11-95 ± 0.30) and subsequently drops off rapidly (see §10.3.4 for error analysis). The bright end slope of the luminosity function is in agreement with theoretical expectations (dashed). The errors bars include both counting uncertainties and incompleteness errors. For a 0.70 M© white dwarf cooling sequence, the limiting magnitude of the cooling white dwarfs provides a white dwarf cooling age of 566 ± Myrs for NGC 2099. This age is in excellent agreement with the main-sequence turn-off age (520 Myrs) for a core-overshooting model. 10.3.3 White Dwarf Luminosity Function and Cooling Age For those objects that have survived the criteria outlined in §10.3.1, we construct a white dwarf luminosity function by binning the objects in 0.50 magnitude intervals and sub-tracting the background field numbers from the cluster field numbers after accounting for the 1.37 aerial difference. Figure 10.5 shows this result after accounting for incom-pleteness corrections as described in §9.1 and summarized in Table 9.1. We find there to be a clear increase in the number of stars as a function of magnitude and then a sharp turn-off at V = 23.5 ( M v = 11.95 ± 0.30) (see Table 10.2). The uncertainty in this value is discussed in detail in the next section. We interpret this turn-off to represent the end Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 79 B - V B - V Figure 10.6: Colour-magnitude diagram shown for the entire cluster content with a superimposed 520 Myrs stellar isochrone (left). The theoretical main-sequence is in excellent agreement with the data for both the red giant clump (top-right) and for the main-sequence from V = 14 to V = 20 (middle-right). The white dwarf population is clearly evident and is shown with respect to a 0.70 M© white dwarf cooling sequence and the corresponding cooling ages (bottom-right- see Figure 10.7 for an expanded figure of this). This Figure, coupled with Figure 10.5 clearly shows that the photometry is fainter than the limiting white dwarf cooling magnitude. of the white dwarf cooling sequence in N G C 2099. The total number of white dwarfs in N G C 2099, found from adding up the bins in the luminosity function, is 50. The bright magnitude slope of the luminosity function is in excellent agreement with the expected slope of the white dwarf luminosity function from theory (Liebert, Dahn, & Monet 1988 [71]). We can use the magnitude of the termination point of the white dwarf cooling sequence to establish a lower limit to the age of N G C 2099 based on white dwarf cooling models (Wood 1994 [123]). Figure 10.6 shows a 0.70 M 0 cooling model superimposed on the cluster colour-magnitude diagram. The age from the limiting magnitude in this cooling sequence is 516 ± Jf| Myrs (see Figure 10.7). This age is a lower limit because Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 80 Figure 10.7: NGC 2099 white dwarf population is shown with superimposed cooling ages. it represents the time taken for the most massive progenitor stars in the cluster to cool to the limiting white dwarf magnitude and does not include the time that these stars spent on the main-sequence. The latter, for 7 M 0 stars, is 50 Myrs, therefore raising the white dwarf cooling age of the cluster to 566 ± }fg Myrs which is in excellent agreement with the core-overshooting turn-off age (520 Myrs). Finally, we note that if we assume a slightly higher mass for the white dwarfs (0.80 M Q ) , the age decreases slightly (530 Myrs) and is in better agreement with the turn-off age. Surprisingly, the cooling ages determined here are insensitive to the stellarity cut (0.50 or 0.75). The peak in the white dwarf luminosity function occurs at M v = 11.95 independent of a 0.50 or 0.75 stellarity cut. The number of objects however, will clearly be less in the 0.75 case. This is in part due to the resolution of the bins used for the white dwarf luminosity function. Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 81 Table 10.2: White Dwarf Luminosity Function V mag No. Stars Error 22.5 6.3 4.2 23.0 18.1 8.6 23.5 26.1 9.2 24.0 -2.5 9.1 24.5 -7.5 7.1 10.3.4 Evaluation of Errors In order to compare the white dwarf cooling age with the main-sequence turn-off age, we must provide a careful account of all errors. As mentioned in §7.2.2, the total error contribution to the true distance modulus includes four terms. Additionally, the error in the limiting white dwarf cooling magnitude must also include the bin uncertainty (±0.25 magnitudes) from Figure 10.5. We can address each of the uncertainties individually and then combine them correctly to account for correlations between the cross terms: 1. ) There is an error in the vertical shift caused by the error in the horizontal shift due to AE(B-V)- This error is proportional to AE{B-V) and to the slope of the main-sequence at the fitting point (S = AV/A(B - V) ~ 5.7). Therefore, A A = SAE{B-v) = 0.17. 2. ) There also exists a judgement uncertainty in the vertical shift needed to super-impose the Hyades main-sequence on the N G C 2099 main-sequence at a fixed value of E ( B - V ) . Under some circumstances, this shift is very hard to determine due to large photometric spread in the main-sequence and the smearing effect of binaries. Given the tightness of the two main-sequences, we estimate this error to be quite small, AB = 0.07. 3. ) The third error we consider is the error in the extinction (AAV) caused by the uncertainties in both the reddening (AE(B-V)) and A R V . This term is evaluated as = A- = A4^t)2 + (m^)]1' (ia2) where we take ARV = 0.20. The above equation gives A c = 0.10. 4. ) The final term in our distance modulus error budget accounts for the small colour shift caused by the metallicity uncertainty of the cluster. As described in §7.2.1, the metallicity is believed to be Z = 0.02. If we assume an uncertainty AZ ~ 0.005 (this Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 82 allows the cluster metallicity to be equal to that of the Hyades), then the corresponding shift in colour is AB-y = AD = 0.03. We combine the above errors by writing the total error as a sum of these terms and then taking the mean and root as shown here: A ( m _ M ) „ = {(AA + AB + Ac + ADf)112 = (A2A + A\ + A2C + A2D + 2AAAB + 2AAAC + 2ABAC + 2AAAD + 2ABAD + 2ACAD)1'2. (10.3) Next, we can eliminate three of the six cross terms, 2AAAB, 2ABAC and 2ABAD, because they are the products of un-correlated errors; they are defined at fixed E(B—V) so AE(B-V) does not enter into these terms. Finally, we note that one of the cross terms, 2AAAc, is negative because the two errors compensate for each other. A n error in AE(B-V) in one of the terms opposes the associated error caused by the AB(B-v) m the other term. Evaluating equation 10.3 with these modifications gives a total error value of 0.16 on the true distance modulus. The error in the limiting white dwarf magnitude can be found by simply adding this distance modulus error (±0.16) in quadrature with the binning error (±0.25). The result gives 0.30, which is the uncertainty used to determine the age uncertainty given above. 10.3.5 Summary of NGC 2099 White Dwarf Studies This project has allowed us to identify 50 white dwarf candidates in N G C 2099. These ob-jects have all passed a stellarity and a statistical subtraction cut. Furthermore, the bright end slope of the white dwarf luminosity function agrees well with theoretical expecta-tions. The turn-over in the white dwarf luminosity function provides an age measurement for the cluster (566 ± J?4, Myrs) which is in excellent agreement with the main-sequence turn-off age (520 Myrs) for a core-overshooting model. A non core-overshooting age of ~300-350 Myrs for N G C 2099 can therefore be ruled out with some certainty. The ap-proach that we have used to estimate these quantities is a purely statistical method that needs further work. Fortunately, new multi-object spectrometers will allow us to simulta-neously measure spectra for a large number of objects. We will eventually obtain spectra for all objects in the faint, blue end of the colour-magnitude diagram before any cuts have Chapter 10. White Dwarf Stars and the Initial-Final Mass Relationship 83 been made. This will then provide a true measure of the number of white dwarfs in N G C 2099. Accurate surface gravities and effective temperatures for the brighter objects will provide crucial mass information which, when coupled with theoretical models, provides progenitor mass information. This will lead to a better understanding of the initial-final mass relationship for white dwarfs. Chapter 11 Conclusions We have described deep (V ~ 25) C C D photometry of a 42' x 28' field centered on two rich open star clusters, N G C 6819 and N G C 2099. For N G C 6819, this photometry indicates both a larger cluster extent (~9.5') and a much richer cluster population (~2900 stars, 2600 M 0 , after incompleteness corrections are applied) than previously estimated. Main-sequence fitting of the un-evolved stars in the cluster indicates a true distance modulus of ( m - M ) 0 = 11.99 ± 0.18 for a reddening value of E ( B - V ) = 0.10. Isochrone fits with up-to-date models are in excellent agreement with the data and comparison with turn-off stars in the cluster provides an age estimation of ~2.5 Gyrs. Measurements of the luminosity function and mass function of the cluster in concentric annuli indicate the cluster to be dynamically evolved. Studies of the cluster C M D suggest clear evidence for mass segregation in N G C 6819. The radial density distribution of the cluster is in good agreement with a single mass King model. Analysis of the faint blue section of the C M D indicates ~21 high probability W D candidates brighter than V = 23.5. Most of these stars are scattered around a 0.7 M Q white dwarf cooling curve. Until we have spectroscopic confirmation of their W D nature, it is premature to determine the initial-final mass relationship for these stars. For N G C 2099, the limit of our photometry is deeper than the limiting magnitude of the coolest white dwarfs in the cluster. Therefore, as expected, the white dwarf luminosity function for the cluster shows a steep turn-over at the termination point of the white dwarf luminosity function (My = 11.95 ± 0.30). After accounting for the main-sequence lifetime of these stars, the cooling age is determined to be 566 ± Jf4. Myrs, and therefore is in excellent agreement with the main-sequence turn-off age (520 Myrs). In order to have consistency between the two ages, core-overshooting models are preferred. The V, B- V C M D for N G C 2099 exhibits a spectacularly rich, long (over 12 magnitudes) and very tightly constrained main-sequence. Main-sequence fitting of the un-evolved stars in the cluster with the Hyades cluster indicates a true distance modulus of ( m - M ) 0 = 84 Chapter 11. Conclusions 85 10.90 ± 0.16 and a reddening value of E ( B - V ) = 0.21 ± 0.03. After accounting for small metallicity differences, the slope changes in the N G C 2099 main-sequence are found to be matched almost perfectly to those in the Hyades cluster. A theoretical stellar isochrone with Z = 0.020 and age = 520 Myrs models several features of the C M D very well, such as slope changes in the upper main-sequence and the red giant clump. The lower main-sequence and the turn-off are found to be slightly redder than the isochrone. The radial distibution of stars in N G C 2099 are in good agreement with a single mass King model. The global luminosity function is found to be flat from M v ~ 2 to My ~ 8 with a subsequent rise for fainter stars. After accounting for stars between our faintest magnitude bin and the hydrogen burning limit, the total cluster population of N G C 2099 is determined to be just under 4000 stars. Although not as severe as for the older cluster N G C 6819, we find some evidence for mass segregation within N G C 2099. The global mass function is shallower than a Salpeter IMF, and when extrapolated to the hydrogen burning limit, provides a lower limit (excluding binaries and stars out to the tidal radius) to the total cluster mass of just over ~2500 M©. The present CFH12K photometry of N G C 6819 and N G C 2099 has allowed us to gain a wealth of information regarding open star clusters, some of which has been discussed in this thesis. Preliminary reductions are already underway for two additional clusters in the survey. With the combined results for many clusters in the survey, we are optimistic that we will complete the global goals of the survey. Toward this effort, further spectroscopic studies are underway for the white dwarf candidates in N G C 6819 and N G C 2099. Bibliography [1] Abt, H.A. , Bolton, C.T. & Levy, S.G. 1972, ApJ , 171, 259 [2] Adams, J.D., Stauffer, J.R., Monet, D.G. , Skrutskie, M . F . & Beichman, C .A . 2001, A J , 121, 4, 2053 [3] Auner, G . 1974, A & A S , 13, 143 [4] Barbon, E.R. & Hassan, S.M. 1974, A & A S , 13, 293 [5] Barkhatova, K . A . , Dronova, V.I . , Pareva, L.I. & Sjasjkina, L.P. 1963, Collect. Works, Volume 1 (Ekaterinburg: Astron. Inst. Urals State Univ.), 3 [6] Barrado y Navascues, D., Stauffer, J.R., Bouvier, J . & Martin, E . L . 2001, ApJ , 546, Iss. 2, 1006 [7] Becker, W. 1948, Astr. Nach., 276, 1 [8] Becker, W. & Svolopoulos, S. 1976, A & A S , 23, 97 [9] Bergeron, P. 1995, PASP, 107, 1047 10] Bertelli, G. , Bressan, A . , Chiosi, C , Fagotto, F. & Nasi, E . 1994, A & A , 106, 275 11] Bertin, E. & Arnouts, S. 1996, A & A S , 117, 393 12] Bessell, M.S., Castelli, F . & Plez, B . 1998, A & A , 333, 231 13] Binney, J . & Tremaine, S. 1987, Galactic Dynamics (Princeton: University Press) 14] Binney, J . & Merrifield, M . 1998, Galactic Astronomy (Princeton: University Press) [15] Bolte, M . 1989, ApJ , 341, 168 [16] Bragaglia, A . et al. 2001, A J , 121, 327 [17] de Bruijne, J.H.J. , Hoogerwerf, R. & de Zeeuw, P.T. 2001, A & A , 367, 111 [18] Burkhead, M.S. 1971, A J , 76, 251 [19] Canterna, R., Geisler, D., Harris, H . , Olszeqski, E . & Schommer, R. 1986, A J , 92, 79 86 Bibliography 87 [20] Canuto, V . M . C . & Mazzitelli, I. 1992, ApJ , 389, 724 [21] Canuto, V . M . C , Goldman, I. & Mazzitelli, I. 1996, ApJ , 473, 550 [22] Castellani, V . , DeglTnnocenti, S. & Prada Moroni, P G . 2001, M N R A S , 320, 1, 66 [23] Claria, J.J., Piatti, A . E . & Lapasset, E. 1998, A & A S , 128, 131 [24] Claria, J.J. 1972, A J , 77, 868 [25] Clemens, D . P 1985, ApJ , 295, 422 [26] Copeland, H. , Jensen, J.O. & Jorgensen, H.E. 1970, A & A , 5, 12 [27] Cudworth, K . M . 1972, A & A , 18, 318 [28] Cuffey, J . & McCuskey, S.W. 1956, ApJ , 123, 59 [29] Cuillandre, J-C. 2001, in preparation [30] D'Antona, F. 1998, ASP Conference Series, 142, 157 [31] DaCosta, G.S. & Freeman, K . C . 1976, ApJ , 206, 128 [32] de la Fuente Marcos, R. 1997, A & A , 322, 764 [33] de la Fuente Marcos, R. 1996, A & A , 314, 453 [34] Foster, D.C., Theissen, A . , Butler, C.J . , Rolleston, W.R.J . , Byrne, P.B. & Hawley, S.L. 2000, A & A S , 143, 409 [35] Francic, S.P 1989, A J , 98, 888 [36] Freytag, B. , Ludwig, H.G. & Steffen, M . 1996, A & A , 313, 497 [37] Galadi-Enriquez, D., Jordi, C. & Trullols, E . 1998, Ap&SS, 263, 307 [38] Garcia-Pelayo, J . M . & Alfaro, E . J . 1984, PASP, 96, 444 [39] Giebeler, H . 1914, Bonn Veroff., 12 [40] Gunn, J .E. & Griffin, R.F. 1979, A J , 84, 753 [41] Hansen, B.M.S. 1999, ApJ , 520, 680 [42] Harrington, P. 1992, S&T, 83, 464 [43] Harris, W.E. , Fitzgerald, M.P. & Reed, B .C . 1981, PASP, 93, 507 Bibliography 88 [44] Hauschildt, P.H., Allard, F., & Baron, E. 1999, ApJ , 512, 377 [45] Hawley, S.L., Tourtellot, J .G. , & Reid, I.N. 1999, A J , 117, 1341 [46] Hi l l , G. & Barnes, J .V. 1971, A J , 76, 110 • [47] Hoag, A . A . , Iriarte, B. , Johnson, H.L., Hallam, K . L . , Mitchell, R .L & Sharpless, S. 1961, Pub. U.S. Naval Observatory, Ser. 2, 17 349 [48] Hogg, A .R . & Kron, G .E . 1955, A J , 60, 365 [49] Ianna, P.A. & Schlemmer, D . M . 1993, A J , 105, 209 [50] Iben, I. Jr. & Renzini, A . 1983, A R A A , 21, 271 [51] Janes, K . A . , Tilley, C. & Lynga, G. 1998, A J , 95, 771 [52] Jeffries, R .D. 1997, M N R A S , 292, 177 [53] Jeffreys, W.H. , III. 1962, A J , 67, 532 [54] Jones, B .F . & Prosser, C.F. 1996, A J , 111, 1193 [55] Kalirai, J.S., Richer, H.B. , Fahlman, G.G. , Cuillandre, J., Ventura, P, D'Antona, F., Bertin, E. , Marconi, G. & Durrell, P. 2001a, A J , 122, 257 [56] Kalirai, J.S., Richer, H.B., Fahlman, G.G. , Cuillandre, J., Ventura, P, D'Antona, F., Bertin, E . , Marconi, G. & Durrell, P. 2001b, A J , 122, 266 [57] Kalirai, J.S., Ventura, P., Richer, H.B., Fahlman, G.G. , Durrell, P., D'Antona, F. & Marconi, G. 2001c, A J , in press [58] Kaluzny, J . & Shara, M . 1988, A J , 95, 785 [59] King, I.R. 1962, A J , 67, 471 [60] King, I.R. 1964, Roy. Obs. Bull . (No. 82), 106 [61] King, I.R. 1966, A J , 71, 276 [62] Koester, D. k Reimers, D. 1996, A & A , 313, 810 [63] Koester, D. & Reimers, D. 1993, A & A , 275, 2, 479 [64] Koester, D. & Reimers, D. 1985, A & A , 153, 260 [65] Koester, D. & Reimers, D. 1981, A & A , 99, L8 Bibliography 89 [66] Landolt, A . U . 1992, ApJ , 104, 340 [67] Lattanzi, M . G . , Massone, G. & Munari, U . 1991, A J , 102, 177 [68] Lee, S.H., Kang, Y . W . & Ann, H.B. 1999, J K A S , 14, 2, 61 [69] Lee, S. & Sung, H . 1995, J K A S , 28, 1, 45 [70] Leggett, S.K., Ruiz, M.T . & Bergeron, R 1998, ApJ , 497, 294 [71] Liebert, J . , Dahn, C.C. & Monet, D .G . 1988, A p J , 332, 891 [72] Lindblad, R O . 1954, St. An. , 18, 1 [73] Lindoff, U . 1972, A & A S , 7, 497 [74] Maitzen, H . M . 1993, A & A S , 102, 1 [75] Marie, M . A . 1992, Ap&SS, 198, 121 [76] Menzies, J .W. & Marang, F. 1996, M N R A S , 282, 313 [77] Mermilliod, J.C., Huestamendia, G., del Rio, G. & Mayor, M . 1996, A & A , 307, 80 [78] Nordlund, J . 1909, Sv. Ark. Math., 5, 17 [79] O'Brien, M.S. , Vauclair, G. , Kawaler, S.D., Watson, T . K . , Winget, D.E. , Nather, R.E. , Montgomery, M . , Nitta, A . , Kleinman, S.J., Sullivan, D.J . , Jiang, X . J . , Marar, T . M . K . , Seetha, S., Ashoka, B .N. , Bhattacharya, J., Leibowitz, E . M . , Hemar, S., Ibbetson, R , Warner, B. , van Zyl, L . , Moskalik, P., Zola, S., Pajdosz, G., Krzesinski, J., Dolez, N . , Chevreton, M . , Solheim, J-E., Thomassen, T., Kepler, S.O., Giovan-nini, O., Provencal, J L . , Wood, M . A . & Clemens, J .C. 1998, A p J , 495, 458 [80] Pena, J .H. & Peniche, R. 1994, Rev. Mexicano Astron. Astrofis., 28, 139 (erratum 32, 193[1996]) [81] Peria, J .H. & Peniche, R. 1994, in I A U Symp. 162, Pulsation, Rotation, and Mass Loss in Early-Type Stars(Dordrecht:Kluwer), 303 [82] Perry, C L , Lee, P D . & Barnes, J .V. 1978, PASP, 90, 73 [83] Perryman, M . A . C , Brown, A . G . A . , Lebreton, Y . , Gomez, A . , Turon, C , de Strobel, G . C , Mermilliod, J . C , Robichon, N . , Kovalevsky, J . & Crifo, F . 1998, A & A , 331, 81 [84] Prosser, C F . & Giampapa, M.S. 1994, A J , 108, 964 Bibliography 90 [85] Prosser, C F . 1993, A J , 105, 1441 [86] Raboud, D. & Mermilliod, J-C. 1998, A & A , 329, 101 [87] Reid, N . 1992, M N R A S , 257, 2, 257 [88] Reimers, D. & Koester, D. 1994, A & A , 285, 451 [89] Reimers, D. & Koester, D. 1988, A & A , 202, 1, 277 [90] Reimers, D. & Koester, D. 1988b, ESO Messenger, 54, 47 [91] Reimers, D. & Koester, D. 1982, A & A , 116, 2, 341 [92] Renzini, A . & Fusi-Pecci, F. 1988, A R A A , 26, 199 [93] Richer, H.B. & Fahlman, G.G. 1996, AIP Conf. Proc. 393, 357 [94] Richer, H.B. , Fahlman, G.G. , Ibata, R.A. , Pryor, C , Bell, R .A . , Bolte, M . , Bond, H.E. , Harris, W.E. , Hesser, J.E., Holland, S., Ivanans, N . , Mandushev, G. , Stetson, P.B. & Wood, M . A . 1997, ApJ , 484, 741 [95] Richer, H.B. , Fahlman, G.G. , Rosvick, J . & Ibata, R. 1998, A p J , 504, L91 [96] Richer, H.B, Hansen, B. , Limongi, M . , Chieffi, A . , Straniero, O. & Fahlman, G .G. 2000, A p J , 529, 318 [97] Rosvick, J . M . & VandenBerg, D. 1998, A J , 115, 1516 [98] Sagar, R. & Griffiths, W . K . 1998, M N R A S , 299, 777 [99] Sagar, R. 1976, Ap&SS, 40, 447 [100] Salpeter, E .E . 1955, ApJ , 121, 161 [101] Sanders, W . L . 1973, A & A S , 9, 213 [102] Sanders, W . L . 1972, A & A , 19, 155 [103] Sanders, W.L . & van Altena, W.F . 1972, A & A , 17, 193 [104] Sanner, J. , Altmann, M . , Brunzendorf, J . & Geffert, M . 2000, A & A , 357, 471 [105] Saumon, D. & Jacobson, S.B. 1999, ApJ , 511, L107 [106] Schaller, G. , Shaerer, D., Meynet, G. & Maeder, A . 1992, A & A , 96, 269 [107] Schneider, H . 1987, A & A S , 71, 531 Bibliography 91 [108] Stetson, P.B. 1992, in IAU Coll. 136, Stellar Photometry - Current Techniques and Future Developments, ed. C.J . Butler & I. Elliot (Cambridge: Cambridge Univ. Press), 291 [109] Sung, H . & Bessell, M.S. 1999, M N R A S , 306, 361 [110] Sung, H. , Bessell, M.S., Lee, H-W, Kang, Y . H . & Lee, S-W. 1999, M N R A S , 310, 4, 982 [111] Tian, K.P . , Zhao, J .L. , Shao, Zh.Y. & Stetson, P.B. 1998, A & A S , 131, 89 [112] Turner, D .G. 1992, A J , 104, 1865 [113] Upgren, A .R . 1966, A J , 71, 8, 736 [114] van den Bergh, S. & Sher, D. 1960, Publ. David Dunlap Obs., 2, 203 [115] Ventura, P., Zeppieri, A . , Mazzitelli, I. & D'Antona, F. 1998, A & A , 334, 953 [116] von Hippel, T. & Gilmore, G. 2000, A J , 120, Iss. 3, 1384 [117] von Hippel, T. & Sarajedini, A . 1998, A J , 116, 1789 [118] von Zeipel, H . & Lindgren, J . 1921, Sv. Vet. H. , 61, N15 [119] Weidemann, V . 2000, A & A , 363, 647 [120] Weidemann, V . 1987, A & A , 188, 74 [121] West, F.R. 1967, ApJS, 14, 359 [122] Widen, R. 1991, in A S P Conf. Ser. 13, The Formation and Evolution of Star Clusters, ed. K . Janes (San Francisco:ASP), 343 [123] Wood, M . A . 1994, A A S Meeting, 185, 4601 [124] Woods, D. & Fahlman, G. 1997, ApJ , 490, 11 [125] Xiong, D.R. 1985, A & A , 150, 133 [126] Zug, R.S. 1933, Lick Obs. Bull . , 16, 130 

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