UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The stellar populations of the globular cluster M55 Mandushev, Georgi 1998

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1999-389405.pdf [ 8.91MB ]
Metadata
JSON: 831-1.0085705.json
JSON-LD: 831-1.0085705-ld.json
RDF/XML (Pretty): 831-1.0085705-rdf.xml
RDF/JSON: 831-1.0085705-rdf.json
Turtle: 831-1.0085705-turtle.txt
N-Triples: 831-1.0085705-rdf-ntriples.txt
Original Record: 831-1.0085705-source.json
Full Text
831-1.0085705-fulltext.txt
Citation
831-1.0085705.ris

Full Text

T H E S T E L L A R POPULATIONS OF T H E G L O B U L A R C L U S T E R M 5 5 By Georgi Mandushev B.Sc. Hon. (Physics) Sofia Univerisy St. Kliment Okhridski M.Sc. (Astronomy) Saint Mary's University  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E O F D O C T O R OF PHILOSOPHY  in T H E F A C U L T Y O F G R A D U A T E STUDIES PHYSICS & A S T R O N O M Y  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH COLUMBIA  October 1998 © Georgi Mandushev, 1998  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.  Physics &; Astronomy The University of British Columbia 129-2219 Main MaU Vancouver, Canada V6T 1Z4  Date:  Abstract  New broad-band, ground-based photometry in four filters (UBVI) for two fields in the sparse, metal-poor Galactic globular cluster M55 (NGC 6809) is presented and analyzed. New values are derived for the reddening (E _ B  distance modulus {{m~M)  v  V  = 0.13 ± 0.02 and E _i = 0.17 ± 0.02), v  = 14.02 ± 0.08) and age (14 ± 1.2 Gyr) of M55. The  main-sequence luminosity function of M55 is found to be different from the luminosity functions of the metal-poor clusters M15, M30 and M92 and this difference is interpreted as a deficiency of low-mass stars by about 50% compared to the other three clusters. The mass function of M55 for masses below O.4A4 is found to be fairly flat and consequently 0  low-mass stars do not dominate the cluster mass. The red giant branch of M55 has been observed from nearly its tip to the subgiant branch. In all passbands the observed luminosity of the red giant clump is lower than the predictions of theoretical models. The ratios of the number of stars on the red giant branch, the horizontal branch and the asymptotic giant branch are found to be in a good agreement with theoretical models. Neither the K-band nor the /-band luminosity functions for the evolved populations in M55 show any significant deviation from the theoretical luminosity functions. In particular, no evidence is found for a deficiency of main-sequence stars compared to the number of stars on the subgiant and giant branches. M55 is the only well-studied, metalpoor cluster for which no discrepancy between observations and canonical luminosity functions is found. A large sample of blue stragglers in the core of M55 is identified and analyzed. It is concluded that the blue stragglers in M55 are born with helium-enriched cores but not  n  envelopes, thus resembling stars that have already evolved away from the main sequence. It is also suggested that the observed blue straggler sequence represents the equivalent of a core helium-enriched main sequence where the blue stragglers spend most of their lives. The observations agree qualitatively with unmixed collisional or merger models, although the former are less likely in the low-density environment of M55.  in  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  ix  Acknowledgements 1  2  xii  Introduction  1  1.1  Globular Clusters and Stellar Populations  1  1.2  The Colour-Magnitude Diagram and the Evolution of Low-mass Stars . .  5  1.3  Structure of Globular Clusters  13  1.4  Earlier Studies of M55  14  The Main Sequence of M55  17  2.1  Observations and Preprocessing  17  2.1.1  18  2.2  2.3  Flat Fields, Overscan Subtraction and Cosmic Ray Removal . . .  Photometry  21  2.2.1  Instrumental Magnitudes  21  2.2.2  Aperture Corrections  22  2.2.3  Transformation to the Standard System  23  2.2.4  Comparison with Earlier Photometry  28  The Color-Magnitude Diagram of M55  31  2.3.1  31  Morphology and principal sequences iv  2.3.2  36  2.4  Photometry of the comparison field  40  2.5  The Reddening and Metalhcity of M55  44  2.6  The Distance and Age of M55  47  2.6.1  Distance Modulus  47  2.6.2  The Age of M55  59  2.7  3  The Stars of the Sagittarius Dwarf Galaxy and the Galactic bulge  Luminosity and Mass Functions for the Main Sequence of M55  63  2.7.1  Completeness Corrections  66  2.7.2  Luminosity Functions  71  2.7.3  Mass Function  79  The Evolved Populations of M55  88  3.1  Observations and Preprocessing  90  3.2  Photometry  92  3.2.1  Instrumental Magnitudes and Aperture Corrections  92  3.2.2  Transformation to the Standard System  93  3.2.3  Quality of the UB VI Photometry: Errors and *  3.3  99  2  The Color-Magnitude Diagram of Evolved Stars  106  3.3.1  110  Fiducial Sequences  3.4  The Horizontal Branch of M55  Ill  3.5  The Red Giant Branch of M55  119  3.6  3.5.1  The Gap at the Base of the Giant Branch and the RGB Clump  .  119  3.5.2  Intrinsic Width of the Red Giant Branch  125  3.5.3  Population Ratios and the Helium Abundance of M55  129  Luminosity Functions for the Evolved Populations in M55  132  3.6.1  133  Artificial Star Tests and Completeness Corrections  v  3.6.2 4  5  Luminosity Functions in V and / : Theory vs. Observations  . . .  137  The Blue Straggler Population of M55  151  4.1  The Blue-Straggler Sample - Definition and Completeness  152  4.2  Radial Distribution  159  4.3  Origin and Evolutionary Status  161  Conclusions  167  5.1  The Main Sequence of M55  167  5.2  The Evolved Populations of M55  169  5.3  The Blue Straggler Stars in M55  172  5.4  Future Directions  173  Appendices  175  A Summary of some important parameters for M55  175  B Publications Related to the Thesis  176  C Photometry Software Used in the Thesis  178  Cl  D A O P H O T II  178  C.2  ALLSTAR  178  C.3  ALLFRAME  178  C.4  Calibration and Transformation Programs  179  D Journal Abbreviations  180  References  181  vi  List of Tables  2.1  Program and comparison field exposure information  20  2.2  List of the observed Landolt standards  25  2.3  Local standards in M55  29  2.4  The fiducial main sequence and subgiant branch of M55  36  2.5  The fiducial main sequence of the Sagittarius Dwarf Galaxy  37  2.6  Galaxy counts in the comparison field  43  2.7  Observed and derived data for nearby subdwarfs  50  2.8  References for the compiled subdwarf data  51  2.9  Comparison of My(RR) estimates from different My(RR) - [Fe/H] relations  58  2.10 Star and galaxy counts in the V band  71  2.11 Star and galaxy counts in the / band  72  2.12 Mass function slopes  84  3.1  Core field exposure information  92  3.2  List of the observed Landolt standards  94  3.3  Local standards in M55  97  3.4  Fiducial points for the giant branch, subgiant branch and the main sequence  112  3.5  Fiducial points for the asymptotic giant branch  113  3.6  Fiducial points for the horizontal branch  113  3.7  Known and Suspected RR Lyrae Stars in the Core Field  115  vii  3.8  Star Counts and Population Ratios  130  3.9  K-band and /-band differential luminosity functions  144  4.1  Photometry for the blue stragglers in the core of M55  157  A.l  Some parameters of M55 as derived in this work  175  vni  List of Figures  1.1  Principal sequences of a globular-cluster colour-magnitude diagram  . . .  7  2.1  The location of the deep field relative to M55  2.2  Finder chart for the local standards in the deep  2.3  Residuals for the Landolt standards  2.4  Color-magnitude diagram for all objects in the deep  2.5  Color-magnitude diagram for the deep field with % restrictions  34  2.6  The principal sequences of M55 and other populations  39  2.7  Colour-magnitude diagram for the comparison  2.8  Plot of the average image-sharpness index r versus V magnitude  2.9  Colour-colour diagram for the central field of M55  19 field  27 30  field  32  2  field  0  41 . . . .  42 45  2.10 Fit of the main sequence of M55 to nearby subdwarfs  54  2.11 Agreement between the shapes of the main sequence and the isochrones .  56  2.12 Isochrone match to the photometry around the turnoff  60  2.13 Comparison between the observed and artificial main sequence  68  2.14 Differences between the input and output magnitudes and colour  69  2.15 Completeness fractions in the V and I passbands  70  2.16 The F-band and /-band luminosity functions  73  2.17 Comparison of the ground-based luminosity function of M55 with HSTbased luminosity functions - faint normalization  75  2.18 Comparison of the ground-based luminosity function of M55 with HSTbased luminosity functions - bright normalization  ix  77  2.19 Comparison of the mass-luminosity relations in V and /  80  2.20 Comparison of low-mass isochrones with the main sequence of M55  . . .  2.21 The mass function of M55  81 83  2.22 More comparisons of the ground-based luminosity function of M55 with HST-ha.sed luminosity functions  86  3.1  Location of the core field relative to M55  91  3.2  Finder chart for the local standards in the core  3.3  Residuals for the Landolt standards  3.4  Magnitude and colour errors as a function of V magnitude  100  3.5  Plot of x versus V magnitude  102  3.6  Rejection criteria in magnitude and colour index  104  3.7  The effect of error and \  3.8  The V, B — V colour-magnitude diagram  107  3.9  The V, V — I colour-magnitude diagram  108  3.10 The V, B — I colour-magnitude diagram  109  field  98  2  2  96  restrictions on the colour-magnitude diagram . 105  3.11 Finder chart for the known and suspected RR Lyrae variables in the core field  116  3.12 Zero-age horizontal branch  fits  120  3.13 Observed and expected width of the red giant branch  127  3.14 Comparison between the "real" and artificial colour-magnitude diagrams  134  3.15 Completeness fractions in the V and / bands  136  3.16 Raw luminosity functions in the V and / bands  139  3.17 Cumulative luminosity functions in the V and I bands  140  3.18 Observed and theoretical /-band luminosity functions  142  3.19 Observed and theoretical F-band luminosity functions  143  x  4.1  The V , B — I colour-magnitude diagram for the core field  153  4.2  Detailed view of the blue-straggler region  155  4.3  Finder chart for the blue stragglers  156  4.4  The radial distribution of the blue stragglers  160  4.5  The blue-straggler distribution and single-star models  163  xi  Acknowledgements  I would like to thank my supervisors, Dr. Harvey Richer and Dr. Greg Fahlman for their support, encouragement and advice. Many thanks go to Peter Stetson for his permission to use  D A O P H O T II, A L L F R A M E  and his other software, as well as for the valuable advice  he has given on many occasions. I am grateful to Don VandenBerg for making his new set of isochorones available in advance of publication and for the many helpful discussions. It a pleasure to thank Peter Bergbusch for providing me with his theoretical luminosity functions prior to publication, and Christine Clement for sending me her latest results on the absolute magnitudes of RR Lyrae variables.  xn  Chapter 1  Introduction  1.1  Globular Clusters and Stellar Populations  The fundamental idea behind the concept of stellar populations is that in the Milky Way and other galaxies there are distinct subsystems of stars whose collective characteristics (kinematics, chemical composition and age) can be very different. The study of stellar populations provides the observational foundations for the theories of galaxy formation, structure and evolution. By observing and analysing the properties of stellar populations, we are trying to understand their origins and how they changed during the various phases of Galactic evolution. In our Galaxy stars and star clusters have been traditionally classified as belonging either to Population I (the disk of the Galaxy) or Population II (the spheroid of the Galaxy, encompassing the bulge and the halo). In many cases this coarse separation of the stellar systems in two populations is sufficient to describe their overall properties: the stars in Population I are confined to a thin (300 to 350 pc), rapidly rotating disk, they have small velocity dispersions, nearly circular orbits and metal abundances similar to that of the Sun. The stars belonging to Population II have highly elongated orbits, high velocities and they form a metal-poor, slowly rotating halo. More than a decade ago it was demonstrated that in our Galaxy, as well as in other spiral galaxies with small bulges, there was another significant population component called the thick disk (van der Kruit k Searle 1982, Gilmore & Reid 1983, Norris 1986), which  1  2  Chapter 1. Introduction  comprises about 5% of the Galactic disk. It has also been argued (Mould 1986, Rich 1993, Rich 1996) that the bulge stars, usually associated with Population II, belong in fact to a distinct, more metal-rich population that does not share common origin with the halo. Globular clusters are remarkably round, compact, self-gravitating collections of thousands to millions of stars. There are about 160 known globular clusters in our Galaxy (Djorgovski Sz Meylan 1993), and globular cluster systems of varying richness have been observed in galaxies beyond the Virgo cluster. Observational and theoretical work in the last few decades has shown that globular clusters are the oldest identifiable objects in the Milky Way, with ages of I O  10  years or more. Most globular clusters are metal-poor  compared to the Sun and have the kinematics and metallicities typical for the extreme Population II (or the halo of the Galaxy). About a quarter of the known globular clusters (most of them inside the solar circle) belong to the intermediate Population II (or the thick disk) of the Milky Way, being more metal-rich and having more circular orbits than their halo cousins (Zinn 1985, Armandroff 1993). One of the important properties of globular clusters and the one that makes them so useful for stellar population studies is that the stars in these clusters are at the same distance, coeval and of nearly identical chemical abundance (e.g., Fahlman et al. 1985, Stetson 1993a). If one is interested in the properties of the halo stars, e.g. luminosity and mass functions, chemical abundance distribution or kinematics, large and uncontaminated samples of extreme Population II stars are difficult to gather and analyze for several reasons. Even large surveys sample only a small volume of space, and at faint magnitudes background galaxies far outnumber halo stars. Distances and chemical compositions can only be estimated crudely from broadband colours and magnitudes, as spectroscopy for individual stars will be prohibitively expensive. While attempts have been made to determine the luminosity function for halo stars in situ (Richer & Fahlman 1992,  Chapter 1. Introduction  3  Dahn et al. 1995, Reid 1995, Reid et al. 1996), for large samples of halo stars it is natural to look at globular clusters, where tens of thousands of Population II stars of common distance, age and metal abundance can be measured simultaneously. A surprisingly large amount of information about any resolved stellar population can be obtained by simply counting the number of stars as a function of their position or brightness. The introduction of CCD (charge-coupled device) detectors in globular cluster studies and the use of computers to measure the coordinates and brightness of individual stars have changed profoundly this old technique. Star counting, which as a method goes at least as far back as Wilham Herschel and his studies of the structure of the Galaxy, can provide important observational input into the origin and evolution of cluster stellar populations. One of the most important applications of star counts is to determine the distribution of stars as a function of magnitude, i.e., to obtain the luminosity function for the observed globular cluster field. The luminosity function of a globular cluster can be split quite naturally in two portions because of the way cluster stars evolve. Below the main-sequence turnoff, all stars are in the same phase of their evolution (core hydrogen burning) and the luminosity of a star is a function only of its mass; thus the main sequence is essentially a mass sequence. The principal usefulness of the main-sequence luminosity function (and the mass function obtained from it) is that it can be used to estimate the shape of the initial mass function and therefore provide information on the physical conditions in which globular clusters formed. Given the close relationship between globular clusters and the halo population of the Galaxy (Carney 1993, Zinn 1996), the unevolved, low-mass stellar content of globular clusters should be similar to that of the field halo stars. This has been one of the motivations behind several observational efforts to obtain the main-sequence luminosity and mass functions for Galactic globular clusters and then extend the results to the halo (Richer et al. 1991, King et al. 1996a, Piotto et al. 1997, King et al. 1998a).  Chapter 1. Introduction  4  The first study suggested that the derived steep mass functions for several globular clusters, if extended to the halo, could mean that a large portion of the halo's mass was in the form of very low-mass stars and that could provide a possible solution to the dark matter problem in the Milky Way and other galaxies. However, the mass functions derived later from observations with the Hubble Space Telescope (King et al. 1996a) were not that steep and, while still rising to the faint limit of the data, would not imply such a large population of low-mass, metal-poor stars in the halo. The issue about the present-day globular cluster mass functions and their relation to the mass function of the galactic halo is not settled yet and is complicated by the internal dynamical evolution of globular clusters and their interaction with the tidal field of the Galaxy. The mass functions of globular clusters can be modified by mass segregation, i.e., the concentration of the heavier stars toward the cluster centre (Pryor et al. 1986, Richer & Fahlman 1989, Piotto 1993) and by the loss of stars (especially low-mass stars) caused by evaporation and disk and bulge shocking (Chernoff & Weinberg 1990, Weinberg 1994, Capriotti & Hawley 1996, Gnedin & Ostriker 1997). Both of these effects lead to a flattening of the mass function, especially near the cluster centre. It is possible in principle to correct for the effects of mass segregation, and in sparse clusters with long relaxation times (such as M55) mass segregation is expected to be relatively weak. The effects of stellar mass loss are much more difficult to correct for and at present can only be crudely estimated (Hut & Djorgovski 1992, Gnedin & Ostriker 1997). Above the main-sequence turnoff, stars evolve rapidly and at an increasing rate, so that the number of stars of a given absolute magnitude is directly proportional to the time spent in that particular phase of their evolution. As a result, the luminosity function of the stars above the turnoff can be used to compare directly the predictions of stellar evolution models to observations (Renzini & Fusi Pecci 1988). This comparison is facilitated by the fact that, unlike the main sequence stars, all evolved stars have similar masses  Chapter 1. Introduction  5  and therefore their luminosity function is not influenced by mass segregation, i.e., the relative number of stars at different evolutionary stages will not depend on where in the cluster they have been observed. For the same reason the luminosity function is insensitive to the slope of the initial mass function. On the other hand, the shape of luminosity function above the turnofF is affected by the age and metal abundance of the cluster and it has been proposed to use the luminosity function of evolved stars to determine the ages and compositions of globular cluster (Paczyriski 1984, see also RatclifF 1987). A direct comparison between the observed and theoretical luminosity functions (Bolte 1994, VandenBerg et al. 1998b, Degl'Innocenti et al. 1997), while more informative, requires knowledge of the distance to the cluster. An alternative approach is to compare the ratios of the number of stars on different branches in the cluster colour-magnitude diagram to those predicted by the theory (Buzzoni et al. 1983, Renzini k Fusi Pecci 1988, Sandquist et al. 1996); this procedure is very robust as long as a sufficient number of stars on each branch are observed. In many places in the thesis the reader is referred to the various branches of the colourmagnitude diagram of a globular cluster and the evolutionary status of the stars on those branches. The next section reviews the morphology of the colour-magnitude diagram and provides a brief outline of stellar evolution as applied to present-day globular clusters. Detailed reviews of the evolution of low-mass stars (those having masses of about two solar masses and below) and its application to star clusters can be found in Hayashi et al. (1962), Iben (1971), Renzini k Fusi Pecci (1988) and Iben (1991), among others.  1.2  The Colour-Magnitude Diagram and the Evolution of Low-mass Stars  The colour-magnitude diagram for a collection of stars is a plot of absolute magnitude versus colour index (the latter being a measure of the star's surface temperature). The  6  Chapter 1. Introduction  terms colour-magnitude diagram and Hertzsprung-Russell diagram are often used as synonyms, although the second term usually refers to a plot of absolute magnitude versus spectral type. In fact, the original diagram by Ejnar Hertzsprung (Hertzsprung 1911) was a plot of absolute visual magnitude versus colour, whereas Henry Russell's version (Russell 1914) was a plot of absolute visual magnitude versus spectral type. Theoreticians, on the other hand, use a version of the Hertzsprung-Russell diagram where luminosity is plotted versus effective temperature. The principal sequences of a typical colour-magnitude diagram for an old globular cluster are illustrated in Figure 1.1. This is a composite diagram made up from the photometry in two different fields in the globular cluster M55, and the relative numbers of stars on the different branches are not correct: for example, the main sequence is much more heavily populated relative to the red giant branch than it appears in Figure 1.1. In this diagram the apparent (not absolute) V magnitude is plotted versus the V — I colour index (apparent magnitudes can be used instead of absolute ones, since all cluster stars can be considered to be at the same distance). Brighter stars have smaller visual magnitudes and cooler stars have larger colour indices. The main sequence is the locus occupied by stars in the longest phase of their lives core hydrogen burning (the conversion of hydrogen into helium in the star's core). For a given chemical composition, the luminosity of a main-sequence star is determined only by its mass: L oc M. , where a ~ 2 below O.57Vf and a ~ 4.5 above 0.5Mq a  0  (Renzini & Fusi Pecci 1988). As the hydrogen in the star's core is slowly being converted into helium, the star becomes hotter and brighter: in the colour-magnitude diagram it moves up and to the left almost parallel to the main sequence. However, as less and less hydrogen remains in the core, the increase in temperature slows down, but the star is still getting brighter: in Figure 1.1 this gradual "peel-off" from the main sequence can be seen clearly for about two magnitudes below the main-sequence turnoff. Eventually all  Chapter 1. Introduction  7  10  AGB.f  15  MSTO 20  25 _j  -1  L  _j  L  V-I  Figure 1.1: A composite colour-magnitude diagram for the stars in twofieldsin the globular cluster M55. Horizontal axis: V-I colour index; vertical axis: apparent V magnitude. The acronyms mean: AGB: asymptotic giant branch; HB: horizontal branch; RGB: red giant branch; SGB: subgiant branch; BSS: blue straggler stars; MSTO: main-sequence turnoff; MS: main sequence.  Chapter 1. Introduction  8  the hydrogen in the central region of the core (about 10% by mass) is exhausted and the star now has a small core of pure helium; such stars occupy the main-sequence turnoff in the cluster colour-magnitude diagram. The colour and luminosity of the main-sequence turnoff depend strongly on chemical composition — lower abundance of metals (elements heavier than helium) and higher helium abundance will make the turnoff brighter and bluer. For fixed abundances of helium and heavy elements, however, the luminosity of the main-sequence turnoff is determined by the age of the cluster — since more massive (and therefore more luminous) stars leave the main sequence sooner, the older the cluster the fainter the turnoff. In the first globular cluster colour-magnitude diagrams reaching below the main-sequence turnoff (Arp et al. 1952, Arp et al. 1953, Sandage 1953) the main sequence terminated so much redder (and fainter) than the known open cluster colour-magnitude diagrams, that it became instantly clear that globular clusters were much older. There are still many uncertainties (both observational and theoretical) in the exact calibration of the turnoff luminosity - age relationship (Chaboyer 1995, VandenBerg et al. 1996), but at present it provides the most reliable means for estimating globular cluster ages and consequently putting a lower limit on the age of the universe. Currently, absolute age determinations for globular clusters are done by matching theoretical isochrones (the loci of stars of the same age but different mass) to the observed turnoff region of the cluster colour-magnitude diagram. This procedure requires knowledge of the distance, reddening and heavy metal abundance of the cluster, and each of these quantities is usually known with an error of at least 10%, and often more. Relative globular cluster ages, on the other hand, can be estimated with a greater precision as they are independent of distance, reddening and, for [Fe/H] < —1.2, metallicity differences (Sarajedini & King 1989, VandenBerg et al. 1990, Stetson et al. 1996). At the point of central hydrogen exhaustion the star's inner core (which now consists  Chapter 1. Introduction  9  of almost pure helium) has shrunk considerably, but its temperature is still too low for the start of helium burning —the conversion of helium into heavier elements such as carbon and oxygen. Most of the luminosity at this point is being generated by hydrogen burning in a thick shell surrounding a small, hot helium core and this is the energy output that continues to drive envelope expansion (Iben 1971). When the envelope expands, luminosity increases only slightly, but the increased radius causes the star to become cooler — in the colour-magnitude diagram the star moves to the right along the subgiant branch. As the star evolves to the red, the thickness of the hydrogen-burning shell rapidly decreases and at the base of the giant branch about 80% of the energy output is produced in a thin shell encompassing only about O.OOIA^©. Theoretical models show that the rate of evolution on the subgiant branch depends on the mass of the star, and hence the subgiant branch luminosity function should contain information about the absolute age of the cluster (as more massive stars leave the main sequence earlier). Recent work on this subject (Bergbusch & VandenBerg 1997) has indicated that it should be possible to estimate globular cluster ages with an accuracy at least as good as that achieved using turnoff luminosities. The event that marks the beginning of the red giant phase is the reappearance of a fully convective envelope that reaches deep into the star's interior. As convection is very effective in transporting to the surface the energy generated by the shell, the star must expand its surface area (and hence luminosity) to accommodate the increased energy flow. In the colour-magnitude diagram the star moves steeply up and slightly to the right — it now follows the red giant branch. As the star is now fully convective (with the exception of the small helium core), its track in the colour-magnitude diagram resembles the Hayashi track of a fully convective protostar contracting towards the main sequence, only in reverse. Throughout the red giant phase the principal source of energy remains hydrogen burning in a very thin (by mass) shell that gradually moves outward.  10  Chapter 1. Introduction  The rate of evolution (determined by how fast the shell burns through the inner half of star's mass) will depend to a large degree on the hydrogen abundance profile established in the earlier phases of evolution; thus the small thickness of the hydrogen-burning shell allows in principle the sampling of this profile (by means of the red giant branch luminosity function) with a resolution of 10~ .M© or better (Renzini Sz Fusi Pecci 1988). For 3  example, late in the red-giant phase the hydrogen-burning shell encounters a discontinuity in the hydrogen profile left by the deeply-penetrating convective envelope when the star was on the lower giant branch (Thomas 1967). This causes a slowing in the rate of evolution and correspondingly a local increase in the number of stars on the red giant branch and a "bump" in the red giant branch luminosity function (Sweigart 1978, Renzini & Fusi Pecci 1988, Fusi Pecci et al. 1990). This so-called "red-giant clump" can be seen on the red giant branch in Figure 1.1 slightly above the level of the horizontal branch. As the star approaches the tip of the red giant branch, the mass of the helium core and the temperature in its centre continue to increase.  Once the core temperature  reaches approximately 10 K, helium ignites via the triple-alpha process: 3He —» C 8  4  1 2  +7  (Burbidge et al. 1957). Since the core is degenerate, the pressure depends only weakly on temperature and therefore the rise in temperature is not followed by core expansion. The triple-alpha process is extremely dependent on temperature (approximately as T ) 30  and so the increased core temperature leads to further energy release and a thermal runaway develops (Iben 1971, Renzini & Fusi Pecci 1988). This so-called "helium flash" marks the end of the red giant branch phase in the star's evolution; the heliumflashlasts only a short time until the high temperature developed in the core lifts the degeneracy, the core expands and cools, and core helium burning is established. In the colour-magnitude diagram the stars burning helium in their cores are found on the horizontal branch. The morphology of the horizontal branch has been one of  11  Chapter 1. Introduction  the most investigated and debated areas in globular cluster research, both by observers (Pusi Pecci et al. 1992, Buonanno 1993, Fusi Pecci et al. 1996 and the references therein) and theoreticians (Rood 1973, Renzini & Fusi Pecci 1988, Iben 1991 and the references therein). The distribution of the stars on the horizontal branch can differ very much between different clusters: there are clusters (M55 among them) where most of the horizontal-branch stars are concentrated to the blue end of the horizontal branch [cf. Figure 1.1). In some clusters most of the stars are found at the red end of the horizontal branch, near the place where it meets the red giant branch, and there are other globular clusters where the horizontal branch is more or less uniformly populated. The horizontal branch is also the place where the short-period pulsating variables of type RR Lyrae are found. They occupy the middle of the horizontal branch, roughly at V — I « 0.6 in Figure 1.1. In some clusters (like M55) there are only a few (or even no) RR Lyr stars, and some clusters contain tens or even hundreds of them. The first horizontal branch models (Iben & Rood 1970, Rood 1973) revealed that (a) horizontal-branch stars must have masses that are much lower (by ~ 25%) than the turnoff stars they descended from, and (b) at least a 10% dispersion in mass was required to reproduce the colour distribution of horizontal-branch stars. The inevitable conclusion was that a variable amount of mass (~ 0.2.M © ± 10%) must be lost after the main-sequence turnoff but before or during helium ignition in order to match the observed horizontal branches. Theoretical models also indicate that the duration of the horizontal branch phase (and hence the observed number of horizontal-branch stars) is a strong function of both helium abundance (Iben & Rood 1969, Renzini 1977) and the extent of mixing in the convective helium-burning core (Renzini 1977, Renzini & Fusi Pecci 1988). It turns out, however, that the ratio R  = A^HB/^VRGB  of the number of stars on the hori-  zontal branch to the number of stars on the red giant branch depends almost exclusively on the helium abundance, whereas the ratio R  2  = TVHB  /NAGB  of the number of stars on  Chapter 1.  Introduction  12  the horizontal branch to the number of stars on the asymptotic giant branch is determined mostly by the extent and mode of central mixing (Renzini 1977, Buzzoni et al. 1983, Renzini k Fusi Pecci 1988, Iben 1991). Both ratios can be determined observationally and at present the value of R provides the best means of estimating helium abundance in globular clusters (Buzzoni et al. 1983, Caputo et al. 1987); the value of R2 is used to constrain the models of horizontal-branch stars. After the supply of helium in the core of a horizontal-branch star is exhausted, heHum continues to burn in a thick shell surrounding the carbon-oxygen core. The stars with more massive envelopes (i.e. with smaller mass losses during the red giant phase) evolve to the right in the colour-magnitude diagram and ascend the asymptotic giant branch, named so because it approaches the red giant branch (Iben k Renzini 1983, Dorman 1992, Vassiliadis k Wood 1993). Horizontal-branch stars with small envelope masses initially evolve almost vertically in the colour-magnitude diagram before joining the asymptotic giant branch (cf. Dorman 1992). The rate of evolution is now greatly accelerated and as a result the asymptotic giant branch is the least populated branch in the colour-magnitude diagram. While on the asymptotic giant branch, the stars pass through two stages: an early stage, where the hydrogen-burning shell that supplied some of the horizontal-branch luminosity is inactive, and a later, so-called thermally-pulsing stage, where the re-establishment of a hydrogen-burning shell leads to the development of thermal instabilities. Towards the end of the latter phase the stars suffer very high mass loss rates that are sufficient to strip the hydrogen-rich envelope; the stars evolve rapidly through the planetary nebula phase and in the colour-magnitude diagram they move far to the left and then descend down into the region of white dwarfs (Iben k Renzini 1983, Renzini k Fusi Pecci 1988, Vassiliadis k Wood 1993). In Figure 1.1 there are also a small number of stars that appear as an extension of the main sequence, bluer and brighter than the turnoff. Discovered first by Sandage in M3  13  Chapter 1. Introduction  (Sandage 1953), these stars are called blue straggler stars or blue stragglers. It is generally accepted now that blue stragglers have formed through a merger of two (or more) lessmassive stars, although the nature and the details of the merger process are still far from certain. The proposed mechanisms include direct stellar collisions and binary-binary collisions (Benz & Hills 1987, Leonard 1996), binary coalescence (Leonard 1996, Mateo 1996) and mass transfer in a binary system (McCrea 1964, Stryker 1993, Mateo 1996). Recent studies (see Mateo 1996 for a review) indicate that blue straggler stars are highly visible tracers of cluster binary populations and their evolution, especially in low-concentration clusters such as M55, where a higher fraction of the primordial binaries is expected to have survived.  1.3  Structure of Globular Clusters  The surface brightness distribution (or the surface stellar density) of most globular clusters can be described by a model specified by the ratio of two lengths: the tidal radius (r ) and the core radius (r ). The core radius was introduced first by King (1962) as an t  c  empirical scale factor and it is approximately equal to the distance at which the surface stellar density of the cluster drops to half of its central value (Peterson &; King 1975, Richstone &; Tremaine 1986). In practice it is determined by fitting King (1966) models to the observed surface brightness profile or to star counts. The tidal radius r is defined t  as the distance at which the surface density of the cluster drops to zero (King 1962). This quantity can be regarded as the physical limit of the cluster, the distance at which the escape velocity is zero and the stars are lost to the Galactic tidal field. Positions and distances in and around globular clusters are often specified in terms of r or r . c  t  The ratio of the tidal and the core radii of a globular cluster can be used as a measure of the degree of its central concentration (King 1962). For the Milky Way globular  14  Chapter 1. Introduction  clusters, the King concentration parameter c = log(r /r ) varies between ~ 0.5 and ~ 2.5 t  c  (Trager et al. 1993). M55 is one of the least centrally concentrated clusters (c = 0.76, Trager et al. 1993) and because of its richness and proximity, it has the largest apparent size of all globular clusters (about 25' in diameter).  1.4  Earlier Studies of M 5 5  M55 is the fifty-fifth object in Messier's catalogue of nebulae and star clusters (Messier 1784) and it is also known under the designations NGC 6809 and C1936-310.  It is  a sparse, metal-poor, moderately bright globular cluster located in the constellation of Sagittarius.  Its J2000.0 right ascension and declination are a = 19 39 59f4 and h  m  8 = — 31°07'44" (Djorgovski &: Meylan 1993); the corresponding galactic coordinates are I = 8?8 and b = -23?4. Despite the low concentration and proximity of M55, the first dedicated photographic studies (Alcaino 1975, Harris 1975 and Lee 1977) did not reach below the main-sequence turnoff. All of them noted the blue horizontal branch and deduced a low metallicity for M55 from the slope and extent of its giant branch. Both Alcaino (1975) and Lee (1977) estimated a reddening of EB-V  = 0.08, the first from comparison with the giant  branches of other, unreddened clusters and the second from the colour-colour diagram of reddened field stars. On the other hand, Kron & Guetter (1976) found an average value of EB-V = 0.16 from six-colour measurements of the integrated light. The most extensive and precise photometry was that of Lee (1977), who derived a distance modulus of (m - M)  v  = 14.05 ± 0.15 assuming that My(HB) = 0.3. He also found a new RR Lyr  variable in addition to the six known before, and the high quality of his photometry allowed him to identify and measure the position of the red giant branch clump. As a result of these studies, by the beginning of the CCD era M55 was known as a nearby,  Chapter 1. Introduction  15  sparse, lightly reddened, metal-poor cluster with a blue horizontal branch. While Lee (1977) constructed a giant branch luminosity function, its value was diminished by the near impossibility of estimating the completeness of photographic photometry in the crowded cluster field (King et al. 1968). The first CCD photometry of M55 was carried out by Penny (1984) who matched theoretical isochrones to the turnoff and the upper main sequence of the cluster and concluded that the discrepancies between theory and observations were reduced compared to photographic photometry. Later Schade et al. (1988) obtained CCD BV photometry in a field north-west of the cluster centre. They found EB-V = 0.14 from comparison with the giant branches of NGC 6752 and M68 and inferred an age of 14 Gyr from isochrone fitting. In another study, Alcaino et al. (1992) carried out four-colour (BVRI) photometry in twofieldsin M55 and used isochrone fits to derive an age of 14-15 Gyr and a reddening of EB-V — 0.16 ± 0.03. In all these investigations, the authors were mostly concerned with the reddening, metallicity and age of M55. As all observed fields were far from the cluster core, the colour-magnitude diagrams were poorly populated above the main-sequence turnoff and no attempts were made to study the evolved populations. Even the deepest photometry (that of Schade et al. 1988) reached only two magnitudes below the turnoff and so not much was known about the main sequence of M55 except that the theoretical isochrones matched its upper portion fairly well. A recent paper by Zaggia et al. (1994) was the first work on M55 that addressed in more detail some properties of its stellar populations. While the main purpose of their study was to carry out star counts along the full radial extent of M55, they also derived a luminosity function for M55 from the main-sequence turnoff to about 2 magnitudes below the turnoff (the faint limit of their photometry). Because of the significant crowding effects and the resultant poor quality of the photometry, the luminosity function was obtained only for stars beyond one core radius. From the main-sequence luminosity  Chapter 1. Introduction  16  function Zaggia et al. (1994) derived a mass function for the upper main sequence (0.8 < A4/A4®  < 0.6) at three different distances from the centre and concluded that there was  evidence for mass segregation. They also argued that the global mass function (i.e., the mass function for the total radial extent of their observations) in that narrow mass range was quite flat, which could indicate substantial stellar mass loss. Zaggia et al. (1994) were the first to identify a significant population of blue straggler stars in M55 and to investigate their radial distribution; they concluded that, unlike other globular clusters, the blue stragglers in M55 were not more centrally concentrated than stars of similar brightness.  Chapter 2  The Main Sequence of M 5 5  This chapter presents the analysis of the deep two-colour (V and /) photometry of a field located at about 2.2 core radii from the cluster centre. The primary goal of the observations in this field was to derive deep main-sequence luminosity and mass functions and to determine the slope of the mass function in different mass ranges, especially at the low-mass end. Since M55 is one of the most open globular clusters (c = 0.76), it is expected to be dynamically young and therefore the low-mass end of its mass function should be close to the initial mass function if there has not been substantial tidal stripping (Richer et al. 1991). The same authors found that for all clusters in their sample the mass functions below ~ 0.4.A4© rose steeply to the faint end of the data. This result implied a large population of very low-mass stars in globular clusters and by association in the halo, but it was based on only six clusters. Recent mass functions derived from HST observations o f a; Cen (Elson et al 1995), 47 Tuc, NGC 6397, M15 and M30 (King et al. 1996a) confirmed that globular cluster mass functions continue to rise to at least ~ 0.1M.&, although maybe not as steeply as found in Richer et al. (1991).  2.1  Observations and Preprocessing  The observations for this project were made by Greg Pahlman and Ian Thompson in 1992 August with the Tektronix 2 CCD at the Cassegrain focus of the 2.5-m du Pont Telescope of the Las Campanas Observatory. This detector contains 1024 x 1024 pixels at a scale of 0"235 per pixel, giving afieldof view of 4' x 4'. The images were taken with 17  18  Chapter 2. The Main Sequence of M55  a gain setting of 2.7e /ADU; the readout noise for this setting was 7.2 e~. A total of 36 _  images (16 in V and 20 in /) through Johnson V and Cousins / filters were obtained of the program field 6'8 southeast of the cluster centre. Figure 2.1 shows the position of the program field overlaid on a DSS (Digitized Sky Survey) image of M55. 1  The program images included 5 short-exposure (60 s and 180 s) frames. In addition, 22 images of a comparison field northwest of M55 were obtained, as well as 66 frames of fields containing Landolt (1992) standard stars. For all frames, the telescope was offset slightly between the exposures (usually 10-30 pixels) in order to minimize the influence of possible cosmetic defects. All frames were obtained under good to excellent seeing conditions (full width at half maximum of 0"7 to 1"2). The coordinates of the program and comparison fields for the epoch J2000.0 are a = 19 40! 0, 8 = —31°02', and h  7l  a = 19 40! l, 8 = —30°36', respectively. The exposure information about the program h  n  and comparison fields is summarized in Table 2.1. 2.1.1  Flat Fields, Overscan Subtraction and Cosmic Ray Removal  The preliminary processing of all images consisted of overscan subtraction followed by flat-fielding. The overscan signal measures the electronics bias level when no photons are counted and it has to be subtracted so that the noise characteristics of the images can be correctly evaluated. Theflat-fieldingcorrects for the different quantum efficiencies of the CCD pixels, as well as for any non-uniform illumination arising from vignetting, uneven thickness of the CCD detector, dust particles on the detector surface etc. For the Las Campanas images I usedflatfieldsobtained by exposing the CCD (through eachfilter)to a uniformly illuminated flat screen inside the dome (so-called dome flats). The screen was T h e Second Epoch Survey of the southern sky was made by the Anglo-Australian Observatory ( A A O ) with the U K Schmidt Telescope. Plates from this survey have been digitized and compressed by the STScI under U . S. Government grant N A G W-2166. Produced under Contract No. NAS5-2555 with the National Aeronautics and Space Administration. 1  Chapter 2.  The Main Sequence of M55  19  Figure 2.1: The location of the deep field relative to M55. The side of the chart is approximately 28', the size of the field is 4' x 4'. North is up and east is to the left. The digitized image of M55 is © 1993-7 by the Anglo-Australian Observatory Board. All Rights Reserved.  Chapter 2.  20  The Main Sequence of M55  Table 2.1: Program and comparison field exposure information  Field Program Program Program Program Program Comparison Comparison Program Program Comparison  UT Date (1992) Aug. 23 Aug. 23 Aug. 23 Aug. 23 Aug. 23 Aug. 24 Aug. 24 Aug. 25 Aug. 25 Aug. 25  Filter I I I V V I V V V I  18x600 60 180 60 180 15x600 300 60 13x600 6x600  Airmass 1.0-1.1 1.2 1.2 1.2 1.2 1.0-1.2 1.3 1.2 1.0-1.1 1.0-1.2  0.9-1.3 1.2 1.2 1.2 1.2 1.0-1.2 1.2 1.0 0.8-1.3 1.1  illuminated by incandescent lamps with colour-compensating filters so that the resultant colour of the screen is close of the colour of the night sky. Both the overscan subtraction and the flat fielding were performed using the appropriate IRAF (Image Reduction and Analysis Facility) routines. The quality of the flat fielding was checked by dividing the individual dome flats by the mean dome flat; the largest peak-to-peak difference was ~ 1.5%, which means that the rms noise contribution from the pixel-to-pixel variations is on the order of 0.7%. Visual examination and several image statistics parameters calculated by IRAF indicated that cosmic ray contamination of the long-exposure (600 s) images was relatively low and therefore I did not attempt cosmic ray removal.  21  Chapter 2. The Main Sequence of M55  2.2  Photometry  2.2.1  Instrumental Magnitudes  The instrumental magnitudes were derived by means of profile-fitting photometry using the latest versions of Peter Stetson's programs FRAME  DAOPHOT  II, A L L S T A R  (Stetson 1987, Stetson 1992, Stetson 1994), as well as his program  and  ALL-  DAOMASTER  (Stetson 1993b). The first step in the reduction process was the derivation of the pointspread function for each frame using  D A O P H O T II.  Between 50 and 70 bright, isolated,  unsaturated stars were chosen in each frame and used to derive the point-spread function. The stellar profiles in all frames were best fitted by a sum of a Gaussian and a Lorentz function: Gaussian : I(r) Lorentz : I(r) V  ;  oc e~ ^ T  oc  2a  — — — — r l + ^/a )/ 2  3  where I{r) is the intensity at a point r pixels from the centre of the stellar image, and a and 8 are parameters of the model stellar profile. Best results were obtained when the point-spread function was allowed to vary quadratically with position in the frame. After the point-spread functions were derived, the output star lists were matched by  ALLSTAR  DAOMASTER  was run for each image and  to create the initial star Ust and  the coordinate transformation file needed to run A L L F R A M E . After the  ALLFRAME  run  has ended, the program leaves copies of the original images where all stars included in the initial list and fit by  ALLFRAME  have been subtracted. The star-subtracted images  were used to search for additional stars missed by the first run of FIND.  DAOPHOT's  routine  Those newly found stars were added to the star Ust and A L L F R A M E was run again,  thus producing for each frame a final star Ust containing the instrumental magnitude, its standard error, the % estimate of the quality of the point-spread function fit for the 2  22  Chapter 2. The Main Sequence of M55  given star and an image-sharpness index measuring how extended an object is compared to the point-spread function. The saturation limit for the long-exposure frames is approximately V = 17.0, i.e., all stars brighter than that have at least a few saturated pixels. For such stars  ALLFRAME  uses the wings of the stellar profile to fit the model point-spread function and a comparison between the magnitudes from the long-exposure and short-exposure frames showed a maximum difference of V i  o n g  —  Khort  ~ —0.04 mag at V ~ 14.5, the saturation limit  for the shortest-exposure images. In the subsequent reductions I used only the shortexposure magnitudes for all stars brighter than V = 17.2, and I believe that even though stars with V < 14.5 are saturated on all frames, their magnitudes as derived from the short-exposure images are no more than ~ 0.05 mag brighter than the true ones. 2.2.2  Aperture Corrections  The instrumental magnitudes returned by  ALLFRAME  are differential magnitudes with  zero-points that differ from frame to frame depending on the exposure time, air mass, seeing, focusing etc. Those magnitudes are derived from a least-squares fit to the stellar profile, but only the pixels within a small radius (usually chosen to be equal to the full width at half maximum of the stellar profile) are used in the fit. This is done so that the well-exposed (and hence having the highest signal-to-noise ratio) pixels carry the largest weight in the profile fit. As some pixels are left out, the instrumental magnitudes returned by A L L F R A M E have to be corrected so that they are on the system of the "total" instrumental magnitudes for the particular frame. These corrections are called aperture corrections and they are derived from synthetic aperture photometry by measuring several isolated, bright stars either through a single, large aperture (thus simulating photoelectric photometry), or through a series of increasing apertures and the construction of growth curves. Thorough discussions of aperture corrections and the growth-curve method can  23  Chapter 2. The Main Sequence of M55  be found in Howell (1989) and Stetson (1990). The aperture corrections were derived using the program D A O G R O W (Stetson 1990). In each program image I selected the brightest and most isolated 30 to 40 stars among those used in the derivation of the point-spread function for that frame. All other stars were subtracted and concentric aperture photometry was obtained for the selected stars. These aperture photometry results were then supplied to  DAOGROW  which returns the  "total" instrumental magnitude and its standard error for each of the selected stars. The aperture correction for a particular frame was obtained by taking the weighted mean of the difference between the "total" magnitude and the profile-fitting  ALLFRAME  magnitude for all selected stars on that frame. As the fields containing Landolt (1992) standards were not crowded, no profile-fitting photometry was necessary for the stars in those fields. Instead,  DAOGROW  was used  to derive directly the total instrumental magnitudes for the standard stars from their aperture photometry. 2.2.3  Transformation to the Standard System  The transformation of the instrumental magnitudes to the standard Vic system was performed in two steps. First, I used observations of faint standard stars selected from the list of Landolt (1992) to derive the zero-points, extinction coefficients and colour terms necessary to transform to the standard photometric system a sample of 68 relatively isolated stars in the program field. On the second step, these 68 stars were used as local secondary standards to calibrate all other program stars. While it is possible to carry out the calibrations in a single step, the use of local standards decreases the frame-to-frame scatter by referring all instrumental magnitudes to a common zero point before the final transformation (Stetson & Harris 1988). Most of the calibration steps were performed using several programs from Peter Stetson's package  CCDPCK,  namely  CCDSTD, CCDAVE  24  Chapter 2. The Main Sequence of M55  and  FINAL.  On each of the three nights the faint standard stars selected from Landolt (1992) were observed at various times during the night. A total of 28 stars in five fields were observed, but not all stars were measured on every night. The list of the standards that were observed and their magnitudes and colour indices are given in Table 2.2. The first step in the calibration procedure was to fit equations of the form v  =  V + ao + a^X -1.25) + a (V-1) + a T  i  =  J +fco+  2  3  - 1-25) + fe (V-7) + 6 T 2  3  independently to the first and second night data. In these equations V and / are the standard magnitudes and v and i are the instrumental magnitudes of the standard stars; X is the airmass and T is the time of mid-exposure relative to the effective midnight. The terms for time dependence were added after a preliminary fit showed clear trends with time in the residuals for both V and / . No other trends in the residuals were noticeable and no additional terms were used in the transformation equations. As the program fields are usually observed at airmass between 1.0 and 1.5, subtracting 1.25 from the airmass makes it more or less centred around zero, thus improving the quality of the fit. The colour coefficients a and b depend on how well the combination telescope-filters2  2  detector matches the standard system so they are expected to be constant in the course of a few days. Therefore, after the first fit was performed, the values of a and b were 2  2  fixed at their weighted means for the two nights and the fit was repeated to solve for the remaining coefficients on those nights. The use of average colour coefficients was also justified by the fact that they were within the errors of the individual nightly values. On the third night only a few standard stars were observed, at different times but at approximately equal airmasses. Therefore, in the calibrations of the third night data the extinction coefficient was taken to be the weighted mean from the first two nights  Chapter 2.  25  The Main Sequence of M55  Table 2.2: List of the observed Landolt standards Star SA110 229 SA110 230 SA110 232 SA110 233 SA110 361 SA110 362 SA110 364 SA110 365 SA110 499 SA110 502 SA110 503 SA110 504 SA110 506 SA110 507 SA95 275 SA95 276 SA95 330 MarkA MarkA 1 MarkA 2 MarkA 3 PG1633+099 PG1633+099 PG1633+099 PG1633+099 PG1633+099 T Phe A T Phe C T Phe D  V  A B C D  13.649 14.281 12.516 12.771 12.425 15.693 13.615 13.470 11.737 12.330 11.773 14.022 11.312 12.440 13.479 14.118 12.174 13.258 15.911 14.540 14.818 14.397 15.256 12.969 13.229 13.691 14.651 14.376 13.118  cry 0 0031 0 0031 0 0032 0 0028 0 0022 0 0021 0 0027 0 0031 0 0031 0 0031 0 0013 0 0021 0 0049 0 0028 0 0061 0 0025 0 0019 0 0040 0 0028 0 0023 0 0025 0 0036 0 0017 0 0025 0 0020 0 0028 0 0022 0 0033  V-I  2.356 1.218 0.889 1.593 0.709 1.803 1.281 2.631 1.273 2.625 0.808 1.482 0.652 1.206 1.944 1.395 2.268 -0.241 0.740 0.751 1.098 -0.212 1.015 1.090 1.138 0.650 0.841 -0.360 1.663  a  v-i  0.0026 0.0050 0.0025 0.0021 0.0029 0.0021 0.0034 0.0029 0.0041 0.0022 0.0080 0.0042 0.0049 0.0025 0.0051 0.0028 0.0048 0.0148 0.0059 0.0045 0.0111 0.0111 0.0020 0.0038 0.0033 0.0032 0.0149 0.0030  Chapter 2.  26  The Main Sequence of M55  and only the zero-points (ao, 60), the colour coefficients (ai, 61) and the time terms (03, fe ) were allowed to vary. After that, new mean colour coefficients were calculated, the 3  fit for the first two nights repeated with a and b fixed to those mean values and new 2  2  extinction coefficients derived. Another fit to the third night's photometry using the new extinction coefficients yielded the final values of the transformation coefficients for all three nights. For the colour coefficients a and b I found mean values of —0.0234 ± 0.0008 and 2  2  -0.0249 ± 0.0012, respectively. The extinction coefficients were  a i  = 0.234 ± 0.003 and  61 = 0.159 ± 0.004 on the first night and a = 0.231 ± 0.007 and b = 0.137 ± 0.007 on the x  x  second night. The most likely explanation for the unusually large values of the extinction coefficients is the high aerosol content of the stratosphere caused by the eruption of Mount Pinatubo in June 1991 (Daniel 1993, Burki et al. 1995). The time terms 6X3 and 63 were of the same sign and had roughly equal magnitudes in both bandpasses (on the order of +0.004 ± 0.0005 mag/hour), indicating that the drift in the zero-points was probably caused by slow variations in the CCD sensitivity during the night. The final transformations were applied to all available observations of the Landolt standards and the program  CCDAVE  was used to obtain their photometric indices on the  standard system. At this stage of the calibrations I also obtained mean magnitudes and colour indices on the standard system for the 68 secondary standards in the program field. The final step in the calibration procedure was to combine the local M55 standards and the Landolt standards in a larger list of standard stars and derive new transformations. Since the program frames span a very limited range in airmass and time, the local standards were used only to improve the zero-points; all other transformation coefficients were kept at their values obtained in the previous stage. After this step I had mean, homogeneous magnitudes and colour indices for the Landolt standards and for 68 stars  Chapter 2. The Main Sequence of M55  27  Figure 2.2: Finder chart for the local standards in the deep field. Star numbers increase with right ascension. The size of the chart is 4' x 4'. North is up and west is to the left.  Chapter 2.  28  The Main Sequence of M55  in the program field. The local standards are identified in Figure 2.2 and their magnitudes and colour indices on the system defined by Landolt's (1992) standards are listed in Table 2.3. Figure 2.3 shows the differences between my photometry of the Landolt standards and their published values. The two labeled stars deserve special mention. The first one, SA 110-362, displays unusually large residuals in both V and V — I. It has only a single observation in Landolt (1992) and being a possible variable it was rejected as a standard after the first transformations were calculated. For this star Landolt gives V = 15.693 and V-I = 1.803, whereas I obtained V = 15.599 + 0.005 and V—I = +1.700±0.010. The second star, T Phe D, also has relatively large residuals: my values of V = 13.162 ±0.003 and V — I = +1.699 ± 0.006 are more than 5<r away from the published photometry, so this star might also be a variable. The average magnitudes and colour indices on the standard system for all program stars were derived using the program FINAL. In addition to the photometric indices and their uncertainties it returns also the mean estimate of the quality of the fit % for each 2  star, as well as the number of V and I frames on which the star was found. The last three quantities were used later to sort out the "good" and the "bad" photometry and to judge the reality of a particular object. 2.2.4  Comparison with Earlier Photometry-  There are two studies of M55 that have photometry of stars in common with the present work. These are the BV photographic photometry of Alcaino (1975) and the BV photographic study by Lee (1977), each having ten stars (not all different) in common with the photometry presented here. In both studies the photographic photometry has been calibrated by means of photoelectric sequences in the field of M55. I found mean differences of  = +0.057 ± 0.028 between the present photometry and that of Alcaino  Chapter 2.  The Main Sequence of M55  Table 2.3: Local standards in M55 Star 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34  V 17.844 18.184 17.848 18.190 17.159 17.328 17.621 17.995 18.055 17.598 17.701 18.234 18.230 17.943 18.024 17.664 19.368 18.064 18.219 17.994 19.513 18.330 16.962 17.706 17.071 17.389 17.725 17.601 17.722 17.093 17.604 17.846 18.084 17.485  (T  V  0 0009 0 0014 0 0014 0 0018 0 0007 0 0017 0 0014 0 0014 0 0012 0 0026 0 0011 0 0011 0 0015 0 0018 0 0012 0 0015 0 0015 0 0018 0 0012 0 0025 0 0040 0 0020 0 0014 0 0015 0 0016 0 0012 0 0014 0 0013 0 0012 0 0017 0 0017 0 0010 0.0012 0.0009  V-I 0.725 0.735 0.878 0.760 0.910 0.854 0.755 0.725 0.723 0.757 0.739 0.721 0.728 1.085 0.722 0.759 2.248 0.745 0.744 0.726 2.521 0.993 0.942 0.738 0.938 0.821 0.638 0.761 1.086 0.926 0.762 0.729 0.721 0.866  CTv-I 0 0021 0 0020 0 0019 0 0025 0 0013 0 0019 0 0021 0 0020 0 0015 0 0033 0 0023 0 0019 0 0021 0 0024 0 0018 0 0023 0 0018 0 0026 0 0020 0 0032 0 0043 0 0024 0 0018 0 0018 0 0019 0 0016 0 0018 0 0015 0 0017 0 0020 0 0020 0 0014 0 0016 0 0014  Star 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68  V 18 242 18 059 18 224 17 943 18 815 18 045 17 720 17 772 17 448 18 223 18 396 18 106 18 064 17 915 18 077 18 192 17 476 18 198 17 176 17 109 18 055 18 228 18 669 18 658 17 549 18 189 18 211 18 234 18 231 17 877 19 240 17 529 17 774 17 936  (TV  0.0011 0.0015 0.0010 0.0013 0.0018 0.0014 0.0012 0.0019 0.0011 0.0016 0.0023 0.0013 0.0017 0.0013 0.0011 0.0024 0.0028 0.0010 0.0018 0.0015 0.0015 0.0012 0.0016 0.0014 0.0012 0.0016 0.0016 0.0012 0.0019 0.0016 0.0018 0.0010 0.0017 0.0011  V-I 0 730 0 717 0 730 0 701 1 804 0 730 0 755 0 802 0 802 0 730 1 007 0 726 0 732 0 726 0 717 0 719 0 869 0 723 0 897 0 886 1 037 0 774 1 921 1 544 0 762 1 155 0 909 1 352 0 843 1 021 1 872 1 104 0 743 0 984  a v-i  0.0014 0.0019 0.0014 0.0017 0.0021 0.0016 0.0016 0.0025 0.0015 0.0020 0.0030 0.0016 0.0021 0.0018 0.0017 0.0029 0.0035 0.0015 0.0021 0.0019 0.0019 0.0016 0.0018 0.0017 0.0016 0.0020 0.0020 0.0016 0.0023 0.0020 0.0023 0.0015 0.0023 0.0017  Chapter 2.  The  30  Main Sequence of M55  -l  1  r~  o.i 0.05  * -I  0  T Phe D  9-  -0.05 110-362  -0.1  J_  J_  11  12  I  13 14 V (Landolt)  I  15  I I  £ L_  16  0.1 0.05  I  T Phe D  -0.05 110-362  -0.1 1 V-I (Landolt)  2  Figure 2.3: Differences between my photometry of Landolt (1992) standards and the published values. The differences AV and A(V-I) are in the sense (this work - Landolt's). The two stars with large residuals are labeled.  Chapter 2.  31  The Main Sequence of M55  (1975), and A V = —0.052 ± 0.022 between the present data and the photometry of Lee (1977), where AV is in the sense (this work — theirs). Schade et al. (1988) also found a difference of AV = —0.030 ± 0.026 (in the same sense) between their CCD photometry and that of Lee. The most likely explanation for these relatively large discrepancies is the use of different sets of filters, detectors and standard stars: the indirect offset of AV « —0.02 ± 0.034 between the present CCD photometry and that of Schade et al. (1988), who used an older list of Landolt's standards, is smaller and within the errors.  2.3  The Color-Magnitude Diagram of M55  2.3.1  Morphology and principal sequences  The colour-magnitude diagram for all ~ 5300 stars in the program field found on at least one V and one I frame, without any selection based on % or photometric errors is 2  shown in Figure 2.4. The main sequence and the subgiant branch are very well defined, as opposed to the handful of stars populating the giant and horizontal branches. An obvious concentration of stars is visible blueward of the main sequence, fainter than V ~ 21, and an excess of stars is noticeable redward of the M55 turnoff at V ~ 18.5, V — I ~ 1.0. These two groups can be attributed to the Sagittarius dwarf spheroidal galaxy (SDG) and the Milky Way bulge, which are discussed in more detail later. Another, somewhat puzzling set of points is seen spread blueward of V — I « 0.7 and below V ~ 21. Since it was not clear what kind of objects might be located in this region of the colour-magnitude diagram, I examined their appearance in several frames and found that almost all of them were residual images left behind in the star-subtracted frames and fitted as real stars on the second run of A L L F R A M E . After some experimenting, it was found that discarding stars that were less than 1.5 pixels from a companion two or more magnitudes brighter eliminated most of the "ghost" stars while leaving real stars  Figure 2.4: Color-magnitude diagram for all objects in the program field found on at least one frame in each colour. No other restrictions have been imposed. Note the group of objects extending blueward of (V — I) « 0.7 and below V « 21.  Chapter 2.  The Main Sequence of M55  33  intact. In order to define the position of the principal sequences in the colour-magnitude diagram of M55, additional selection criteria were imposed on the sample of stars plotted in Figure 2.4: only stars found on at least three frames in each filter and with a value of x < 1-5 were retained for this purpose. The resultant colour-magnitude diagram is 2  plotted in Figure 2.5. Since the program field is located more than two core radii from the center of M55, the horizontal branch and the upper giant branch are poorly populated and no attempt was made to define fiducial lines for those sequences. However, a few features in the upper colour-magnitude diagram of M55 are worth mentioning. The most interesting one is the clump of stars located blueward of the giant branch at V ~ 15.2, V — I & 0.85, about 1.5 mag below the horizontal branch. I examined carefully the appearance of each star in the V and I frames and did not notice anything that would affect their photometry — they all appear to be normal, unblended stars, far from heavily saturated objects or cosmetic defects. It is unlikely that they are field stars for at least two reasons: (a) the colour-magnitude diagram of the comparison field has only a single star at that location, and (b) similar clumps of stars are clearly visible in the same location in the colour-magnitude diagrams of Lee (1977), Penny (1984), Schade et al. (1988) and Mateo et al. (1996), who observed different M55 fields. These objects can be tentatively identified with the "yellow straggler" stars observed in globular clusters such as E3 (Hesser et al. 1984) and M15 (Stetson 1994) and evidently present in the colour-magnitude diagram of M92 shown in Figure 15 of Stetson & Harris (1988). In the case of E3 and M15 however, the distribution of the yellow stragglers suggests that many of them are blends of horizontal-branch and red-giant stars or of subgiants and red giants. In Figure 2.5, as well as in the much richer upper colour-magnitude diagrams of Schade et al. (1988) and Mateo et al. (1996), the yellow stragglers are confined within a relatively small region of the colour-magnitude diagram. Given the small number of  Chapter 2.  34  The Main Sequence of M55  V-I  Figure 2.5: Color-magnitude diagram for all objects in the program field found on at least three frames in each colour and having % < 1.5. The members of close pairs (< 1.5 pixels) two or more magnitudes fainter than their companions were also deleted. Note the prominent SDG sequence and the excess of stars redward of the main-sequence turnoff. 2  Chapter  2.  35  The Main Sequence of M55  horizontal-branch and giant-branch stars in this field, it is unlikely that more than one of the yellow stragglers in Figure 2.5 is a blend. It is quite possible that these are red horizontal branch stars belonging to the bulge of the Milky Way, as they have the appropriate colour indices and apparent magnitudes (see, e.g., Holtzman et al. 1998). There are also two stars lying on the extension of the main sequence, one close to the turnoff and the other about 1.5 mag below the blue horizontal branch; these are most likely blue stragglers stars (see Chapter 4). There is a single star on the M55 horizontal branch that falls in the RR Lyrae region. The photometry does not show any variability, at least within a time interval of two hours. Both Alcaino (1975) and Lee (1977) have observed this star (their star numbers 63 and 4217, respectively) and did not note any variability either. Non-variable stars are known to exist in the RR Lyr instability strip (see e.g. Silberman & Smith 1995), but with the available data it is not possible to determine whether this is an RR Lyr variable or not. The fiducial main sequence and turnoff region of M55 were obtained by means of a mode-finding program which searches for the greatest density of points over colour. The handful of stars on the subgiant branch and the base of the giant branch were simply divided in a few groups and the mean magnitude and colour of each group was calculated. A cubic spline was used to construct a smooth curve through all resulting points and that curve, tabulated in the first column of Table 2.4, was adopted as the fiducial sequence of M55. The second column contains the dispersion  <TV-I  around  the main-sequence ridge line, calculated as 1.4826 times the median absolute deviation (Whittaker & Robinson 1924) in the corresponding magnitude interval. The use of the median of the absolute deviations instead of the usual mean square of the deviations is justified by the non-Gaussian distribution of the photometric errors (see section 2.7). The apparent magnitude and the colour of the main-sequence turnoff as obtained  Chapter 2.  The Main Sequence of M55  36  Table 2.4: The fiducial main sequence and subgiant branch of M55 V 16.80 17.00 17.20 17.40 17.60 17.80 18.00 18.20 18.40 18.60 18.80 19.00 19.20 19.40  V-I 0.953 0.936 0.902 0.830 0.756 0.729 0.721 0.725 0.734 0.746 0.761 0.780 0.801 0.825  o~v-i 0.011 0.011 0.011 0.012 0.012 0.013 0.013 0.014 0.015 0.015 0.016 0.017 0.018 0.019  from thefiducialline are: V  TO  2.3.2  V 19 60 19 80 20 00 20 20 20 40 20 60 20 80 21 00 21 20 21 40 21 60 21 80 22 00 22 20  V-I 0.853 0.883. 0.914 0.946 0.981 1.021 1.062 1.103 1.146 1.193 1.245 1.297 1.347 1.395  V 22 40 22 60 22 80 23 00 23 20 23 40 23 60 23 80 24 00 24 20 24 40 24 60 24 80  a v-i 0.021 0.022 0.023 0.025 0.026 0.028 0.029 0.030 0.031 0.033 0.035 0.036 0.038 0.039  V-I 1.441 1.486 1.528 1.567 1.605 1.643 1.678 1.708 1.734 1.762 1.786 1.803 1.815  av-i 0 041 0 043 0 045 0 048 0 051 0 054 0 058 0 063 0 068 0 075 0 085 0 103 0 130  = 18.03 ± 0.05 and ( V - I ) o = +0.72 ± 0.01. T  The Stars of the Sagittarius Dwarf Galaxy and the Galactic bulge  Besides the obvious main sequence of M55, there is another prominent concentration of stars in Figure 2.5, blueward of the lower main sequence of M55. In their observations of an M55 field Mateo et al. (1996) noted a similar sequence at the same location in the colour-magnitude diagram and identified it as consisting of main-sequence and turnoff stars associated with the Sagittarius dwarf spheroidal galaxy (Ibata et al 1994). Fahlman et al. (1996), using the photometry presented here, compared the sequence in Figure 2.5 with the bulge population observed in the field of the globular cluster M4. They demonstrated that the stars in this sequence are too faint to be attributed to the bulge and also argued that they belong to the Sagittarius dwarf galaxy. The ridge line for the SDG sequence seen in Figure 2.5 is listed in Table 2.5. A  Chapter 2.  37  The Main Sequence of M55  Table 2.5: The fiducial main sequence of the Sagittarius Dwarf Galaxy V 21.160 21.814 22.338 22.929 23.460 24.113 24.642  V^T  0.787 0.773 0.795 0.833 0.921 1.074 1.205  comparison with the fiducial main sequence for M55 (Table 2.4) shows that the mainsequence turnoff of SDG is about as blue as the turnoff of M55. The interpretation of this fact depends on the metallicity of SDG. Both Mateo et al. (1995) and Sarajedini Sz Layden (1995) argued that the dominant stellar populations in SDG are significantly more metalrich than M55, and it follows therefore that SDG must be younger than M55 — Fahlman et al. (1996) estimated from isochrone fits that M55 is 2 to 4 Gyr older than SDG. This age estimate raises interesting questions about the star formation history of the SDG: this dwarf spheroidal has the "very young" globular cluster Ter 7 (Buonanno et al. 1994), the "young" cluster Arp 2 (Buonanno et al. 1995) and the "old" clusters Ter 8 and M54 (Ortolani & Gratton 1990, Sarajedini & Layden 1995, Montegriffo et al. 1998). It seems quite likely that most of the SDG field population (including the stars seen in Figure 2.5) formed after Ter 8 and M54 (Sarajedini & Layden 1995, Layden & Sarajedini 1997). Mateo et al. (1996) also drew attention to the excess of stars redward of the M55 turnoff at V ~ 18.0, V — Itt  1.0, which is clearly visible in Figure 2.5 as well, and which they  identified as the red clump on the giant branch of SDG. Since the line of sight to M55 passes through the outer bulge of the Milky Way, one would expect to see traces of the bulge population in addition to the Sagittarius Dwarf  Chapter 2.  The Main Sequence of M55  38  stars. The predicted location of the bulge stars, as well as the principal sequences of SDG and M55 are shown in Figure 2.6 by solid lines. The position of the bulge sequence and the offsets required to place it in the M55 CMD were taken from Fahlman et al. (1996) (see their Figure 2). The 10-Gyr, [Fe/H] = -0.79 isochrone from VandenBerg k Bell (1985), which Fahlman et al. (1996) found to fit well the main sequence of SDG, is shown by a dashed line. From the overlap with the bulge sequence it can be concluded that bulge stars not only contribute to the excess of stars redward of the M55 turnoff, but will be mixed with the M55 main sequence as well. In fact, the slight "puff-up" of the M55 main sequence at V ~ 20 is most likely caused by the presence of bulge stars. Assuming that the surface density of the bulge stars obeys de Vaucouleur's law with r  e  « 2.5 kpc, one can estimate the number of bulge stars in Figure 2.5 by counting  the stars within 2<r of the bulge ridge line in the colour-magnitude diagram of M4 (Fahlman et al. 1996) and correcting the counts for the different distance of the tangent point of the line of sight to the galactic center. I found that the expected number of bulge stars for 18.5 < V < 22.0 is Nb = 67 ± 15, where the uncertainty includes only the Poisson errors. While only a rough estimate, the low expected number of bulge stars confirms the visual impression that the Milky Way bulge does not contribute significantly to the density of stars on the main sequence of M55. As far as the stars of the Sagittarius Dwarf are concerned, one can expect a contribution from SDG giants to the M55 main sequence around V — 20.7 (see Figure 2.6). Adopting an apparent distance modulus of (m — M)v ~ 17.7, metalhcity of [Fe/H] fa —0.79 and an age of ~ 12 Gyr for SDG (Fahlman et al. 1996), I used a theoretical luminosity function from Bergbusch & VandenBerg (1992) to obtain an estimate of ~ 3 red giants in the interval 20.4 < V < 20.9. It is clear that the stars from the Sagittarius dwarf galaxy will make a negligible contribution to the main-sequence luminosity function of M55 both around V = 20.7 and below V = 24.5.  Chapter 2.  39  The Main Sequence of M55  15  h  20  25  V-I  Figure 2.6: The principal sequences of M55, the Sagittarius dwarf galaxy and the predicted location of the bulge stars. Thefiducialsequences are marked by solid fines. The 10-Gyr, Y = 0.20, [Fe/H] = -0.79 isochrone from VandenBerg & Bell (1985) is shown by a dashed Une.  Chapter 2.  2.4  40  The Main Sequence of M55  Photometry of the comparison field  In order to correct the luminosity function of M55 for field-star contamination, a comparison field 0?5 north of the programfieldwas observed. The images of the comparison field were reduced and the photometry calibrated in the same way as those of the program field and Figure 2.7 shows the colour-magnitude diagram of the comparison field. The Sagittarius dwarf galaxy sequence is easily discernible in this plot as well, however the photometry is of poorer quality because of the fewer images and less total integration time. The main sequence and turnoff stars of the Galactic bulge form a noticeable "wall" to red of where the turnoff of M55 would be, at V — I K, 0.9 and V brighter than 21. In this plot the fiducial sequence of M55 and the SDG sequence are shown by solid fines, and the location of the bulge stars is shown by a dashed line. It is clear from the colour-magnitude diagram that field stars contribute little to the main-sequence luminosity function of M55 and therefore will not affect the locus of the fiducial MS of the cluster. In addition to foreground and background stars, it is expected that some number of background galaxies will be present in both the comparison and the program fields. An estimate of the contribution of faint galaxies can be obtained from the high-latitude, deep CCD galaxy counts of Woods et al. (1995). Their V-band and J-band relations predict the following number of background galaxies expected in the deep M55 field: log AV = (0.41V - 7.68) mag  -1  and log JV} = (0.32/- 5.14) mag , respectively, without -1  corrections for reddening. The actual number of galaxies will be different because of the increased absorption at the lower galactic latitude of M55 and the possibility of galaxy clustering. Another way to estimate the degree of contamination from galaxies is to use the average image-sharpness index r returned by 0  ALLFRAME  (see Stetson &; Harris 1988).  Chapter 2.  41  The Main Sequence of M55  V-I  Figure 2.7: The colour-magnitude diagram of the comparison field. All objects with X < 1.5 are plotted. Solid lines mark the location of the M55 main sequence and the Sagittarius sequence from Figure 2.6. Note the vertical "wall" formed by bulge stars to the red of the M55 turnoff. The predicted location of the bulge sequence is shown by a dashed line. 2  Chapter 2.  42  The Main Sequence of M55  2  h  o  0 0  °° <V> 0  4 g°° O °° °°  0° ° o  °on  80  -2  25  20  15  Figure 2.8: Plot of the average image-sharpness index r versus V magnitude for all objects in the comparisonfieldwith % < 1.5. Note the excess of objects with positive values of ro0  2  Chapter 2.  The Main Sequence of M55  43  Table 2.6: Galaxy counts in the comparison field  V 21-22 22-23 23-24 24-25  2  W  13 27 76 62  Figure 2.8 shows a plot of r field with x  N (V) 9 24 62 159  N(V)  0  /  20-21 21-22 22-23 23-24  N(I)  N (I)  12 39 107 2  22 46 97 203  W  versus V magnitude for all objects in the comparison  < 1-5- It is clear that at faint magnitudes (V > 21.5, / > 21) there  are more objects with positive r than with negative ro- Since r is expected to be 0  0  symmetrically scattered around zero independently of magnitude, one can assume that all excess objects with r  0  > 0 in a given magnitude interval are galaxies. A t brighter  magnitudes the restriction % < 1.5 seems to eliminate galaxies quite well, and in any 2  case the number of galaxies drops rapidly with decreasing magnitude. In each magnitude interval fainter than V = 21, I counted the number of objects N  r+  with positive sharpness  indices and the number of objects N _ with negative ones. The difference N r  r+  — N_ r  then yields an estimate of the number of galaxies in the field. Table 2.6 gives the galaxy counts as determined by the described technique and the expected number of galaxies Nw(V)  and Nw(I) (corrected for the adopted reddening in this field, see section 2.5) as  given by the Woods et al. (1995) relations. Except for very faint magnitudes where the incompleteness is large, there is a good agreement between the observed and predicted counts and therefore I used the latter to correct the observed luminosity function of M55 for galaxy contamination. It should be noted that the numbers in Table 2.6 refer to the total counts (actual and predicted), without any regard where the object is in the colourmagnitude diagram. The galaxy corrections to the observed luminosity function of M55, on the other hand, were calculated by taking into account only the galaxies inside the 3cr  Chapter 2.  44  The Main Sequence of M55  strip used to construct the luminosity function of M55 (see section 2.7 for more details).  2.5  The Reddening and Metallicity of M55  Having an accurate value for the reddening for M55 is crucial for the purposes of distance determination and comparison with theoretical models. Reddening is often specified by the colour excess EB-V = (B—V) — (B—V)o, where B—V is the observed colour index and (B — V)o is the unreddened or intrinsic colour index. Another often used colour excess is Ev-i  and the choice of course depends on the filters used for the photometry.  The early reddening determinations for M55 were based on its integrated colours (Kron & Mayall 1960, van den Berg 1967, Kron & Guetter 1976) and photoelectric photometry of field stars (Lee 1977) and they all gave a value of EB-V around 0.08. However, the later studies of Schade et al. (1988) and Buonanno et al. (1989) found EB-V = 0.14 from a comparison of the colour-magnitude diagram of M55 with the colour-magnitude diagrams of clusters with well-determined reddening. The reddening maps of Burstein Sz Heiles (1982) and the compilation of Peterson (1993) also support a higher value for EB-V  (~ 0.13 and 0.11, respectively).  A better estimate for the reddening towards M55 can be obtained using the new highquality UBVI photometry of the evolved populations in M55 presented in Chapter 3. One possible approach is to use the U, B and V photometry of the blue horizontal branch stars to obtain the reddening from the colour-colour diagram. Figure 2.9 presents the U—B vs. B—V plot for stars in the core of M55 (see Chapter 3 for details about the photometry). Only stars with % < 1.4, O~B-V < 0.05 and O-JJ-B < 0.05 are shown here and the solid line 2  is the two-colour intrinsic relation for main-sequence Population I stars (Johnson 1966). The derivation of colour excesses from such a diagram is fairly straightforward provided that the slope of the interstellar reddening line X = ETJ-B/EB-V  is known. I adopted  Chapter 2.  45  The Main Sequence of M55  B-V  Figure 2.9: Colour-colour diagram for all stars in the central field of M55 with % < 1.4, (JB-V < 0.05 and ou-B < 0.05. Solid line: the intrinsic relation for main-sequence Population I stars. Dashed line: the same relation reddened by EB-V = 0.13. The arrow shows the direction of reddening for the adopted value of EU-B/EB-V = 0.72. 2  Chapter 2.  46  The Main Sequence of M55  — 0.72  ETJ-B I EB-V  and using only stars with B — V < 0.1 (to minimize the effects of  metallicity) I obtained E -'v — 0.13 ± 0.02. It should be noted that while this "average" B  value of X = 0.72 is often used, the slope of the reddening Une varies from one region of the sky to another (FitzGerald 1970, Turner 1989, Turner 1994). For small colour excesses however, this effect is not very important: in the case of M55, a change in X from 0.65 to 0.8 resulted in a change of only 0.006 mag in EB-VOnce the colour excess in B—V is known, it can be converted to Ey-i using pubUshed ratios. Dean et al. (1978) derived Ey-i  EB-V IEv-i  (1986) Table 3 gives Ey-ij'EB-V  = 1.250EB-V,  whereas Taylor's  = 1.271 for stars around spectral type AO. The value  given by Dean et al. (1978) is for stars earUer that B2, so it is sUghtly less than what would be appropriate for the cooler (B5-A0) stars on the blue horizontal branch of M55. Adopting the mean of the two determinations, Ey-i/EB-V factor from E -v B  —  1-26 as the conversion  to E y _ , I found E -i = 0.16 ± 0.02 for the field of M55. 7  V  Another approach that aUows the determination of Ey-i directly is the technique developed by Sarajedini (1994). It consists of incrementing a starting value of Ev-i (usually Ey-i = 0) until the values of [Fe/H] from the two equations [Fe/H] =  9.668(V-/) -10.64  (2.1)  [Fe/H] =  -0.9367AVi. + 0.2606  (2.2)  0ltf  2  are equal. In these equations (V—I)o, is the unreddened value of the colour of the giant g  branch at the level of the horizontal branch and AV1.2 is the distance in V between the horizontal branch and the red giant branch at (V—I)o = 1.2. It should be noted that what Sarajedini calls "the level of the horizontal branch" is essentiaUy V(RR): the average visual magnitude of the cluster RR Lyrae variables. In Chapter 3 I found V(HB) = 14.46 and (V — I) = 1.065, leading to the simultaneous determination of [Fe/H] = —1.98 and g  Ey-i  = 0.18 ± 0.01. This value of E -i V  impUes E -v B  =  0.14 if E -i/E -v V  B  =  1-26 is  Chapter 2.  The Main Sequence of M55  47  used. The error estimate for Ey-i came from the mean quoted rms scatter of ±0.06 dex in the above equations. The weighted mean of the two independent determinations is EB-V  — 0.13 ± 0.02 and Ey-i — 0.17 ± 0.02 and these are the values that I adopted as  the best estimates of the reddening towards M55. The value [Fe/H] = —1.98 obtained from equations (2.1) and (2.2) agrees very well with the recent determinations of the metalhcity of M55 by Geisler et al. (1992) and Minniti et al. (1993), who obtained [Fe/H] = —1.95 from Washington photometry and high-dispersion spectroscopy of M55 giants. This agreement could be spurious, however, as the coefficients in equations (2.1) and (2.2) were derived using met alii cities on the old Zinn & West (1984) scale, on which M55 has a metaUicity of —1.82. The unweighted mean of all four determinations is [Fe/H] = —1.92, which I adopted as the value for the metaUicity of M55. All of the metaUicities discussed so far have been derived from observations of red giant stars in M55. There are some indications, however, that such observations may give too high a value of [Fe/H] compared to the metaUicity obtained from spectroscopy of subgiant or main-sequence stars (King et al. 1998b). The important implications of this discrepancy and how it affects the derived distance to M55 are discussed in the next section.  2.6 2.6.1  The Distance and Age of M55 Distance Modulus  The best current estimate of the distance to M55 is that of Schade et al. (1988), who derived an apparent distance modulus of (m — M)y = 14.10 from a match of the observed horizontal branch of the cluster to theoretical zero-age horizontal branch models. This approach reUes on the accuracy of (a) the theoretical horizontal branch models and  Chapter 2.  The Main Sequence of M55  48  (b) the adopted transformations from luminosity and effective temperature to absolute magnitude and colour index. It is preferable, however, to obtain a distance estimate independent of any theoretical assumptions (as much as that is possible) and then compare theory to observations. Direct measurements of the distance to M55 by means of trigonometric parallaxes are not yet possible and therefore one has to rely on more indirect methods. One the most direct and reliable approaches is the technique of main-sequence fitting that is used extensively in distance determinations for open clusters. It was probably Baade (1948) who first suggested the use of this method to find the distances to globular clusters (see Sandage 1986), but the first applications (Sandage 1970) had to wait until reliable parallaxes and deep enough photometry on the main sequence became available. In its present-day form this method consists of fitting the fiducial main sequence of the cluster to a sample of nearby subdwarfs with well-determined parallaxes, metaUicities and accurate photometry. Aside from parallax errors, the most uncertain element in this method as applied to globular clusters is the correction of the observed subdwarf colours for the different metal abundances: it requires knowledge of the metaUicities of the stars and the cluster (typical errors ±0.10 dex) and the use of theoretical models to derive the actual corrections. This is the only step where theoretical models are used and as these are differential corrections, they are fairly insensitive to smaU changes or uncertainties in the models. I have coUected from the literature new observational data (paraUaxes, photometry and metaUicities) for a total of 16 subdwarfs which are Usted in Table 2.7. Table 2.8 gives the references for the compiled quantities. From left to right the columns in Table 2.7 contain: HD or BD number, the Hipparcos catalogue number (HIP), V and V—I on the Cousins system and their standard errors, the colour index adjusted to the metaUicity of M55, the adopted metaUicity of the subdwarf, the absolute paraUax and its standard error  Chapter 2.  49  The Main Sequence of M55  (both in milliarcseconds), and the absolute magnitude and its standard error. Details on the how the entries were compiled or calculated follow. The source for all parallaxes was the Hipparcos Catalogue (Perryman et al. 1997a, Perryman et al. 1997b). For a subdwarf to be included in the sample, I required that Cir/ir  < 0.20 in order to minimize the selection biases (Smith 1987, Oudmaijer et al. 1998).  For several stars some of the original photometry was done on the Johnson system and in those cases I used the relations in Carney (1983a) to transform the photometry to the Cousins system before calculating the mean values given in Table 2.7. As a rule, there was a good agreement between the photometric data from the different sources, resulting in standard errors of the mean below 0.01 mag in V and V — I for most stars. According to Ryan (1992), several of the stars in Table 2.7 (HD numbers 25329, 134439, 134440, 194598 and 201891) are slightly reddened (E -v B  = 0.01), with only  BD +66 268 having a larger reddening of Eg-v = 0.04. Therefore the colour indices and absolute magnitudes of those stars were corrected by the appropriate amounts — the colours by A(V — I) = 1.26EB-V and the absolute magnitudes by AMy = — 3.1EB-VBefore performing the main-sequence fit, the observed subdwarf colours must be adjusted to account for the metaUicity difference between each individual star and M55. Colour corrections were kindly derived for aU subdwarfs except HD 140283 by Don VandenBerg using a new set of a-enriched isochrones (VandenBerg et al. 1998b). The corrections were calculated as the difference between the colour of the star (at its observed My) and the colour it would have if it were located on the [Fe/H] = —1.92 isochrone. These differences were then appUed to the dereddened colours of the subdwarfs to obtain the final, corrected colour indices given in the seventh column of Table 2.7. The metaUicity of HD 140283 was too low to calculate the colour offset directly, but I assumed that it wiU have an offset close to that of the next most metal-poor star of similar absolute mag-  Chapter 2.  50  The Main Sequence of M55  o Si' b o  1>- o  o o  oo  T-H  o o  00  T-H  to  as  CN  CO CN  as CN  o  o  o  o  o o  LO  00  CO CO  T-H  as  o o  T-H  T-H  o  CN T-H  CO  o'  o  o  oo o o o  o  o  o  o  1—1 o  o  r-H r-H  T-H  CN CO oo a s CN CO t ~ r~ r— CN CO CO CO a s CO o T-H T-H o r-H T-H r - H r-H o  LO  t-H  as CN  CO CN  CM  LO  L6 CO o  o  CO  o  oo  o  o  o  o  LO  T-H  o  as r-H CO  LO  CN CO 0 0  o  o o  co  o  LO  CO r-H T-H o o o o T—1  o  00  CN CN oo CO CO CO o oo CO* CO  o CN  t--  T-H  r-H T - H CO o r-H CO o CO L O CO  oq CO CO a s CO CO co CO  LO  LO  o  CO CO T-H  ^  O  LO  o  LO  a s CO r— CO CO CO oo L O  o  o  o  T-H  O  o  o  o  o  o  o  T-H  CN  O  co o  T-H  T-H  T-H  CN CN  as  CO O O CN L O  as o  LO  CO T-H  LO  <-<  T3 rO  oo  CO  >.  LO  T-H  CN T-H  R3 1)  n  .O  «3 ^3 T3  >  CN CD  CM  1  IS  o i  CO oo CN CN CN  LO  oo o  r-H  T-H  1  as  as o  as o r-H o  T-H  SH  CO o CO CO a s o CO oo CO OS a s L O L O L O oo CO OS  o  o  o  o  LO  CO CO o T-H o o o  r-H CO CO CO CO  T-H  CO r-H  T-H  r-H  r-H  CO  CN  r-H r-H O O T-H L O o o LO as CO 0 0 T-H L O CO CN T-H r-H CN r-H r-H CN r-H r-H r H  1 1 1 1  1  o  OO  LO  CN  o  as  1 1 1  1  1  LO  LO  CN CN t CO L O  o  o  o  o  LO  oo a s  o o  o o  CO O O  CO  o  o  as  T-H  LO  CN  CN CO CN  1  1  o  1 1  ]>-  CO CN  LO  r-H  o  CO CO CO a s 0 0 oo L O 1>- oo  o  o  o  CD  -a  T-H  Cl  o o  d  CD > H <D to  o  o o  CN  r-H L O T-H o  o o  o o  o  o  o  r-H  as  T-H  as  T-H  co o  CO  00  as as  LO  o  o  o  o  o  o  CN  o  CN  CN t -  o o  o o  o o  o o  o o  o o  o o  T-H  CN LO  o CO o 1^ o T-H o  I  LO  CN  LO  T-H  T-H  T-H  LO  r-H o CO a s o  as  D  et?  CO LO  co T - H CN o t— CO  CN CO 0 0  00  00  oo a s a s  p-H  H  o  LO  T-H  o CS o  0 0 as o T-H CO CO CN CO CO 0 0 a s oo CO a s  o  o o  CN  o o  o  CN CO  o  o  i>-  CO CN r-H CN o  o o o  CN  o o o  T-H  o o  r-H  CN  o o  T-H  o o  o o  LO  O O  o o  r-H oo CN CN CO as  LO  o  o o  o o  o  r-H CN  CN  a s CO CN L O CO CO CO oo  CO CO  o o  o  O o  oo o  o  o  oo o o o  oo 00 CN T - H o o o o o o o  o  oo  00  L O CO T - H CO CO a s as oo  CN o T—1 r-H  PH  r-H  w  Q Q W  Tt* L O T-H C3s OS r-H OS L O T-H  LO  oo  T-H  00  CO ^t-  as o  CN a s CO o  OS L O r-H CN  CO  oo  a s CN CO CO OS CO t~ o L O CO  CO J> L O CO. GO a s  CO co LO CO CO CO t > CO CN CN C3S co oo LO CO as  ^  CO a s t ~ OS co r-H •"-I o •vo CO 0 0 GX> <M o o co T-H r-H >-i T - H  o  C3S  CO r-H  CN as  o o  T-H  as LO  CO  o  oo  CO  rH as o r-H CN  T-H  00  T-H  LO  CO  T-H  as  oo  T-H  CO CO  T-H  r-H  00  LO  r-H  GO CO oo  as  o  T-H  CO L O CN a s CO oo CO CO  + +  Chapter 2.  The Main Sequence of M55  51  Table 2.8: References for the compiled subdwarf data HD/BD 19445 25329 64090 74000 84937 103095 108177 122196 134439 134440 140283 181743 194598 201891 +66 268 +38 4955  Photometry 5,6,10,11,12 6 5,6,11 4,5,9,10,11 5,6,10,11,12 5,6,11,12,16 5,10,11,12 4,10,13 4,6,8,10,13 4,6,8,13 4,5,6,10,11,12 3,9,11,13 5,8,9 5,8,9,10 5 5,6  [Fe/H] 3,7,11,18,19 7,11,15 1,3,7,11,15 1,3,7,11,17,18 3,7,10,17,18,19,20 7,11,14 7,10,11,17,18 3,7,10,19,20 7,10,11 7,11,15 1,7,10,18,19,20 3,7,19 7,11,17,18,19,20 3,7,10,11 1,2,11,17,18 1,2  (1) Axer et al. 1994; (2) Axer et al. 1995; (3) Beers et al. 1990; (4) Carney 1980; (5) Carney 1983b; (6) Carney k Aaronson 1979; (7) Cayrel de Strobel et al. 1992; (8) Dean 1981; (9) Eggen 1987; (10) Laird 1985; (11) Laird et al. 1988; (12) Lanz 1986; (13) Ryan 1992; (14) Smith et al. 1992; (15) Spiesman k Wallerstein 1991; (16) Taylor 1986; (17) Tomkin et al. 1986; (18) Tomkin et al. 1992; (19) Zhao k Magain 1990; (20) Zhao k Magain 1991; nitude (HD 122196). Since HD 140283 was not used in the lit anyway (it turned out to be a subgiant, see the discussion below), any small error in its colour index is not important. The use of the a-enriched isochrones is supported by the measurements of Gratton et al. (1997), who found an average a-enhancement of [a/Fe] = 0.26 ± 0.08 for the metal-poor field subdwarfs. Because of the necessity for colour corrections, accurate metal abundances for the subdwarfs (especially the more metal-rich ones) are almost as important as well-determined parallaxes. When the literature was searched for metallicity data on the subdwarfs, preference was given to [Fe/H] values derived from high-dispersion spectroscopy or similar  Chapter 2.  52  The Main Sequence of M55  techniques. As a result the different determinations agreed fairly well and the uncertainty for most of the [Fe/H] values in Table 2.7 is on the order of 0.1 dex. Recently, Pont et al. (1998) reported high-quality metallicity determinations from Coravel spectra for a sample of subdwarfs from the Hipparcos Catalogue. For the five stars that I have in common with his sample, the agreement in metaUicities is excellent. The largest difference is 0.12 dex for HD 108177 (Pont et al. 1998 give [Fe/H] = -1.92 for that star) and for the remaining four stars the differences are below 0.05 dex. The absolute magnitudes listed in Table 2.7 were calculated directly from the parallaxes and then corrected for the biases present in the Hipparcos data. These biases were taken from Pont et al. (1998), who argued that the absolute magnitudes calculated from the Hipparcos parallaxes do not require the application of the traditional Lutz-Kelker corrections (Lutz & Kelker 1973, Hanson 1979). They carried out extensive Monte Carlo simulations of the selection effects present in the Hipparcos data and concluded that a correction of AMy = +0.064 should be applied to the absolute magnitudes of metal-poor stars ([Fe/H] ~ —2), and a smaller correction of AMy = +0.011 to the more metal-rich ([Fe/H] ~ — 1) subdwarfs. The subdwarfs in my sample can be divided into three groups centred around [Fe/H]  —1.9, [Fe/H] « —1.5 and [Fe/H] « —1.3, and  so I used AMy = +0.064 for the stars in the first group, A M y = +0.038 for the stars in the second group and AMy = +0.027 for the subdwarfs in the third, most metal-rich group. The last two values were found from a linear interpolation between the metal-rich and metal-poor values of Pont et a/.(1998). After the absolute magnitudes were calculated, several of the subdwarf candidates (marked by italics in the table) turned out to be evolved stars. They were not included in the main sequence fit to avoid uncertainties associated with their unknown age. It should be noted that for subgiants the photometric bias is of different size and in the opposite direction of that for unevolved stars (see Pont et al. 1998). The values of My  Chapter  2.  53  The Main Sequence of M55  for the subgiants in Table 2.7 include the corrections recommended by Pont et al. (1998). After the fiducial main sequence of M55 was shifted blueward by Ey-i  = 0.17, a  weighted least-squares fit to the subdwarf data (Table 2.7, columns 7 and 12) yielded an apparent distance modulus of (m — M)y = 14.02 ± 0.08.  This corresponds to a  true distance modulus of (m — M) = 13.62 ± 0.10 or a distance of 5300 pc assuming 0  that Ry = 3.1 (Ry being the ratio of total to selective absorption in the V passband, Ry = Ay IEB-V)1/<T , where a 2  2  In the main-sequence fit the weight of each data point was set to consists of four components added in quadrature: the standard error  of the parallax measurements ev, the photometric errors in the apparent V magnitude (cry) and colour index (cry_j), and an additional component of 0.02 mag coming from the uncertainty of the reddening estimate. The values of a(My) in the last column of Table 2.7 reflect only the first two error sources (ov and ay). The best fit of the M55 main sequence to the subdwarfs is shown in Figure 2.10, where the vertical and horizontal error bars correspond to the values of aM and ay_j, v  respectively. One of the stars used in the fit, BD +66 268, is listed as a suspected nonsingle star in the Hipparcos Catalogue. If it is a binary star, it could appear as much as 0.75 mag brighter than the more massive component in the system. Pont et al. (1998) have estimated that on the average, binaries will be 0.375 mag brighter than a single star of the same colour index. If this number is adopted for BD +66 268, the derived distance modulus is reduced by ~ 0.01 mag but the use of such a correction is not warranted until the duplicity of BD +66 268 has been established more reliably. In general, one would expect to see about six binaries among the 12 unevolved subdwarfs in in Figure 2.10, and of those about three could be <> 0.75 mag above the single-star metal-poor main sequence (see Pont et al. 1998). Undetected binaries are expected in any subdwarf sample and this means that the derived distance modulus will always be biased towards larger values. In Figure 2.10 the evolved subgiant stars (not used in the fit) are shown by open circles.  Chapter 2.  54  The Main Sequence of M55  (v-D  c  Figure 2.10: Thefiducialmain sequence of M55 fitted to the sample of subdwarfs discussed in the text. The unevolved subdwarfs to which the main sequence of M55 was fitted are shown by filled circles. The subgiant stars (not used in the fit) are marked by open circles. Thefiducialmain sequence and subgiant branch of M55 are shown by a dashed fine.  Chapter 2.  The Main Sequence of M55  55  As can be seen, they all appear fainter than the M55 subgiant branch, which could be caused by inappropriate colour corrections (as they depend on the assumed age, 14 Gyr in this case). Taken at face value, the position of the metal-poor subgiants implies that the sample of subdwarfs that I used is about 2 Gyr older than M55. Pont et al. (1998), on the other hand, obtained a good agreement between the subgiants in their sample and the subgiant branch of M92, and concluded that the subdwarfs and M92 were coeval. This is a somewhat disturbing contradiction, since three of the subgiants in Table 2.7 are also in the list of Pont et al. (1998) (who used the same theoretical isochrones to calculate their colour corrections). The main reason for the discrepancy appears to be the larger values for the subgiants' reddening adopted by Pont et al. (1998). They used E -v B  = 0.01, 0.05 and 0.04 for HD 84937, HD 122196 and HD 140283, respectively,  whereas I assumed that these are all unreddened stars. If the higher reddenings were adopted, these stars would he on or very close to the subgiant branch of M55. The fourth subgiant, HD 74000, has one of the most uncertain parallaxes and less significance should be attached to its position in the colour-magnitude diagram. It appears that at least for some subdwarfs (and metal-poor subgiants) the metallicity and reddening uncertainties outweigh the uncertainty associated with the parallax errors and more effort should be spend on obtaining accurate photometry and colour excesses for these important stars. Another possible source of uncertainty in the subdwarf colour corrections is the shape of the isochrones used to derive them. Figure 2.11 (which has the same vertical scale as Figure 2.10) shows that the models used to derive the colour corrections predict the shape of the main sequence of M55 very well. On the other hand, Figure 2.10 indicates that there is a good agreement between the shape of thefiducialmain sequence of M55 and the subdwarf sequence. One can conclude therefore that any possible uncertainties introduced by the shape of the isochrones would be negligible compared to the parallax and metallicity errors.  Chapter  2.  56  The Main Sequence of M55  V-I  Figure 2.11: The [Fe/H] = -1.84, [ct/Fe] = 0.3 isochrone overlaid on the main sequence of M55. No offsets other than the ones indicated in the plot have been applied to the photometry or the isochrones. Note the excellent agreement between the shapes of the main sequence and the isochrone.  Chapter 2.  57  The Main Sequence of M55  As mentioned in the preceding section, globular cluster metallicities are usually obtained from spectroscopic observations of cluster red giant stars. When the colours of the subdwarfs are adjusted to the mono-metallicity sequence defined by the cluster's [Fe/H] value, it is implicitly assumed that the metallicity of the cluster as derived from spectroscopy of its red giants is the same as would be obtained if the cluster's main-sequence stars were observed (as it is the main sequence that is fit to the subdwarfs). It is well known that red giant stars in some globular clusters show enhanced abundances of a number of heavy elements (Sneden et al. 1991, Kraft 1994), probably as a result of the deep mixing during the red giant branch evolution.. It is quite possible, therefore, that the surface abundances of the unevolved main sequence stars are intrinsically different from those found in the evolved stars on the red giant branch. In a recent spectroscopic study, King et al. (1998b) found that the abundances derived from stars near the turnoff of the globular cluster M92 were lower by a factor of two than those derived from its red giants. If the same were true for M55, it would have a true metallicity of [Fe/H] « —2.2. This new value will reduce the distance estimate somewhat, although at such low metal abundances the position of the main sequence in the V, V — I plane becomes almost insensitive to variations in metallicity — the colour corrections for the most metal-rich subdwarfs will increase by 0.015 mag, which translates into a drop in the distance modulus of ~ 0.06 mag. In order to compare the distance to M55 derived here with the predictions of the few My(RR) - [Fe/H] calibrations, I adopted [Fe/H]  M55  = -1.92 and V(HB) = 14.46 for the  apparent visual magnitude of the horizontal branch (see Chapter 3). The corresponding absolute magnitude of the horizontal branch is then My (HB) = My(RR) = 0.44 ± 0.09, where the often used symbol My(RR) denotes the mean magnitude of the RR Lyr stars. The error estimate for My(HB) was found by adding in quadrature the standard error of the V(HB) determination and the error of the apparent distance modulus.  Chapter 2.  58  The Main Sequence of M55  Table 2.9 lists the brightness of the horizontal branch predicted from the My (RR) metallicity relations cited in the first column. The second column of the table gives the expression used to calculate the values in the third column, and the last column lists the differences between the observed and predicted values of My (HB). Four of the relations in Table 2.9 (Sandage 1993a, Clement 1996, Gratton et al 1997 and Chaboyer et al. 1998) predict My (HB) that is within lcr of the value derived in this work. The predictions of Reid (1997), Gratton et al. (1997) and Chaboyer et al. (1998) are based on parallaxes from the Hipparcos Catalogue. The value in Clement (1996) is based on a new approach of using hydrodynamic pulsation models (Simon & Clement 1993) to predict the luminosity of RR Lyr stars from their light curve parameters. This method was recently applied to NGC 2298 (Clement et al. 1995) and M9 (Clement & Shelton 1996) and yielded an average value of My(RR) = 0.49 for the two clusters. Since their respective metallicities (-1.81 and -1.78) and horizontal-branch morphology types ((B- R)/(B+  V + R) = 0.93  and 0.87) are nearly identical to those of M55 ([Fe/H] = -1.92, (B - R)/(B + V + R) = 0.93), one would expect the three clusters to have very similar horizontal branches and hence comparable values of My(RR), which is what is observed. Table 2.9: Comparison of My(RR) estimates from different My(RR) - [Fe/H] relations Reference Carney et al (1992) Sandage (1993a) Fernley (1994) Chaboyer et al. (1996) Clement (1996) Reid (1997) Gratton et al. (1997) Chaboyer et al. (1998)  Relation 0.15[Fe/H] + 1.01 0.30[Fe/H] + 0.94 0.21[Fe/H] + 0.97 0.20[Fe/H] + 0.98 0.27[Fe/H] + 0.97 My(HB) = 0.22 at [Fe/H] = -1.9 0.22[Fe/H] + 0.82 0.23[Fe/H] + 0.83  My (RR) 0.72 0.36 0.57 0.60 0.45 0.22 0.40 0.39  Difference -0.28 +0.08 -0.13 -0.16 -0.01 +0.22 +0.04 +0.05  Chapter 2.  2.6.2  The Main Sequence of M55  59  The Age of M55  With the distance modulus, reddening and metallicity of M55 known, one can compare the observed location of the main-sequence turnoff with stellar models and derive an age estimate for the cluster. I have chosen to use the new set of isochrones by VandenBerg et al. (1998b) because their theoretical models include many improvements over the old VandenBerg & Bell (1985) set: new opacities, reaction rates, equation of state and colour transformations, as well as more representative helium abundance. Also, since the new models allow for non-solar ratios of the a-elements (0, Ne, Mg, S etc.), they are much more realistic as far as the chemical mixture is concerned. The isochrone set for [Fe/H] = —1.84, which best matches the metallicity of M55, has been calculated for three values of the a-enrichment, namely [a/Fe] = 0.0, 0.3 and 0.6. As there are no direct measurements of the amount of a-enrichment in M55, one has to refer to observations in other clusters in order to choose an appropriate value of [a/Fe]. Many spectroscopic studies of globular cluster giants (Sneden et al. 1991, Carney 1996 and the references therein) have shown that [a/Fe] is usually between 0.2 and 0.4 (around [Fe/H] ~ —2) and rarely as high as 0.5. I adopted therefore [a/Fe] = 0.3 as the most plausible value for M55. Figure 2.12 shows a comparison between the M55 photometry around the mainsequence turnoff and a set of isochrones with the following chemical composition parameters: helium abundance Y = 0.236, heavy metal abundance [Fe/H] = —1.84 and a-enrichment [a/Fe] = 0.3.  The isochrones are for ages of 12, 14, and 16 Gyr and  they have been shifted by the distance modulus and reddening indicated on the right: E -i V  = 0.17 and (m - M)  v  = 14.02. Only the region around the MSTO and the sub-  giant branch is shown because their locations are most age-sensitive and allow for better comparison between theory and observations.  Chapter 2.  60  The Main Sequence of M55  V-I  Figure 2.12: A comparison between the photometry around the turnoff and a set of a-enriched isochrones from VandenBerg et al. (1998b). The parameters of the isochrones are listed in the upper right corner. All isochrones have been shifted by the adopted apparent distance modulus and reddening, but no other shifts have been applied. The ages of the isochrones increase from top to bottom.  Chapter 2.  61  The Main Sequence of M55  The 14 Gyr isochrone seems to match best the turnoff of M55, so an initial estimate of the age of M55 is ~ 14 Gyr. For the purposes of age determination, one is interested in the luminosity of the turnoff, My(TO). This quantity is independent of the uncertain effective temperature - colour index transformations and is much less dependent on model parameters such as the equation of state, the treatment of convection, etc. (VandenBerg et al. 1996). In order of increasing age (12, 14 and 16 Gyr), the isochrones shown in Figure 2.12 predict the following values of My(TO): 3.80, 3.98 and 4.09, with uncertainties on the order of ±0.05 mag. Using Vro = 18.03 ± 0.05 and (m - M)  v  = 14.02 ± 0.08, one obtains My(TO) = 4.01 ± 0.09 for M55. This value is  matched best by the 14 Gyr isochrone, whose turnoff is at My(TO) = 3.98 ± 0.05. An error estimate for the age of M55 can be obtained as follows.  Assuming that  the absolute magnitude of the main-sequence turnoff depends only on the age and metallicity of the cluster, the uncertainty AT of the cluster age can be found from (AT)  2  ~ [^^AMy(TO)]  2  + [ ^ A [ F e / H ] ] , where AMy(TO) is the uncertainty 2  in the turnoff luminosity and A [Fe/H] is the uncertainty in [Fe/H] . The values of the partial derivatives as obtained from the models of VandenBerg et al. (1998b) are QM^TO)  ~ °-  3 8a n d  IfvSJ ~ ~ - ' 0  14  a n df o r  v(TO)  AM  =  ±0.09 and A [Fe/H] « ± 0 . 1  one finds A T PH ±1.2 Gyr. This value is based only on the adopted errors in the metallicity of M55 (±0.1), the cluster distance modulus (±0.08), and the apparent turnoff magnitude (±0.05); it does not take into account any uncertainties in the models or possible systematic errors in the subdwarf calibration and as such should be considered only a lower limit. The present age estimate of 14 ± 1.2 Gyr is about 2<r away from the 10-12 Gyr range advocated by some recent studies (Reid 1997, Gratton et al. 1997, Chaboyer et al. 1998) and it is important to understand why. Except for the a-element abundance, the isochrone parameters that I adopted (metallicity and helium abundance) are well supported  Chapter 2.  62  The Main Sequence of M55  by observations (the helium content of the M55 stars is determined in Chapter 3). Using isochrones with a higher a-enhancement will lead to lower ages, although such isochrones match very poorly the subgiant branch and the lower giant branch of M55. The 12-Gyr, [a/Fe] = +0.6 isochrone from VandenBerg et al. (1998b) predicts M (TO) v  = 4.03 + 0.05,  which agrees with the "observed" value about as well as the 14-Gyr, [a/Fe] = +0.3 isochrone (see above). The poor agreement between the 12-Gyr, [a/Fe] = +0.6 isochrone and the subgiant and giant branches of M55 can be reduced or eHminated by adopting a higher value for the mixing-length parameter used in the models (see Figure 3 in VandenBerg 1983). On the other hand, the value of My(TO) for M55 depends critically on the adopted apparent distance modulus of M55, and hence on the way the absolute magnitudes of the subdwarfs were determined. Aside from the obvious uncertainties associated with errors in the photometry, reddening and metal abundances of the subdwarfs, one has to consider the nature of the biases in the subdwarf sample and how they were treated. Reid (1997) used the traditional Lutz-Kelker corrections (which for his sample ranged from AMy  K  = 0 to AMy  K  = —0.4) and arrived at subdwarf distances that were ~ 0.3 mag  higher than the earlier values. As a result, he obtained average ages of ~ 10 Gyr for his sample of metal-poor globular clusters. Gratton et al. (1997) neglected any bias corrections altogether, as their simulations of the Hipparcos subdwarf sample indicated very small negative biases (~ —0.002 mag). As a result, their globular cluster distance moduli were ~ 0.1 mag smaller than Reid's (1997) values and they obtained higher ages (11.8 ± 1.2 Gyr) for the oldest metal-poor clusters. Chaboyer et al. (1998) also did not apply any corrections to the magnitudes of their subdwarfs and obtained ages similar to those of Gratton et al. (1997). It should be noted that Chaboyer et al. (1998) used only three clusters of intermediate metallicity, together with other distance indicators, to calibrate the distance scale and derive a mean  Chapter 2.  The Main Sequence of M55  63  age of 11.5 ± 1.2 Gyr for a sample of metal-poor globular clusters. My estimate of 14±1.2 Gyr for age of M55 is based on the subdwarf analysis of Pont et al.  (1998), who found positive, metallicity-dependent biases in the parallaxes of the Hip-  parcos  subdwarf sample. However, if I adopt the approach of Gratton et al. (1997) and  Chaboyer et al. (1998) (that is, no bias corrections at all), the distance modulus of M55 is increased to (m - M)  v  = 14.07 ± 0.08 (Afy(TO) = 3.96 ± 0.09) and the best estimate  for the age of M55 remains 14 Gyr. Thus, it appears that M55 is intrinsically older than many of the other metal-poor globular clusters. This result, combined with the finding by Pont et al. (1998) that M92 is also about 14 Gyr old, implies that the conflict between the ages of the oldest globular clusters and the age of the Universe inferred from recent determinations of the Hubble constant HQ (Hamuy et al. 1996, Madore et al. 1998) remains unresolved.  2.7  Luminosity and Mass Functions for the Main Sequence of M55  As discussed briefly in Chapter 1, the main-sequence luminosity function and the mass function derived from it can be used as powerful tools for exploring the dynamics of globular clusters, the physical conditions of star formation and the connection between the stellar content of globular clusters and that of the halo. From the observations and analysis presented so far, M55 has emerged as one of the handful of globular clusters where the main sequence can be studied from the turnoff to about seven magnitudes fainter. As a result the luminosity function and mass function of M55 can be explored over a wide range in luminosity and mass, unlike a number of studies where various correlations with metaUicity, position in the Galaxy and other parameters have been claimed for mass function slopes derived over a fairly narrow (~ 0 . 3 ^ © ) mass range  Chapter 2.  The Main Sequence of M55  64  (McClure et al. 1986, Piotto 1991, Capaccioli et al. 1993, Djorgovski et al. 1993). Without a knowledge of the shape of the mass function at lower masses it could be next to impossible to understand the cause of cluster-to-cluster differences in the mass-function slopes — are they primordial, random or the result of subsequent dynamical evolution? Another interesting question related to the low-mass stellar content of M55 is raised by the study by Pryor et al. (1991), who derived a central mass-to-light ratio of 1.3 ± 0.4 and a global mass-to-light ratio of 1.7 ± 0 . 5 for this cluster. While these numbers depend on the adopted low-mass cutoffs and other details of their models, they nonetheless imply that M55 should have only a small number of low-mass stars (i.e., stars with A4/A4Q  ^ 0.4). Deep mass functions should be able to provide critical tests of such  dynamical studies, for it is well known that even perfect fits to the surface brightness profile and the radial velocity profile do not constrain the mass and mass-to-light ratios that well (Merritt 1987, Meylan k Pryor 1993). Deep mass functions are also needed to verify the results of dynamical globular cluster models based on proper motion analysis (e.g.,  Leonard et al. 1992a), as the latter are not that much model-dependent and hold  the promise of more reliable cluster masses and mass-to-light ratios. The study by Leonard et al. (1992a), as well as an earlier work by Lupton et al. (1987) suggested that a large portion of a globular cluster's mass is in the form of low-mass stars. Richer et al. (1991) made the same suggestion, although from a different point of view — they found that in all clusters in their sample the mass function below 0.4A4© rose steeply to the faint end of the data, implying that very low-mass stars could make a significant contribution to the total mass of the cluster. Similar results were obtained from Hubble Space Telescope  observations (King et al. 1996a, Piotto et al. 1997, King et al. 1998a),  although the slope of the mass function below O^A'f© was not as steep as found by Richer et al. (1991). In none of the clusters, however, was there any indication that the mass function turns over at the low-mass end, even though some of the mass functions reached  Chapter  2.  65  The Main Sequence of M55  within ~ O.O3A^0 of the theoretical hydrogen-burning limit. The main-sequence luminosity function is obtained by counting the stars on the main sequence, usually in apparent magnitude bins of uniform width. If the distance to cluster is known, one can write the luminosity function in terms of the absolute magnitude, as all cluster stars can be considered to be at the same distance: dN  =  $(M )dM v  v  where dN is the number of stars in an absolute visual magnitude interval of width dMy, and $(My) is the differential V luminosity function. It is possible, of course, to derive a luminosity function in other passbands (B, I etc.) as well. While counting stars sounds simple, there are two important issues to be considered before the luminosity function can be used in a meaningful way: what is the contribution of background stars and galaxies to the luminosity function, and how complete is the luminosity function at fainter magnitudes, where many stars are missed because they are either masked by the sky noise or too close to other, brighter stars (Stetson 1991, Fahlman 1993). Both effects require corrections to the raw luminosity function and the derivation of these corrections is explained in detail later in this section. Once the completeness-corrected and background-subtracted luminosity function has been determined, it can be converted to a mass function since =  \$(M )dM \ v  v  \§(M)dM\,  where $(A4) is the differential mass function defined in a way similar to the differential luminosity function: dN  =  $(M)dM.  It is often written in a power-law form $(M)  oc  M~  {1+x)  (2.3)  Chapter 2.  66  The Main Sequence of M55  although it is not always possible to represent the observed globular-cluster mass functions with a single power law — the value of the mass spectral index x is often different for the different mass intervals (see, e.g., Richer et al. 1991). The transformation of the luminosity function to a mass function requires the knowledge of how mass and luminosity are related for the stars on the main sequence - the so-called mass-luminosity relation: $(A4) where dMy/dAA  =  $(My)  dM  v  1M  is the slope of the mass-luminosity relation. For Population I stars  the mass-luminosity relation is well known from measurements of nearby binary stars (Kroupa et al. 1990, Henry & McCarthy 1993).  For the metal-poor stars of Popula-  tion II, however, the mass-luminosity relation is unknown and one has to rely on the one derived from theoretical models. As most of the models seem to represent the main sequence fairly well (at least for AA J> 0.2A4®),  the theoretical mass-luminosity relation  for Population II stars in that mass range is probably not very far from the true one.  2.7.1  Completeness Corrections  It is well known that crowding and the increasing contribution of the sky background reduce the discovery probability for faint stars (Stetson 1991). One of the major advantages of working with CCD images is that artificial stars can be added to the images (as one already has a model star — the point-spread function) and then one can calculate the fraction of added stars that has been recovered. In order to estimate the completeness of the data at faint magnitudes, I generated eight sets of artificial stars with magnitudes following a theoretical luminosity function (Bergbusch & VandenBerg 1992) in the interval 21 < V < 26 and colours given by the fiducial main sequence of M55 (linearly extrapolated for V > 24.8). Each set contained 270 stars which were placed at random  Chapter 2.  67  The Main Sequence of M55  locations in the program images using the  routine  DAOPHOT  ADDSTAR.  The resultant  eight sets of 36 images were reduced in exactly the same way (described in section 2.2) as the original program images. The only difference between the original and the artificial image sets was the ~ 5% increase in the crowding in the artificial images and I assumed that this slight increase will have a negligible effect on the artificial star photometry. Figure 2.13 shows a comparison between the colour-magnitude diagram of the observed lower main sequence of M55 (left panel) and the same area of the colour-magnitude diagram for the recovered stars (right panel). The artificial stars plotted here were subject to the same restriction as the real stars, namely % < 1.5 and each star discovered on 2  at least three frames in each filter. One can see that the main sequence of the recovered artificial stars closely resembles the appearance of the real cluster sequence, which means that the additional crowding caused by the artificial stars did not affect their photometry. The artificial-star main sequence starts at V = 21.0 as for brighter stars the recovery fraction is near unity and I did not add any stars brighter than V = 21.0. Figure 2.14 shows the differences between the measured and input Fand I magnitudes and colour versus the output magnitude for all recovered artificial stars found on at least three frames in each colour and with % < 1.5. It is apparent that there is an excess of 2  stars with large negative residuals (i.e. stars measured brighter and bluer), contrary to what one would expect for normally-distributed photometric errors. Note also that the difference between the input and output magnitudes can exceed 0.5 mag, especially for the fainter stars. Therefore, besides the usual method of calculating the completeness simply as the ratio / = n /n , out  given magnitude bin and n  o u t  in  where n- is the number of artificial stars put in a m  is the number of stars recovered in the same bin, I also  used the approach suggested by Drukier et al. (1988) which accounts for the effect of "bin jumping" — the fact that the measured magnitude of an artificial star may be quite different from its input magnitude. I found, however, that the differences between  Chapter 2.  68  The Main Sequence of M55  V-I  V-I  Figure 2.13: Comparison between the observed lower main sequence of M55 and the main sequence of the recovered artificial stars  Chapter 2.  The Main Sequence of M55  69  Figure 2.14: Differences between the input and output magnitudes and colour versus the output magnitude for all recovered artificial stars found on at least three frames in each colour and having % < 1.5 2  Chapter  2.  70  The Main Sequence of M55  I  Figure 2.15: Completeness fractions in the V and / passbands. The hand-drawn solid line shows the adopted completeness fractions at a given magnitude. the corrected counts for the two methods were within the Poisson uncertainties and subsequently used only the completeness ratios / = n /n- , out  m  as found for each magnitude  bin, to correct the observed counts. Figure 2.15 shows plots of the completeness ratios f(V) and /(/) as a function of magnitude. The hand-drawn smooth curves represent the adopted completeness fractions and the error bars show the formal la errors calculated from the assumption that n /:  a\f)  = /(l -  f)/n . in  o u t  are drawn from a binomial distribution with a probability  Chapter 2.  2.7.2  71  The Main Sequence of M55  Luminosity Functions  Because of the presence of bulge and SDG stars in the program field, the M55 luminosity function was constructed by counting only the stars within 3o-y-i of the fiducial main sequence, where the values of cry-i were taken from Table 2.4. Since this restriction will undoubtedly reject some true cluster members as well, it was imposed also on the recovered artificial stars before calculating the completeness corrections, and on the counts of field stars and galaxies in the comparison field. In order to evaluate the completeness of the comparisonfieldphotometry, six sets of 270 artificial stars per set were generated and the images with the artificial stars reduced in the same fashion as the original frames. As in the case of the program field, the completeness corrections were then calculated as f' , 1  where / = n  out  /n . in  Table 2.10: Star and galaxy counts in the V band V  17.25 17.75 18.25 18.75 19.25 19.75 20.25 20.75 21.25 21.75 22.25 22.75 23.25 23.75 24.25 24.75  TV 21 47 69 77 128 152 154 160 186 190 211 258 329 516 470 397  f(V)  N  N  N  1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.991 0.975 0.949 0.916 0.873 0.815 0.750 0.659 0.503  0 0 0 0 0 0 0 1 1 2 4 6 10 15 24 39  0 0 0 2 2 4 6 2 7 6 5 8 15 23 53 56  21 47 69 75 126 148 148 158 182 192 221 282 379 650 636 694  s  h  0  <r(N ) 0  5 7 8 9 11 12 13 13 14 15 16 20 25 37 46 64  log $ 1.623 1.973 2.140 2.176 2.401 2.471 2.471 2.501 2.561 2.583 2.646 2.751 2.880 3.114 3.104 3.143  Chapter 2.  72  The Main Sequence of M55  The final star counts in V and I are given in Table 2.10 and Table 2.11, respectively. For each magnitude bin the tables give the raw cluster counts N and their completeness / , the expected number of faint galaxies 7V as calculated from the relations of Woods et g  al. (1995), corrected for field size and reddening, the number JVb of field stars (corrected for incompleteness), the final counts No corrected for incompleteness and background contamination, and their formal error <x(JVo). The last column of the two tables contain log  the logarithm of the luminosity function of M55 in units of stars per unit magnitude  interval. The errors of the final, corrected counts were found by adding in quadrature the individual errors of the cluster andfieldcounts, the completeness corrections and the galaxy counts. The latter were estimated from the uncertainties quoted by Woods et al. (1995). Table 2.11: Star and galaxy counts in the I band I  N  16.39 16.99 17.52 17.99 18.44 18.87 19.29 19.70 20.09 20.47 20.84 21.23 21.64 22.05 22.49 22.94  12 46 70 79 126 153 151 162 187 184 217 262 329 520 503 411  f(I)  1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.990 0.972 0.944 0.919 0.879 0.835 0.770 0.667 0.486  N  s  0 0 0 0 0 0 0 2 3 3 4 6 8 12 17 25  iVb 0 0 0 2 2 3 7 2 7 6 8 4 13 27 66 93  No  <r(N )  12 46 70 77 124 150 144 160 183 185 224 289 373 636 672 728  4 7 8 9 11 12 13 13 14 15 17 19 24 36 46 63  Q  log $ 1.266 1.914 2.161 2.223 2.448 2.548 2.542 2.603 2.677 2.698 2.767 2.863 2.962 3.174 3.184 3.194  The 7-band and /-band luminosity functions of M55 are plotted in Figure 2.16,  Chapter 2.  73  The Main Sequence of M55  Figure 2.16: The F-band and /-band luminosity functions (top and bottom panels, respectively). Open circles show the raw counts, and filled circles show the counts corrected for incompleteness and field star and galaxy contamination. Note the dip in the luminosity function at My tt 4.8 (Mi tt 4.2) and the plateau around M ~ 5.5 - 7.5. v  Chapter 2.  74  The Main Sequence of M55  where the open and filled circles show the raw and corrected counts, respectively, per unit magnitude interval. The absolute magnitudes were calculated using (m — M)y = 14.02 and Ev-i = 0.17. The F-band luminosity function has a distinct plateau centred at My ~ 6.5 which is also seen to some extent in the /-band luminosity function at Mi ~ 5.5. Such a flattening around My fa 6.5 is a common feature in the luminosity functions of many globular clusters (cf. McClure et al. 1986, Piotto et al. 1997); what seems to be unusual in the case of M55 is that such a wide plateau is typically seen in clusters more metal-rich than M55. Another interesting detail in the luminosity function is the conspicuous dip at My fa 4.8 (Mj fa 4.2).  It is seen in the luminosity functions of  other clusters as well, however, as opposed to the plateau mentioned above, theoretical models still fail to reproduce it (Bergbusch 1990). Perhaps the most significant feature of the observed luminosity function of M55 is that after a steep rise it levels off at My fa 9.8 (Mi fa 8.3) and remains nearly flat to the faint limit of the data. In this respect the M55 luminosity function is similar to the deep luminosity functions of M13 (Drukier et al. 1988), M15 and M30 (Piotto et al. 1997) and at Cen (Elson et al. 1995), where the flattening occurs at My fa 10 ± 0.5 or Mj fa 8.5 ± 0.5. In Figures 2.17 and 2.18 the /-band luminosity function of M55 (which is groundbased) is compared to the Hubble Space Telescope luminosity functions of M15, M30 and M92 taken from Piotto et al. (1997). It is clear that the faint limit of the M55 data is just a few tenths of a magnitude brighter than the point where the luminosity function turns over and starts a rapid decline. The deep ground-based luminosity functions never reached that point, the point where the changing slope of the mass-luminosity relation causes the drop in the luminosity function. Note also that there is no evidence that the ground-based faint-end completeness corrections for M55 have been over-estimated, as has been suggested for other clusters (King et al. 1996b). In the first plot (Figure 2.17) the luminosity functions of M15, M30, M55 and M92  Chapter 2.  The  3.5  75  Main Sequence of M55  h  caO CO  '1 <0 w  2.5  o o  QO O  1.5  10  Figure 2.17: Comparison of the ground-based /-band luminosity function of M55 with the HST-based luminosity functions of M15, M30 and M92. The luminosity functions are normalized over the whole magnitude range of the main sequence of M55. With this normalization, M55 exhibits a relative excess of bright (i.e., high-mass) stars.  Chapter  2.  The Main Sequence of M55  76  were normalized so that all four clusters have equal number of stars in the magnitude interval 3.25 < Mi < 9.25, that is, over the whole range of the observed main sequence in M55. Since there are much more faint stars than bright ones, this is essentially a normalization over the faint end of the luminosity function. With this normalization, the luminosity function of M55 shows a relative excess of bright stars over the magnitude range extending from the turnoff (at Mj = 3.4) to about 2.5 mag fainter. In the second plot (Figure 2.18) the luminosity functions of the four clusters were normalized so that they have equal number of stars in the magnitude interval 4.0 < Mj < 5.0, that is the luminosity functions were aligned at the bright end. In this case the luminosity function of M55 is lower than the other three by a factor of ~ 1.6 for Mj > 6, in other words M55 shows a deficiency of faint stars relative to M15, M30 and M92. M55 is the most metal-rich of the four clusters whose luminosity functions are shown in Figures 2.17 and 2.18 — [Fe/H] of -1.9 as opposed to [Fe/H] « -2.1 for the other three. Still the difference is small enough to assume that the same mass-luminosity relation applies to all four clusters and therefore they should have luminosity functions with similar shapes. The comparisons between the luminosity functions suggest, however, that M55 either is deficient in low-mass stars or over-abundant in high-mass stars, depending on how the luminosity functions are normalized. One can simply accept that the differences observed in Figure 2.17 and Figure 2.18 (to the extend that they are real) are a consequence of the different star formation conditions in the proto-clusters than became M55, on one hand, and M15, M30 and M92, on the other hand. If all four cluster formed with similar initial mass functions, however, about the only mechanism that could cause an excess of high-mass stars while keeping the faint ends of the luminosity functions in agreement is mass segregation. A relative excess of high-mass stars in the luminosity function of M55 is expected if mass segregation in the observedfieldis stronger than the mass segregation (if any) in the fields of M15, M30 or  Chapter 2.  The Main Sequence of M55  77  Figure 2.18: Comparison of the ground-based J-band luminosity function of M55 with the ilST-based luminosity functions of M15, M30 and M92. The luminosity functions are normalized over the magnitude interval 4.0 < Mi < 5.0. With this normalization, M55 exhibits a relative deficiency of faint (i.e., low-mass) stars.  Chapter  2.  78  The Main Sequence of M55  M92. However, the M55 field is more than two core radii from the cluster centre and the estimated relaxation time in the field makes it unlikely that a significant mass segregation has taken place.  2  Thus, insofar as the ~ 2<r excess of bright stars seen in Figure 2.17 can  be considered significant, one could conclude that it was artificially created by adopting the faint-end normalization instead of aligning the luminosity functions at the bright end. If one adopts the bright-end normalization, then the most straightforward interpretation of Figure 2.18 is that the luminosity function of M55 shows a deficiency of faint stars, produced either by primordial differences in the initial mass function, or by a preferential loss of low-mass stars (A4 < 0.6A4®).  M55 is deep in the disruption zones in the  "vitality diagrams" of Weinberg (1994) and Gnedin & Ostriker (1997), which indicate the likelihood of a cluster being destroyed by tidal and bulge shocks within a Hubble time (~ 10  10  years). As M55 is a very sparse (though massive) cluster and its orbit  takes it within at least ~ 5 kpc from the centre of the Galaxy (its current location), it is quite possible that M55 has lost some portion of its low-mass stellar population via evaporation and tidal and bulge shocks. The small concentration of M55 suggests that an earlier loss of stars may also have occured — large core radius and small concentration are characteristics of clusters that may have experienced an expansion driven by a mass loss via stellar winds and supernova explosions in the first few 10 years. As this mass s  loss causes the tidal radius of the cluster to shrink and the core to expand, low-mass stars are preferentially lost through the tidal boundary of the cluster (Elson et al. 1987, Chernoff 1988). It is plausible, therefore, to suggest that the apparent deficiency of faint stars implied by Figure 2.18 reflects a preferential loss of low-mass stars caused by a combination of evaporation and tidal and bulge shocks. To summarize, if M55 and the other three clusters used in the comparison started 2  Djorgovski (1993) gives ~ 2.5 Gyr for the central relaxation time of M55. The stellar density in the  observed M55 field is about 40 times as small as that in the cluster centre, and relaxation time scales roughly as the reciprocal of the number density (Spitzer 1987).  Chapter 2.  The Main Sequence of M55  79  out with similar luminosity functions, it would be very difficult to find a mechanism that would explain the excess of bright stars in the absence of mass segregation. On the other hand, there is a plausible physical mechanism (loss of stars via evaporation and tidal stripping) that can be used to interpret the deficiency of low-mass stars implied by Figure 2.18.  It is possible, of course, that the differences between the luminosity  functions of M55, on one hand, and M15, M30 and M92, on the other hand, are simply a reflection of primordial differences in the initial mass functions of these clusters.  2.7.3  Mass Function  In order to convert the luminosity functions into a mass function, a mass-luminosity relation is required. I considered four different sources of mass-luminosity relations for low-mass stars, namely Bergbusch & VandenBerg (1992), D'Antona & Mazzitelli (1996), Alexander et al. (1997) and Baraffe et al. (1997). The models of Bergbusch & VandenBerg (1992) do not include V — I colours so they were used only in the conversion of the Vband counts to a mass function. The four mass-luminosity relations are compared in Figure 2.19, where the top panel shows the mass - Mr laws and the bottom panel shows the mass - My laws. The mass-luminosity relation of D'Antona &; Mazzitelli (1996) deviates from the other three and below Q.bM.Q it predicts too high a luminosity at a given mass (or too low a mass at a given absolute magnitude) compared with the other mass-luminosity relations. In Figure 2.20 suitable V,V—I isochrones from the models of D'Antona & Mazzitelli (1996), Alexander et al. (1997) and Baraffe et al. (1997) are overlaid on the colourmagnitude diagram of M55, where the photometry of M55 was shifted by the adopted distance modulus of (m — M)v = 14.02 and reddening of E y - i = 0.17. The choice of the isochrones was made as follows. D'Antona &z Mazzitelli (1996) tabulated models only for Z = 0.0001 and Z = 0.001, which are too metal-poor and too metal-rich, respectively,  Chapter 2.  80  The Main Sequence of M55  1  0.8  0.6  0.4  0.2  \og(Jl/M ) Q  Figure 2.19: Comparison of the mass-luminosity relations in V (bottom) and / (top). The letters mean: A - Alexander et al. (1997); B - Baraffe et al. (1997); D - D'Antona & Mazzitelli (1996); BV: Bergbusch & VandenBerg (1992).  Chapter 2.  The Main Sequence of M55  81  Figure 2.20: The isochrones of Alexander et al. (1997; A), Baraffe et al. (1997; B) and D'Antona & Mazzitelli (1996; D) overlaid on the colour-magnitude diagram of M55. All isochrones are for stars with a heavy metal abundance of Z ~ 0.0006. The models of Baraffe et al. (1997) reproduce well the main sequence of M55 both in shape and location.  Chapter 2.  The Main Sequence of M55  82  for a comparison with M55 (Z ss 0.0005 if [Fe/H] = -1.9 and [a/Fe] = 0.3 are adopted). Therefore I did a crude interpolation to derive an isochrone corresponding to Z = 0.0005, and this is the one shown in Figure 2.20. Both Alexander et al. (1997) and Baraffe et al. (1997), on the other hand, have tabulated Z = 0.0006 isochrones and I used those directly. It is clear that the models of Baraffe et al. (1997) show the best match with the main sequence of M55. Since they are able to predict both the shape and the location of the main sequence correctly, one can assume that they should also be able to predict the right mass-luminosity relation, and therefore I chose to use their models to derive the mass function of M55. It should be noted that all present models of low-mass stars have not been tested rigorously yet, and that at the very low-mass end (AA < 0.15A4©) they all show significant discrepancies with observations.. While this is outside of the mass range explored in this work, it should be kept in mind that until the mass-luminosity relation for Population II stars is determined observationally, the globular-cluster mass functions will remain somewhat uncertain. On the other hand, when comparisons are made between clusters of similar metaUicities, the uncertainties in the mass-luminosity relation are expected to play a minor role. The mass function of M55 derived from the I luminosity function and using the mass-luminosity relation from Baraffe et al. (1997) is shown in Figure 2.21 by fiUed circles connected with a solid line. The mass function derived form the V counts is almost identical and is not shown. For comparison I have also plotted the mass function derived from the V luminosity function using the mass-luminosity relation of Bergbusch & VandenBerg (1992) (dash-dot line) and the mass functions obtained from the I counts using the mass-luminosity relations from D'Antona & MazziteUi (1996) (dotted Hne) and Alexander et al. (1997) (dashed Une). The range of the mass function shapes and slopes in this plot gives an idea of the range of uncertainty in the present-day mass-luminosity  Chapter 2.  83  The Main Sequence of M55  1  0.8  0.6  0.4  0.2  \og(Ji/Ji ) Q  Figure 2.21: The mass function of M55 in the mass range 0.2 <> A4/MQ & 0.8 derived from the /-band luminosity function using the mass-luminosity relation from Baraffe et al. (1997) (filled circles connected by solid lines). The mass function obtained from the V-band luminosity function and the mass-luminosity relation of Bergbusch k VandenBerg is shown by a dash-dot line. Also shown are the mass functions converted from the / counts using the mass-luminosity relations from Alexander et al. (1997) (dashed line) and D'Antona k Mazzitelli (1996) (dotted line).  Chapter 2.  84  The Main Sequence of M55  relations. In Table 2.12 I summarize the the slopes of the mass functions derived from the different models in two important mass ranges: 0.5 £ AA/AAQ  £ 0.8 (nine points) and  AA < 0.4.M© (the last four points). Thefirstmass interval has been used often to search for correlations between the mass spectral index x and other cluster parameters (see, e.g., McClure et al. 1986, Richer et al. 1991 and Djorgovski et al. 1993). Because of the upturn of the mass function below 0.4A4©, its slope in this mass range will determine the contribution of low-mass stars to the total mass of the cluster — it is clear from Eq. 2.3 that any value of x > 1 will produce an infinitely large cluster mass as AA —> 0. While the low-mass cutoff of the mass function is almost certainly not zero, for values of x > 2 very low-mass stars will dominate the cluster mass. Table 2.12: Mass function slopes X  X  Mass-luminosity Relation Bergbusch k VandenBerg (1992) D'Antona k Mazzitelli (1996) Alexander et al. (1997) Baraffe et al. (1997)  0.5 £ .M/M© £ 0 . 8 0.5 ± 0 . 3 0.8 ± 0 . 2 0.6 ± 0 . 3 0.8 ± 0 . 2  M < 1.4 1.2 1.1 0.7  0.4A4© ± 0.2 ± 0.2 ±0.2 ±0.2  One can see that below 0.4A4© the mass-luminosity relation of Baraffe et al. (1997) produces a mass function with the shallowest slope (x = 0.7 ± 0.2). The average slope in that mass range is x ~ 1.1, however the value obtained from Baraffe et al. (1997) should be given preference since their models are the only ones that match the main sequence of M55. In the range 0.5 £ AA/AA  @  £ 0.8 the mass functions predicted by the different  models agree fairly well (as seen also from the errors in the slopes), therefore one can consider x = 0.7 ± 0.2 to be a representative value for the slope in that mass range. It may appear then that the whole mass function should be well represented by a power law with x PS 0.7, but this is not so: excluding the highest-mass point, a linear fit to the mass  Chapter 2.  85  The Main Sequence of M55  function in the interval 0.2 <> A4/A4@ < 0.8 has % = 15. Since the probability that a 2  value of x this large occured by chance is only ~ 10 , it appears unlikely that the mass 2  -4  function of M55 can be represented by a single power law of the form $(A4)  A4~( \  oc  1+x  If the goodness-of-fit estimate is ignored and the whole mass range 0.2 < M.JA4®  <  0.8  is fitted by a single power law, the best value of the mass spectral index over that interval is x = 0.4 ± 0.1. The only other estimate of the slope of the mass function in M55 comes from Zaggia et al (1997), who obtained x = 0.7 ± 0 . 2 in the interval 0.6 < M/M  @  < 0.8  from observations of stars between two and six core radii (2r < r < r ) from the cluster c  t  centre. The excellent agreement of the two high-mass estimates (I obtained x = 0.7 ± 0 . 2 for 0.5 ^ A4/MQ  ^ 0.8) indicates that the mass function of M55 beyond two core radii  has not been modified by mass segregation. If the mass function were affected by mass segregation, the x derived in this work (from a field at a distance of 2.2 core radii) should have been significantly smaller than the value of Zaggia et al. (1997), whose value is based on a much wider radial coverage extending from two core radii to the tidal radius of M55. Does the value x — 0.7 for the low-mass stars in M55 indicate a depleted low-mass population? One can compare this value with the results from the HST-hased. luminosity functions of M15, M30, M92 and NGC 6397 (Piotto et al. 1997). The first three clusters were discussed earlier, but now in Figure 2.22 I have added the luminosity function of NGC 6397. Its luminosity function is even lower than that of M55, and Piotto et al. (1997) have argued that this cluster has also experienced loss of low-mass stars, apparently stronger than in M55. Looking at Figure 2.22, one would expect M55's value of the mass spectral index x for low-mass stars to be between those of NGC 6397 and the other three clusters and indeed, Piotto et al. (1997) derive x PS 0 below 0.4A4© for NGC 6397, and a: PS 1 for M15, M30 and M92.  Chapter 2.  The Main Sequence of M55  86  Figure 2.22: Same as Figure 2.18, but now with the luminosity function of N G C 6397 added.  Chapter 2.  The Main Sequence of M55  87  It should be pointed out again that all luminosity functions discussed here were derived from observations in a single field in each cluster and there is always the possibility that mass segregation has changed the relative numbers of faint and bright stars. The effects of mass segregation are judged best by means of a detailed dynamical modelling of multi-field observations, which are not currently available for M55. However, in all clusters the programfieldswere well beyond the half-mass radius and it is unlikely, then, that mass segregation has affected the luminosity functions to the degree seen in Figure 2.22. The mass and luminosity functions of M55, together with the earlier results for NGC 6397 (Piotto et al. 1997) seem to provide the first evidence that the low-mass stellar populations of massive globular clusters can be modified by evaporation, tidal stripping and bulge shocks. It is worth emphasizing that the evidence comes mostly from the luminosity functions of these two clusters and not only from the much more uncertain mass functions. Why is NGC 6397 more deficient in faint stars than M55? If the clusters had different star formation histories, metaUicity did not play a role as both have identical metal abundances ([Fe/H] = —1.9). Their orbits are unknown, although they have very different radial velocities (+20 km/s for NGC 6397 and +175 km/s for M55, Pryor & Meylan 1993). As noted by Piotto et al. (1997), NGC 6397's radial velocity could mean that the cluster has been around the bulge and the disk of the Milky Way more often and therefore has experienced more tidal stripping. Also, NGC 6397 and M55 must have had a very different dynamical history, as NGC 6397 is a core-coUapsed cluster (Trager et al. 1993) whereas M55 is at the opposite end of the central concentration scale. Since core-coUapsed clusters experience expansion of the envelope after the core coUapse (Elson et al. 1987), it is possible that the low-mass stars in the envelope of NGC 6397 have become easier to strip away during the passages through the Galactic disk or near the bulge.  Chapter 3  The Evolved Populations of M 5 5  This chapter presents the analysis of the four-colour UBVI photometry of a field in the core of M55. The primary goal of the observations in this field was to study the evolved stars in M55 and to search for objects with peculiar colours which are sometimes seen in the central regions of globular clusters. Such objects are thought to form during close stellar encounters that should be much more frequent in the denser clusters cores. There is some evidence now (Fusi Pecci et al. 1992, Buonanno et al. 1997) supporting the suggestion of Buonanno et al. (1985) that the higher stellar densities introduce modifications in the core stellar populations. Because of the low central concentration of M55, it is possible to do precise photometry at the very centre of the cluster and the stellar populations in its core can be easily resolved and studied from the ground. The observations described here are the first CCD observations of the evolved populations in M55 that cover the whole magnitude range from below the main-sequence turnoff to the tip of the red giant branch. As discussed in Chapter 1, the luminosity functions of the red giant branch and the subgiant branch, as well as the distribution of stars on the horizontal branch and the ratios of the number of stars on the various branches can provide important tests of stellar evolution theory that are not possible by fitting isochrones to the main sequence and the turnoff of the cluster (Paczyriski 1984, Renzini &; Fusi Pecci 1988). The luminosity function above the turnoff is directly related to the rate of stellar evolution, which in its turn is influenced by the stellar structure developed during the preceding evolutionary stages (Renzini & Fusi Pecci 1988). Therefore  88  Chapter 3.  The Evolved Populations of M55  89  the luminosity function of evolved stars is particularly sensitive to the processes occuring in the deep stellar interiors and several studies in the last decade have confronted the observed and predicted luminosity functions with the purpose of testing our understanding of the structure and evolution of globular cluster stars. Stetson (1991) compared the combined luminosity function for three metal-poor clusters (M68, NGC 6397 and M92) with theoretical luminosity functions similar to those published later by Bergbusch & VandenBerg (1992). The comparison was prompted by a suggestion by Faulkner & Swenson (1988) (see also Faulkner & Swenson 1993) that an accumulation of the hypothetical weakly-interacting massive particles (WIMPs) in the cores of old stars could change their evolution. The calculations predicted that stars with WIMPs in their cores would leave the main sequence earlier, with some hydrogen still left in the core. This unconsumed hydrogen will lengthen the stay of the stars on the subgiant branch as they burn it off before moving on to the red giant branch, and longer duration of the subgiant phase means more stars in the corresponding bins of the luminosity function.  Curi-  ously enough, Stetson (1991) found that for the three clusters in his sample there were about 50% more stars on the subgiant branch than predicted by the "canonical" theoretical luminosity functions. Later Bolte (1994) derived the luminosity function of another metal-poor globular cluster, M30 (NGC 7099), from the tip of the giant branch to about 2.5 magnitudes below the turnoff. He concluded that, depending on the normalization used, there was either an excess of subgiant and red giant stars (at the ~ 5<r level), or theory predicted too many stars (about 20% more) below the main-sequence turnoff. The luminosity function of the evolved stars in M30 was studied also by Bergbusch (1996), who reached the same conclusion — that the relative number of subgiant and red giant stars compared to the number of turnoff stars was higher than predicted by the theoretical luminosity functions. On the other hand, the luminosity functions of two more metal-rich clusters, NGC 288 (Bergbusch 1993) and M5 (Sandquist et al 1996), showed  Chapter 3. The Evolved Populations of M55  90  no significant discrepancies between theory and observations. It should be noted that while the luminosity function of NGC 288 was somewhat poorly determined because of the small number of red giants in that cluster, the luminosity function of Sandquist et al. (1996) for M5 was based on more than 20,000 stars and therefore their conclusions should be very robust. A hypothesis emerging from all these studies is that metal-poor ([Fe/H] ~ —2) globular clusters show an excess of subgiant and red giant stars compared to the number of turnoff stars, whereas more metal-rich clusters ([Fe/H] ~ —1.3 for NGC 288 and M5) do not exhibit such anomalies. It is therefore one of the main goals of this chapter to derive a luminosity function from below the main-sequence turnoff to the upper red giant branch of M55 and compare it to theoretical predictions.  3.1  Observations and Preprocessing  The observations analyzed here were made by Greg Fahlman and Ian Thompson in 1992 August with the Tektronix 2 CCD at the Cassegrain focus of the 2.5-m du Pont Telescope of the Las Campanas Observatory. Except for the filters used, the instrument setup was exactly the same as that described in detail in Chapter 2. A total of 30 images through Johnson UBV and Cousins / filters were obtained of a program field centred at the cluster core. The images were taken under excellent seeing conditions — on most frames the full width at half maximum of the stellar profile is ~ 0''8 and there are a few frames with full width at half maximum of 0"65. The exposure information for the core field frames is summarized in Table 3.1, and Figure 3.1 shows the position of the program field overlaid on a Digitized Sky Survey image of M55. The preliminary processing of the images followed the steps explained in Chapter 2 — first the overscan columns in all frames were fitted by a polynomial and subtracted from  Chapter 3.  The Evolved Populations  of M55  91  Figure 3.1: The location of the core field relative to M55. The side of the chart is approximately 28', the size of the field is 4' x 4'. North is up and East is to the left. The digitized image of M55 is © 1993-7 by the Anglo-Australian Observatory Board. All Rights Reserved.  Chapter 3. The Evolved Populations of M55  92  Table 3.1: Core field exposure information UT Date Aug. Aug. Aug. Aug. Aug.  Filter  24 24 24 24 24  U U B V I  Exposure 00  2x120 2x300 5x40 11x10 10x5  Airmass  Seeing  1.1 1.0 1.0 1.0 1.0  (") 1.1 1.0 0.8 0.8 0.7  the frame, and then the images wereflat-fieldedusing dome flats obtained in each of the four filters.  3.2  Photometry  3.2.1  Instrumental Magnitudes and Aperture Corrections  The derivation of the instrumental magnitudes was carried out in a way almost identical to that described in Chapter 2, by using Peter Stetson's suite of programs ALLSTAR - ALLFRAME.  DAOPHOT  -  Since the central field is much more crowded, however, it was  more difficult to select isolated stars for obtaining the point-spread function. As a result, only about 35 - 40 stars on each frame were used for that purpose but that number was still large enough to ensure the quality of the point-spread function. Although the exposure times were only 5 s in I, several of the brightest red giants were slightly saturated on even the worst-seeing I frames. None of them was saturated on the V, B or U frames, however. As only the few central pixels were saturated in the best-seeing I frames, the magnitudes returned by A L L F R A M E should be close to the true ones. Four of these stars have \ giants, but not by much.  2  values that are somewhat higher than the fainter red  Chapter  3.  The Evolved  Populations  93  of M55  The aperture corrections were derived in the same way as described in Chapter 2. In each program image I selected the brightest and most isolated ~ 30 stars among those used for obtaining the point-spread function for that frame. All other stars were subtracted and concentric aperture photometry was obtained for the selected stars. These aperture photometry results were then supplied to  DAOGROW  and the returned aperture  corrections were averaged to obtain the mean aperture correction for the given frame.  3.2.2  Transformation to the Standard System  Similar to the deep-field photometry, the transformation of the instrumental magnitudes to the standard UBVIc  system was performed in two steps. First, I used observations  of faint standard stars selected from the Ust of Landolt (1992) to derive the zero-points, extinction coefficients and colour terms necessary to transform to the standard photometric system a sample of 36 relatively isolated stars in the program field. On the second step, these 36 stars were used as local secondary standards to caUbrate aU other program stars. The reasons for this two-step process were explained in Chapter 2 and are also discussed in Stetson &; Harris (1988). On the night of 1992 August 24, a total of 15 standard stars in four fields were observed. Two of these stars, SA110-362 and T Phe D were not used in the caUbrations because of their large residuals in the fit (see the caUbration section of Chapter 2). AU remaining stars were used for the V and / caUbrations, but only six were observed in U and B. It is possible therefore that the U and B transformations are more uncertain than the V and / caUbrations. The Ust of standards stars that were observed on the night of 1992 August 24 and their magnitudes and colour indices are given in Table 3.2. The first step in the calibration procedure was to fit equations of the form u  =  U + a + a (X 0  1  -1.25) + a (U-B)  + a (U-B)  2  2  3  )  +  aT 4  Chapter 3. The Evolved Populations of M55  94  Table 3.2: List of the observed Landolt standards Star SA110 229 SA110 230 SA110 232 SA110 233 SA110 361 SA110 362 SA110 364 SA110 365 MarkA MarkA 1 MarkA 2 MarkA 3 T Phe A T Phe C T Phe D  V 13.649 14.281 12.516 12.771 12.425 15.693 13.615 13.470 13.258 15.911 14.540 14.818 14.651 14.376 13.118  o0 0031 0 0031 0 0032 0 0028 0 0022 v  0 0021 0 0027 0 0019 0 0040 0 0028 0 0024 0 0028 0 0022 0 0033  B -V 1 910 1 084 0 729 1 281 0 632 0 632 1 133 2 261 -0 242 0 609 0 666 0 938 0 793 -0 298 1 551  0~B-V  0.0091 0.0050 0.0028 0.0034 0.0022 0.0067 0.0091 0.0018 0.0090 0.0031 0.0034 0.0046 0.0024 0.0030  U-B 1.391 0.728 0.147 0.812 0.035 0.035 1.095 1.895 -1.162 -0.014 0.096 0.651 0.380 -1.217 1.871  V-I 2.356 1.218 0.889 1.593 0.709 1.803 1.281 2.631 -0.241 0.740 0.751 1.098 0.841 -0.360 1.663  0~U-B  0 0225 0 0116 0 0045 0 0070 0 0029 0 0088 0 0313 0 0038 0 0136 0 0046 0 0105 0 0071 0 0043 0 0118  b = B + bo + b^X -1.25) + b (B-V) + b (X-1.25)(B-V) 2  =  v  i  V + c + (X-1.25) 0  Cl  = / + do + d (Xx  +c (V-1)+ 2  1.25)+ d (V-1)+ 2  3  0~v-l 0 0026 0 0050 0 0025 0 0021 0 0029 0 0021 0 0034 0 0048 0 0148 0 0059 0 0045 0 0032 0 0149 0 0030  +bT 4  cT 3  dT 3  to the photometry of the standard stars. In these equations U, B, V and I are the standard magnitudes and u, b, v and i are the instrumental magnitudes of the standard stars; X is the airmass and T is the time of mid-exposure relative to the effective midnight. The terms for time dependence, the quadratic term for the U — B colour index and the second-order extinction coefficient for B—V were added after the preliminaryfitsshowed clear trends in the respective residuals. No other trends in the residuals were noticeable and no additional terms were used in the transformation equations (in particular, no quadratic term for B — V was necessary). For consistency, I used the values of the extinction coefficients c\ and di, as well as the colour terms c and d as found from the 2  2  Chapter 3.  The Evolved  Populations  of M55  95  above equations, not the average values determined earlier in Chapter 2 (although those values were nearly the same). The final transformations were applied to all observations of the Landolt's standards on that night, as well as to the 36 local standards in the core field. After that both sets were combined in a larger list and the zero points of the transformations were again solved for, resulting in a list of homogeneous magnitudes and colour indices for the Landolt's standards and for the 36 local standards in the program field. The local standards are identified in Figure 3.2 and their magnitudes and colour indices on the system defined by Landolt's (1992) standards are listed in Table 3.3. Figure 3.3 shows the differences between my photometry of the Landolt standards and their published values. The removal of T Phe D reduced the range of the B — V indices of the standard stars to B — V ~ 1.0 for the reddest of them. On the other hand, the six reddest red giants in the corefieldhave B—V tt 1.4 (see the next section) and so their B—V colours could be more uncertain. These six red giants aside, the B—V colours of all cluster stars are below B — V ~ 1.1, so the calibration covers the whole range in B—V for the stars in this field. The same problem exists for the U—B colour indices as well, in the sense that the reddest standard star has U — B tt 0.7. This will again affect only the six reddest giants, which have U—B tt 1.2; the U—B indices of all other cluster stars fall within the range of the U — B calibrations. Unlike the photometry in Chapter 2, I did not attempt detailed comparison with the earlier photographic photometry of Lee (1977), whose field overlaps with the core field. The comparison in Chapter 2 showed significant differences between the two data sets and little new information would be gained from further comparison. The CCD photometry of Schade et al. (1988) and Alcaino et al. (1992), on the other hand, was carried out in three fields to the north of M55 that do not overlap with the core field discussed here, and so a direct comparison with their photometry is not possible.  Chapter 3.  The Evolved Populations  96  of M55  ® 33  13 s r  V'V © * - .  14  22 •0 -25®-  12  17  . O •  9  8  *  •.-".<$>.'  •  31 ' **  35  !>  •15/ : . • '®> 4  36©  27  •  :  . 24 .  « ® 028-.  ® 34  ir .16  Figure 3.2: Finder chart for the local standards in the core field. North is up and west is to the left. Star numbers increase with right ascension. The size of field is 4' x 4'.  Chapter 3.  The Evolved Populations  97  of M55  Table 3.3: Local standards in M55 St ar 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36  V  <7  14.725 14.790 13.354 12.665 13.065 13.678 13.084 13.659 14.054 13.018 13.569 12.695 13.040 14.000 13.173 12.744 12.529 13.314 13.142 14.742 12.570 14.679 13.817 12.551 13.511 15.152 12.238 14.947 14.633 13.148 14.119 13.410 12.967 13.375 13.286 13.763  0.0035 0.0029 0.0022 0.0048 0.0021 0.0023 0.0018 0.0028 0.0036 0.0023 0.0021 0.0033 0.0027 0.0047 0.0050 0.0048 0.0044 0.0019 0.0027 0.0035 0.0037 0.0040 0.0025 0.0044 0.0018 0.0032 0.0018 0.0024 0.0035 0.0022 0.0018 0.0030 0.0024 0.0036 0.0030 0.0023  V  B—V 0.081 0.130 0.964 1.071 1.004 0.759 0.999 0.798 0.871 1.015 0.838 1.054 0.996 0.675 0.977 1.057 0.973 0.865 0.985 0.115 1.108 0.093 0.740 1.041 0.841 0.050 1.191 0.078 0.042 0.914 0.654 0.950 1.012 0.961 0.961 0.731  a -v 0.0049 0.0052 0.0039 0.0071 0.0042 0.0033 0.0052 0.0045 0.0057 0.0051 0.0045 0.0054 0.0046 0.0056 0.0068 0.0067 0.0077 0.0038 0.0038 0.0057 0.0059 0.0069 0.0048 0.0064 0.0036 0.0054 0.0065 0.0061 0.0064 0.0044 0.0047 0.0056 0.0042 0.0054 0.0052 0.0044 B  U—B -0.052 0.050 0.431 0.663 0.499 0.176 0.532 0.200 0.283 0.536 0.224 0.616 0.483 0.069 0.485 0.647 0.511 0.335 0.526 0.093 0.735 0.039 0.168 0.589 0.267 -0.124 0.860 -0.084 -0.138 0.394 0.079 0.449 0.593 0.474 0.462 0.186  0 0043 0 0052 0 0038 0 0067 0 0052 0 0033 0 0054 0 0041 0 0055 0 0051 0 0045 0 0059 0 0044 0 0036 0 0055 0 0050 0 0094 0 0039 0 0037 0 0071 0 0062 0 0060 0 0045 0 0053 0 0051 0 0049 0 0086 0 0061 0 0062 0 0097 0 0058 0 0058 0 0055 0 0064 0 0080 0 0049  V—I 0.166 0.227 1.164 1.267 1.209 1.006 1.202 1.058 1.117 1.210 1.067 1.263 1.197 0.935 1.208 1.267 1.195 1.125 1.188 0.223 1.288 0.185 1.002 1.257 1.082 0.123 1.355 0.139 0.140 1.138 0.931 1.166 1.227 1.174 1.188 0.992  ay-i 0.0047 0.0043 0.0027 0.0055 0.0026 0.0028 0.0024 0.0034 0.0046 0.0026 0.0026 0.0036 0.0032 0.0050 0.0054 0.0056 0.0046 0.0026 0.0031 0.0047 0.0043 0.0050 0.0029 0.0052 0.0024 0.0052 0.0033 0.0043 0.0054 0.0031 0.0026 0.0036 0.0030 0.0040 0.0036 0.0034  0.1 0.05  ,  i  i  98  of M55  i  i  i  i  |  i  i-  i  i  |  i  111111111  The Evolved Populations  -  Chapter 3.  E-  0 -0.05  • •  ',  -0.1  ,  •  1  ,  13  12  .  .  1i  •  1  - .  i  14  V 0.1  I  *  :  i  1  i  1  i  :  16  15  (Landolt) i  i  |  .  ,  ,  i  |  .  .  I  1  1  1  1  0.05  cq. <  S  0  *  -0.05 =  -0.1  ,  ,  i  ,  I  ,  ,  :  ,  -0.5  B-V 0.1  1  1  1  1  1  1.5  0.5  .  1  1  1  1  1  1  1  1  (Landolt) |  1  1  1  1  !  |  I  1  1  1 1.  1  1  1  1  I  0.05  -0.05 -  -0.1  1  1  1  1  1  1  1  1  1 ,  1  .  I  1  1  1  1  1  1  1  1  1  1  1  1  i ?  1  1  1  1  0.5  0  U-B 0.1  1  -0.5  •1.5  *  i  I I  xs  0  (Landolt) 1  1  1  I  1 1 1 1 1 1 1 1 1  • I  1  I  0 -0.05 -0.1  T  E—i =,  ,  1 ,  ,  ,  ,  1  •  ,  ,  ,  1  V-I  n  0.05  ,  1  ,  ,  ,  ,  :  2  (Landolt)  Figure 3.3: Differences between my photometry of Landolt (1992) standards and the published values. A l l differences are in the sense (this work — Landolt's).  Chapter 3.  3.2.3  The Evolved Populations  99  of M55  Quality of the UBVI Photometry: Errors and %  2  Before presenting the various colour-magnitude diagrams that can be made from the UBVI  data, I would like to discuss briefly the photometric results and the selection  criteria that I used to separate the "good" from the "bad" data. For each star,  ALLFRAME  returns three parameters that can be used to judge the quality of the photometry for that star: the magnitude standard error, which reflects the internal and external errors (estimated from the readout noise and photon statistics, and the frame-to-frame scatter, respectively), the \  2  goodness-of-fit measure (which is determined from the profile fit),  and the number of iterations that were required for the fit to converge. A large number of required iterations is an indication that the profile of the object is difficult to fit with the model point-spread function (Stetson, private communication) and this quantity can be used to reject non-stellar or severely blended objects at faint magnitudes, where selection based on % is not so effective. After the instrumental magnitudes are transformed to 2  the standard system, it is convenient to express the photometry for each star in terms of one magnitude and three colour indices, which I selected to be V, B — V, U — B and V—I.  Their errors can be written as, for example, cr _ B  v  = a -\-o- , B  v  where a  B  and a \ are  the variations of the corresponding magnitudes. Identical expressions can be written for the other colour indices as well and it is clear that the individual magnitudes and their errors can be easily found from the colour indices and their errors. In addition, for each star it is known on how many frames it was found, and this could be useful for deciding the reality of faint stars. In Figure 3.4 I show a plot of the magnitude and colour errors as a function of V magnitude. There are 10313 stars in each panel — all the stars that have been found and measured on at least one frame in each filter. In addition to the "traditional" colour indices B — V, U — B and V — I, the bottom panel of this figure shows the errors in the  Chapter 3.  The Evolved Populations  of M55  Chapter 3.  The Evolved Populations  101  of M55  colour index B — I = (B — V) + (V — I) whose usefulness will be discussed later. One can see that, first, the photometry is of very good quality, a result of the excellent seeing and sampling of the stellar profile allowed by the use of the Cassegrain focus. Second, fainter than V J> 17.5, some stars have magnitude and colour errors that are too large for their brightness — while the median V error at V = 19 is cry = 0.019, there are many stars that have cry <; 0.05 at that magnitude. Essentially all stars with abnormally large photometric errors are fainter than V « 17.5, the magnitude at which the number of stars as a function of magnitude starts to increase rapidly. There is little doubt that the photometry of such stars is compromised by crowding and I decided to remove them from the sample. One could speculate that stars with anomalous errors will also have large x values 2  and that it should be possible to clean the star sample by restricting it only to stars with small values of x - Unfortunately, this is not so. The top panel of Figure 3.5 shows 2  a plot of x versus V magnitude for all 10313 stars. There are some peculiarities in 2  the distribution of x with magnitude, for example the slight upward trend both fainter 2  and brighter than V ~ 17, as well as the curved shape of the plot for 14 £ V £ 17. Nevertheless, most stars have % ~ 1 for the whole magnitude range, which is what is 2  expected if the ALLFRAME error parameters are chosen well and if the adopted gain and signal-to-noise ratio are correct. In order to discard the stars with too high a % (which 2  means bad profile fit), I adopted a cutoff value of % = 1.4 and accepted all stars with % 2  below that value.  1  2  However, that did not change much the appearance of the error plots  in Figure 3.4. While many stars with large errors around V ~ 18.5 (corresponding to the X "hump" in the top panel of Figure 3.5) disappeared, most of them remained, especially 2  fainter than V ~ 18.5. Therefore I decided to apply a selection criterion that rejects all The value 1.4 is somewhat arbitrary. It was chosen so that the bulk of faint stars and nearly all bright stars were below the cutoff value in Figure 3.5. This also applies to the 2<7 ed cutoffs discussed later. x  m  Chapter 3.  The Evolved  Populations  H  h  H  102  of M55  H  !-  h  H  h  H  h  X  J  L_  20  14  12  7  Figure 3.5: Plot of % versus F magnitude for all 10313 stars (top) and for the stars that have errors less than 2<r simultaneously in V and V — I (bottom). Note the reduced number of stars with \ > 1.4 in the lower panel. If the A L L F R A M E error parameters are chosen well, and if the adopted gain and signal-to-noise ratio are correct, % should be around one (solid line). The dashed line marks the adopted cutoff of % < 1-4. 2  med  2  2  2  Chapter 3.  The Evolved Populations  103  of M55  stars having errors that are more than twice the median error at a given magnitude. For that purpose I divided the error plots in Figure 3.4 into 0.5-mag bins, found the median error <x d in each bin and then fitted a relation of the form me  <r  =  med  a^-™?'**  to the median errors for each magnitude or colour index.  The values of a\ and a  2  were determined from the fit, and the power index (3 was 2.5 for <xy and 2.0 for the median errors of the colour indices. In Figure 3.6, the adopted dependence of 2<r d on me  V magnitude is shown by a dashed line in each panel. A l l stars having errors below the dashed lines and % < 1.4 were accepted as "good" stars; the remaining stars were 2  discarded. The error selection criterion was relaxed for the brighter stars (V < 17.0), which are much less affected by crowding. The value of O~B-I of B-V,  is not directly available since the caUbrations were done in terms  U — B and V — I. The value of B-I  but the exact relation between  O~B-V  hand, is unknown. I adopted o~ -i B  a_ B  I  =  is obtained by adding B-V  and <7y_j, on one hand, and  and  V-I,  on the other  O~B-I,  B-v + °v-i ~~ ^ vi which is obtained directly from  a  a  = a + aj. The exact way of calculating a -i is not that crucial, however, since B  B  the rejection decision is based on <x i and not on the actual value of me<  The lower panel of Figure 3.5 shows a plot of \  2  &B-I-  versus V magnitude for aU 9401  stars that passed the error selection rule simultaneously in V and V — I (the results are similar if other colour indices are used). One can see that now the number of stars with x > 1-4 has been substantially reduced and the ~ 250 stars that remain can be 2  eUminated using the \  2  < 1-4 cutoff.  As iUustrated in Figure 3.7, this combination  of c-cUpping and % rejection appears to work reasonably well in discarding stars with 2  lower-quaUty photometry. The top panel shows the V, V — I colour-magnitude diagram for all 10313 stars; no restrictions have been imposed on ay, <rv-i or \''• The lower panel  Chapter 3.  The Evolved Populations  0.3  104  of M55  ~i  1 r  0.2 0.1 0 0.3  H  1 1  H  1— 1 1 h  h  -+—H  0.2 0.1 0 0.3  0.2 0.1 0 0.3  0.2 0.1 0 0.3  0.2 0.1 0 _I  12  14  i  i_  16  18  20  V  Figure 3.6: Magnitude and colour errors as a function of V magnitude. The dashed lines mark the adopted rejection criterion for each panel.  Chapter 3.  The Evolved  Populations  -0.5  0  105  of M55  0.5  1.5  1  2  V-I Figure 3.7: Illustration of the effect of error and % restrictions on the appearance of the colour-magnitude diagram. Upper panel - no restrictions applied; lower panel - only stars with % < 1.4 and cr < 2<j in V and V — I are plotted. 2  2  med  Chapter 3.  The Evolved Populations  106  of M55  shows the same colour-magnitude diagram but only stars with x < 1-4 and magnitude 2  and colour errors less than 2cr d are plotted. Similar results are achieved for the other me  colour indices.  3.3  The Color-Magnitude Diagram of Evolved Stars  With photometry in four filters it is possible to construct several different colour-magnitude diagrams, depending on what colour index was chosen for the abscissa and what magnitude for the ordinate. The two most commonly used colour indices in globular cluster studies are B—V and V—I, and in Figures 3.8 and 3.9 I have shown the V, B—V and V, V — I colour-magnitude diagrams, respectively, for the central field of M55. In these diagrams I have plotted only the stars that pass the cr < 2<7 a test simultaneously me  in magnitude and colour (B — V or V — I), and have x < 1.4. The only exception was 2  made for four of the brightest red giants (shown by star symbols in Figure 3.9), which have x ~ 1-6 because they are saturated in the I frames (but their magnitudes and 2  colours in the V, B—V diagram are unaffected). For comparison, Figure 3.10 shows the  V, B — I colour-magnitude diagram, with the same restrictions on photometric errors and x applied. The red giant branch and the main sequence in this plot are tighter 2  and exhibit less scatter, especially when compared to the V, B — V diagram. This is not surprising, as the wide wavelength coverage makes the B—I colour index about four times more sensitive to variations in effective temperature compared to the V — I and B — V indices (Stetson 1993a). In terms of the colour-magnitude diagram morphology, it means that two sequences with a fixed effective temperature difference (the physical quantity measured by the colour index) will be separated easier and more accurately in B — I than in the other two colours (compare the separation of the asymptotic and red giant branches in the V, V — I and V, B — I diagrams). I have chosen, therefore, the V, B — I  Figure 3.8: V, B—V colour-magnitude diagram for all 9096 stars having errors less than 2<r both in V and B-V, and % < 1.4. The blue and red edges of the instability strip are shown by dotted lines. 2  med  Chapter 3.  The Evolved Populations  i  1  108  of M55  1  1  1  1  1  1  1  1  1  1  1  UJ  I  [  1  i  i  i  1  1  1  1  r  M55  9148  stars . ' sa  I  I  — 0.5  I  0.5  0  E  1^ 1  I  1.5  1 2  V-I  Figure 3.9: V, V — I colour-magnitude diagram for all 9148 stars having errors less than 2cr both i n V and V-I, and \ < 1-4. The four red giants w i t h % > 1.4 are shown by star symbols. The blue and red edges of the instability strip are shown by dotted lines. 2  med  2  Chapter 3.  The Evolved Populations  T  r  1  1  1  1  i  109  of M55  1  1  1  1  1  1  1  i  i  >  i  |  i  r  #  M55  9137 stars  . ..  B-I  Figure 3.10: V, B-I colour-magnitude diagram for all 9137 stars having errors less than 2cr both in V and B-I, and % < 1.4. The four red giants with % > 1-4 are shown by star symbols, the known RR Lyr stars are marked by open triangles and the suspected RR Lyr variables are shown by plus signs. The blue and red edges of the instability strip are shown by dotted lines. 2  med  2  Chapter 3.  The Evolved Populations  of M55  110  colour-magnitude diagram as the primary tool for studying the evolved populations in M55. None of the colour-magnitude diagrams shows the presence of objects with unusual colours which are sometimes found one to three magnitudes above the horizontal branch and to the blue of the red giant branch. The origin of some of these stars, especially in dense clusters, is probably connected to the frequent encounters and envelope stripping that may occur in dense cluster cores. Others are most likely stars evolving from the extreme blue horizontal branch to the asymptotic giant branch or from the tip of the asymptotic giant branch to the white dwarf region (see, e.g., Saviane et al. 1998). There are several very blue (but faint) stars in the Figure 3.10 (around B—I  0.2 and V fa 20)  whose origin is uncertain. The most interesting one is the single star at B—I — —0.5 and V = 18.6 which may be a central star of a planetary nebula on its way to the white dwarf sequence. Such stars are extremely rare because the timescale of the planetary nebula white dwarf transition is very short. It appears to be a real object, as it is blue on all colour-magnitude diagrams (including those in U), and in all frames that I examined it on it looks like a normal star, without cosmetic defects or bright stars around it. The nature of this object could be confirmed from spectroscopy and if it is really on its way to becoming a white dwarf, it could be useful for matching the theoretical tracks of very young white dwarfs (of which there are so few) to observations. It is possible, of course, that this is a very blue field star, although such stars must be a rarity in the field as well.  3.3.1  Fiducial Sequences  Different algorithms were used in finding the fiducial lines for the different sequences in the colour-magnitude diagrams. Since blending of stellar images produces a noticeable redward bias in the richly-populated main sequence and subgiant branch (see, e.g., the  Chapter 3.  The Evolved  Populations  of M55  111  lower main sequence in Figure 3.10), the fiducial points for those sequence were determined by finding the mode of the colour distribution in 0.2 - 0.5 mag-wide magnitude bins. For the subgiant branch — which is almost horizontal in all colour-magnitude diagrams — I found the mode of the magnitude distribution in colour bins. For the red giant branch, asymptotic giant branch and the horizontal branch, I simply used the weighted means. Some single points around the tip of the giant branch and on the asymptotic giant branch were adopted as fiducial points because of the scarcity of stars on those branches. The fiducial points for all three colour-magnitude diagrams are listed in Table 3.4 (the red giant branch, subgiant branch and main sequence), Table 3.5 (the asymptotic giant branch) and Table 3.6 (the horizontal branch). Since stars brighter than V = 12.2 are saturated in all I images, I did not list any fiducial points brighter than that in the V—I and B — I colour-magnitude diagrams. Only the V, B — V fiducial points go all the way to the tip of the red giant branch.  3.4  The Horizontal Branch of M 5 5  M55 has a prominent blue horizontal branch with an extended blue tail reaching more than two magnitudes below the level of the horizontal branch. Such tails are occupied by the horizontal-branch stars with the lowest masses (Dorman 1992) and are not unusual in clusters with blue horizontal branches (see, e.g., Borissova et al. 1997). As low-mass horizontal-branch stars evolve rapidly up and to the right in the colour-magnitude diagram, the horizontal branch has a somewhat fuzzy appearance caused by the stars that have already moved up. Nevertheless, one can clearly see a well-defined lower envelope which is identified with the zero-age horizontal branch, the place where stars settle after the helium flash and start burning helium in their cores. In all three colour-magnitude diagrams I have shown the approximate locations of the  Chapter 3.  The Evolved Populations  of M55  112  Table 3.4: Fiducial points for the giant branch, subgiant branch and the main sequence V  11.468 11.655 12.224 12.656 13.095 13.391 13.589 13.890 14.240 14.490 15.001 15.228 15.515 15.738 15.978 16.206 16.480 16.748 16.904 17.038 17.148 17.262 17.334 17.453 17.587 17.762 18.016 18.283 18.449 18.659 19.005 19.323 19.520 19.866  B-V  1.526 1.393 1.189 1.079 0.988 0.947 0.922 0.889 0.857 0.834 0.800 0.783 0.762 0.751 0.739 0.727 0.712 0.700 0.692 0.679 0.667 0.637 0.610 0.555 0.522 0.496 0.488 0.495 0.503 0.520 0.545 0.577 0.604 0.651  V  12.223 13.095 13.489 13.951 14.182 14.460 14.769 15.029 15.315 15.640 16.076 16.353 16.652 16.927 17.189 17.303 17.418 17.563 17.852 18.149 18.451 18.748 19.050 19.346 19.643 19.892  V-I  1.358 1.199 1.153 1.104 1.087 1.065 1.042 1.024 1.004 0.987 0.964 0.952 0.934 0.920 0.893 0.860 0.801 0.750 0.710 0.702 0.718 0.741 0.764 0.797 0.832 0.865  V  12.224 12.637 13.180 13.626 14.020 14.409 14.878 15.198 15.646 15.994 16.412 16.743 17.038 17.242 17.356 17.456 17.562 17.655 17.761 18.007 18.304 18.555 18.859 19.148 19.549 19.848  B-I  2.547 2.356 2.178 2.058 1.980 1.915 1.846 1.804 1.746 1.708 1.670 1.634 1.594 1.525 1.433 1.352 1.287 1.250 1.217 1.187 1.200 1.224 1.269 1.306 1.398 1.491  Chapter 3.  The Evolved Populations  113  of M55  Table 3.5: Fiducial points for the asymptotic giant branch V  12.524 13.161 13.319 13.587 13.770 14.060  B-V  1.052 0.903 0.863 0.817 0.745 0.668  V  12.524 13.161 13.319 13.587 13.770 14.060 14.359  V-I  1.245 1.144 1.123 1.073 1.001 0.932 0.830  V  12.524 13.240 13.587 13.770 14.160  B-I  2.297 2.016 1.890 1.746 1.591  Table 3.6: Fiducial points for the horizontal branch V  16.159 15.917 15.633 15.315 14.930 14.736 14.579 14.557 14.467  B-V  -0.022 -0.015 -0.003 +0.023 +0.082 +0.117 +0.162 +0.215 +0.256  V  16.159 15.917 15.621 15.367 15.160 14.930 14.727 14.709 14.626 14.573 14.423  V-I  0.045 0.078 0.076 0.096 0.112 0.145 0.174 0.225 0.275 0.334 0.415  V  B-I  14.449 14.558 14.626 14.721 14.728 14.959 15.179 15.380 15.723 16.159 16.615  +0.663 +0.563 +0.446 +0.353 +0.272 +0.225 +0.150 +0.113 +0.066 +0.023 -0.011  Chapter 3.  The Evolved  Populations  of M55  114  instability strip, the place where the variable stars of type RR Lyrae are found . In B—V, I adopted the edges given in Sandage (1990) for M15, a cluster with a metaUicity similar to that of M55. Their colours were transformed to V—I and B—I using the appropriate zero-age horizontal branch models of VandenBerg et al. (1998b). The RR Lyr gap edges were put in the observed colour-magnitude diagrams by using the values of EB-V and Ev-i  found in Chapter 2.  Three of the six RR Lyr stars listed in Helen Sawyer Hogg's (1973) catalogue appear in the core field studied here; they are marked by triangles in Figure 3.10 and in order of increasing B — I index their catalogue numbers are V4, V6 and V5. Three more stars on the horizontal branch of M55 are located in the instability strip in the V, B — V colour-magnitude diagram, but the reddest of these three is just outside the RR Lyr gap in the V, V — I and V, B — I colour-magnitude diagrams. These stars are marked by plus signs in Figure 3.10 and unless their magnitudes or colours are greatly in error, they should be RR Lyr variables too. Two of the stars have been observed by Lee (1977) (his star numbers 4520 and 2526) but he did not indicate any variabiHty. The location of the known and suspected RR Lyr stars relative to the instability strip should be regarded as approximate only since their colours vary during the pulsation cycle; the edges taken from Sandage (1990), on the other hand, are based on the colours of the variables averaged over the pulsation period. Therefore the fact that in the B—V diagram one of the known RR Lyr stars is outside of the blue edge, and that one of the suspected RR Lyr stars is to the red of the red edge in the other two colour-magnitude diagrams is of little importance. It should be noted that none of the M55 RR Lyr variables have accurate photometry and period determinations (the periods of King & Bruzual 1976 were derived from fairly crude photometry) and it would be of considerable interest to do that. Unfortunately, the short time span of the images used here also did not aUow searching for variability or the determination of any periods. The known and suspected RR Lyr variables in  Chapter 3.  The Evolved  Populations  of M55 .  115  Table 3.7: Known and Suspected RR Lyrae Stars in the Core Field Star V4 V5 V6 SI S2 S3  V  14.290 14.653 14.427 14.568 14.426 14.438  B-V  0.326 0.416 0.283 0.439 0.451 0.502  U-B  0.029 0.051 0.005 0.011 -0.009 0.021  V-I  B-I  0.461 0.614 0.521 0.722 0.686 0.753  0.787 1.030 0.804 1.161 1.137 1.255  M55 are shown in the finder chart in Figure 3.11, where the known variables are labeled with the numbers from Helen Sawyer Hogg's (1973) catalogue (V4, V5 and V6), and the suspected RR Lyr stars are labeled SI, S2 and S3. It must be emphasized that these stars are suspected of being RR Lyr variables solely based on their location in the colour-magnitude diagram; the photometry analyzed here does not allow confident variability search. The photometry for the six known or suspected RR Lyr stars is listed in Table 3.7 and from their colour indices it is easy to find which is which in the V — I and B — I colour-magnitude diagrams. The small number of RR Lyrae stars in M55, the lack of accurate photometry for them and the total absence of horizontal-branch stars to the red of the instability strip precludes the determination of the level of the horizontal branch by the usual methods of finding the mean magnitude of the ensemble of RR Lyr stars or averaging the magnitudes of the stars on both sides of the RR Lyr gap. Instead, I tried two approaches to estimate V(HB), the apparent visual magnitude of the horizontal branch. For the first one, I averaged the V magnitudes of the six stars fainter and to the blue of the blue edge of the instability strip. These stars appear to form a natural lower envelope and from them I obtained V(ZAHB) = 14.55 ± 0.01, where V(ZAHB) stands for the visual apparent magnitude of the zero-age horizontal branch. On the  Chapter 3.  The Evolved Populations  of M55  116  Figure 3.11: Finder chart for the known (V4, V5 and V6) and suspected (SI, S2 and S3) R R Lyrae variables in M55.  Chapter  3.  The Evolved  Populations  of M55  117  average V(RR) < V(ZAHB) because horizontal branch stars evolve upwards in the colour-magnitude diagram, and therefore I applied a correction to V(ZAHB) to bring it to the level of V(RR). Carney et al. (1992) and Sandage (1993b), using the same data set, derived slightly different offsets between the level of the zero-age horizontal branch and the mean level of the horizontal branch: V(RR) = V(ZAHB) - 0.05[Fe/H] - 0.20 (Carney et al 1992) and V(RR) = V(ZAHB)-0.05[Fe/H]-0.16 (Sandage 1993b). Using [Fe/H] 5 = -1.92, I found V(RR) - V(ZAHB) = AV = -0.10 from the first expresM5  sion, AV = —0.06 from the second one and adopted the average A V = —0.08. Then the apparent magnitude of the horizontal branch is V(HB) = V(RR) = 14.47 ± 0.03, where the uncertainty was estimated from Figure 4 in Carney et al. (1992). The second method was much more straightforward: I simply averaged the V magnitudes of the four presumably constant stars just outside of the blue edge of the instability strip, finding V(RR) = 14.42 ± 0.05. Both values for V(RR) agree reasonably well, so I took the weighted mean to arrive finally at V(RR) = 14.46 ± 0.03. This is the value that was used in Chapter 2 to derive M y (RR) and to compare it to the predictions of various distance calibrations. This is also the value that was used to derive an independent estimate for the reddening and metallicity of M55 by means of Sarajedini's (1994) technique (see Chapter 2). A quantitative measure of the distribution of the horizontal-branch stars can be provided by the number ratio (B — R)/(B + V + R) (Lee et al. 1990), as well as by several other parameters introduced by Fusi Pecci et al. (1993) and Buonanno et al. (1997). Among the latter, (B — V) k, the dereddened colour of the peak of the horizontalpea  branch distribution, and the index B2/(B + V + R) can be measured from the V, B — V colour-magnitude diagram, but the other parameters are difficult to derive without transforming Figure 3.8 to the unknown scale used in their work. From Figure 3.8 I found the following numbers: B = 85, V = 6, R = 0, and, using EB-V = 0.13, B2 = 53. Here  Chapter 3.  The Evolved Populations  of M55  118  B is the number of stars to the blue of the RR Lyr gap, V is the number of variables (RR Lyr stars), R is the number of stars to the red of the instability strip and B2 is the number of horizontal-branch stars bluer than (B — V)o = —0.02. With these numbers the horizontal-branch morphology indices are (B — R)/(B -f V + R) = 0.93 and B2/(B  + V + R) = 0.58. The last number differs significantly from the one given by  Buonanno et al. (1997) who found B2/(B + V + R) = 0.15 for M55. The reason for the discrepancy is that they used the photographic photometry of Lee (1977) where the long blue tail is almost non-existent. In a series of papers (Buonanno et al. 1985, Fusi Pecci et al. 1992, Fusi Pecci et al. 1996 and Buonanno et al. 1997), Roberto Buonanno, Flavio Fusi Pecci and their collaborators have advanced the idea that cluster environment is one of the factors that determines the morphology of the horizontal branch. In particular, they have suggested and presented evidence that in general, only clusters with high total mass and high central density have long blue tails in their horizontal branches, whereas sparse clusters have compact blue horizontal branches. The results presented here indicate that M55 deviates from this rule — it has a prominent blue tail in its horizontal branch, but on the other hand it is one of the least centrally concentrated clusters. This could mean that the relation high central density =}> blue tails is not that straightforward, or it could indicate that the blue tail in the horizontal branch of M55 is a remnant from some earlier event that modified the populations in this cluster. The presence of a prominent blue straggler component (discussed in more detail in Chapter 4) also suggests that M55 may have experienced a period of strong stellar interactions and that the morphology of its horizontal branch reflects to some degree the dynamical history of the cluster. However, neither the star counts (Zaggia et al. 1997) nor the velocity measurements (Pryor et al. 1991) in M55 seem to have found traces of unusual events in the history of the cluster. Another problem with the "dynamical history" explanation is that the whole post-turnoff stellar  Chapter 3.  The Evolved  Populations  119  of M55  evolution is quite fast (~ 10 years), and so the mechanism responsible for producing 8  extended blue tails must be working now — in a low-density environment. In conclusion, I compare in Figure 3.12 the observed horizontal branch of M55 and two sets of zero-age horizontal branch models. The top panel shows the models of VandenBerg et al. (1998b) for [Fe/H] = -1.84, Y = 0.2356 and [a/Fe] = 0.3, shifted by the distance modulus and reddening listed in the caption. The bottom panel shows the oxygenenriched models of Dorman (1992) for [Fe/H] = -2.03, Y = 0.235 and [O/Fe] = 0.7, again shifted by the appropriate distance modulus and reddening. In addition to the location of the zero-age horizontal branch, Dorman's models show also the evolution off the horizontal branch. One can see that the models of VandenBerg et al. (1998b) match superbly the red and lower envelopes of the horizontal branch of M55, and Dorman's (1992) models are only slightly bluer and more luminous. The lower asymptotic giant branch of M55 is also matched very well by Dorman's tracks of stars evolving away from the horizontal branch. The range of stellar masses corresponding to the observed extension of the horizontal branch is from 0.56A4© (the faintest tip of the blue tail) to 0.66A4© at the blue edge of the instability strip. There does not appear to be any red horizontal-branch stars, although one or two of the stars near the bottom of the asymptotic giant branch could be high-mass horizontal-branch stars.  3.5 3.5.1  T h e R e d G i a n t B r a n c h of M 5 5 T h e G a p at the Base of the G i a n t B r a n c h and the R G B  Clump  Even a casual visual inspection of the red giant branch in Figure 3.10 reveals a fairly obvious clump of stars at V = 14.25 and an apparent gap in the star distribution at the base of the giant branch. Both features are present in the other colour-magnitude diagrams as well; the first can be identified with the so-called "red giant branch clump"  Chapter 3.  The Evolved Populations  120  of M55  T — i — i — | — i — I — I — I — | — i — i — I — I — | — i — i — I — I  |  i  r  B-V  Figure 3.12: Comparison between the observed horizontal branch of M55 and horizontal branch models of Vandenberg et al. (1998), top panel, and Dorman (1992), bottom panel. Both sets of models have been shifted by (m — M)y = 14.02, EB-I = 0.30 (top) and E -v — 0.13 (bottom). The bottom panel shows also the evolutionary tracks of horizontal-branch stars for selected masses. B  Chapter  3.  The Evolved  Populations  121  of M55  and the second with the gap seen in the giant branches of some metal-poor globular clusters: NGC 6752 (Lee k Cannon 1980), NGC 288 (Buonanno et al. 1984) and M30 (Bolte 1994). The existence of gaps in the red giant branch was suggested for the first time by Sandage et al. (1968), and was quickly followed by claims of gaps in the giant and horizontal branches of other clusters. Demarque et al. (1988) offered a possible physical explanation (rapidly rotating stellar cores) and Bahcall k Yahil (1972) suggested that statisticalfluctuationsin the star counts could give rise to the observed gaps. On the other hand, later observations of NGC 288 (Bergbusch 1993) and M30 (Bergbusch 1996) did not confirm the earler claims of gaps at the base of the giant branch in these two clusters. This raises the question how significant is the gap at the base of the red giant branch in M55; I attempted to answer it by estimating the probability that a deficiency of stars as big as seen in the colour-magnitude diagram in Figure 3.10 can occur by chance. For that purpose I counted the number of stars iVe, NG and TVp in three adjacent magnitude bins of equal width: 16.89 < V < 17.04, 17.04 < V < 17.19 and 17.19 < V < 17.34, where the middle bin is centred on the gap, and NB, iVc and JVF are the numbers, respectively, in the bin brighter than gap, in the gap and in the bin fainter than the gap. The three bins span a sufficiently small magnitude range and can be assumed to be equally complete. I counted only stars within 0.05 mag of the giant branch ridge fine tabulated in Table 3.4 and found the following numbers: N  B  = 70, NQ = 22 and  JVjF = 39. Since the luminosity function of M55 is nearly linear around V = 17 (see Figure 3.16), one can expect (iVs + Np)/2 = 54.5 stars in the "gap" bin, instead there are only 22 there. Assuming that counting stars in the colour-magnitude diagram is a Poisson process, the probability that 22 stars are observed when 54.5 are expected is 5 x 10 . -7  So formally, the presence of the gap at the base of the red giant branch of  M55 is highly significant. One should note, however, that the bin selection (position and  Chapter 3.  The Evolved Populations  of M55  122  width) maximizes the statistical significance of the gap. W h e n stars are counted i n wider bins (as i n the giant-branch luminosity function later) the gap may disappear completely. Since the core field studied here samples only about 70% of the red giants in the cluster, the discrepancy between observed and expected counts could be reduced somewhat if the whole cluster is observed. The only other C C D observations of the lower giant branch of M 5 5 are those of Zaggia et al. (1997), who covered a wider area in and around the cluster core. A n examination of their colour-magnitude diagram reveals a gap at the base of the giant branch that is very well defined on the red side, but somewhat smeared on the blue side, where the photometry appears to be affected by blending. A s they have sampled a larger fraction of the cluster's giants ( ~ 85%), the existence of this feature in their colour-magnitude diagram lends support to the suggestion that a gap i n the star distribution exists at the base of the giant branch of M55.  A n y such gap, i f real, must correspond t o a  temporary increase of the rate of evolution, however at present time there appears t o be no explanation for such a phase of rapid evolution. Demarque (1988) has suggested that a possible solution might be found in stellar evolution models that incorporate helium diffusion and convective and rotational mixing. Such models have been matched to observations of open clusters (Chaboyer et al. 1995), but not to globular clusters and so it is not clear yet whether these models can explain the gaps i n the giant branches. The other interesting feature on the red giant branch of M 5 5 is the clump of stars at V ~ 14.25. The existence of this feature (called the R G B clump) was predicted by Thomas (1967) and it arises from a slow-down in the rate of evolution caused by the passage of the hydrogen-burning shell through the chemical composition discontinuity left behind by the deepest penetration of the convective envelope. W h e n a star is near the base of the giant branch, the convective envelope reaches into regions that were  Chapter 3.  The Evolved Populations  123  of M55  previously burning hydrogen and are therefore helium-rich, as opposed to the hydrogenrich material brought in by the envelope. A discontinuity is developed between the two regions of different molecular weight and when the shell passes through this discontinuity, its mean molecular weight is reduced, causing a temporary drop in luminosity and a slowing of the rate of hydrogen burning. On the red giant branch this pause in the evolution is manifested by a clump of stars whose luminosity depends on the metaUicity and the mass (and hence age) of the stars in the clump. Generally speaking, the RGB clump is not a prominent feature and it was not found confidently until the study of 47 Tuc by King et al. (1985). The identification of the clump is easier in metal-rich clusters (such as 47 Tuc) because it is at lower luminosity and hence in a more populated part of the red giant branch. In 47 Tuc King et al. (1985) found the clump 0.5 mag fainter than the horizontal branch, whereas in M55 it is 0.3 mag brighter than the horizontal branch. The exceUent quality of the giant-branch photometry in M55 makes it easy to pick out the clump visually, but usuaUy it can only be identified using either the integrated or differential luminosity functions (Fusi Pecci et al. 1990). The red giant branch luminosity function of M55 is studied in detail later in the thesis and the visual identification of the clump made in the colour-magnitude diagram is confirmed there. In the colour-magnitude diagram the RGB clump consists of 13 closely-clustered stars, whose weighted mean magnitudes are and  /dump  Vciump  = 14.236 ± 0.007,  = 13.151 ± 0.007. The values of  -B iump c  -B ium c  P  = 15.094 ± 0.007  and I \ump were derived from the c  mean colours of the clump stars. For comparison, theoretical luminosity functions calculated by Peter Bergbusch (Bergbusch & VandenBerg 1998) from the [Fe/H] = —1.84, [a/Fe] = +0.3, 14-Gyr models of VandenBerg et al (1998b) predict that the RGB clump should be located at My = 0.00, which corresponds to  Vciump  = 14.02 if the distance  modulus derived in Chapter 2 is used. What could be the reasons for this 0.2 mag  Chapter 3.  The Evolved Populations  124  of M55  difference? It was already mentioned that the luminosity of the clump is determined by the heavy metal abundance Z, the helium abundance Y, and the age of the cluster. For the last two quantities, the required changes to remove the discrepancy are too large — M55 must have Y < 0.2 (cf. Figure 5 of King et al. 1985) or an age about 18 Gyr (as lower stellar mass implies lower giant branch luminosity) in order to account for the difference. An error in the distance modulus is unlikely to be an explanation either, since the turnoff also will become brighter — the luminosity of the clump relative to the turnoff is A V j ^ p = —3.79 ± 0.05, whereas the theoretical luminosity function predicts A y J ^  p  = —3.99, i.e., the discrepancy remains. As far as metallicity is con-  cerned, [Fe/H] for M55 appears fairly well determined (see Chapter 2), however the degree of a-enrichment of the cluster is unknown. Throughout this work I have assumed [a/Fe] = +0.3 which is typical for the metal-poor clusters in which this value has been determined observationally (Carney 1996). It is possible, however, that the stars in M55 are even more abundant in a-elements compared to the Sun, and have, say, [a/Fe] = +0.6. The theoretical luminosity function for [a/Fe] = +0.6 (Bergbusch & VandenBerg 1998) does indeed predict a clump centred at My = 0.23 (V = 14.25), in perfect agreement with its observed luminosity. The relation between Z and M £  l u m p  from Fusi Pecci et al.  (1990) — which is based on observations of the RGB clump in 11 globular cluster — also predicts that My of the clump will drop by 0.2 mag when the metal abundance is raised from Z = 0.0005 ([Fe/H] = -1.9, [a/Fe] = +0.3) to Z = 0.0009 ([Fe/H] = -1.9, [a/Fe] = +0.6). A higher a/Fe ratio will also account fully for the ~ 0.1-mag deviation of M55 from the said relation. According to Fusi Pecci et al. (1990), in NGC 6397 (which has [Fe/H] identical to that of M55) the RGB clump is 0.40 ± 0.16 magnitudes brighter than the cluster horizontal branch. With the value of V(ZAHB) = 14.55 found earlier, the corresponding magnitude difference for M55 is AV^J^p = —0.31. This deviation almost disappears, however, if V(ZAHB) = 14.60 is adopted for M55 from the fit of the  Chapter 3.  The Evolved Populations  of M55  125  zero-age horizontal branch models of VandenBerg et al. (1998b) (see Figure 3.12). It is possible, of course, that some combination of higher age and higher a-elements abundance causes the discrepancy between the observed and predicted luminosities of the RGB clump. If all these parameters are correct, however, some changes to the theoretical models may have to be considered. Among these, convective overshoot appears to be the preferred mechanism for lowering the luminosity of the RGB clump (King et al. 1985). Convective overshoot is the term that refers to the penetration of the bottom of the convective envelope into the radiative region, beyond the boundary at which the adiabatic and radiative gradients are equal. The amount of overshooting in low-mass stars is thought to be small and is usually ignored in standard stellar models. Calculations show that when convective overshoot is included, the hydrogen-burning envelope passes through the composition discontinuity earlier and therefore at lower luminosity, resulting in a substantial drop in the RGB-clump luminosity (King et al. 1985, Alongi et al. 1991, Girardi et al. 1996). In summary, the most likely explanations for the 0.2 mag difference between the theoretical and observed luminosities of the red giant clump in M55 are an a-enhancement larger than the adopted value of [a/Fe] = +0.3, or a modest amount of convective overshoot at the bottom of the convective envelope, or some combination of the two. When the metal abundance of M55 is known accurately, it should be possible to constrain the amount of overshooting better, similar to the observational constraints placed on overshooting in the convective cores of the more massive open-cluster stars (Stothers 1991, Rosvick & VandenBerg 1998, Pols et al. 1998).  3.5.2  Intrinsic Width of the Red Giant Branch  It is well known that the abundance of elements heavier than helium strongly affects the observed colour of the various branches of the cluster colour-magnitude diagram —  Chapter 3.  The Evolved Populations  126  of M55  in particular, for a fixed age the giant branch of a metal-rich cluster is redder than the giant branch a of more metal-poor cluster. This is true for the main sequence and the turnoff as well, however their colours are affected by the helium abundance and less sensitive to metaUicity — from Figure 8 of Bergbusch & VandenBerg (1992) one can see that the change in B — V for the upper main sequence is about half of that for the red giant branch as [Fe/H] varies between —0.5 and —2.3. One can use then the observed width of the giant branch as an indicator of the chemical homogeneity within the cluster (Renzini 1977, Sandage &; Katem 1983), since a giant-branch width beyond that explained by photometric errors wiU provide an upper limit on the metaUicity spread in the cluster. To estimate the observed width of the red giant branch, I calculated, in each colour index, the distribution of the colour residuals from the giant branch ridge line for aU stars in the magnitude range 14.0 < V < 17.0 and within ±0.1 mag of the ridge line. These distributions, binned in 0.005 mag in colour, are shown as histograms in the corresponding panels of Figure 3.13; their best-fitting Gaussian distributions are shown by solid lines. The giant branch width expected from photometric errors alone was estimated by finding the median colour error for aU stars in the interval 14.0 < V < 17.0 and then constructing a Gaussian distribution with a standard deviation equal to that 1.4826 times median error. I found median colour errors of a ed{B—V) = 0.00741, cr ed(V—I) = 0.00712 and m  o~ ed(B—I) m  m  = 0.00764, resulting in the curves drawn by dotted lines in Figure 3.13. This  is an approximate procedure; the proper treatment would be to calculate the expected distribution of the residuals by taking into acount the distribution of the colour errors (Sandage & Katem 1983). However, as pointed out by them, simply adopting the mean error gives identical results and is entirely adequate. It is clear from Figure 3.13 that photometric errors alone cannot explain the whole width of the giant branch, that is, the giant branch of M55 may have a non-zero intrinsic  Chapter 3.  The Evolved Populations  of M55  127  Figure 3.13: Comparison between the observed and expected widths of the giant branch. The histograms show the observed distribution of the colour residuals i n the corresponding colours. Dotted lines show the width expected from photometric errors alone and solid lines show the best Gaussian fit to the colour residuals. E a c h panel is labelled w i t h the standard deviation of the best-fitting Gaussian distribution.  Chapter 3.  The Evolved Populations  128  of M55  width. It is possible to obtain an estimate of this intrinsic width by deconvolving the observed distributions of the colour residuals A(B — V), A(V — I) and A(B — I) (solid lines) and the photometric error distributions (dotted lines). widths are: a (B-V) I  The resulting intrinsic  = 0.012, <7j(V-I) = 0.011 and cr (B — I) = 0.014. These are I  upper limits on the intrinsic widths expected from a spread in the heavy metal abundance in M55, since additional scatter could be caused by rotation, blending, differential reddening, binaries and possibly other sources. From the isochrones of Bergbusch & VandenBerg (1992) I estimated d(B - V)/d([Fe/H]) o-i(B-V)  « 0.12/dex at M  v  = 2, so from  — 0.012 the upper limit on the chemical inhomogeneity in M55 is 0.10 dex (a  factor of ~ 1.26). This value is in a good agreement with the estimates for other globular clusters, for example M92 (0.18 dex, Stetson 1993a), NGC 288 (0.07 dex, Stetson 1993a), NGC 6171 (0.13 dex, Ferraro tt al. 1991). It is worth mentioning that Smith &: Norris (1983) found small variations in the abundances of CN and CH in M55, but it is unclear whether these variations reflect primordial differences (e.g., mild carbon enrichment from supernovae) or surface abundance differences resulting from the first deep penetration of the convective envelope in the beginning of the red giant branch phase. This is an important question, as the magnitude of any primordial chemical inhomogeneities in globular clusters puts strong constraints on the theories of cluster formation (see, e.g., the review of Lin & Murray 1991). For example, the upper limit of ~ 0.1 dex on the metallicity spread in today's globular clusters implies, among other things, that (a) they formed in gas clouds that were somehow almost completely homogenized after an initial enrichment phase; (b) the degree of this initial enrichment varied by a factor of a few hundred (corresponding to the observed cluster-to-cluster metallicity spread); and (c) the star formation epoch lasted only a short time (less than the evolution timescale of massive stars), otherwise younger stars would have been contaminated by the products of the first-generation supernovae. The last point also follows from the fact that cluster  Chapter 3.  The Evolved Populations  129  of M55  stars are coeval, that is within a few percent no age spread is observed within a cluster (Stetson 1993a). All of these and other conditions have to be met by any theory that attempts to explain globular cluster formation. 3.5.3  Population Ratios and the Helium Abundance of M 5 5  It was already mentioned that above the main-sequence turnoff, the number of stars in a given evolutionary phase (shell hydrogen burning, core helium burning etc.) is directly proportional to the duration of that phase. Theoretical lifetime ratios depend on several assumptions, including the helium abundance, the presence and extent of semi convection on the horizontal branch, the structure of the helium core and the extent of overshooting. Comparisons of theoretical duration ratios to the observed population ratios allow then not only fairly sensitive tests of the model assumptions, but in some cases make it possible to constrain some unknown or poorly known model parameters, the best examples being helium abundance and the treatment of semiconvection (Renzini b, Fusi Pecci 1988). Four useful population ratios are  R — -/VHB/^VRGB  and  R!  =  NUB/(NRGB  +  N^GB),  which depend almost exclusively on the helium abundance (Buzzoni et al. 1983, Caputo et al.  1987), andii?i =  N^GB/NRGB  and  R  2  =  NAGB/-^HB>  which have been used to  confirm the existence of a semiconvective zone at the outer boundary of the heliumburning core in horizontal-branch stars and to study the degree of central mixing in such stars (Buzzoni et al. 1983, Buonanno et al. 1985, Renzini & Fusi Pecci 1988). In these expressions  TVHB  is the total number of stars on the horizontal branch,  number of stars on the giant branch branch, and  ./VAGB  brighter  NRGB  is the  than the level of the zero-age horizontal  is the number of stars on the asymptotic giant branch.  All four ratios involve the brightest stars in the cluster and so completeness and field contamination are rarely a problem. The artificial star tests described in the next section indicate that brighter than V = 16 the photometry is complete, and judging from the  Chapter 3.  The Evolved  Populations  of  130  M55  Table 3.8: Star Counts and Population Ratios Population or Ratio NHB NRGB NAGB R R! Ri R2  Number or Value 91 61 13 1.49 1.23 0.21 0.14  Error 9.5 7.8 3.6 0.25 0.19 0.07 0.04  V, V — I diagram of the comparison field (see Chapter 2) no background corrections are necessary. I used the V, B—I colour-magnitude diagram (Figure 3.10) to count the stars, since it has the tightest sequences and best separation of the upper red giant branch and the asymptotic giant branch. Adopting V^ZAHB) = 14.55 , I found the numbers 2  and population ratios given in Table 3.8, where the uncertainties for the counts are the Poisson errors. The values of R for 15 globular clusters tabulated by Buzzoni et al. (1983) plus x  the value for NGC 5466 from Buonanno et al. (1985) range from 0.16 to 0.28 with a weighted mean of R — 0.21 ± 0.02, a value identical to that derived here for M55. The x  weighted mean value of R , calculated from the counts given in the same two papers, is 2  R = 0.144±0.009, again identical to the value of R for M55 found here. As discussed in 2  2  Buonanno et al. (1985) and Renzini &; Fusi Pecci (1988), the ratio R changes by a factor 2  of eight for models with or without semiconvection, and thus puts a very strong constraint on horizontal-branch models. Models with the so-called "canonical" semiconvection predict lifetime ratios ^AGB/^RGB = 0.2 and ^AGBAHB = 0.14 (Renzini & Fusi Pecci 1988) As mentioned earlier, the fit of the zero-age horizontal branch models of VandenBerg et al. (1998b) to the observed horizontal branch yields V ( Z A H B ) = 14.60. Lowering V ( Z A H B ) by 0.05 mag adds only a single star to the giant branch sample and does not affect the calculated ratios. 2  Chapter 3.  The Evolved Populations  131  of M55  and therefore it can be argued that the population ratios ^AGB/^VHB  A^AGB/^RGB  = 0.21 ± 0.02 and  = 0.14 ± 0.01 provide conclusive evidence that semiconvection develops in  horizontal-branch stars. The helium abundance of M55 can be estimated from the number ratios R and R' using the calibrations from Buzzoni et al. (1983): Y(R)  =  0.176 + 0.380log R  Y(R')  =  0.204 + 0.457 log R',  =  0.168  and from Caputo et al. (1987): Y(R . ) 0 9  + 0.461  log  R., 0 9  where R0.9 has been derived from clusters having B/(B + R) = 0.9. Since M55 has B/(B  + R) ~ 1 this is the appropriate relation to use. Using the ratios from Table 3.8,  I obtained Y(R) Y(R')  =  0.242±S:SS,  =  0.245+ °; °3 3 , 2  o  9  o  Y(Ro. ) = 0.2481SZ. 9  The weighted mean of these three estimates is  Y  = 0.24.tomli  which agrees with the  mean values of Buzzoni et al. (1983) (Y = 0.23 ± 0.02) and Caputo et al. (1987) (Y == 0.24 ± 0.01). It should be mentioned that all three calibrations are based on fairly old sets of horizontal-branch models which do not include oxygen or a-enhancement, unlike the newer models of Dorman (1992) and VandenBerg et al. (1998b) (for which no R calibration has ever been done). Still, the above values agree very well with recent estimates of the primordial helium abundance Yp from observations of extragalactic metal-poor H II regions (Olive et al. 1991, Olive Sz Steigman 1995, Izotov et al. 1997), which all give  Chapter 3.  The Evolved Populations  Yp — 0.24 ± 0.01.  of M55  132  Together all these values appear to support the predictions of big  bang nucleosynthesis theories that Yp J> 0.235 (Denegri et al. 1990, Walker et al. 1991, Mathews et al. 1993) although observational and systematic errors still do not allow the determination of Yp to the third decimal place, which would be needed to falsify current big bang models.  3.6  Luminosity Functions for the Evolved Populations in M55  It was discussed briefly i n the beginning of this chapter that the luminosity functions of several metal-poor globular clusters show excess of subgiant and red giant stars relative to the number of stars around the turnoff when compared to theoretical l u m i nosity functions (Stetson 1991, Bolte 1994, Bergbusch 1996, VandenBerg et al. 1998a). O n the other hand, the luminosity functions of more metal-rich clusters like N G C 288 (Bergbusch 1993) and M 5 (Sandquist et al. 1996) appear to agree well w i t h the predicted ratios. This problem remains unresolved and i t is very interesting to see whether M 5 5 also displays such a discrepancy. A n y new data i n support of a disagreement between the observed and theoretical luminosity functions will increase the possibility that either current stellar models do not predict correctly the rate of evolution on the red giant branch or additional physical mechanisms acting on the main sequence must be considered. A possible solution to the main-sequence - giant-branch discrepancy has been proposed recently by VandenBerg et al. (1998a), who showed that a satisfactory agreement between theory and observation is achieved i f the interiors of cluster stars are rapidly rotating (in the case of M30 the best match is for periods of rotation between seven and ten days). Interior rotation seems to affects the star loci i n the colour-magnitude  diagram  by observationally undetectable amounts (<J 0.01 mag), while the decrease of the rate of evolution on the giant branch is enough to raise that part of the luminosity function.  Chapter 3.  The Evolved Populations  of M55  133  The luminosity functions for the evolved populations also allow exploration of the constraints that the subgiant branch and the giant branch provide. The most prominent feature of the luminosity function above the turnoff is the sudden jump (or break) caused by the rapid increase in the number of stars from the base of the giant branch to the turnoff. This part of the luminosity function (more specifically its slope and location) is very sensitive to variations in chemical abundance and age as it shows the largest deviations when these parameters are changed (Paczynski 1984, Ratcliff 1987). It appears that currently the principal obstacle to a better use of the subgiant break is the small number statistics — many of the break's features can only be constrained when the luminosity function is sampled in 0.1 mag-wide bins or finer, and this requires ~ 10  3  or more stars on the subgiant branch only. Other features in the luminosity function, notably the RGB clump (which in the luminosity function becomes a "bump", the socalled RGB bump) also hold the promise of providing important information about the cluster. As noted earlier, the luminosity of the RGB bump is sensitive to the degree of overabundance of a-elements relative to the Sun, as well as to the amount of convective overshoot (Fusi Pecci et al. 1990, Alongi et al. 1991).  3.6.1  Artificial Star Tests and Completeness Corrections  The artificial star tests were carried out in two stages. On the first stage I added 10330 artificial stars to all 30 U, B, V and / images. In order to speed up the accumulation of a sufficient number of giant-branch stars, on the second stage I used only the 73-band and /-band frames for the tests, thus reducing several times the amount of computer time required for the calculations. A total of 11552 stars (4332 on the giant branch and 7220 on the main sequence) were added to the five B and ten / frames used on the second stage. Since only the 5-band and /-band images were used, I carried out a separate run of A L L F R A M E to provide a comparison to the output from the second set of tests. In  Chaptei  3.  The Evolved Populations  of M55  Figure 3.14: Comparison between the "real" V, V—I colour-magnitude diagram (left) and the colour-magnitude diagram of the recovered artificial stars (right).  Chapter 3.  The Evolved  Populations  135  of M55  particular, the magnitude, colour and % rejection criteria were redefined in terms of the 2  / magnitude so that they could be used on the output of the tests. Figure 3.14 shows a comparison between the observed colour-magnitude diagram of the cluster (left panel) and the same area of the colour-magnitude diagram for the recovered stars (right panel) from thefirstset ( UBVI) of artificial star tests. The artificial stars plotted here were subject to the same restriction as the real stars, namely % < 1.4 2  and the magnitude and colour errors of each star had to be less than twice the median error at that magnitude. One can see that the excess of stars found in the "real" colour's  magnitude diagram above the subgiant branch and to the blue of the red giant branch is seen also in the artificial colour-magnitude diagram and can be explained by blending of subgiant and lower-giant branch stars. The completeness factors in the V and / bands f(V) and /(/) were calculated as the ratio / and n  = n  o u t  o  u  t  /ni , n  where n- is the number of artificial stars put in a given magnitude bin in  is the number of stars recovered in the same bin. It was discussed briefly in the  preceding chapter that this procedure may not be very precise, as some stars are recovered in bins that are different from the one they were put in. However several discussions of completeness corrections (Stetson & Harris 1988, Bergbusch 1993) have indicated that the differences between this simple-minded approach and more sophisticated methods are small and within the errors of the completeness corrections themselves, at least when the field is not terribly crowded. I came to the same conclusion in Chapter 2 where I used both the method of Drukier et al. (1988) and the simple / — n Plots of the completeness factors /  = n  o  u  t  / n i  n  out  / n i corrections. n  as a function of magnitude are displayed  in Figure 3.15. The top panel shows the completeness corrections in the V-band, which were calculated using only the first set of artificial star tests (since the second set did not include F-band images).  The two lower panels in Figure 3.15 show the /-band  completeness fraction as derived from the UBVI tests (middle panel) and the BI tests  Chapter 3.  The Evolved Populations  of M55  136  Figure 3.15: The completeness factors in V (top) and I (middle and bottom). The solid lines show the adopted completeness factors as a function of magnitude. See text on the difference between the two /-band plots.  Chapter 3.  The Evolved Populations  of M55  137  (bottom panel). All three plots show that the recovered sample is slightly over-complete in the magnitude range 17 ^ V ^ 18.5 (16.5 ^ I ^ 17.5). This can be explained by the combination of three factors: a steep rise of the luminosity function in that interval, the tendency of crowding to make stars appear brighter and the fact that the subgiant branch is fairly steep both in V and I. The adopted completeness factors / as a function of magnitude are shown by solid lines in Figure 3.15. The values of / in the /-band are virtually identical for the two sets of artificial star experiments, except for the slightly different degree of over-completeness above the turnoff. Since the giant-branch and subgiant-branch number statistics is better for the second set of artificial star tests, for the /-band counts I adopted the relation displayed in the bottom panel. However, choosing the completeness relation in the middle panel (that is, the one defined by the UBVI experiments) will not change the /-band luminosity function by any appreciable amount. From the plots in Figure 3.15 one can conclude that the bright star photometry in the core field is complete to about / = 18 and V = 18.5, but the completeness drops rapidly after that. The average over-completeness in the ~ 1 mag interval around the turnoff is about 2% and it will not have a serious impact on the conclusions that are drawn from the luminosity function. 3.6.2  Luminosity Functions in V and / : Theory vs. Observations  Before deriving the luminosity functions I restricted the region in the colour-magnitude diagram where the stars were actually counted to a strip around the giant branch and main sequence with a width determined by the scatter around the fiducial sequences. Because the main sequence is wider to the red of the ridge fine, the width of the strip below the turnoff was increased on the red side to include the stars that are scattered further to the red than to the blue. Naturally, the same restriction was imposed on the  Chapter 3.  The Evolved Populations  138  of M55  counts in the artificial-star colour-magnitude diagram before the completeness corrections were derived. When the deep luminosity functions for the two-core-radii field were derived in Chapter 2, special care was taken to exclude from the counts field stars and galaxies. No such corrections were applied to the present data, as essentially all bright stars in the colourmagnitude diagram are cluster stars, and the contribution of galaxies is practically zero. Some field stars are certainly present in the colour-magnitude diagram, mostly to the red of the giant branch and the main sequence, but they were excluded since only stars around the principal sequences were counted. In any case, the colour-magnitude diagrams of both the comparisonfield(Chapter 2) and the corefieldindicate that field stars will make a negligible contribution (<C 1%) to the luminosity function in any magnitude bin and therefore no corrections for field star/galaxy contamination were applied. The resultant raw (i.e. not corrected for incompleteness) differential luminosity functions in the V and / bands are shown in Figure 3.16, where $(V) and *$(/) are the number of stars per unit magnitude interval. In this plot solid dots mark the part of the luminosity function which is estimated to be complete, while open circles mark the bins where the completeness is below 1. In both bands the RGB bump (at V = 14.25 and I = 13.15) is very prominent, deviating by about 3<r from the adjacent bins. The gap at the base of the giant branch is also seen as a dip in the /-band luminosity function (at / = 16.1), but because it is so narrow, it is smeared in the other luminosity function. Another confirmation of the visual identification of the RGB bump can be seen in Figure 3.17, where I have plotted the bright (i.e., complete) part of the cumulative (or integrated) V and / luminosity functions $c(V) and $c(/)- The cumulative luminosity function in, say, the V"-band is defined by  M V n )  =  Em)  Chapter 3.  The Evolved Populations  of M55  139  Figure 3.16: The raw luminosity functions in the V and / bands. Note the prominent RGB bump at V = 14.25 and / = 13.15 and the dip at I = 16.1 corresponding to the gap at the base of the giant branch.  Chapter 3.  The Evolved Populations  of M55  140  Figure 3.17: The cumulative luminosity functions in the V and I bands as derived from the raw (but complete) counts. Note the change in the slope (dotted lines) of the luminosity function on the two sides of the RGB jump.  Chapter 3.  The Evolved  Populations  of M55  141  where V is the V magnitude of the n-th bin and $(Vi) is the number of stars in the n  i-th. bin. In other words, $c(V) is simply the total number of stars in all bins up to, but not including the bin at magnitude V. In Figure 3.17 the RGB bump is revealed by the jump in the level and the change in the slope of the cumulative luminosity function at V  14.3 and I ?s 13.2. The change in slope is indicated by the dotted lines which  are least-squares fits to the luminosity function level before and after the jump. This method of locating the RGB bump was first proposed by Deborah Crocker and Robert Rood (as quoted by Fusi Pecci et al. 1990) and with the exception of a few metal-rich clusters (where the clump can be identified from the colour-magnitude diagram or the differential luminosity function), this has been the only way to find the RGB bump in metal-poor clusters (see Fusi Pecci et al. 1990 for more details). The completeness-corrected /-band and V-band luminosity functions are shown in Figure 3.18 and Figure 3.19, respectively and listed in Table 3.9. They were obtained by dividing the raw star counts in each filter by the appropriate completeness relation shown in Figure 3.15. In the upper panel of each figure the observed luminosity function is compared to theoretical luminosity functions for a fixed chemical abundance ([Fe/H] = -1.84 and [a/Fe] = +0.3), but different ages: 12, 14 and 16 Gyr. In the lower panel, the observed luminosity function is compared to theoretical luminosity functions for the preferred metaUicity and age of M55 ([Fe/H] = —1.84 and 14 Gyr, respectively), but with different a/Fe enhancement: [a/Fe] = +0.3 and [a/Fe] = +0.6.  The theo-  retical luminosity functions (Bergbusch & VandenBerg 1998) were calculated from the latest models of VandenBerg et al. (1998) and were kindly provided in advance of publication. The absolute magnitudes in the theoretical luminosity functions were converted to apparent magnitudes using the distance moduli derived in Chapter 2: (TO — M)y = 14.02 and (TO — M)i = 13.85, where (TO — M)i was obtained from (TO — M)v = 14.02 and the adopted colour excesses Ev-i  =  0.17. The position of the observed main-sequence  Chapter 3.  The Evolved Populations  142  of M55  I  r  12 Gyr 14 Gyr 16 Gyr [Fe/H] = -1.84 [a/Fe] = +0.3  O  2  H  h  H  1 h  H  h  [a/Fe] = +0.3 [a/Fe] = +0.6 [Fe/H] = -1.84 Age = 14 Gyr  o  2  12  14  18  Figure 3.18: The differential luminosity function in the J-band corrected for incompleteness. Top panel: the observed luminosity function is compared to theoretical luminosity functions for [Fe/H] = -1.84, [a/Fe] = +0.3 and ages of 12, 14 and 16 Gyr. Bottom panel: the observed luminosity function is compared to theoretical luminosity functions for the preferred age and iron abundance of 14 Gyr and [Fe/H] = —1.84, and a-enrichment of [a/Fe] = +0.3 and +0.6. The location of the main-sequence turnoff is marked by TO.  Chapter 3.  The Evolved Populations  12  14  143  of M55  16  18  20  V  Figure 3.19: The differential luminosity function in the F-band corrected for incompleteness. Top panel: the observed luminosity function is compared to theoretical luminosity functions for [Fe/H] = -1.84, [a/Fe] = +0.3 and ages of 12, 14 and 16 Gyr. Bottom panel: the observed luminosity function is compared to theoretical luminosity functions for the preferred age and iron abundance of 14 Gyr and [Fe/H] = —1.84, and a-enrichment of [a/Fe] = +0.3 and +0.6. The location of the main-sequence turnoff is marked by TO.  Chapter 3.  The Evolved Populations  144  of M55  Table 3.9: V-band and /-band differential luminosity functions V 12.725 12.975 13.225 13.475 13.725 13.975 14.225 14.475 14.725 14.975 15.225 15.475 15.725 15.975 16.225 16.475 16.725 16.975 17.225 17.475 17.725 17.975 18.225 18.475 18.725 18.975 19.225 19.475 19.725  log $ ( V ) 1 079 1 301 1 204 1 301 1 204 1 447 1 806 1 602 1 681 1 806 1 748 1 903 2 033 2 158 2 079 2 246 2 282 2 394 2 622 3 028 3 161 3 292 3 353 3 410 3 479 3 567 3 616 3 658 3 687  -1<7  0.374 0.257 0.301 0.257 0.301 0.206 0.125 0.165 0.148 0.125 0.135 0.110 0.093 0.079 0.088 0.071 0.068 0.058 0.044 0.027 0.023 0.020 0.018 0.017 0.016 0.014 0.014 0.013 0.013  + l<r 0.198 0.161 0.176 0.161 0.176 0.139 0.097 0.119 0.110 0.097 0.103 0.088 0.076 0.067 0.073 0.061 0.059 0.052 0.040 0.025 0.022 0.019 0.018 0.017 0.015 0.014 0.013 0.013 0.012  /  11.625 11.875 12.125 12.375 12.625 12.875 13.125 13.375 13.625 13.875 14.125 14.375 14.625 14.875 15.125 15.375 15.625 15.875 16.125 16.375 16.625 16.875 17.125 17.375 17.625 17.875 18.125 18.375 18.625 18.875  log$(/) 0.623 1.447 1.204 1.204 1.079 1.380 1.806 1.556 1.556 1.778 1.833 1.643 1.944 2.121 2.134 2.093 2.225 2.358 2.309 2.573 2.858 3.048 3.212 3.332 3.417 3.500 3.568 3.662 3.720 3.793  -la 1.618 0.206 0.301 0.301 0.374 0.228 0.125 0.176 0.176 0.130 0.121 0.156 0.104 0.083 0.082 0.086 0.073 0.062 0.066 0.047 0.034 0.027 0.022 0.019 0.017 0.016 0.015 0.013 0.012 0.011  +lo0 296 0 139 0 176 0 176 0 198 0 149 0 097 0 125 0 125 0 100 0 094 0 114 0 084 0 070 0 069 0 072 0 062 0 054 0 057 0 043 0 031 0 025 0 021 0 018 0 017 0 015 0 014 0 013 0 012 0 Oil  Chapter 3.  The Evolved Populations  of M55  145  turnoff is also marked nn all plots. The theoretical luminosity functions shown in Figures 3.18 - 3.19 were all calculated for a value of the mass spectral index of x = 0.0, but the choice of x makes little difference for the magnitude range explored here. The luminosity functions were normalized to the observed star counts by making the predicted number of stars between the base of the red giant branch and the RGB clump equal to the observed number of stars in the same luminosity interval. This particular way of normalization was chosen because: (a) the counts are complete there and the number statistics is good; and (b) the slone of the luminosity function between the base of the giant branch and the RGB clump is insensitive to any of the input model parameters (Bergbusch 1990), and therefore this part of the giant branch is especially suitable for comparison between models and observations. In the following discussion I will be referring primarily to the F-band luminosity function in Figure 3.19, but most of the comments apply to the /-band luminosity function as well.  The RGB  Bump  The enhancement in the luminosity function caused by the clump of stars on the red giant branch of M55 is easily seen in all luminosity functions, but especially well in V where the scatter in the bright part of the luminosity function is smallest. The RGB bump was discussed in more detail earlier and here I only compare its observed position with that predicted by the theoretical luminosity functions. It is clear that in both bands the [Fe/H] = —1.84, [a/Fe] = +0.3 luminosity function predicts too high a luminosity for the bump and varying the age of M55 will not improve the agreement much. Adopting a higher age will have a greater impact on the luminosity of the subgiant break than on the luminosity of the bump, thus increasing the disagreement between the observed and theoretical luminosity function around the break.  Chapter 3.  The Evolved Populations  146  of M55  One can see that increasing the a-enhancement from [a/Fe] = +0.3 to [a/Fe] = +0.6 appears to remove all of the discrepancy between the observed and predicted luminosity of the bump. This is only partly true, however, since increasing the a-elements abundance also makes the turnoff fainter, so that for the same age, the distance-independent magnitude difference A V ™ between the bump and the turnoff is reduced but not comb  p  pletely eliminated. In the F-band, the observed value is A V ^ ^ p — —3.79 + 0.05, while the 14-Gyr, [a/Fe] = +0.3 luminosity function predicts A V ™ = -3.99 and the 14-Gyr, b  p  [a/Fe] = +0.6 luminosity function predicts A V ^ ^ = —3.89. The agreement is no better in the /-band, where the observed difference is A / j ^ p = —4.17 + 0.05 and the predicted values are A / ^ n = -4.47 for [a/Fe] = +0.3 and AI^ p  p  = -4.33 for [a/Fe] = +0.6.  This means that the RGB bump discrepancy is not completely resolved by adopting a higher degree of a-enrichment and, as discussed earlier, convective overshoot may have to be invoked to remove the disagreement completely. Still, the comparison between the theoretical and observed luminosity functions suggests that M55 may be more abundant in a-elements than other clusters of similar metallicity and it would certainly be very interesting to determine the a-enhancement in M55 spectroscopically and compare it with the value suggested by the luminosity function comparison.  The Subgiant Break  The sharp increase of the number of stars between the base of the giant branch and the main-sequence turnoff creates a prominent feature in the luminosity function which is called the subgiant break. This nearly vertical part of the luminosity function is the most sensitive to variations in the chemical abundance (metaUicity and helium content), age and distance modulus (Paczyriski 1984, Ratcliff 1987, Degl'Innocenti et al. 1997). In particular, the luminosity of the break varies with age and, if the distance to the  Chapter 3.  The Evolved Populations  147  of M55  cluster and its metaUicity are known, it could be a better age estimator than theoretical isochrones (Paczynski 1984). In addition, the slope of the break is sensitive to metaUicity and the height of the jump is determined by the helium abundance (Ratcliff 1987). A U these advantages, however, are diminished somewhat by the nearly horizontal nature of the subgiant branch in the B and V colour-magnitude diagrams. Even when using narrow magnitude bins the whole magnitude range of the subgiant branch is contained within only a few of them and its structure is difficult to study in detail. The subgiant branch is most vertical in the /-band colour-magnitude diagrams, leading to a fairly shallow slope in the luminosity function, as can be seen in Figure 3.18. There does not appear to be any substantial disagreement between the shapes and locations of the subgiant region in the theoretical and observed luminosity functions, both in V and / . This is an indication that the adopted values for the age, metaUicity, distance and helium abundance of M55 are consistent with each other. There is also no evidence in the V and / luminosity functions for the so-caUed subgiant excess that has been observed in some metal-poor clusters (Stetson 1991, Bolte 1994). From the tiny subgiant peak right at the top of the break to the turnoff, the observed and predicted numbers agree very weU in both bands.  The Overall Agreement Between Theory and  Observations  Neither the V-band nor the /-band luminosity functions show a significant deviation from the theoretical luminosity functions, with the possible exception of the R G B bump. In particular, there is no evidence for a deficiency of main-sequence stars compared to the number of stars on the subgiant and red giant branch. In M30, for example, the main sequence is depressed by 0.1-0.2 dex (or 20%-50%) when the theoretical luminosity function is normalized to the giant branch; if the normalization is done on the main sequence, then there is an excess of giant stars compared to the models (Bolte 1994,  Chapter 3.  The Evolved Populations  of M55  148  Bergbusch 1996). No such depression is apparent in Figures 3.18 or 3.19, although one may argue that when observations are compared to the V-band, [a/Fe] = +0.6 luminosity functions, a slight deficiency is seen fainter than the turnoff. Overall, the agreement between the observed and theoretical luminosity functions is quite good and it is hard to imagine a reasonable change in the input parameters (age, metallicity or a-enhancement) that will create the discrepancy seen in the luminosity functions of clusters like M30, M68, M92 and NGC 6397. The fact that the shape of the theoretical luminosity function changes very slightly when the basic parameters ([Fe/H], [a/Fe] and age) are varied within reasonable limits (Stetson 1991, VandenBerg et al. 1998a) makes it nearly impossible to reproduce the main-sequence deficiency unless the completeness corrections derived here are terribly wrong (by 20% or more). The same statement, however, can be made regarding the luminosity function of, say, M30 (the best studied of the listed clusters) — it would be very difficult to adjust the parameters of the model luminosity functions so that the giant-branch - main-sequence discrepancy disappears. In a recent paper, VandenBerg et al. (1998a) showed that theoretical luminosity functions derived from evolutionary models that include core rotation agree very well with the observed luminosity function of M30. The preliminary results indicate that the amount of core rotation that is need to reconcile theory with observations is fairly small and has only a small influence on the magnitudes and colours of the stars, that is, in the colour-magnitude diagram the location and width of the red giant branch and the main sequence remain the same. The latter implies that the location of the isochrones also does not change, that is the age - turnoff luminosity relation is not affected appreciably. If core rotation is indeed the cure for the luminosity function anomalies in metalpoor clusters, then the lack of any substantial discrepancy between canonical models and observations in M55 implies that either the stars in M55 do not have rotating cores or their  Chapter 3.  The Evolved  Populations  of M55  149  angular speeds are too low to affect the rate of evolution. One can ask how is M55 different from the other four clusters in which main-sequence deficiencies have been found, and in which presumably all stars have rotating cores. While M55 is a metal-poor cluster like all of them, it is very different structurally and hence may have had a different dynamical history. Of the four clusters, M30 and NGC 6397 are core-collapsed clusters, and M92 and M68 are centrally concentrated, massive clusters (Djorgovski & Meylan 1993). If one attempts to explain how the lack of stellar core rotation in M55 is coupled to its structure and dynamics, one possible place to look would be the distribution of angular momentum in the cluster. Since globular clusters as a whole rotate slowly, most of the 3  initial angular momentum of the proto-cluster cloud must have been transferred or lost by several mechanisms: mass loss from stellar winds and the ejection of the left-over gas, mass loss from stars leaving the cluster (probably the dominant mechanism) and redistribution of the angular momentum between the individual stars and their orbital motion. It is possible that different angular momentum loss/transfer mechanisms were at play in M55 and the more concentrated clusters, resulting in the suggested difference in rotation rates. Since angular momentum in individual stars can be transferred only outward (and may be lost) and not inward, the redistribution of the angular momentum must have happened during the very early history of the cluster. If indeed there is a connection between the angular momentum retained in individual stars and the structural properties of the cluster, it may be not that easy to detect observationally because of the large number of stars that have to be observed in order to assign some quantitative measure of stellar rotation in a given cluster. On the other hand, if such a connection exists, the presence of luminosity function discrepancies may well be related to the structural properties of the cluster. This is again a suggestion that has to confirmed observationally. 3  T h i s suggestion is due to Peter Bergbusch.  Chapter 3.  The Evolved Populations  of M55  150  Even though there is theoretical support for the relation between metaUicity and steUar rotation (Deliyannis et al. 1989), it should be noted that the connection between metaUicity and presence or absence of luminosity function anomalies is rather tenuous at the present, as it is based on four metal-poor (M30, M69, M92 and N G C 6397) and two metal-rich (M5 and N G C 288) clusters. M55 appears to break this rule but without good-quality luminosity functions for the turnoff and the evolved populations in more clusters it is early to draw any firm conclusions that the unusual luminosity functions of some metal-poor clusters are related to their metaUicity.  Chapter 4  The Blue Straggler Population of M 5 5  In the colour-magnitude diagrams presented in Chapter 3 (see, e.g., Figure 3.10) there is a prominent component that appears to be an extension of the main sequence to bluer colours and brighter magnitudes than the cluster's main-sequence turnoff. If these were normal cluster stars, they should have evolved away from the main sequence about 9xl0 years ago, and yet they are still there, as if they lag behind the other cluster stars 9  in their evolution. Discovered for the first time in the globular cluster M3 by Sandage (1953), they were named blue straggler stars or blue stragglers because of the appearance that they straggle behind on the main sequence. In the same paper that announced their discovery, Sandage suggested that these could be completely mixed stars which the recent calculations of Stromgren had predicted should remain on the main sequence, unlike the unmixed models of Schwarzschild that evolved rapidly to the red as hydrogen was exhausted in the core. The hypothesis that blue stragglers are single stars whose lifetimes have been prolonged by internal mixing is still considered plausible, although much of the observational evidence in support of mixing applies also to some of the models involving binary coalescence or mass transfer (see Livio 1993 and Stryker 1993 for a detailed review of blue-straggler models). Most of the nearly 700 blue stragglers observed in globular clusters so far have been found in sparse clusters (Ferraro et al. 1995), although recent high-resolution Hubble Space Telescope and ground-based observations (Paresce 1993, Yanni et al. 1994, Ferraro et al. 1997) show that blue stragglers are found in abundance in more concentrated  151  Chapter 4.  The Blue Straggler Population  152  of M55  clusters as well. It is generally accepted that blue stragglers have formed through a merger of two (or more) less-massive stars, although the nature and the details of the merger process are still far from certain. Recent reviews of the proposed merger mechanisms — direct stellar collisions, binary coalescence, mass transfer in a binary system or binary-binary collisions — can be found in Livio (1993), Stryker (1993), Leonard (1996) and Mateo (1996). The discovery of eclipsing variables among the blue stragglers in NGC 5466 (Mateo et al. 1990) provided the first direct evidence that blue stragglers are closely linked to binary stars. Since then more eclipsing binaries have been found among the blue stragglers in both young and old clusters (see Mateo 1996 for a summary). It is clear now that blue stragglers are highly visible tracers of cluster binary populations and their evolution, especially in low-concentration clusters such as M55, where a higher fraction of the primordial binaries is expected to have survived. The first observations of blue stragglers in M55 appear to be those of Sarajedini (1993), who reported the discovery of five blue stragglers.  Zaggia et al. (1994) also  noted the presence of blue stragglers in the central region of M55, although they did not investigate their properties. In both cases either B, V or V, I filters were used, and as was noted earlier in Chapter 3, these filter combinations produce colour indices that have a lower temperature resolution than the B — I colour index. This makes it difficult to separate the fainter blue stragglers from the turnoff stars and as a result only the brightest and bluest of the cluster blue stragglers are discovered in such surveys.  4.1  The Blue-Straggler Sample — Definition and Completeness  In Figure 4.11 have reproduced for convenience the V, B—I colour-magnitude diagram for the core field of M55 (Figure 3.10 from Chapter 3). A prominent blue-straggler sequence can be seen extending from ~ 0.7 mag below the main-sequence turnoff to ~ 2 magnitudes  Chapter 4.  The Blue Straggler Population  153  of M55  Figure 4.1: V, B — I colour-magnitude diagram for all 9137 stars having errors less than 2<r d both in V and B—I, and % < 1.4. The four red giants with % > 1.4 are shown by star symbols, the known RR Lyr stars are marked by open triangles and the suspected RR Lyr variables are shown by plus signs. The blue and red edges of the instability strip are shown by dotted lines. This is an exact copy of Figure 3.10 reproduced here for convenience. 2  me  2  Chapter  4.  The Blue Straggler Population  of M55  154  brighter than the turnoff. In order to put the blue stragglers selection on a more objective basis, I calculated the dispersion CTB-I around the ridge line (Table 3.4) as a function of magnitude and then compiled an initial Ust of blue stragglers candidates consisting of all stars brighter than V = 18.65 and with B — I indices more than 2><TB-I away from the ridge line. This approach ensured that most of the M S and turnoff stars scattered into the blue stragglers domain by photometric errors were avoided when making up the blue-straggler list. T h e blue-straggler region of the C M D is shown i n more detail i n Figure 4.2, where the dashed line marks the adopted separation between the blue stragglers candidates and the bulk of the turnoff stars. A s seen there, I have left out the objects above the main-sequence turnoff that could be possible blends of two turnoff stars. A few stars were removed from the initial list because they were too far to the red of the blue stragglers region in the B — V and V — I C M D s .  Also, two more blue stragglers candidates were  excluded as they did not converge in less than 200 iterations in A L L F R A M E , indicating that their images were either non-stellar (e.g. faint blue galaxies) or severely blended. I n the end, a careful visual inspection of the remaining blue stragglers candidates ensured that all of them looked "normal" and their photometry was not compromised by obvious cosmetic defects, diffraction spikes and the like. The final blue straggler sample, a total of 76 stars, is plotted i n Figure 4.2 by larger symbols (dots and circles) and listed i n Table 4.1. A finder chart for the blue stragglers in the core field is shown i n Figure 4.3, where the stars marked by squares are the supra blue stragglers discussed later i n this chapter. A natural question is how many of the stars identified here as blue straggler stars are i n fact field stars which only appear to be cluster blue stragglers. One possible source of background contamination is the Galactic bulge since M55 is projected against its outer, low-density portion (see Chapter 2). However, the bulge stars would be located redward  Chapter 4. The Blue Straggler Population of M55  155  Figure 4.2: The final BSS sample is shown by larger dots and empty circles. The dashed line marks the adopted separation between the blue stragglers candidates and the turnoff stars.  Chapter 4. The Blue Straggler Population of M55  156  Figure 4.3: Finder chart for the 76 blue stragglers in the core of M55. Stars marked by squares are the supra blue stragglers discussed in the text. Star numbers increase with right ascension. The field shown here is 4' x 4 ' . North is up and west is to the left.  Chapter  4.  The Blue Straggler Population  of M55  157  Table 4.1: Photometry for the blue stragglers in the core of M55.  #  V  1 16.158 2 17.324 3 18.112 4 17.906 5 17.123 6 18.711 7 16.476 8 17.171 9 18.046 10 18.215 11 17.456 12 17.717 13 17.111 14 17.566 15 17.792 16 15.932 17 16.893 18 18.567 19 18.620 20 18.170 21 17.383 22 18.301 23 18.308 24 16.953 25 17.571 26 18.657 27 18.325 28 17.030 29 18.131 30 17.500 31 18.383 32 15.866 33 18.055 34 17.241 35 17.460 36 17.594 37 16.853 38 16.873  B-V  0.311 0.252 0.370 0.442 0.346 0.513 0.246 0.359 0.455 0.409 0.459 0.444 0.210 0.379 0.410 0.138 0.196 0.448 0.451 0.407 0.300 0.457 0.434 0.355 0.453 0.540 0.428 0.226 0.442 0.491 0.343 0.449 0.380 0.254 0.429 0.450 0.380 0.277  U-B  - 0 206 - 0 014 - 0 126 - 0 121 - 0 032 +0 079 +0 067 - 0 050 - 0 102 - 0 113 - 0 137 - 0 148 +0 061 - 0 109 - 0 165 - 0 075 +0 075 - 0 208 - 0 123 - 0 198 - 0 021 - 0 151 - 0 200 +0 021 - 0 146 - 0 033 - 0 223 +0 080 - 0 160 - 0 037 - 0 154 - 0 041 - 0 159 +0 050 - 0 108 -0.124 -0.044 -0.016  V-I  #  0.526 0.484 0.627 0.625 0.452 0.613 0.332 0.464 0.619 0.703 0.626 0.655 0.525 0.515 0.507 0.259 0.275 0.595 0.646 0.607 0.493 0.661 0.600 0.538 0.640 0.588 0.673 0.442 0.621 0.611 0.650 0.562 0.537 0.374 0.593 0.638 0.562 0.474  39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76  V  18.155 17.237 16.694 17.799 17.089 18.147 17.873 18.038 17.371 17.202 17.371 16.997 17.978 17.485 18.536 16.707 17.941 17.823 18.133 16.174 16.561 18.353 16.770 17.896 17.831 18.345 16.858 18.275 17.988 17.348 18.023 16.636 16.659 17.856 17.066 18.051 18.413 16.413  B-V  0.439 0.411 0.260 0.451 0.423 0.451 0.458 0.419 0.333 0.326 0.281 0.132 0.457 0.455 0.412 0.259 0.452 0.412 0.466 0.226 0.284 0.462 0.232 0.338 0.384 0.454 0.273 0.461 0.429 0.376 0.415 0.259 0.260 0.395 0.323 0.454 0.473 0.243  U- -B  - 0 161 - 0 125 +0 050 - 0 137 - 0 113 - 0 155 - 0 149 - 0 165 - 0 025 - 0 018 - 0 066 +0 064 - 0 032 - 0 137 - 0 218 +0 278 +0 033 - 0 108 - 0 114 +0 123 +0 152 - 0 143 +0 060 - 0 114 - 0 068 - 0 153 +0 038 - 0 124 - 0 112 - 0 049 - 0 128 +0 103 +0 106 - 0 122 - 0 002 - 0 119 - 0 154 +0. 080  V-I  0 597 0 611 0 357 0 643 0 654 0 611 0 653 0 500 0 397 0 366 0 503 0 233 0 564 0 661 0 663 0 340 0 663 0 607 0 637 0 381 0 371 0 595 0 366 0 545 0 457 0 591 0 368 0 614 0 592 0 491 0 546 0 354 0 343 0 557 0 456 0 656 0 591 0 329  Chapter 4.  The Blue Straggler Population  of M55  158  of the M55 turnoff (see Figure 2.6) and therefore would contribute little to the mostly blue stars in the blue-straggler sequence. In addition, the V, V — I colour-magnitude diagram of the comparison field north of M55 (Figure 2.7) shows no stars at all at the location of the blue-straggler region. One can conclude therefore that there is a negligible field star contamination of the blue-straggler sample. As seen in Figure 4.1, most of the blue stragglers in M55 form a surprisingly tight sequence extending from V ~ 18.6 to V ~ 16, and I shall refer to it as the blue-straggler main sequence. Excluding the four brightest stars and the two stars to the blue of the blue-straggler main sequence, the blue-straggler sample can be divided into two groups: the blue-straggler main sequence itself (solid dots) and the group of ten stars located ~ 0.75 mag above it (circles). The separation between the two groups is somewhat arbitrary at the red edge and it is possible that the few reddest blue stragglers are in fact normal turnoff stars. These ten stars (which I named supra blue stragglers) occupy a location suggesting that at least some of them may be binary blue stragglers and I will discuss this hypothesis later. Another interesting feature of the blue-straggler main sequence is the apparent gap in the blue-straggler distribution at V ~ 17.5. The core field studied here covers only the central one core radius of M55 so this gap may not exist if a larger area were surveyed. It is worth mentioning, however, that Sarajedini (1992) found a gap at the same location (My ~ 3.6) in the combined blue-straggler luminosity function of NGC 5897 and NGC 6101. The completeness of the core field photometry was investigated in Chapter 2 and it was shown there that in the B, V and I bands the data were complete to ~ 0.5 mag below the turnoff. It should be clear from the colour-magnitude diagram, however, that around the turnoff and fainter the completeness of the blue-straggler sample will be determined mostly by the difficulty of separating bona fide blue stragglers from turnoff stars and not by the photometric incompleteness (crowding and sky noise). Therefore I estimated the  Chapter 4.  The Blue Straggler Population  of M55  159  completeness of the blue-straggler sample by generating 1500 artificial blue stragglers in ten separate experiments. The input V magnitudes were uniformly distributed between 15.85 and 18.70, and the colours of the artificial stars were assumed to be represented by a straight line through the blue-straggler region. From these experiments I estimated that the completeness / of the blue-straggler sample is ~ 1 for stars brighter than V — 17.5 and / Ri 0.8 at V — 18.1, where most of the incompleteness is caused by the scatter of blue stragglers into the turnoff region. Fainter than V ~ 18.3 the values of / become rather uncertain as an increasing fraction of the added blue stragglers is recovered to the red of the dashed line shown in Figure 4.2. Thus, while one can clearly see blue stragglers fainter than the main-sequence turnoff of M55, their completeness is difficult to estimate since it is not known what fraction of the faint blue stragglers lies redward of the line used to select them in the colour-magnitude diagram.  4.2  Radial Distribution  Several studies of blue straggler stars in globular clusters have found them to be more centrally concentrated than the subgiant branch and red giant branch stars of similar brightness, first in NGC 5466 (Nemec & Harris 1987) and subsequently in many other clusters (Stryker 1993). In Figure 4.4 I compare the cumulative radial distributions for three groups of stars in the core field: 53 stars from the blue-straggler main sequence, 1880 turnoff and subgiant branch stars, and the ten stars that were provisionally named supra blue straggler stars(open circles in Figure 4.2). For this comparison I used only stars with V < 18.25 since the fainter blue stragglers are not as complete as the turnoff stars of the same brightness. A one-sided, two-sample Kolmogorov-Smirnov test applied to the cumulative distributions indicated that the 63 blue stragglers in the whole sample are more centrally concentrated than the SGB stars at the 95% significance level, a  Chapter 4. The Blue Straggler Population of M55  i  1  1  1  1  1  160  1 r  r(arcsec)  Figure 4.4:  T h e cumulative radial distributions for the total blue-straggler  sample  (dashed line), the turnoff and subgiant stars in the same magnitude, range as the blue stragglers (solid line), and the supra blue straggler stars (dotted line)  Chapter 4.  The Blue Straggler Population  of M55  161  result that is similar to the probabilities obtained in other studies. The difference in the radial distribution implies that the blue stragglers in M55 are more massive than the ~ 0.75Af© subgiants and turnoff stars, as would be expected if they were formed by the merger of two less massive stars. If this is indeed the explanation for their stronger central concentration, it implies also that most of the mergers should be old enough so that mass segregation can change their radial distribution noticeably. What is also interesting is that the ten supra blue straggler stars follow the distribution of the "normal" blue stragglers; this runs counter to the earlier suggestion that these ten stars might be binary blue stragglers. On the other hand, the artificial-star experiments showed that only a few percent of the recovered blue stragglers are scattered by blends and photometric errors to the supra blue-straggler region, that is, it is unlikely that those ten stars are all blends of two fainter stars. It is also possible that these are either older, less-massive blue stragglers that have evolved away from the main sequence, or they have lower envelope helium content compared with the rest of the blue stragglers. Binary systems consisting of a blue straggler and a main-sequence star have been found in other clusters, a typical example being NJL 5 in u> Cen (Helt et al. 1993). Such systems could have formed in binary-binary collisions or by merger of the close pair in a hierarchical triple, as proposed by Leonard k Fahlman (1991) and Leonard (1996).  4.3  Origin and Evolutionary Status  It is now almost universally accepted that blue stragglers are formed through a merger of two less massive stars, either by mass transfer/coalescence in a binary system or by direct stellar collisions (Benz k Hills 1987, Mateo et al. 1990, Leonard k Fahlman 1991, Leonard 1996, Mateo 1996). Recent studies by Lombardi et al. (1996), Sandquist et al. (1997), Sills et al. (1997) and Ouellette k Pritchet (1998) have focused on the amount  Chapter 4.  The Blue Straggler Population  of M55  162  of mixing during the merger process and how it affects the merger products' Kfetimes and location in the colour-magnitude diagram. These newer simulations indicate that, contrary to what was assumed before, colHsional remnants are not well mixed and have composition profiles similar to those of the parent stars; the models of Sandquist et al. (1997) found that this was true for binary mergers as well. In addition, Sandquist et al. (1997) followed the evolution of both mixed and unmixed mergers and suggested that because of their hydrogen-rich cores, fully mixed blue stragglers should populate a relatively narrow locus along the zero-age main sequence, as opposed to unmixed mergers which were predicted to spend the larger fraction of their lifetimes away from the zero-age main sequence. Similar conclusions were reached also by Ouellette & Pritchet (1998). In Figure 4.5 I compare the distribution of the blue stragglers in the colour-magnitude diagram with the zero-age main sequence and a 4-Gyr isochrone from Bertelli et al. (1994), whose models were calculated for Z = 0.0004 and scaled solar abundances of the a-elements. This is a good approximation to an a-enhanced composition for M55 with [Fe/H] = —1.9 and [a/Fe] = 0.3. The zero-age main sequence (solid line) and the 4-Gyr isochrone (dotted Une) have been shifted so that the turnoff of the 14 Gyr isochrone from the same set coincides with the cluster turnoff. While this shift gives different values for the distance modulus and reddening of M55 than the one adopted here, I am more interested in the location of the blue stragglers relative to the cluster main sequence, and therefore smaU absolute colour and magnitude errors in the models are unimportant. The 4-Gyr isochrone is not used to derive an age for the blue stragglers, but is used to show that the blue stragglers in M55 closely resemble sUghtly evolved stars, that is stars with enhanced core heUum content (see below). Indeed, some blue stragglers could be very young, and some faint ones could be a few 10 years old. 9  One can see that, unUke the blue stragglers in many other clusters, the majority of the blue stragglers in M55 form a relatively narrow sequence similar to the single-age  Chapter 4. The Blue Straggler Population of M55  163  B-I  Figure 4.5: The blue stragglers distribution in the colour-magnitude diagram compared to the zero-age main sequence (solid line) and a 4 Gyr isochrone (dotted line) for single stars with the metaUicity of M55.  Chapter 4.  The Blue Straggler Population  of M55  164  population of ordinary cluster stars. This small width indicates that most blue stragglers are in their longest-lived evolutionary stage, presumably core hydrogen burning. It is unlikely that these are completely mixed merger/collisional remnants, as the fully-mixed models of Sandquist et al. (1997) spend much of their life close to the zero-age main sequence and so one should see a concentration of blue stragglers near the zero-age main sequence, something that is clearly not observed for the brighter blue stragglers in M55. These are found much higher than the zero-age main sequence and resemble stars that have already evolved away from the main sequence. I conclude therefore that most of the blue straggler stars in M55 have helium-enriched cores, but not envelopes, similar to what is predicted for unmixed merger remnants (Sandquist et al 1997, Sills et al. 1997). It follows then that the observed blue-straggler sequence represents a core helium-enriched main sequence and that the newly-formed blue stragglers should begin their life at its lower envelope. As can be seen in Figure 4.5, for the less massive blue stragglers this lower envelope approaches the single-star zero-age main sequence, in agreement with the scenario outlined in Sandquist et al. (1997) — the progenitors of the low-mass blue stragglers are low-mass single stars with little core helium enrichment and therefore one should find the former close to the zero-age main sequence. It should be noted, however, that the bright blue stragglers observed in M55 are much higher above the zeroage main sequence than predicted by Sandquist et al. (1997) for the unmixed massive blue stragglers. Adopting the 4-Gyr isochrone in Figure 4.5 as the blue-straggler main sequence lower envelope, onefindsthat it is about 0.5 mag brighter than the zero-age main sequence at (B — V)o = 0.18, whereas the zero-age unmixed models of Sandquist et al. (1997) are more luminous than the zero-age main sequence by 0.25 mag at (B—V)o = 0.04 and by much less (~ 0.1 mag) at (B — V)o = 0.18. This discrepancy suggests that at a given mass, the M55 blue stragglers have a larger core helium content than predicted by the models of Sandquist et al. (1997).  Chapter  4.  The Blue Straggler Population  of M55  165  In Figure 4.5 there are several bright blue stragglers that are not on the blue-straggler main sequence. Two or three of them he on or very close to the zero-age main sequence and may well be fully-mixed mergers/collisional remnants. Since such blue stragglers are expected to have long main-sequence lifetimes, their small number indicates that complete mixing is a very rare event. The bright object located on the extension of the blue-straggler main sequence is probably a massive unmixed remnant, and three more blue stragglers appear to be on the subgiant branch. Since the brightest stars on the blue-straggler main sequence are all evolved objects, they are not the most massive blue stragglers. Their mass is probably close to the turnoff mass of the 4-Gyr isochrone (~ 1.1A4®),  while the two presumably unmixed blue stragglers near the zero-age main  sequence (at V ~ 17.0) would have masses of ~ 1.3A4©. The fact that we do not see more massive objects also supports the conclusion that most blue stragglers in M55 are unmixed merger/mass transfer remnants: such objects are formed with high core helium content and therefore should evolve rapidly to the red giant branch. As Sandquist et al. (1997) found that little or no mixing occurs in either the collisional or binary merger case, it is difficult to distinguish observationally between the two scenarios in the way suggested by Bailyn & Pinsonneault (1995). Given the low central density of M55, however, it is unlikely that the blue stragglers in M55 are products of direct single-single stellar collisions of the type considered by Benz & Hills (1987). Without information on the numbers and the properties of binary blue stragglers in M55 it is difficult to choose between the other possible formation scenarios — binary coalescence, mass transfer in a binary system (Mateo et al. 1990) or binary-binary collisions (Leonard & Fahlman 1991, Leonard et al. 1992b). If any of these scenarios always results in a mixed remnant, however, it should be dismissed as a possible formation mechanism for most blue stragglers in M55. Of course, it is also possible that more than one mechanism is at work in M55, as implied by the apparent presence of one or two fully-mixed blue  Chapter  4.  The Blue Straggler Population  of M55  166  stragglers in the colour-magnitude diagram. Several formation mechanisms have been proposed also for the blue stragglers in M3 (Ferraro et al. 1993, Sigurdsson et al. 1994, Ferraro et al 1997) and M67 (Leonard 1996). The isochrone in Figure 4.5 suggests that some of the stars seen to the blue of the red giant branch, as well as the few stars at the base of the asymptotic giant branch may be the descendants of massive blue stragglers. This question has been discussed extensively by Fusi Pecci et al (1992) and here I note only that this population of "blue" giant-branch objects may also include blends (or even binaries) in which one component is a red giant and the other one is a turnoff star or, less likely, a blue straggler.  Chapter 5  Conclusions  In this thesis I have presented and analyzed photometry from two fields in the metalpoor globular cluster M55 with the purpose of understanding its stellar content and relating it to several issues in contemporary astrophysics, the most important of which include: the halo mass function and its relation to the dark matter problem; globular cluster evaporation and destruction; the ages of globular clusters and the age of the universe; observational tests of stellar evolution theory and structure and evolution of stellar mergers. The mainfindingsof this work are summarized in the next three sections and Appendix A presents in a condensed form the values of important parameters for M55 derived in this work.  5.1  The Main Sequence of M 5 5  From the U — B vs. B — V colour-colour diagram and independently using Sarajedini's (1994) technique, I derived a reddening of E -v B  = 0.13 ± 0.02 and E -i V  = 0.17 ± 0.02,  with the values from the two determinations in excellent agreement. The second method also yielded a metaUicity estimate of [Fe/H] = —1.98, also in very good agreement with recent spectroscopic values of [Fe/H] = —1.95. These are the first direct determinations of the reddening in the direction of M55 and I beUeve that the high quality of the photometry makes them fairly reliable. A distance modulus of (m — M)v — 14.02 ± 0.08 was derived by fitting the main sequence of M55 to a sample of 12 nearby subdwarfs with weU-determined parallaxes 167  Chapter 5.  168  Conclusions  from the Hipparcos Catalogue.  This determination depends critically on the nature of  the biases in the Hipparcos sample; I adopted the corrections derived in Pont et al. (1998), whose results are backed by extensive Monte-Carlo simulations of the Hipparcos data. If no corrections are applied to the absolute magnitudes of the subdwarfs (the approach advocated by Gratton et al. 1997 and Chaboyer et al. 1998), the distance modulus of M55 is (m-M)v  = 14.07 ±  0.08.  An estimate of 14 ± 1.2 Gyr for the absolute age of M55 was obtained using the absolute magnitude of the main-sequence turnoff, a robust indicator which is independent of the model colour calibrations. The same result is obtained even if the larger value of (m — M)v — 14.07 for the distance modulus of M55 is used and it suggests that M55 is intrinsically older than the majority of globular clusters for which Hipparcos  distances  and ages have been derived. This age of M55, combined with the finding by Pont et al. (1998) that M92 is also about 14 Gyr old, implies that the conflict between the ages of the oldest globular clusters and the age of the Universe inferred from recent determinations of the Hubble constant HQ remains unresolved. The main-sequence luminosity function of M55 is different from the luminosity functions of the metal-poor clusters M15, M30 and M92. Depending on the way the four luminosity functions are normalized, M55 either has an excess of bright stars on the upper main sequence (3.5 £ Mi <> 5.5) at the ~ 2o~ level, or a pronounced deficiency of faint stars for Mi > 6 at the ~ 5<r level. Mass segregation is unlikely to cause the differences between the luminosity functions, since the observed fields in all four clusters are located well beyond the half-mass radius. Under the assumption that all for clusters started out with similar luminosity functions, the difference between the luminosity function of M55 and the other three clusters can be interpreted as a deficiency of low-mass stars caused by loss of stars through evaporation and tidal shocks. The alternative explanation — that M55 developed an excess of bright stars in the absence of mass segregation seems rather  Chapter 5.  Conclusions  169  unphysical. Together with earlier results on the luminosity function of NGC 6397, M55 appears to provide evidence that the low-mass stellar populations of massive globular clusters can be depleted, most likely by evaporation, tidal stripping and disk and bulge shocks. It is also possible that the differences in the luminosity functions are primordial, reflecting different star formation conditions or different early evolution of the cluster populations. At ~ 2 core radii the mass function of M55 is fairly flat, with a slope at the low-mass end of x = 0.7 ± 0.2. This value implies that while low-mass stars make a significant contribution to the total mass of the cluster, they are not the dominant mass component. This result, combined with the data on similarly shallow mass function slopes in clusters observed with the Hubble Space Telescope, puts in doubt the suggestion that very lowmass stars can provide the solution to the dark matter problem of the halo. In the field of M55 there are two other distinct populations, namely stars from the Sagittarius dwarf galaxy and a sparse sample of stars belonging to the Galactic bulge. Neither the Sagittarius dwarf nor the bulge stars contribute significantly to the mainsequence luminosity function of M55.  5.2  The Evolved Populations of M 5 5  The horizontal branch of M55 has a predominantly blue morphology, with an extended blue tail reaching about two magnitudes below the level of the horizontal branch. The presence of a long blue tail contradicts the suggestion by Fusi Pecci et al. (1992) and Buonanno et al. (1997) that such tails are related to a high-density cluster environment and are found only in centrally concentrated, massive clusters. Three of the stars on the horizontal branch are within or near the RR Lyr instability strip but are not known variables. I have listed them as suspected RR Lyr stars and  Chapter 5.  170  Conclusions  provided a finder chart to facilitate their further observation. The suggestion that they are RR Lyr variables is based only on their location in the colour-magnitude diagram and not on detected variability. The red giant branch of M55 has been observed from nearly its tip to the subgiant branch. The red-giant clump (RGB clump) was confidently identified both visually (in the colour-magnitude diagram) and in the luminosity function. The mean apparent magnitudes of the RGB clump are 7 /dump  M  cium  = 14.236 ± 0.007, / i  d ) i m p  c l u m p  = 15.094 ± 0.007 and  = 13.151 ± 0.007, corresponding to absolute magnitudes of M P  =  Q  9  4  ±  0  o  g  a  n  d  M  cium  P  =  _  Q  7  Q  ±  Q Qg  I n  a l l  pass  L U M P Y  = 0.22 ± 0.08,  bands, the observed  luminosity of the clump is lower than what is predicted by theoretical models. Expressed in terms of the distance-independent differences  AV^unp  a n  d  A-^ciSnp>  the discrepancy  is 0.2 - 0.3 mag for the [a/Fe] = +0.3 models and 0.1 - 0.15 mag for the more a-rich [a/Fe] = +0.6 models. I conclude that the RGB clump discrepancy is not completely resolved by adopting a higher degree of a-enhancement and convective overshoot may have to be invoked to remove the disagreement completely. The width of the red giant branch of M55 was measured for the first time and it was found that photometric errors alone cannot account for the full width of the giant branch, that is the giant branch of M55 has a non-zero intrinsic width. I derived intrinsic widths of o-i(B-V) = 0.012, <rj(V-J) = 0.011 and ai(B-I) = 0.014, which put an upper limit of 0.10 dex on the chemical abundance spread among the stars of M55. The ratios of the number of stars on the red giant branch, the horizontal branch and the asymptotic giant branch were found to be in a good agreement with theoretical models, in particular they support the canonical treatment of semiconvection in horizontal-branch stars. The helium abundance of M55 was determined by means of the R method as calibrated by Buzzoni et al. (1983) and Caputo et al. (1987). The helium mass fraction in the cluster stars was found to be Y = 0.24iooi9, which is in excellent  Chapter 5.  Conclusions  171  agreement with the estimates of the primordial helium abundance (Ip = 0.24±0.01) and supports the use of the Y = 0.2356 evolutionary models of VandenBerg et al. (1998b). The most important result from the study of the giant-branch and turnoff luminosity functions is that neither the V-band nor the /-band luminosity functions show any significant deviation from the theoretical luminosity functions. In particular, there is no evidence for a deficiency of main-sequence stars compared to the number of stars on the subgiant and red giant branchs. I also found no evidence in the luminosity function for the so-called subgiant excess that has been observed in some metal-poor clusters — in both V and / the observed and predicted numbers agree very well through the whole subgiant region. The only noticeable discrepancy between theory and observations is in the location of the red giant branch bump — theoretical models consistently predict a magnitude for the RGB bump that is ~ 0.2 mag brighter than the observed one. It has been suggested recently that one way to achieve agreement between the numbers of giant-branch stars and turnoff stars in metal-poor clusters is to include core rotation in stellar evolutionary models (VandenBerg et al. 1998a). This approach works very well in the case of M30, however in M55 the absence of discrepancy between observations and canonical models (without rotation) implies that the stars in M55 have cores that rotate too slowly to affect evolution on the giant branch. Since M55 is structurally very different from the other four clusters in which the giant-branch - main-sequence discrepancy has been found, it is possible that the lack of significant core rotation in M55 is somehow connected to its dynamical history. For example, the distribution of angular momentum between orbital motion and stellar rotation could have been different in clusters that today we classify as "strongly" or "weakly" centrally concentrated. Ultimately, the suggestion that stellar rotation and cluster structure are connected can be confirmed or rejected observationally by (a) determining stellar rotation rates in globular clusters of similar metallicity but different central concentration, and (b) investigating whether the  Chapter 5.  Conclusions  172  giant-branch - main-sequence discrepancy in low-metallicity clusters is related to their structural parameters. As it stands now, M55 is the only metal-poor cluster in which the relative numbers of giant-branch and turnoff stars agree with the predictions of canonical evolutionary models.  5.3  The Blue Straggler Stars in M 5 5  I have identified and presented UBVI photometry for 76 blue stragglers in M55. Most of them form a tight sequence extending from 0.6 mag below the main-sequence turnoff to about 2 mag brighter than the turnoff. I believe that M55 is the first globular cluster in which blue stragglers have been confidently identified fainter than the cluster turnoff and for which a nearly complete sample brighter than the turnoff exists (with the possible exception of M3). I found that, as in many other clusters, the blue stragglers in M55 are more concentrated towards the cluster centre compared to the subgiants of similar brightness. Taking into account the small width of the blue-straggler sequence and its location relative to the cluster zero-age main sequence, I conclude that the blue stragglers in M55 are born with helium-enriched cores but not envelopes, thus resembling stars that have already evolved away from the main sequence. The degree of enrichment appears to increase with mass, as implied by the widening gap between the cluster zero-age main sequence and the blue-straggler main sequence as one goes to higher luminosities. The second important conclusion is that the observed blue straggler sequence represents the equivalent of a core helium-enriched main sequence where the blue stragglers spend most of their lives. The observations agree qualitatively with the unmixed collisional/merger models of Sandquist et al. (1997) and I conclude that the majority of  Chapter 5.  Conclusions  173  the blue stragglers in M55 are unmixed binary mergers or mass-transfer remnants. Collisional origin cannot be ruled out completely, especially from binary-binary collisions which could be common in sparse clusters. The absence of information on the frequency of binary blue stragglers in M55 makes it difficult to choose a specific merger scenario, but the homogeneity of the blue stragglers sample suggests that either a single formation mechanism is dominant in M55, or the variety of formation routes produce blue stragglers with uniform properties.  5.4  Future Directions  While this thesis has presented a fairly comprehensive study of M55, several questions either remain unanswered or need to be addressed in more detail by additional work. Some of the problems that I consider most important include: • The differences between the luminosity functions of M55 and other clusters. Evidence for or against intrinsic variations in the shape of the global luminosity function can be provided by a detailed dynamical modelling of multi-field observations. Deep photometry in two additionalfields,one far into the envelope of M55 and another interior to the present 2-core radii field should allow a robust determination of the global luminosity and mass functions of M55. A more distant and ambitious goal would be to explore the cluster-to-cluster variations of the main-sequence luminosity functions, particularly in clusters inside the solar circle where tidal stripping and bulge shocks are predicted to be particularly strong. From an observer's point of view this a very challenging project because many of the bulge clusters suffer from a significant background contamination and deep luminosity functions may be very difficult to derive.  Chapter 5.  Conclusions  174  • Is M55 the only metal-poor globular cluster whose luminosity function agrees with the theoretical luminosity function? One can attempt to answer this question by deriving luminosity functions for large samples of evolved and turnoff stars in metalpoor clusters of different central concentration. While this would be a rather big project, the advent of large-format CCD cameras should allow its completion in a reasonable amount of time. • Related to the previous question is the problem of the rotation rates of globularcluster stars, in M55 as well as other clusters. The pioneering work of Peterson (1983, 1985a, 1985b) has shown that horizontal-branch stars in globular cluster rotate with velocities of up to v sinz ~ 30 km/s. The suggestion that the agreement between theory and observations in M55 may be produced by the absence of rotation in the cluster stars can be checked by measuring the rotation rates for stars on the blue horizontal branch of M55. • Search for eclipsing and pulsating variables among the blue stragglers in M55. Similar searches in other globular clusters have found that ~ 20% of the their blue stragglers are eclipsing binaries or SX Phe stars (short-period pulsating variables), although there are clusters where repeated systematic searches have not found any variable stars. The photometric properties, relative frequencies and period distribution of the potential eclipsing binaries provide important information on the origin and evolution of the blue stragglers, as well as on the fraction of primordial binaries in M55. SX Phe stars are potentially useful as distance indicators and their periods and period changes can provide independent estimate of the blue-straggler masses.  Appendix A  Summary of some important parameters for M 5 5  In the following table I present a summary of some important parameters for M55 as they have been derived or improved upon in this thesis. Table A . l : Some parameters of M55 as derived in this work Value ± l c r 0.13 ± 0 . 0 2 0.17 ± 0 . 0 2 14.02 ± 0 . 0 8 13.85 ± 0 . 0 8 13.62 ± 0 . 1 0 18.03 ± 0 . 0 5 17.32 ± 0 . 0 5 18.52 ± 0 . 0 5 14.46 ± 0 . 0 3 14.55 ± 0 . 0 5 0.93 ± 0 . 1 0.52 ± 0 . 0 8 1.49 ± 0 . 2 5 1.23 ± 0 . 1 9 0.21 ± 0 . 0 7 0.14 ± 0 . 0 4 -1.92 ± 0 . 1 0 0.24 ± 0 . 0 2 14 ± 1.2 Gyr  Parameter EB-V EV-I (m-M)v  (m-M)j (m-M)o VTO ITO BTO  V(HB) V(ZAHB) (B-R)/(B+V+R) B2/(B+V+R) R R' Ri  R2  [Fe/H] Y  Age  175  Appendix B  Publications Related to the Thesis  PreUminary results from Chapter 2 and Chapter 4 of this thesis have appeared in the following refereed publications: 1. On the Blue Straggler Population of the Globular Cluster M55, by G. I. Mandushev, G. G. Fahlman, H. B. Richer & I. B. Thompson, 1997, AJ, 114, 1060. 2. A Photometric  Study of the Globular  Cluster M55,  by G. I. Mandushev, G. G.  Fahlman, H. B. Richer & I. B. Thompson, 1996, AJ, 112, 1536. 3. The Main-Sequence  Stars of the Sagittarius  Dwarf  Galaxy,  by G. G. Fahlman,  G. I. Mandushev, H. B. Richer, I. B. Thompson & A. Sivaramakrishnan, 1996, ApJ, 459, L65. In addition, poster papers containing results from the thesis were presented at the Canadian Astronomical Society meetings in Penticton (May 1995), Kingston (June 1996) and Edmonton (1997). The results on the distance, age and the main-sequence mass function of M55 presented in the thesis differ from the ones published in Mandushev et al. (1996). In that work, the distance of (m — M)v = 13.90 and the age of 16-Gyr were based on the preHipparcos,  ground-based parallaxes of the subdwarfs used in the distance determination.  The use of the Hipparcos parallaxes in the thesis resulted in an increase of the distance estimate for M55 and as a consequence, a reduction of the age of the cluster from 16 to 14 Gyr. 176  Appendix  B.  Publications  Related to the Thesis  177  As far as the mass function of M55 is concerned, a shallower slope for the lowmass end was derived in the thesis — x fa 1 as opposed to the value of as PS 1.6 obtained in Mandushev et al. (1996). This difference arises from the use of different massluminosity relations —in Mandushev et al. (1996) the composite mass-luminosity relation of Fahlman et al. 1989 was used because of the lack of theoretical mass-luminosity relations for metal-poor, low-mass stars. By the time the thesis was being written, several mass-luminosity relations for low-mass stars had been published (see Chapter 2 for a discussion on the choice of the mass-luminosity relation used in the thesis).  Appendix C  Photometry Software Used in the Thesis  This appendix presents a brief description of the digital photometry and calibration software used in the thesis. The programs were developed by Peter Stetson at the Dominion Astrophysical Observatory and a detailed description of the programs can be found in the relevant papers cited in the data-reduction sections of Chapter 2.  C l  DAOPHOT II  This is a suite of programs for finding stars in the digital images and measuring their coordinates and brightness by means of aperture photometry. It was also used for deriving the point-spread function of the stellar images, as well as for adding artificial stars to the frames.  C.2  ALLSTAR  This is a program for performing profile-fitting photometry in crowded fields (such as in most globular clusters). The brightness of the stars is measured by fitting the pointspread function to the stellar profiles (which may be partially blended).  C.3  ALLFRAME  This is a program which, similarly to ALLSTAR, is used for profile-fitting photometry in crowded fields. It has the advantage of fitting the stellar profile simultaneously on  178  Appendix  C. Photometry  Software Used in the Thesis  179  all frames on which a particular star has been found. This results in a more precise photometry, especially in very crowded fields.  C.4  Calibration and Transformation Programs  These are the programs in the "CCD Package" distributed by Peter Stetson and include DAOMATCH and DAOMASTER for cross-referencing the stars in several frames of the same field, DAOGROW for growth-curve analysis of aperture photometry, as well as the following programs for collectiong observational information and calibrating the profile-fitting photometry: COLLECT, CCDLIB, CCDSTD, CCDAVE and FINAL.  Appendix D  Journal Abbreviations  Abbreviation  Journal Name  AJ  The Astronomical Journal  A&A  Astronomy & Astrophysics  A&AS  Astronomy & Astrophysics Supplement Series  ApJ  The Astrophysical Journal  ApJS  The Astrophysical Journal Supplement Series  ARA&A  Annual Review of Astronomy and Astrophysics  BAAS  Bulletin of the American Astronomical Society  MmSAI  Memorie della Societa Astronomica Italiana  MNASSA  Monthly Notices of the Astronomical Society of South Africa  MNRAS  Monthly Notices of the Royal Astronomical Society  PASP  Publications of the Astronomical Society of the Pacific  RMxAA  Revista Mexicana de Astronomia y Astrofisica  180  References  [Alcaino 1975]  Alcaino, G. 1975, A&AS, 22, 193  [Alcaino et al. 1992]  Alcaino, G., Liller, W., Alvarado, F., k Wenderoth, E. 1992, AJ, 104, 190  [Alexander et al. 1997]  Alexander, D. R., Brocato, E., Cassisi, S., Castellani, V., Ciacio, F., k Degl'Innocenti, S. 1997, A&A, 317, 90  [Alongi et al. 1991]  Alongi, M., Bertelli, G., Bressan, A., k Chiosi, C. 1991, A&A, 244, 95  [ArmandrofT 1993]  Armandroff,  T. E. 1993,  Galaxy Connection,  in The Globular Cluster -  ed. by G. Smith & J. Brodie (ASP,  San Francisco), ASP Conf. Series, 48, 48 [Arp et al. 1952]  Arp, H. C , Baum, W. A., k Sandage, A. 1952, AJ, 57, 4  [Arp et al. 1953]  Arp, H. C , Baum, W. A., k Sandage, A. 1953, AJ, 58, 4  [Axer et al. 1994]  Axer, M., Fuhrmann, K., k Gehren, T. 1994, A&A, 291, 895  [Axer et al. 1995]  Axer, M., Fuhrmann, K., k Gehren, T. 1995, 300, 751  181  AkA,  182  References  [Baade 1948]  Baade, W. 1948, PASP, 60, 230  [Bahcall k Yahil 1972]  Bahcall, J. N., & Yahil, A. 1972, ApJ, 177, 647  [Bailyn k Pinsonneault 1995]  Bailyn, C. D., k Pinsonneault, M. H. 1995, ApJ, 439, 705  [Baraffe et al. 1997]  Baraffe, I., Chabrier, G., Allard, F., k Hauschildt, P. H 1997, A&A, 327, 1054  [Beers et al. 1990]  Beers, T. C., Preston, G. W., Shectman, S. A., k Kage, J. A. 1990, AJ, 100, 849  [Benz k Hills 1987]  Benz, W. k Hills, J. G. 1987, ApJ, 323, 614  [Bergbusch 1990]  Bergbusch, P. A. 1990, AJ, 100, 182  [Bergbusch 1993]  Bergbusch, P. A. 1993, AJ, 106, 1024  [Bergbusch 1996]  Bergbusch, P. A. 1996, AJ, 112, 1061  [Bergbusch k VandenBerg 1992] Bergbusch, P. A., k VandenBerg, D. A. 1992, ApJS, 81, 163 [Bergbusch k VandenBerg 1997] Bergbusch, P. A., k VandenBerg, D. A. 1997, AJ, 114, 2604 [Bergbusch k VandenBerg 1998] Bergbusch, P. A., k VandenBerg, D. A. 1998, in preparation [Bertelh et al. 1994]  Bertelli, G., Bressan, A., Chiosi, C., Fagotto, F., k Nasi, E. 1994, A&AS, 106, 275  183  References  [Bolte 1994]  Bolte, M . 1994, A p J , 431, 223  [Borissova et al. 1997]  Borissova, J . , Catelan, M . , Spassova, N . , k Sweigart, A. V . 1997, A J , 113, 692  [Buonanno 1993]  Buonanno, R. 1993, in The Globular Cluster - Galaxy Connection,  ed. by G . Smith k J . Brodie (ASP, San  Francisco), A S P Conf. Series, 48, 131 [Buonanno et al. 1984]  Buonanno, R., Corsi, C. E., Fusi Pecci, F., Alcaino, G . , k Liller, W . 1984, A p J , 277, 220  [Buonanno et al. 1985]  Buonanno, R., Corsi, C. E . , k Fusi Pecci, F . 1985, AkA,  [Buonanno et al. 1989]  145, 97  Buonanno, R., Corsi, C. E . , k Fusi Pecci F. 1989, A & A , 216, 80  [Buonanno et al. 1994]  Buonanno, R., Corsi, C. E . , Fusi Pecci, F . , Fahlman, G. G., k Richer, H . B . 1994, A p J , 430, L121  [Buonanno et al. 1995]  Buonanno, R., Corsi, C. E . , Fusi Pecci, F . , Richer, H . B . , k Fahlman, G . G . 1995, A J , 109, 650  [Buonanno et al. 1997]  Buonanno, R., Corsi, C , Bellazzini, M . , Ferraro, F . R., k Fusi Pecci, F . 1997, A J , 113, 706  [Burbidge et al. 1957]  Burbidge, E . M . , Burbidge, G . R., Fowler, W . A . , k Hoyle, F . 1957, Revs. Modern Phys., 29, 547  [Burki et al. 1995]  Burki, G., Rufener, F . , Burnet, M . , Richard, C , Blecha, A . , k Bratschi, P. 1995, A & A S , 112, 383  184  References  [Burstein k Heiles 1982]  Burstein, D., k Heiles, C. 1982, AJ, 87, 1165  [Buzzoni et al. 1983]  Buzzoni, A., Fusi Pecci, F., Buonanno, R., k Corsi, C. E. 1983, A&A, 128, 94  [Capaccioli et al. 1993]  Capaccioli, M., Piotto, G., k Stiavelli, M. 1993, MNRAS, 261, 819  [Capriotti k Hawley 1996]  Capriotti, E. R., k Hawley, S. L. 1996, ApJ, 464, 765  [Caputo et al 1987]  Caputo, F., Martinez Roger, C , k Paez, E. 1987, A&A, 183, 228  [Carney 1980]  Carney, B. W. 1980, AJ, 85, 38  [Carney 1983a]  Carney, B. W. 1983a, AJ, 88, 610  [Carney 1983b]  Carney, B. W. 1983b, AJ, 88, 623  [Carney 1993]  Carney, B. W. 1993, Connection,  in The Globular Cluster - Galaxy  ed. by G. Smith k J. Brodie (ASP, San  Francisco), ASP Conf. Series, 48, 234 [Carney 1996]  Carney, B. W. 1996, PASP, 108, 900  [Carney k Aaronson 1979]  Carney, B. W., k Aaronson, M. 1979, AJ, 84, 867  [Carney et al. 1992]  Carney, B. W., Storm, J., k Jones, R. V. 1992, ApJ, 386, 663  [Castellani et al. 1997]  Castellani, V., Ciacio, F., Degl'Innocenti, S., Fiorentini, G. 1997, A&A, 322, 801  185  References  [Cayrel de Strobel et al 1992]  Cayrel de Strobel,  G., Hauck, B., Francois, P.,  Thevenin, F., Friel, E., MermiUiod, M., & Borde, S. 1992, A&AS, 95, 273 [Chaboyer 1995]  Chaboyer, B. 1995, ApJ, 444, L9  [Chaboyer et al. 1995]  Chaboyer, B., Demarque, P., & Pinsonneault, M. H . 1995, ApJ, 441, 876  [Chaboyer et al. 1996]  Chaboyer, B., Demarque, P., & Sarajedini, A. 1996, ApJ, 459, 558  [Chaboyer et al. 1998]  Chaboyer, B., Demarque, P., Kernan, P. J., Sz Krauss L. M. 1998, ApJ, 494, 96  [Chernoff 1988]  Chernoff, D. F. 1988, in Globular Cluster Systems in ed. by J. Grindley & A. G. D. Philip (Reidel,  Galaxies,  Dordrecht), p. 283 [ChernofT & Weinberg 1990]  Chernoff, D. F., & Weinberg, M. D. 1990, ApJ, 351, 121  [Clement 1996]  Clement, C. M. 1996, BAAS, 189, #81.13  [Clement & Shelton 1996]  Clement, C. M., & Shelton, I. 1996, 112, 618  [Clement et al. 1995]  Clement, C. M., Bezaire, J., & Giguere, D. 1995, A J , 110, 2200  [Dahn et al. 1995]  Dahn, C. C , Liebert, J., Harris^ H. C , & Guetter, H. H. 1995, in The Bottom of the Main Beyond,  Sequence and  ed. by C. G. Tinney (Springer, Berlin), p. 239  186  References  [Daniel 1993]  Daniel, J. S. 1993, PhD Thesis, Michigan Univ., Ann Arbor  [D'Antona k Mazzitelli 1996]  D'Antona, F., k Mazzitelli, I. 1996, ApJ, 456, 329  [Dean 1981]  Dean, J. F. 1981, MNASSA, 40, 14  [Dean et al. 1978]  Dean, J. F., Warren, P. R., k Cousins, A. W. J. 1978, MNRAS, 183, 569  [Degl'Innocenti et al. 1997]  Degl'Innocenti, S., Weiss, A., k Leone, L. 1997, A&A, 319, 487  [Deliyannis et al. 1989]  Deliyannis, C. P., Demarque, P., k Pinsonneault, M . H. 1989, ApJ, 347, L73  [Demarque 1988]  Demarque, P. 1988, in Globular Galaxies,  Cluster Systems  in  ed. by J. Grindley k A. G. D. Philip (Reidel,  Dordrecht), p. 121 [Demarque et al. 1972]  Demarque, P., Mengel, J. G., k Sweigart, A. V. 1972, ApJ, 173, L27  [Denegri et al. 1990]  Denegri, D., Sadoulet, B., k Spiro, M. 1990, Rev. Mod. Phys., 62, 1  [Djorgovski 1993]  Djorgovski, S. 1993, in Structure and Dynamics Globular  Clusters,  of  ed. by S. Djorgovski k G. Meylan  (ASP, San Francisco), ASP Conf. Series, 50, 373 [Djorgovski k Meylan 1993]  Djorgovski, S., k Meylan, G. 1993, in Structure Dynamics  of Globular  Clusters,  and  ed. by S. Djorgovski  187  References  k G. Meylan (ASP, San Francisco), ASP Conf. Series, 50, 325 [Djorgovski et al. 1993]  Djorgovski, S., Piotto, G., k Capaccioli, M. 1993, AJ, 105, 2148  [Dorman 1992]  Dorman, B. 1992, ApJS, 81, 221  [Drukier et al. 1988]  Drukier, G. A., Fahlman, G. G., Richer, H. B., k VandenBerg, D. A. 1988, AJ, 95, 1415  [Eggen 1987]  Eggen, 0. J. 1987, AJ, 93, 393  [Elson et al. 1987]  Elson, R., Hut, P., k Inagaki, S. 1987, ARA&A, 25, 565  [Elson et al. 1995]  Elson, R. A. W., Gilmore, G. F., Santiago, B. X., k Casertano, S. 1995, AJ, 110, 682  [Fahlman 1993]  Fahlman, G. G. 1993, in The Globular Cluster - Galaxy Connection,  ed. by G. Smith k J. Brodie (ASP, San  Francisco), ASP Conf. Series, 48, 117 [Fahlman et al. 1985]  Fahlman, G. G., Richer, H. B., k VandenBerg, D. A. 1985, ApJS, 58, 225  [Fahlman et al. 1989]  Fahlman, G. G., Richer, H. B., Searle, L., k Thompson, I. B. 1989, ApJ, 343, L49  [Fahlman et al. 1996]  Fahlman, G. G., Mandushev, G., Richer, H. B., Thompson, I. B., k Sivaramakrishnan, A. 1996, ApJ, 459, L65  188  References  [Fall k Rees 1977]  Fall, S. M., k Rees, M. J. 1977, MNRAS, 181, 37p  [Faulkner k Swenson 1988]  Faulkner, J., k Swenson, F. J. 1988, ApJ, 329, L47  [Faulkner k Swenson 1993]  Faulkner, J., k Swenson, F. J. 1993, ApJ, 411, 200  [Fernley 1994]  Fernley, J. 1994, A&A, 284, L16  [Ferraro et al. 1991]  Ferraro, F. R., Clementini, G., Fusi Pecci, F., k Buonanno, R. 1991, MNRAS, 252, 357  [Ferraro et al. 1993]  Ferraro, F. R., Fusi Pecci, F., Cacciari, C., Corsi, C , Buonanno, R., Fahlman, G. G., k Richer, H. B. 1993, AJ, 106, 2324  [Ferraro et al. 1995]  Ferraro, F. R., Fusi Pecci, F., k Bellazzini, M. 1995, A&A, 294, 80  [Ferraro et al. 1997]  Ferraro, F. R., Paltrinieri, B., Fusi Pecci, F., Cacciari, C , Dorman, B., Rood, R. T., Buonanno, R., Corsi, C. E., Burgarella, D., k Laget, M. 1997, A&A, 324, 915  [FitzGerald 1970]  FitzGerald, P. M. 1970, AkA, 4, 234  [Freeman 1996]  Freeman, K. C. 1996, in Formation Halo . . . Inside and Out,  of the  Galactic  ed. by H. Morrison k A. Sa-  rajedini, (ASP, San Francisco), ASP Conf. Series, 92, 3 [Fusi Pecci et al. 1990]  Fusi Pecci, F., Ferraro, F. A., Crocker, D. A., Rood, R. T., k Buonanno, R. 1990, A&A, 238, 95  [Fusi Pecci et al. 1992]  Fusi Pecci, F., Ferraro, F. A., Corsi, C. E., Cacciari, C , k Buonanno, R. 1992, AJ, 104, 1831  189  References  [Fusi Pecci et al 1993]  Fusi Pecci, F., Ferraro, F. A., Bellazzini, M., Djorgovski, S., Piotto, G., k Buonanno, R. 1993, AJ, 105, 1145  [Fusi Pecci et al 1996]  Fusi Pecci, F., Bellazzini, M., Ferraro, F. A., Buonanno, R., k Corsi, C. E. 1996, in Formation of the Galactic Halo ...Inside and Out, ed. by H. Morrison k A. Sarajedini, (ASP, San Francisco), ASP Conf. Series, 92, 221  [Geisler et al 1992]  Geisler, D., Minniti, D., k Claria, J. J. 1992, A J , 104, 627  [Gilmore k Reid 1983]  Gilmore, G., k Reid, N. 1983, MNRAS, 202, 1025  [Girardi et al 1996]  Girardi, L., Bressan, A., Chiosi, C , Bertelli, G., k Nasi, E. 1996, A&AS, 117, 113  [Gnedin k Ostriker 1997]  Gnedin, 0 . Y., k Ostriker, J. P. 1997, ApJ, 474, 223  [Gratton et al 1997]  Gratton, R. G., Fusi Pecci, F., Carretta, E . , Clementini, G., Corsi, C. E., k Lattanzi, M. 1997, ApJ, 491, 749  [Hamuy et al 1996]  Hamuy, M., Phillips, M. M., Suntzeff, N. B., Schommer, R. A., Maza, J., k Aviles, R. 1996, AJ, 112, 2398  [Hanson 1979]  Hanson, R. B. 1979, MNRAS, 186, 875  [Harris 1975]  Harris, W. E. 1975, ApJS, 29, 397  [Hayashi et al 1962]  Hayashi, C , Hoshi, R., k Sugimoto, D. 1962, Progress of Theoretical Physics Suppl., No. 22  190  References  Helt et al. 1993]  Helt, B. E., Jorgensen, H. E., King, S., k Larsen, A. 1993, A&A, 270, 297  Henry k McCarthy 1993]  Henry, T. J., k McCarthy, D. W., Jr. 1993, A J , 106, 773  Hertzsprung 1911]  Hertzsprung, E. 1911, Publ. Astrophys. Obs. Potsdam, No. 63  Hesser et al. 1984]  Hesser, J. E., McClure, R. D., Hawarden, T. G., Cannon, R. D., von Rudloff, R., Kruger, B., k Egles, D. 1984, PASP, 96, 406  ;Hogg 1973]  Hogg, H. B. S. 1973, Publ. David Dunlap Obs., 3, No. 6  Holtzman et al. 1998]  Holtzman, J. A., Watson, A. M., Baum, W. A., Grillmair, C. J., Groth, E. J., Light, R. M., Lynds, R., k O'Neil, E. J., Jr. 1998, AJ, 115, 1946  HoweU 1989]  HoweU, S. B. 1989, PASP, 101, 616  Hut k Djorgovski 1992]  Hut, P., k Djorgovski, S. 1992, Nature, 359, 806  Tbata et al. 1994]  Ibata, R. I., Gilmore, G., k Irwin, M. G. 1994, Nature, 370, 194  Tben 1971]  Iben, I., Jr. 1971, PASP, 83, 697  Iben 1991]  Iben, I., Jr. 1991, ApJS, 76, 55  Iben k Renzini 1983]  Iben, I., Jr., k Renzini, A. 1983, ARA&A, 21, 271  Iben k Rood 1969]  Iben, I., Jr., k Rood, R. T. 1969, Nature, 223, 933  191  References  [Iben k Rood 1970]  Iben, I , Jr., k Rood, R. T. 1970, A p J , 161, 587  [Izotov et al. 1997]  Izotov, Y . I., Thuan, T. X . , Lipovetsky, V . A . 1997, ApJS, 108, 1  [Johnson 1966]  Johnson, H . L. 1966, A R A & A , 4, 193  [King 1962]  King, I. R. 1962, A J , 67, 471  [King 1966]  King, I. R. 1966, A J , 71, 64  [King k Bruzual 1976]  King, I. R., k Bruzual, G. A . 1976, A & A , 50, 459  [King et al. 1968]  King, I. R., Hegemann, E . , Jr., Hodge, S. M . , k White, R. E. 1968, A J , 73, 456  [King et al. 1985]  King, C. R., Da Costa, G . S., k Demarque, R 1985, A p J , 299, 674 King, I. R., Cool, A . M . , k Piotto, G. 1996, in Forma-  [King et al. 1996a]  tion of the Galactic Halo .. .Inside and Out, ed. by H .  Morrison k A . Sarajedini (ASP, San Francisco), A S P Conf. Series, 92, 277 King, I. R., Piotto, G., Cool, A . M . , Anderson, J . , k [King et al. 1996b]  Sosin, C. 1996, in Science with the Hubble Space Telescope  - IT, ed. by P. Benvenuti, F . Macchetto k E .  Schreier, p. 297 King, I. R., Anderson, J . , Cool, A . M . , k Piotto, G . [King et al. 1998a]  1998, A p J , 492, L37  192  References  [King et al. 1998b]  King, J. R., Stephens, A., Boesgaard, A. M., k Deliyannis, C. R 1998, AJ, 115, 666  [Kraft 1994]  Kraft, R. R 1994, PASP, 106, 503  [Kron k Guetter 1976]  Kron, G. E., k Guetter, H. H. 1976, AJ, 81, 817  [Kron k MayaU 1960]  Kron, G. E., k Mayall, N. U. 1960, AJ, 65, 581  [Kroupa et al. 1990]  Kroupa, P., Tout, C. A., k Gilmore, G. 1990, MNRAS, 244, 76  [Laird 1985]  Laird, J. B. 1985, ApJS, 57, 389  [Laird et al. 1988]  Laird, J. B., Carney, B. W., k Latham, D. L. 1988, AJ, 95, 1843  [Landolt 1992]  Landolt, A. U. 1992, AJ, 104, 340  [Lanz 1986]  Lanz, T. 1986, A&AS, 65, 195  [Layden k Sarajedini 1997]  Layden, A. C , k Sarajedini, A. 1997, ApJ, 486, L107  [Lee 1977]  Lee, S.-W. 1977, A&AS, 29, 1  [Lee 1993]  Lee,  Y.-W.  Connection,  1993,  in  The  Globular  Cluster -  Galaxy  ed. by G. Smith k J. Brodie (ASP, San  Francisco), ASP Conf. Series, 48, 142 [Lee k Cannon 1980]  Lee, S.-W., k Cannon, R. D. 1980, JKAS, 13, 15  [Lee et al. 1990]  Lee, Y.-W., Demarque, P., k Zinn, R. 1990, ApJ, 350, 155  193  References  [Lee et al. 1994]  Lee, Y.-W., Demarque, P., k Zinn, R. 1994, ApJ, 423, 248  [Leonard 1996]  Leonard, P. J. T. 1996, in The Origins, Evolution, Destinies of Binary Stars in Clusters,  and  ed. by E. Milone  k J.-C. MermiUiod (ASP, San Francisco), ASP Conf. Ser., 90, 337 [Leonard k Fahlman 1991]  Leonard, P. J. T., k Fahlman, G. G. 1991, AJ, 102, 994  [Leonard et al. 1992a]  Leonard, P. J. T., Richer, H. B., k Fahlman, G. G. 1992, A J , 104, 2104  [Leonard et al. 1992b]  Leonard, P. J. T., k Linnell, A. P. 1992, AJ, 103, 1928  [Lin k Murray 1991]  Lin, D. N. C , k Murray, S. D. 1991, in The Formation and Evolution  of Star Clusters,  ed. by K. Janes (ASP,  San Francisco), ASP Conf. Series, 13, 55 [Livio 1993]  Livio, M. 1993, in Blue Stragglers, ed. by R. A. Saffer,  (ASP, San Francisco), ASP Conf. Series, 53, 3 [Lombardi et al. 1996]  Lombardi, J. C , Jr., Rasio, F. A, k Shapiro, S. 1996, ApJ, 468, 797  [Lupton et al. 1987]  Lupton, R. H., Gunn, J. E . , k Griffin, R. F. 1987, A J , 93, 1114  [Lutz k Kelker 1973]  Lutz, T. E., k Kelker, D. H. 1973, PASP, 85, 573  [Madore et al. 1998]  Madore, B. F., Freedman, W. L., Silbermann, N., Harding, P., Huchra, J., Mould, J. R., Graham, J. A.,  194  References '->  Ferrarese, L., Gibson, B . K . , Han, M . , Hoessel, J . G . , Hughes, S. M . , Ilhngworth, G . D., Phelps, R., Sakai, S., k Stetson, P. B . 1998, Nature, 395, 47 [Mathews et al 1993]  Mathews, G. J., Schramm, D. N . , k Meyer, B . S. 1993, A p J , 404, 476  [Mateo 1996]  Mateo, M . 1996, in The Origins, Evolution, tinies of Binary Stars in Clusters,  and Des-  ed. by E . F . Milone  k J.-C. Mermilhod, (ASP, San Francisco), A S P Conf. Ser., 90, 346 [Mateo et al 1990]  Mateo, M . , Harris, H . C , Nemec, J . , k Olszewski, E . W. 1990, A J , 100, 469  [Mateo et al 1995]  Mateo, M . , Udalski, A . , Szymariski, M . , Kaluzni, J . , Kubiak, M . , k Krzeminski, W . 1995, A J , 109, 588  [Mateo et al 1996]  Mateo, M . , Mirabal, N . , Udalski, A . , Szymariski, M . , Kaluzni, J . , Kubiak, M . , Krzeminski, W . , k Stanek, K . Z. 1996, A p J , 458, L13  [McClure et al 1986]  McClure, R. D., VandenBerg, D . A . , Smith, G . H . , Fahlman, G . G., Richer, H . B . , Hesser, J . E . , Harris, W. E., Stetson, P. B . , k Bell, R. A . 1986, A p J , 307, L49  [McCrea 1964]  McCrea, W . H . 1964, M N R A S , 128, 147  [Merritt 1987]  Merritt, D. 1987, A p J , 313, 121  195  References  [Messier 1784]  Messier, C. 1784, Catalogue des nebuleuses et des amas d'etoiles, Connaissance des Temps, p. 227  [Meylan k Pryor 1993]  Meylan, G., &; Pryor, C. 1993, in Structure and Dynamics of Globular Clusters,  ed. by S. Djorgovski k G.  Meylan (ASP, San Francisco), ASP Conf. Series, 50, 31 [Minniti et al. 1993]  Minniti, D., Geisler, D., Peterson, R. C , k Claria, J. J. 1993, ApJ, 413, 548  [Montegriffo et al. 1998]  Montegriffo, P., Bellazzini, M., Ferraro, F. R., Martins, D., Sarajedini, A., k Fusi Pecci, F. 1998, MNRAS, 294, 315  [Mould 1986]  Mould, J. 1986, in Stellar Populations,  ed. by C. A.  Norman, A. Renzini k M. Tosi (Cambridge Univ. Press, Cambridge), p. 9 [Nemec k Harris 1987]  Nemec, J. M., k Harris, H. C. 1987, ApJ, 316, 172  [Norris 1986]  Norris, J. 1986, in The Galaxy, ed. by G. Gilmore and B. CarsweU (Reidel, Dordrecht), p. 297  [Olive k Steigman 1995]  Olive, K. A., k Steigman, G. 1995, ApJS, 97, 49  [Olive et al. 1991]  Olive, K. A., Steigman, G., k Walker, T. P. ApJ, 380, LI  [Ortolani k Gratton 1990]  Ortolani, S., k Gratton, R. 1990, A&AS, 82, 71  [Oudmaijer et al. 1998]  Oudmaijer, R. D., Groenewegen, M. A. T., k Schrijver, H. 1998, MNRAS, 294, L41  196  References  [Ouellette k Pritchet 1998]  Ouellette, J . A . , k Pritchet, C. J . 1998, A J , 115, 2539  [Paczynski 1984]  Paczyrlski, B . 1984, A p J , 284, 670  [Paresce 1993]  Paresce, F. 1993, in Blue Stragglers, ed. by R. A . Saffer, (ASP, San Francisco), A S P Conf. Series, 53, 30  [Penny 1984]  Penny, A . J . 1984, in Observational lar Evolution  Theory,  Tests of the Stel-  ed. by A . Maeder k A . Renzini  (Reidel, Dordrecht), p. 157 [Perryman et al. 1997a]  Perryman, M . A . C , et al. 1997, The Hipparcos and Tycho Catalogues,  [Perryman et al. 1997b]  E S A SP-1200 (Noordwijk, E S A )  Perryman, M . A . C , Lindegren, L . , Kovalevsky, J . , Hog, E., Bastian, U . , Bernacca, P. L . , Creze, M . , Donati, F . , Grenon, M . , Grewing, M . , van Leeuwen, F . , van der Marel, H . , Mignard, F . , Murray, C. A . , Le Poole, R. S., Schrijver, H . , Turon, C , Arenou, F . , Froeschle, M . , Petersen, C. S. 1997, A & A , 323, L49  [Peterson 1993]  Peterson, C. J . 1993, in Structure Globular Clusters,  and Dynamics  of  ed. by S. Djorgovski k G . Meylan  (ASP, San Francisco), A S P Conf. Series, 50, 337 [Peterson 1983]  Peterson, R. C. 1983, A p J , 275, 737  [Peterson 1985a]  Peterson, R. C. 1985, A p J , 289, 320  [Peterson 1985b]  Peterson, R. C. 1985, A p J , 294, L35  [Peterson k King 1975]  Peterson, C. J . , k King, I. R. 1975, A J , 80, 427  197  References  [Piotto 1991]  Piotto, G . 1991, in The Formation Star Clusters,  and Evolution of  ed. by K . Janes (ASP, San Francisco),  A S P Conf. Series, 13, 200 [Piotto 1993]  Piotto, G. 1993, in Structure and Dynamics of Globular Clusters,  ed. by S. Djorgovski k G. Meylan (ASP, San  Francisco), A S P Conf. Series, 50, 233 [Piotto et al. 1997]  Piotto, G., Cool, A . M . , King, I. R. 1997, A J , 113, 1345  [Pols et al. 1998]  Pols, 0 . R., Schroder, K.-P., Hurley, J . R., Tout, C. A . , k Eggleton, P. P. 1998, M N R A S , 298, 525  [Pont et al. 1998]  Pont, F., Mayor, M . , Turon, C , k VandenBerg, D. A . 1998, A & A , 329, 87  [Pryor et al. 1991]  Pryor, C , McClure, R. D., Fletcher, J . M . , k Hesser, J. E. 1991, A J , 102, 1026  [Pryor k Meylan 1993]  Pryor, C , k Meylan, G . 1993, in Structure and Dynamics of Globular  Clusters,  ed. by S. Djorgovski k  G. Meylan (ASP, San Francisco), A S P Conf. Series, 50, 357 [Pryor et al. 1986]  Pryor, C , Smith, G . H . , k McClure, R. D . 1986, A J , 92, 1358  [Ratcliff 1987]  Ratcliff, S. 1987, A p J , 318, 196  [Reid 1995]  Reid, I. N . 1995, in The Bottom of the Main Sequence and beyond,  ed. by C. G . Tinney (Springer, Berlin),  198  References  p. 307 [Reid 1997]  Reid, I. N. 1997, AJ, 114, 161  [Reid et al. 1996]  Reid, I. N., Yan, L., Majewski, S. R., Thompson, I., k Smail, I. 1996, AJ, 112, 1472  [Renzini 1977]  Renzini, A. 1977, in Advanced Stages in Stellar Evolution, ed. by P. Bouvier k A. Maeder (Geneva Obs., Geneva), p. 149  [Renzini k Fusi Pecci 1988]  Renzini, A., k Fusi Pecci, F. 1988, ARA&A, 26, 199  [Rich 1993]  Rich, R. M. 1993, in The Globular Cluster - Galaxy Connection,  ed. by G. Smith k J. Brodie (ASP, San  Francisco), ASP Conf. Series, 48, 287 [Rich 1996]  Rich, R. M. 1996, in Formation . . . Inside and Out,  of the Galactic  Halo  ed. by H. Morrison k A. Sarajedini  (ASP, San Francisco), ASP Conf. Series, 92, 24 [Richer k Fahlman 1984]  Richer, H. B., k Fahlman, G. G. 1984, ApJ, 277, 227  [Richer k Fahlman 1989]  Richer, H. B., k Fahlman, G. G. 1989, ApJ, 339, 178  [Richer k Fahlman 1991]  Richer, H. B., k Fahlman, G. G. 1991, in The ation and Evolution  of Star Clusters,  Form-  ed. by K. Janes  (ASP, San Francisco), ASP Conf. Series, 13, 120 [Richer et al. 1991]  Richer, H. B., Fahlman, G. G., Buonanno, R., Fusi Pecci, F., Searle, L., k Thompson, I. B. 1991, ApJ, 381, 147  199  .References  [Richer k Fahlman 1992]  Richer, H . B., k Fahlman, G. G . 1992, Nature, 358, 383  [Richstone k Tremaine 1986]  Richstone, D. 0., k Tremaine, S. 1986, A J , 92, 72  [Rood 1973]  Rood, R. T. 1973, A p J , 184, 815  [Rosvick k VandenBerg 1998]  Rosvick, J . M . , k VandenBerg, D . A . 1998, A J , 115, 1516  [Russell 1914]  Russell, H . N . 1914, Popular Astronomy, 22, 275  [Ryan 1992]  Ryan, S. G. 1992, A J , 104, 1144  [Sandage 1953]  Sandage, A . 1953, A J , 58, 61  [Sandage 1970]  Sandage, A . 1970, A p J , 162, 841  [Sandage 1986]  Sandage, A . 1986, A R A & A , 24, 421  [Sandage 1990]  Sandage, A . 1990, A p J , 350, 603  [Sandage 1993a]  Sandage, A . 1993, A J , 106, 703  [Sandage 1993b]  Sandage, A . 1993, A J , 106, 719  [Sandage k Katem 1983]  Sandage, A., k Katem, B . 1983, A J , 88, 1146  [Sandage k Wallerstein 1960]  Sandage, A . , k WaUerstein, G . 1960, A p J , 131, 598  [Sandage et al. 1968]  Sandage, A . , Katem, B . , k Kristian, J . 1968, A p J , 153, L129  [Sandquist et al. 1996]  Sandquist, E . L . , Bolte, M . , Stetson, P. B . , k Hesser, J. E . 1996, A p J , 470, 910  200  References  [Sandquist et al. 1997]  Sandquist, E. L., Bolte, M., k Hernquist, L. 1997, ApJ, 477, 335  [Sarajedini 1992]  Sarajedini, A. 1992, AJ, 104, 178  [Sarajedini 1993]  Sarajedini, A. 1993, in Blue Stragglers, ed. by R. A. Saffer, (ASP, San Francisco), ASP Conf. Series, 53, 14  [Sarajedini 1994]  Sarajedini, A. 1994, AJ, 107, 618  [Sarajedini k King 1989]  Sarajedini, A., k King, I. R. 1989, AJ, 98, 1624  [Sarajedini k Layden 1995]  Sarajedini, A. k Layden, A. C. 1995, AJ, 109, 1086  [Saviane et al. 1998]  Saviane, I., Piotto, G., Fagotto, F., Zaggia, S., Capaccioli, M., k Aparicio, A. 1998, A&A, 333, 479  [Schade et al. 1988]  Schade, D., VandenBerg, D. A., & Hartwick, F. D. A. 1988, AJ, 96, 1632  [Searle k Zinn 1978]  Searle, L., k Zinn, R. 1978, ApJ, 225, 357  [Sigurdsson et al. 1994]  Sigurdsson, S., Davies, M. B., k Bolte, M. 1994, ApJ, 431, L115  [Silberman k Smith 1995]  Silberman, N. A., k'Smith, H. A., 1995, AJ, 109, 1119  [Sills et al. 1997]  Sills, A., Lombardi, J. C , Jr., Bailyn, C. D., Demarque, P., Rasio, F. A., k Shapiro, S. L. 1997, ApJ, 487, 290  [Simon k Clement 1993]  Simon, N. R., k Clement, C. M. 1993, ApJ, 410, 526  [Smith 1987]  Smith, H. 1987, A&A, 188, 233  201  References  [Smith k Norris 1983]  Smith, G. EL, k Norris, J. 1983, PASP, 95, 635  [Smith et al. 1992]  Smith, G., Lambert, D. L., k Ruck, M. J. 1992, A&A, 263, 249  [Sneden et al. 1991]  Sneden, C., Kraft, R. P., Prosser, C. F., k Langer, G. E. 1991, AJ, 102, 2001  [Spiesman k Wallerstein 1991]  Spiesman, W. J., k Wallerstein, G. 1991, AJ, 102, 1791  [Spitzer 1987]  Spitzer, L. 1987, Dynamical Clusters  Evolution  of  Globular  (Prinston Univ. Press, Prinston)  [Stetson 1987]  Stetson, P. B. 1987, PASP, 99, 191  [Stetson 1990]  Stetson, P. B. 1990, PASP, 102,932  [Stetson 1991]  Stetson, P. B. 1991, in The Formation of Star Clusters,  and  Evolution  ed. by K. Janes (ASP, San Francisco),  ASP Conf. Series, 13, 88 [Stetson 1992]  Stetson, P. B. 1992, in Astronomical Software and Systems I,  Data  Analysis  ed. by D. M. Worrall, C.  Biemesderfer, k J. Barnes (San Francisco, ASP), ASP Conf. Ser., 25, 297 [Stetson 1993a]  Stetson, P. B. 1993, in The Globular Cluster - Galaxy Connection,  ed. by G. Smith k J. Brodie (ASP, San  Francisco), ASP Conf. Series, 48, 14 [Stetson 1993b]  Stetson, P. B. 1993, in Stellar Photometry  -  Cur-  rent Techniques and Future Developments, ed. by C. J.  202  References  Butler k I. Elliot (Cambridge University, Cambridge), p. 291 [Stetson 1994]  Stetson, P. B . 1994, PASP, 106, 250  [Stetson k Harris 1988]  Stetson, P. B . , k Harris, W . E . 1988, A J , 96, 909  [Stetson et al. 1996]  Stetson, P. B . , VandenBerg, D . A . , k Bolte, M . 1996, PASP, 108, 560  [Stothers 1991]  Stothers, R. B . 1991, A p J , 383, 820  [Stryker 1993]  Stryker, L. L . 1993, PASP, 105, 1081  [Sweigart 1978]  Sweigart, A . V . 1978, in The HR Diagram,  ed. by  A. D. G . Philip and D . Hayes (Reidel, Dordrecht), p. 333 [Taylor 1986]  Taylor, B . J . 1986, ApJS, 60, 577  [Thomas 1967]  Thomas, H.-C. 1967, ZA, 67, 420  [Tomkin et al. 1986]  Tomkin, J . , Sneden, C , k Lambert, D . L . 1986, A p J , 302, 415  [Tomkin et al. 1992]  Tomkin, J . , Lemke, M . , Lambert, D. L . , k Sneden, C. 1992, A J , 104, 1568  [Trager et al. 1993]  Trager, S. C , Djorgovski, S., k King, I. R. 1993, in Structure  and Dynamics  of Globular  Clusters, ed. by  S. Djorgovski k G. Meylan (ASP, San Francisco), A S P Conf. Series, 50, 347  203  References  [Turner 1989]  Turner, D. G. 1989, A J , 98, 2300  [Turner 1994]  Turner, D. G. 1989, R M x A A , 29, 163  [VandenBerg 1983]  VandenBerg, D. A . 1983, ApJS, 51, 29  [VandenBerg k Bell 1985]  VandenBerg, D. A . , k Bell, R. 1985, ApJS, 58, 561  [VandenBerg et al. 1990]  VandenBerg, D. A . , Bolte, M . , k Stetson, R B . 1990, A J , 100, 445  [VandenBerg et al. 1996]  VandenBerg, D. A . , Bolte, M . , k Stetson, R B . 1996, A R A & A , 34, 461  [VandenBerg et al. 1998a]  VandenBerg, D. A., Larson, A . M . , De Propris, R. 1998, PASP, 110, 98  [VandenBerg et al. 1998b]  VandenBerg, D. A . , Swenson, F. J . , Rogers, F. J . , Iglesias, C. A . , k Alexander, D. R. 1998, in preparation  [van den Berg 1967] [van der Kruit k Searle 1982] [Vassiliadis k Wood 1993] [Walker 1994] [Walker et al. 1991]  van den Berg, S. 1967, PASP, 84, 306 van der Kruit, P., k Searle, L . 1982, A & A , 110, 61 Vassiliadis, E., k Wood, P. R. 1993, A p J , 413, 641 Walker, A . R. 1994, A J , 108, 555 Walker, T. P., Steigman, G., Kang, H.-S., Schramm, D. M . , k Olive, K . A . 1991, A p J , 376, 51  [Weinberg 1994]  Weinberg, M . D. 1994, A J , 108, 1414  204  References  [Whittaker & Robinson 1924]  Whittaker, E . T., k Robinson, G . 1924, The Calculus of Observations,  [Woods et al. 1995]  (Blackie k Son, London)  Woods, D., Fahlman, G. G., k Richer, H . B . 1995, A p J , 454, 32  [Yanni et al. 1994]  Yanni, B . , Guhathakurta, P., Schneider, D. P., k Bachall, J . N . 1994, A p J , 435, L59  [Zaggia et al. 1994]  Zaggia, S. R., Piotto, G., k Capaccioli, M . 1994, M m SAI, 65, 667  [Zaggia et al. 1997]  Zaggia, S. R., Piotto, G., k Capaccioli, M . 1997, A & A , 327, 1004  [Zhao & Magain 1990]  Zhao, G., k Magain, P. 1990, A & A , 238, 242  [Zhao & Magain 1991]  Zhao, G., k Magain, P. 1991, A & A , 244, 425  [Zinn 1985]  Zinn, R. 1985, A p J , 293, 424  [Zinn 1993]  Zinn, R. 1993, in The Globular Cluster - Galaxy Connection,  ed. by G. Smith & J . Brodie (ASP, San Fran-  cisco), A S P Conf. Series, 48, 39 [Zinn 1996]  Zinn,  R. 1996, in Formation  Halo . . . Inside and Out,  of  the  Galactic  ed. by H . Morrison & A . Sara-  jedini (ASP, San Francisco), A S P Conf. Series, 92, 211 [Zinn & West 1984]  Zinn, R., & West, M . J . 1984, ApJS, 55, 45  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085705/manifest

Comment

Related Items