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Searching for variability in the globular cluster Messier 4 Ferdman, Robert Daniel 2002

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S E A R C H I N G FOR V A R I A B I L I T Y IN T H E G L O B U L A R C L U S T E R MESSIER 4 by Robert Daniel Ferdman B.Sc.(Physics), Trent University, 2000 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF SCIENCE in The Faculty of Graduate Studies (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E ^ N I V E R S I T Y ^ / F BRITISH C O L U M B I A October 3, 2002 © Robert Daniel Ferdman, 2002 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics and Astronomy The University Of British Columbia Vancouver, Canada 11 Abstract Time-series data taken with the Hubble Space Telescope (HST) of a field six core radii (~ 5') from the centre of the globular cluster Messier 4 (M4) have been analysed in search of variable objects. These observations consisted of 98 x 1300 s exposures in the F606W (V) filter, and 148 x 1300 s exposures in the F814W (I) filter, covering a period of about 10 weeks in early 2001. Various selection criteria were employed to select candidate variable stars. These included: selection of outliers in the R M S and variability index vs. magnitude distributions, as well as selection of stars in the colour-magnitude diagram of the cluster that lie above the main sequence (MS) and between the MS and white dwarf sequence. White dwarf stars that lie within the empirical ZZ Ceti instability region were also selected as candidates. Period searches were then performed on the selected candidates using the phase dispersion minimization (PDM) technique. The reliability of the P D M search results for this dataset was tested using synthetic light curves of eclipsing binary star and sinusoidal light curves at a few different periods. Results from this analysis showed that there are probably no eclipsing binary stars or periodic variables with periods on the order of a half a day, down to limiting magnitudes of V ~ 25 and I ~ 24. This is consistent with the absence of contact binaries such as the W Ursae Majoris systems. However, one candidate variable star does show a increase in brightness of ~ 0.1 magnitudes in both bandpasses; this rise in amplitude seems to last for a few days. If this object is periodic, its period is too long to detect given the time sampling of the data. Possible explanations concerning the nature of this object include a binary system with a white dwarf primary and a low-mass main sequence secondary, or a B Y Draconis variable star. Both of these objects can exhibit nonperiodic flaring or low amplitude, periodic variability in the light curve due to starspots. i i i Contents Abstract i i Contents i i i List of Tables v List of Figures vi Acknowledgements vi i i 1 Introduction 1 1.1 Light variations in stars 2 1.1.1 Eclipsing Binary Stars 2 1.1.2 Pulsating Variables 10 1.1.3 Supernovae 12 1.2 Expectations 13 1.3 Thesis Overview 17 2 Observations and Data Reduction 19 2.1 Observations 19 2.2 Data Reduction 21 2.2.1 Preprocessing and Combined-Image Photometry 21 2.2.2 Single-Image Photometry 22 2.2.3 Magnitude Offset Calculations 22 2.2.4 Data Point Rejection 23 iv 3 Data Analysis 27 3.1 Light curves 27 3.2 Candidate selection 27 3.2.1 Statistical Outliers 29 3.2.2 Main Sequence Outliers 34 3.2.3 ZZ Ceti Candidates 36 3.3 Periodicity Analysis 40 3.4 Reliability Tests 45 3.5 Searches for Non-Periodic Variable Stars 50 3.5.1 Median-Smoothed Light Curves 50 3.5.2 Searching for Supernovae 52 4 Results 64 4.1 Overview of Remaining Candidate 64 4.2 Light Curve of Candidate Star 66 4.3 Phase Dispersion Minimization Analysis 70 4.4 Reliability Tests . 76 5 Discussion 84 5.1 A Lack of Variability 84 5.2 Speculations on the Remaining Candidate 86 5.3 The Future 89 5.4 Closing Remarks 90 Bibliography 91 V List of Tables 3.1 RMS outlier candidates 31 3.2 Variability index candidates 36 3.3 Main sequence outlier candidates 38 3.4 ZZ Ceti variable star candidates 40 3.5 Median smoothed light curve candidates 54 3.6 Supernova candidates 57 4.1 Artificial star test results: Algol-type in J 80 4.2 Artificial star test results: Algol-type in V 80 4.3 Artificial star test results: W UMa-type in 7 81 4.4 Artificial star test results: W UMa-type in V 81 4.5 Artificial star test results: Sinusoid in I 82 4.6 Artificial star test results: Sinusoid in V 82 5.1 ZZ Ceti variable star coordinate list 90 List of Figures 1.1 Examples of eclipsing binary light curves 4 1.2 Roche geometry and equipotential surfaces 7 1.3 Physical classification of close binary systems 8 1.4 White dwarf variable stars on the H-R diagram 11 1.5 Hubble diagram using type la supernovae 14 2.1 Transmission curves for the F606W and F814W filters on HST 20 2.2 Example of magnitude offset calculation 24 2.3 R M S magnitude distributions 26 3.1 Example light curve 28 3.2 R M S magnitude distributions and candidate variable stars 30 3.3 Variability index distributions and candidate variable stars 35 3.4 Colour-magnitude diagram showing main sequence outlier candidates . . 37 3.5 Colour-magnitude diagram showing ZZ Ceti candidates 39 3.6 Example of a periodic light curve 43 3.7 Example of P D M analysis 44 3.8 Artificial Algol-type light curves 47 3.9 Artificial W UMa-type light curves 48 3.10 Artificial sinusoidal light curves 49 3.11 Comparison between photometry performed on real and artificial star images 51 3.12 Candidate variable stars from median smoothing analysis 53 3.13 Supernova candidates from magnitude differences between epochs . . . . 56 vii 3.14 Image subtraction of galaxies on the PG chip 60 3.15 Image subtraction of galaxies on the WF2 chip 61 3.16 Image subtraction of galaxies on the WF3 chip 62 3.17 Image subtraction of galaxies on the WF4 chip 63 4.1 Location of remaining candidate on P C chip 65 4.2 Location of remaining candidate on M4 colour-magnitude diagram . . . . 67 4.3 Light curve of remaining candidate 68 4.4 Light curve of remaining candidate after performing aperture photometry 69 4.5 Example of P D M analysis on a star in WF3 71 4.6 Example of P D M analysis on a star in WF4 72 4.7 Phase dispersion minimization analysis of remaining candidate 74 4.8 P D M analysis of remaining candidate after aperture photometry 75 4.9 Example P D M analysis on a synthetic W U M a light curve 77 4.10 Limiting magnitudes for successful period recovery of artificial binary star periods 83 5.1 Decomposition of candidate star 88 V l l l Acknowledgements First and foremost I would like to thank my supervisor, Dr. Harvey Richer. This thesis would not have been possible without his encouragement and support. I am also very much indebted to him for giving me the opportunity to conduct and present my research in many places around the world, and to collaborate with experts in the field. I would like to extend a very special thanks to Dr. Jaymie Matthews, James Brewer, and Jason Rowe for putting up with my tireless questioning, and for providing me with a great deal of insight into the analysis portion of my work. Many thanks to Dr. Brad Gib-son for introducing me to various techniques used in variable star searches, and helping me implement them during my time in Australia. Thanks also to Dr. Robert E . Wilson for providing me with an up-to-date version of the Wilson-Devinney code. I am very grateful to Harvey, Jaymie, and Stephane Courteau for their reassurance in my times of self-doubt and for their advice regarding my future in astronomical research. A big thanks to all the graduate students in the astronomy group at U B C , who provided a real sense of comradery and who have proven to be great friends as well as colleagues. A great many thanks is due to the Physics department at Trent University, and especially to Dr. James Jury, for giving me my first experience in the world of scientific research. It is hard to see myself where I am now without the high quality education that I was lucky enough to receive from them. Last, but by no means least, I would like to thank my family, my friends (all over Canada), and those people very special to me who constantly encourage me and always let me know how much they believe in me. 1 Chapter 1 Introduction The driving force, that originally led me to accept this project as my masters thesis research was the exciting prospect of analysing images, taken with the Hubble Space Telescope. A n added incentive was the opportunity to be part of a larger research cam-paign that would attempt to answer questions of fundamental astrophysical importance. However, the search for variability in M4, and particularly for close binary systems, is an endeavor of great importance in its own right. It is well known that binary stars are important for the determination of stellar masses. In addition, knowledge of the popu-lation of these objects in globular clusters (GCs) is fundamental to understanding their evolution and dynamical history, which is a topic that is not yet fully understood. Even though the data were not originally collected for the purpose of searching for variable stars, it did have potential for the discovery of such objects, since it included 246 separate observations of the same field over about two and a half months. The images represented the deepest ever imaging of a globular cluster, reaching the faintest part of the white dwarf sequence ever observed. In the context of this project, it allowed for the study of variability within magnitude ranges that have seldom been explored. Before describing the data and its analysis, it is useful to provide a brief summary of some concepts which are relevant to this study, in particular those pertaining to variable stars. In addition, the motivation behind conducting searches for these objects, and their importance to different areas of study, will be discussed. Expectations for this project based on past searches will also be considered. Chapter 1. Introduction 1.1 Light variations in stars 2 The first recorded observation of a variable star took place in 1595, that being the long-period variable o Ceti, better known as Mira. Since then, variable stars have been and continue to be a much studied and important phenomenon in several areas of astro-nomical research. Besides being extremely fascinating objects in themselves, they are vital to furthering our understanding in research areas as diverse as stellar populations, cosmology, and the evolution of stars, clusters, and galaxies. The term variable star is a general one, referring to any star that exhibits variations in its brightness, spectrum, or radial velocity over time. A description of the classes of variable stars relevant to this study will be discussed in what follows, and the importance of studying each of these classes to this project will be outlined. There exist many ways of classifying variable stars. The types most relevant to this thesis are eclipsing variables (mostly consisting of eclipsing binary stars), pulsating variables, and eruptive objects. The type of variability observed in a star can usually be determined through analysis of its light curve: a plot of a star's magnitude over time. The period and shape of the light curve can usually distinguish immediately the type of variable star that one has observed. In addition, it can provide a great deal of insight into the physical characteristics of the star or system observed. 1.1.1 Eclipsing Binary Stars As the name suggests, an eclipsing binary is a system that contains two or more stars in orbit around their mutual centre of mass, and whose orbital plane is inclined such that one star will periodically be eclipsed by its companion in the line of sight of the observer, resulting in an overall dimming of the system. The amount of dimming that occurs depends on several factors, including the mass ratio of the two objects, their sizes and surface temperatures, as well as the orbital inclination and semimajor axis of the system. Eclipsing binary stars can be further divided into a few main categories, and again, there are different ways of performing this classification. One method involves grouping Chapter 1. Introduction 3 different types according to their physical properties. Another means of classifying a given star is through the shape of its light curve. This is the method of classification that will be employed in what follows; the physical properties of eclipsing binaries will be discussed in the next subsection. There are three principal groups of eclipsing binary stars based on the shape of their light curves. Each of these groups is named after the first star discovered in each class, as adopted by the General Catalogue of Variable Stars (GCVS) [1]. However, classifying a star as one of these types by no means implies that they have the same physical characteristics as the prototype star [2]. The Algol-type, or E A binaries have light curves with eclipses that are clearly defined and have distinct start and end times. Light variations are virtually non-existent outside the eclipses. The periods of these stars are typically on the order of days. The ft Lyrae-type, or E B eclipsing binaries have light curves that also show well-defined eclipses, but with significant light variation outside the eclipses. Periods for this type are also on the order of days. The W Ursae Majoris-type, or E W binary systems, exhibit a continuous variation in magnitude. There is little or no distinction between points in the orbital period that are within or outside an eclipse. In most cases of W Ursae Majoris (W UMa) stars, the light curve shape indicates an extremely close binary system, whose components are strongly distorted from being spherical in shape, and have periods that are typically < 0.7 days. Examples of these three classes of eclipsing binary light curves are shown in Figure 1.1. Most eclipsing binary systems are not resolvable into their component stars. In par-ticular, those with stars only a few stellar radii apart are called close binary stars, and are typically separated by 1-20 RQ. In these systems, the tidal torques that are exerted on each star by its companion cause the binary to evolve such that the rotational axis of each star has an inclination of 0° with respect to the orbital plane, with orbits that are synchronised (Prot — Porb) and circularised (e = 0). In the 1950s the seminal work of Kopal [3] described properties of close binary stars very accurately by using the Roche model to interpret the light curves of these objects, rather than the more simplistic ge-Chapter 1. Introduction 4 -02 0 0.2 0.4 0.6 0.8 1 -0.2 0 0.2 0.4 0.6 0.8 1 IP •« "i---.V:::.i:..:;:; : =^:t;i"::- J EW -0.2 , 0 0.2 0.4 0.8 phase Figure 1.1: Examples of typical eclipsing binary star light curves. Top: A n Algol (EA) light curve. Middle: A 0 Lyrae (EB) light curve. Bottom: A W U M a (EW) light curve [2]. Chapter 1. Introduction 5 ometrical model that was used until that time [4]. In doing this work, he was able to classify close binary systems as detached, semidetached, and contact binary stars. The properties of each of these groups will be summarized, but first, a brief description of the Roche model is necessary. The Roche Model The Roche Model describes gravitational potentials in a two-body system, and assumes that the system is already synchronised and circularised. First, a reference frame is defined such that the origin coincides with the centre of the primary (the more massive companion in the system) and is corotating with that star (see Figure 1.2, top). The separation a of the companions is defined to be of unit length for convenience. The distance of an arbitrary point P(x, y, z) from the primary is given by r\ = (x2+y2+z2)1/2, and its distance from the other star (the secondary) is given by r2 = [(x — l)2 + y2 + z2]1^2. The gravitational potential $ at P(x, y, z) is the sum of the point-mass potentials of each companion and the rotational potential of the system, which is given by Gmi Gm2 OJ2 m2 \ o x -1 — T + y (mi + m2) (1.1) 1*1 r2 where mi and m2 are the masses of the primary and secondary, repectively, and u is the rotational velocity of the system. For a circularised, synchronous orbit of period P , u> can be recast through use of Kepler's Third Law as: 2 u ' = ( t ) ' = G ( " V m 2 > ° G ( " " + T O ; ) ' ( 1 ' 2 ) since a = 1. Substituting this and the mass ratio of the secondary to the primary, q = m2/mi, into equation 1.1, and defining the quantity $ n = —2$/G(mi + m2), the following expression is obtained: 2 , 2 9 , / _ q V , ^ ( l + gjrx (l + q)r2 \ (l + q)J is the normalized form of the potential, which can then be calculated for any point P(x,y,z) around the two companions. Through equation 1.3, one can find a series of Chapter 1. Introduction 6 equipotential surfaces. The bottom portion of Figure 1.2 shows an example of a cross section of these surfaces through the orbital plane of a binary system system with unequal masses. One can see that close to the position of each star, these surfaces are circular, or very near so, and as one moves away from each star, these surfaces become increasingly distorted. The points L 1 ; L 2 , etc. are called Lagrangian points. A test mass placed at any of these points experiences no net force acting on it, however a slight displacement of the mass will cause it to fall back into the potential well of the system, and so the Lagrangian points are referred to as being unstable. The point at which the surfaces surrounding the stars intersect is called L\, or the inner Lagrangian point. The equipotential surfaces that touch at L\ define the limiting volumes for each star in the binary system, and are referred to as the Roche lobes. The size of the Roche lobes depends mainly on two variables: the mass ratio q of the system, and the ratio of each star's radius to the separation between the companions (R/a). As the primary star expands due to its usual evolutionary process, it may eventually fill its Roche lobe. At this time, the star ceases to evolve as a single object with constant mass; matter can then be transferred easily through L\ to the secondary. In addition, L2 and L 3 , called the outer Lagrangian points, which are located on either side of each companion, can provide other escape routes for matter from each star. Mass loss and transfer is not only caused when stars expand to meet the Roche lobe. The Roche lobe itself can contract to meet the star's surface. This can be caused by effects such as magnetic braking, whereby a magnetically active low-mass main sequence star interacts with its stellar wind to lose angular momentum, causing the system to decrease in separation and period [2]. Angular momentum loss in close binary systems is thought to be one of the chief mechanisms preventing core collapse in globular clusters. This will be discussed in more detail in the next subsection. As stated earlier, the Roche Model was the basis for Kopal to group close binary systems into three classes: The detached binary systems are those whose component stars are situated well within their respective Roche lobes, and remain roughly spherical in shape. Semidetached binary systems are those in which one companion star fills its Chapter 1. Introduction 7 P(x,y,z) Figure 1.2: Top: Illustration of Roche geometry. Bottom: A n example cross section of Roche equipotential surfaces along the orbital plane of a close binary system [5]. Chapter 1. Introduction 8 Roche lobe, while the other does not. In Contact binary systems, both components completely fill, or even overfill their Roche lobes. Here, strong deviations from sphericity occur, and in the case of overfull Roche lobes, the two component stars share a common atmospheric envelope. Diagrams illustrating these three classes of close binary star are shown in Figure 1.3. Figure 1.3: Illustration of the classification scheme for close binary systems introduced by Kopal [3]. Top left: Detached binary system; stars are well within their respective Roche lobes. Top right: Semidetached binary system; one com-panion star has filled its Roche lobe. Bottom: Contact binary system; both stars have overfilled their Roche lobes and share a common envelope [2]. Chapter 1. Introduction 9 Binary Stars and Globular Clusters In the context of this study, the discovery of close binary systems in M4 would aid in understanding how the cluster has evolved over time. Even a small fraction of binary stars in a cluster can contribute greatly in preventing or delaying core collapse of the cluster. A full treatment of this topic can be found in Hut et al. [6]; a brief description of the mechanism is given below. There are two relevant time scales when describing core collapse. The first is the so-called half mass relaxation time, which is the typical time it takes a star to cross the half mass radius of the cluster, which is typically 106 years. Globular clusters tend to be in a stable equilibrium on this time scale. The other time scale of interest is the two-body, or thermal timescale. This is the typical time it takes for two-body encounters to redistribute the energy of the system between its component stars. This takes place in a timeframe on the order of 106 - 108 years, depending on the initial central concentration of the cluster. At this point, the central density of the cluster begins to undergo a rapid increase due to mass segregation, eventually producing a central cusp in its radial density profile. To halt this runaway collapse, some heating source must be available at the centre of the cluster. The main source of this heating is believed to be binary stars. When a single star approaches a close binary system, it can be temporarily captured by the system, or even exchanged with one of the original constituents of the binary. The result is that the binary will harden, meaning it will lose angular momentum and decrease in period and separation. This causes some of its energy to be transferred into the kinetic energy of the escaping star, so that the single star will leave the encounter with more energy than it had entering it [6], [7]. The existence of many blue stragglers in globular clusters, some of which are generally believed to be the product of close binary mergers, further supports the belief that binary stars are the principal heating source aiding in the prevention of a globular cluster from undergoing core collapse. Binary stars can often be recognised in the colour-magnitude diagram (CMD) of a cluster. In the case of binary systems comprised of similar main sequence (MS) stars, one Chapter 1. Introduction 10 would expect that since they are not resolvable as separate objects, they would appear on the C M D as a single star of the same colour, but twice as bright (i.e. 0.75 magnitudes brighter). Thus, a second sequence parallel to the MS would be formed by binary stars. In general, there should be no preference for equal-mass binary systems, but it has been shown that systems with components of unequal mass would produce a second sequence as well [8]. In globular clusters, observations of second sequences are rare. Even in cases where one is found, many stars in the sequence are actually optical doubles, due to the effects of crowding in globular cluster images. However, binary stars have been found in globular clusters. Contact binaries have been shown to form a sequence on globular cluster CMDs that extends uniformly on both sides of the MS turn-off; in fact, W U M a type eclipsing binaries are among the most common type of short-period variable stars found in GCs [9]. 1.1.2 Pulsating Variables Pulsating stars periodically vary in brightness due to expansions and contractions of the star itself. For example, the Cepheid variables have periods that typically range from 1-50 days. These stars become pulsationally unstable when they evolve onto a region of the Hertzsprung-Russell (H-R) diagram called the Instability Strip. This region defines a combination of luminosity and temperature which favours a state of pulsation rather than a stable equilibrium between gravitational forces and outward pressure in the star. This results in standing sound waves that resonate within the star,.causing it to pulsate radially. Other stars lying on the instability strip include, but are not limited to, R R Lyrae (P < 1 day) and 5 Scuti stars (P = 1-3 hours). These stars generally have magnitudes that are too bright to be included in this dataset (stars with magnitudes V < 19 and / < 17.5 are saturated on the images) . However, the instability strip does extend downward to the white dwarf (WD) sequence of the H-R diagram, and there do exist W D stars lying within this region that have been observed to be variable. These stars are named DAV (DA refers to the spectral type of these stars, in which hydrogen Chapter 1. Introduction 11 absorption lines are prominent, and V refers to the star being variable), or ZZ Ceti stars, after the first such star discovered. The periods of these stars are relatively short, about 100-1000 s, and typically vary with an amplitude on the order of a tenth of a magnitude. This dataset does include this region of the white dwarf sequence, and so a search for these stars is possible. Figure 1.4 shows the H-R diagram and the positions of some of these variable stars on it, including the ZZ Ceti stars. I 1 1 1 l I 5.0 4.5 4.0 3.5 logTeff Figure 1.4: Illustration of the location of white dwarf instability regions on the H-R diagram. Only stars within the DAV (ZZ Ceti) instability region are likely included in the M4 dataset [10]. The study of DAV stars is of great importance for several reasons: they provide a Chapter 1. Introduction 12 glimpse into the underlying structure of white dwarf stars, which is in itself an interesting topic. More generally, being degenerate stars, ZZ Ceti variables allow reserachers to study matter under extreme physical conditions, and through asteroseismology, they provide a laboratory for testing theories regarding the behaviour of degenerate matter. In addition, the star formation history of the Galaxy can be probed to very early times through interpretation of the properties of halo white dwarfs [10] (see also [11]). Stars that lie on the instability strip of the H-R diagram are not the only ones that undergo pulsation. In fact, there are several instability regions in which variable stars reside. The same is true for white dwarf stars. In addition to the ZZ Ceti stars, there are the D B V white dwarf variables (helium absorption lines are prominent), as well as the D O V (stars entering the white dwarf stage of evolution) and P N N V (planetary nebula nuclei) variable stars, which are associated with the birth of white dwarf stars, and thus are of much higher temperature [5]. Stars of brightness corresponding to these other types of variable white dwarf stars are not included in this dataset since they would be saturated on the images. 1.1.3 Super novae Supernovae (SNe) belong to the class of variable stars called eruptive objects, and are far and away the most luminous of all stars, at least briefly. They occur when a star violently explodes, and at the peak of their luminosity they can outshine their host galaxy. There are many classes of supernovae, but in general, most fall into one of three categories, based on their spectra. SNe which show hydrogen lines in their spectra are referred to as type II supernovae. Those that do not show prominent hydrogen lines in their spectra are called type I supernovae. Type I SNe are further subdivided into type la supernovae, which show S i + absorption lines in their spectra, and type lb supernovae, which do not show such lines. Based on the physical characteristics of these different classes, type lb SNe are more akin to type II supernovae. These events occur due to gravitational collapse and rebound in high mass stars. It is believed that type Chapter 1. Introduction 13 lb supernovae are related to the deaths of the most massive stars, which have lost their hydrogen envelopes, and thus show no hydrogen lines in their spectra. The progenitors of type la SNe are thought to be white dwarf stars which have gained enough mass to pass the Chandrasekhar limit of 1.4 M©, either by the merger of two white dwarfs, or by the accretion of matter onto a white dwarf in a close binary system with a main sequence companion. The importance of studying supernovae is severalfold. For one, SNe are the principal supplier of heavy elements and energy to the interstellar medium, making their study critical to the understanding the overall chemical evolution of the Universe. In addition, they are closely related to the star formation rate in galaxies, especially in the case of type II and lb supernovae, which occur relatively quickly after the formation of their progenitor stars. Type la supernovae have been shown to be reliable standard candles. Unlike other standard candles such as Cepheid variable stars, they can be seen at cosmological distances, making them useful for a variety of research endeavors. For example, they can be used to measure distances to high-redshift galaxies, thus aiding in the study of galactic evolution. However, the use of type la SNe as standard candles has in recent years been most publicized due to studies that claim to show that the Universe is accelerating in its expansion, and is characterized by a cosmology with a low matter density and a non-zero cosmological constant (see Figure 1.5). This implies the existence of a vacuum energy density with negative pressure, otherwise known as "dark energy", the driving force behind the accelerating expansion of the Universe [13], [12]. The ability of this dataset to probe very faint magnitudes allows for the examination of numerous background galaxies for signs of a supernova event. Expectations regarding the discovery of SNe in this dataset will be discussed in the following section. 1.2 Expectations As discussed earlier in this chapter, eclipsing binary stars are known to exist in globular clusters. For M4 in particular, seven W U M a binary stars thought to be cluster members Chapter 1. Introduction 14 Redshift Figure 1.5: Hubble diagram showing the measurements of luminosity distances of type la supernovae at various redshifts, taken from various research campaigns to measure these distances. It is believed that the supernovae at high redshift are fainter and thus more distant than expected for a decelerating Universe, suggesting that the Universe is accelerating in its expansion, and implying the existence of a non-zero cosmological constant [12]. Chapter 1. Introduction 15 were found by Kaluzny et al. [14] in an 8.8' x 8.8' field centred on the cluster with apparent magnitudes reaching as faint as V = 17.69. The field observed in the current dataset is well within this area. However, unlike the above study, the magnitude range that can be probed for variability in the current study ranges from V ~ 20 — 26, but there is no reason to expect that the existence of binaries is exclusive to bright magnitudes. In a recent study of the globular cluster 47 Tucanae (47 Tuc), which probed magnitudes as faint as V = 25, 11 detached eclipsing binaries and 15 W U M a systems were discovered in a sample of 46422 main sequence stars, resulting in an observed binary frequency of 0.056%. [15]. In this study of M4, 2102 stars have been analysed. If no assumptions are made about the dependence of the distribution of binary systems with cluster radius, one would expect that only one or two binary stars should be discovered in the field observed based on the observed frequency in 47 Tuc. This already small number is decreased if it is assumed that the majority of binaries are expected to reside close to the cluster centre, since the M4 field was taken at about 6 core radii from the cluster centre. The picture becomes even more grim considering that only 9 of the 15 W U M a eclipsing binaries found in the 47 Tuc study are below the main sequence turn-off point. However, 71 B Y Draconis variables, which are thought to be members of binary systems, were also discovered, making up 0.15% of the sample. This converts to a slightly more optimistic expectation of 3 or 4 of these objects being discovered in the M4 field. As of 2001, only 31 ZZ Ceti stars have been discovered [16]. In searching for ZZ Ceti variables using this dataset, it is not expected that many will be found. The exposure time of each observation in this study is 1300 s, which is on the order of the typical pulsation period observed for these stars. This does not allow for the sampling of the star's light curve at various stages of its pulsation cycle. One would then expect that photometry of these stars would yield an average magnitude, with very little or no signs of variability, thus making it difficult to classify them as ZZ Ceti variables, even though in reality they may be present in the sample. As a way of gauging the number of supernova events that are expected to be seen in this dataset, a comparison is made to a recent study of the Hubble Deep Field (HDF) Chapter 1. Introduction 16 in which two supernovae were discovered. As with M4, the H D F was observed with the W F P C 2 camera on HST, and just as deep, with a limiting magnitude of I ~ 28 [13], [17], [18]. However, unlike the HDF, the images of M4 are taken close to the Galactic plane and ecliptic, and contain zodiacal light, as well as many stars. This results in much higher background levels compared to the HDF. This significantly decreases the number of observable faint galaxies in the field. As will be seen in §3.5.2, 170 galaxies were identified in the M4 images. By contrast, ~ 2100 galaxies were identified in the H D F [19]. Scaling the discovery of 2 supernovae in H D F to the current dataset, it is estimated that ~ 0.08 extragalactic SNe would be expected in this set of images. It is not expected that any SNe will be found within the cluster itself; stars residing in M4 are very old relative to the lifetime of a star that eventually explodes in a type II supernova. The discovery of a type la supernova is a possibility, but a very narrow one; the rate of type la SNe occuring in the Milky Way is thought to be of order one per 267 years [7]. There also exists the possibility of discovering variable objects which have not been discussed here, such as cataclysmic variables, flare stars, transiting planetary-sized ob-jects, or perhaps a new species of variable star. Cataclysmic variables (CVs) are a special class of binary system with a white dwarf star as the primary and a low-mass main sequence star as the secondary. Here, the white dwarf is accreting matter from the secondary, which has filled its Roche lobe. Included as types of CVs are classical and dwarf novae. These objects are very unpredictable, making it difficult to quantify how many are expected to be seen in the M4 field. However, in globular clusters, which have a high frequency of stellar interactions, it is expected that an abundant population of CVs would be found [20]. A relatively small sample of stars may limit this abundance in the current observations, but the possible existence of CVs in this dataset is not discounted. Flare stars are M or K-type stars with very strong magnetic fields, which undergo occasional and sudden outbursts of energy, causing their brightness to increase tem-porarily. Flare stars include the BY Draconis (BY Dra) variables. These stars are binary systems which are chromospherically active, and show enhanced magnetic activ-ity, which is generally manifested visually as flaring or low-amplitude periodic variablity Chapter 1. Introduction 17 due to starspots. As touched upon earlier, in the binary frequency study of 47 Tuc, 71 B Y Dra variables were discovered with periods ranging from 0.4-10 days in the magni-tude range V ~ 14 - 24.5, comprising 0.15% of their sample of stars [15]. This converts into an expectation of finding about 3 or 4 B Y Dra stars in the M4 field, assuming the distribution of these stars is more or less constant with cluster radius. The main science driver for the above-mentioned study of 47 Tuc was to define the frequency of close-in, gas giant planets orbiting main sequence stars [21]. This study reported a null result, even though simulations that were carried out clearly show excellent sensitivity to transits. However, those observations were centred on the core of 47 Tuc, which may be too hostile an environment for planets or protoplanetary disks to survive. The M4 images were taken far from the core of the cluster, where the number density of stars is much smaller, and may be a more suitable environment for planets to form. On the other hand, most extrasolar planets are found to be orbiting stars with relatively high metallicity. M4 is a more metal poor globular cluster than 47 Tuc, perhaps indicating that a smaller fraction of planets will be found in the current dataset. If the null result from 47 Tuc is any indication, it is expected that in the present study of M4, which contains only ~ 4.5% of the number of stars in the 47 Tuc dataset, no planetary transits will be found. 1.3 Thesis Overview This thesis describes time-series photometry performed on stars observed over a period of about 10 weeks with the Hubble Space Telescope, in a search for variability in the globular cluster M4. Chapter 2 deals with the data and its reduction. Chapter 3 discusses the analysis of the reduced data. This includes the description of methods used to select candidate variable stars, period searches performed on those candidates, tests of the reliability of these period searches using artificial light curves, and searches for non-periodic variables, with particular attention paid to the search for supernovae in background galaxies. Chapter 4 presents results obtained from the data analysis and Chapter 1. Introduction 18 reliability tests. Finally, chapter 5 discusses these results and any possible implications they may have. 19 Chapter 2 Observations and Data Reduction The Hubble Space Telescope (HST) has been very successful at providing data for anal-ysis of objects and fields which are very difficult with ground-based telescopes. The advantages of the superior resolution of HST over its ground-based counterparts is evi-dent in many past research endeavours. One example is the imaging of the Hubble Deep Field (HDF; [19]). This required both very deep imaging and high precision photometry, which the HST was instrumental in providing. Another recent program which made full use of the above-mentioned advantages of HST was the deepest ever imaging of a globular cluster, specifically Messier 4 (M4). The primary purpose of the images was to search for the faintest (and thus coolest and oldest) white dwarf stars in the cluster, in order to find the end of the white dwarf sequence, and thus age the cluster itself, constraining the age of the Universe. For this study, two separate epochs of imaging were performed. The first series of data was taken during cycle 4 (1995; see [22]), and the second series during cycle 9 (2001), which contained the majority of the data that now exists for the cluster [18], [23]. It is this second epoch of observations which provided the data for this variability search. 2.1 Observations The cycle 9 observations of M4 were taken over 123 orbits with the Wide Field Planetary Camera (WFPC2) on HST, covering a period of about 10 weeks in early 2001. These im-ages were taken in two filters: F606W and F814W, whose transmission curves are shown in Figure 2.1 [24]. These filters will henceforth be referred to as V and I, respectively, Chapter 2. Observations and Data Reduction 20 U 1 1 1 1 I I I I I I I I I I I I I I i I 2000 4000 6000 8000 104 MA) Figure 2.1: Transmission curves for the F606W (left) and F814W (right) filters aboard HST [24]. Chapter 2. Observations and Data Reduction 21 for simplicity (these are their Johnson-Cousins counterparts, though their corresponding transmission curves are not identical). This data set consists of 98 x 1300 s exposures obtained in V, and 148 x 1300 s exposures obtained in I, all taken in the same field of M4, about six core radii 1 (~ 5') from the cluster centre. These observations make up a time-series dataset that can be used to investigate the presence of variable objects in this particular field of M4. The existence of data in two filters is useful for two reasons: (i) it allows for the verification of variability in two inde-pendent bandpasses; and (ii) it allows for the investigation of particular populations of stars which are easily recognised in the colour-magnitude diagram (CMD) of the cluster. These include possible binary sequence members, or stars which fall along the ZZ Ceti instability strip, for example. 2.2 Data Reduction 2.2.1 Preprocessing and Combined-Image Photometry The images were preprocessed according to recipes given in Stetson [25] and Stetson et al. [26]. Crowding was not an issue on these images, due to the distance from the cluster centre, as well as the high resolution of the HST images. A master star list, frame-to-frame coordinate transformations, and photometry of the combined image of the field was completed by Brewer et al. [27], which resulted in colour-magnitude diagrams that reach apparent magnitudes of V ~ 30 and I ~ 28. These extremely faint data allow for the study of variability in this globular cluster down to magnitudes which have rarely been explored. 1The core radius of a globular cluster is defined as the radius at which the surface brightness is half its central value. Chapter 2. Observations and Data Reduction 22 2.2.2 Single-Image Photometry To create time-series data for each star found in the M4 field, it was necessary to perform photometry on each individual exposure. To accomplish this, the following steps were taken: (1) the now-known frame-to-frame transformations were used to create a combined median image of the field for each chip on W F P C 2 . This was performed using the M O N T A G E 2 software written by Peter Stetson, which transforms the coordinates of each individual image to match a reference coordinate system, and then produces a combined image. In order to remove as much cosmic ray contamination as possible, it was specified that a pixel should appear in at least six of the frames in order to be included in the final combined image. (2) Once the combined median image was created, aperture photometry was performed on it using D A O P H O T . Subsequently, point spread function (PSF) photometry was performed using the A L L S T A R program, by employing PSFs built specifically for each chip and filter on W F P C 2 by Peter Stetson [28]. (3) The output photometry and frame-to-frame transformations were then provided to A L L F R A M E , again written by Stetson, in order to perform PSF photometry on each individual exposure [29],[30]. This software automates the PSF photometry of each individual exposure by using the frame-to-frame coordinate transformations. It also derives star centroids for each original image, and facilitates cross-identification of any given star between frames by assigning identification numbers to each star, which are consistent through all individual frames. Once this single-image photometry was completed, the location of any given star and its accompanying photometry was specified for every single exposure taken in the M4 field. 2.2.3 Magnitude Offset Calculations Since HST is above the Earth's atmosphere, it is not subject to seeing effects experienced by its ground-based counterparts. However, it is susceptible to slight focus changes due to the telescope moving in and out of the Earth's shadow as it orbits, causing temperature fluctuations (an effect known as breathing). As a result, the full width Chapter 2. Observations and Data Reduction 23 half-maximum (FWHM) of the PSF of stars on the frames can vary slightly between individual images. The result is that small offsets in the magnitudes calculated from performing PSF photometry will exist from frame to frame. These must be corrected before performing any analysis on the data. To do this, the magnitudes that are output by A L L F R A M E for every image on a given chip were compared to those of a chosen reference image, which was chosen to be the first image taken in the data set2. The offset was calculated by taking the weighted mean of the magnitude difference for each star common to both frames. Thus, for a given image, the offset to be added to each magnitude calculated from PSF photometry would where the summation is performed over all the stars in each frame. The error on the magnitude offset is then 1/y/w. This calculation is equivalent to fitting a straight line through a plot of m vs. m r ef, and obtaining the y-intercept. A n example of such a plot is shown in Figure 2.2. 2.2.4 Data Point Rejection The usual way of rejecting data points consists of fitting the distribution of magnitudes of a given star to a Gaussian profile, and throwing out any data points at a magnitude greater than 3<r or so from the mean of the distribution, and perhaps reiterating this process a number of times. When looking for variability, however, objects searched may include those which vary greatly in magnitude, such as eclipsing binary stars with deep minima, or supernovae. By performing this so-called uka clipping" type of data 2 Use of the combined median image as a reference was also explored, but there turned out to be a negligible difference in the final magnitudes and magnitude errors obtained, compared with using a single exposure as the reference image. be: (2.1) where (2.2) Chapter 2. Observations and Data Reduction 24 28 CO 6 CD m 3 26 24 •3 2 2 tut) cd S 20 18 - i 1 r i 1 r - i 1 r - n I r V* 1 • • • v'-/ / S _ l I l _ -1 I L_ 18 20 22 24 26 magnitude (reference image) 28 Figure 2.2: Plot of stellar magnitudes computed for one image compared to that of the chosen reference image, both taken in the F814W (I) filter on the WF2 chip with W F P C 2 . The line plotted is the best fit to the distribution. The y-intercept of this line is the calculated magnitude offset between the images. Chapter 2. Observations and Data Reduction 25 point rejection, data points that may be rejected could actually be real effects due to a significantly varying object. For this study, a "bad data point" is considered to be one which has a magnitude error which is significantly larger than the rest of the data points for the corresponding star. To illustrate how "significantly large" was defined, Figure 2.3 shows plots of root-mean-square magnitude (RMS or a) vs. mean magnitude in I and V for each chip. It can been seen that for all of these chips and filters, the scatter of data points begins to increase significantly at a well defined "elbow" in the distribution, after which point the RMS grows rapidly with fainter magnitudes. With a few exceptions, the maximum scatter of data points for all the stars found in the data set is approximately ~ 0.09-0.1 magnitudes. To be conservative, this is the value chosen as the cutoff magnitude error, and any data point with an error above this value was rejected as a bad data point. Thus, there are no data points used which have an error larger than the RMS scatter of almost any star in the dataset. The remaining data were used to create time-series magnitude profiles of each star found, and analysis was then performed to determine if any of these stars demonstrated variability. Chapter 2. Observations and Data Reduction 26 0 . 1 0 . 0 5 tT o o.i 0 . 0 5 0 0 . 1 0 . 0 5 0 0 . 1 0 . 0 5 0 b 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - PC, / : • v ' -i-. • - T ' i " 1—r r—r • 1 "ft- f t " " " ' " n i i i 1 i — 1 1 | 1 1 1 | I 1 1 | 1 1 1 | 1 I 1 | 1 - WF2, I : : : t 1—|—1—1—1—j—1—1—1—|—1—1—1—1—1—1—1—|—r— - WF3, J . : L,. ••!—h~t~-f-r- 1 — f ' V i T i " r T T T i i l l -= H — ( - H — i | i i i — \ — \ — i — i — \ — \ — i — i — ] — \ — - WF4, I -' •;~-t"7 ' r ''I' 1 T "r "'t 1 " . ' " T ^ f - *Ti i l l -1 8 20 22 24 26 1 8 20 22 24 26 I I | I I I | 1 1 1 | 1 1 1 | 1 1 1 | 1 - PC, v ••' : : : 7 ;/ i —r i r r • i 1 i * ' i t i I I I I I i 1 i —1 i i 1 i i i 1 i i i 1 i i i 1 i i i j -1 - WF2- v .-.: v " 1 • • i v ' * -i i i i 1 i i i l i i i I i i i i — f^—1 1 1—1 1 | 1 1 1—|—1 1—1 1—1 .• 1 1 j . - . 1 - WF3, V . -r • t 4 f ~-• ••>*. -i J -- t 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1- 1 1 — :+H—|—1—1—1—1—1—1—1—|—1—1—1—|—HH—1—f-H— : WF4, V . . " ; , V -.. .'. A*.? \ . : — •*?' — - t i " T i i i T i i i I i i i 1 i i i 1 i — 2 0 2 2 2 4 2 6 2 8 2 0 2 2 2 4 2 6 2 8 magnitude Figure 2.3: Graph of RMS magnitude as a function of magnitude for stars found in each chip of W F P C 2 in the M4 images. Each distribution has a well-defined "elbow" above which the scatter of data points for the stars rises sharply. The distribution reaches a maximum in each case at an RMS of approximately 0.1 magnitudes. The chip and filter are labelled in the top left corner of each plot. 27 Chapter 3 Data Analysis 3.1 Light curves After reducing the data and rejecting bad data points as described in the previous chap-ter, variability searches were conducted on the light curves of the stars in the data set. As discussed in §1.1, a light curve is the set of data points representing a given star's magnitude over time. A n example of a light curve from this dataset is given in Figure 3.1. 3.2 Candidate selection It would be very inefficient to search for periodicity in the more than 2000 stars that are present in the dataset. It is naturally expected that almost all of the stars in the sample are not variable stars, so instead, different criteria for choosing candidate stars were employed in order to narrow down the number of candidate variable stars. First, any star that contained in its light curve less than six data points was immediately rejected as a candidate. This may seem like a rather generous cutoff value, but when searching for possible supernova events, six points could be enough to determine if it is indeed a viable candidate. Stars were excluded from the search list if: • The image of the star is saturated on the C C D ; • The star is too close to the edge of the image; • The star image is affected by light from a nearby saturated star; or Chapter 3. Data Analysis "i 1 1 1 1 1 1 1 1 1 1 r r 19.4 19.6 19.8 M M M i ?! i M 20 h 19.4 19.6 19.8 20 _i i i i_ 2150 2155 2160 ~i 1 1 1 1 1 1 1 r i i n I _ l I I l _ 2210 2215 2220 MJD - 49791 Figure 3.1: Example light curve for a star in WF4, including magnitude errors. Chapter 3. Data Analysis 29 • The star is partially or exactly aligned with a diffraction spike of a bright or satu-rated star. The remaining candidates are presented in tables accompanying each subsection. 3.2.1 Statistical Outliers One way to spot potential variable stars is to use the magnitude dispersion of a given star's light curve relative to the mean magnitude of that star. First, a simple identification was made of outliers in standard deviation (RMS magnitude) as a function of magnitude. Secondly, the variability index statistic, which is robust against data corrupted by cosmic rays and warm pixels, was used. Both methods involve selecting stars that do not follow an otherwise well-defined distribution. RMS Magnitude Outliers As discussed and shown in §2.2.4, the RMS magnitude is a well-defined function of magnitude, which almost every star follows. There are a number of stars, however, which deviate from this distribution. These stars have a larger scatter than is expected for stars of similar magnitude, and are potential variable star candidates. These R M S magnitude outlier stars are shown in Figure 3.2, and candidates chosen by using this method are listed in Table 3.1. In this and other tables, coordinates of the objects listed are based on work on M4 by Ibata et al. [31], and coordinates for stars not included in that study were calculated using the C C M A P and C C T R A N tasks on the I R A F analysis package. Chapter 3. Data Analysis 30 0.08 0.06 0.04 0.02 0.08 0.06 0.04 0.02 0 0.08 0.06 0.04 0.02 0 0.08 0.06 0.04 0.02 0 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . r PC, / • -_ - o I - ° . ••• — - o : - Q. d?J j> ._0 *n i i i i i i i i i i i i i i i -. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 . r WF2, J ° r o ° z. o ° <: ~ o o o _ ; : * • -= ->~r ,.°&,.°&fy -l l t l l l l l l l l l l l f l ~"~' rl—1—|—1—1—1—|—1—1—1—1—1—1—1—|—1—H r WF3, I ' '• -_ - o • -- 0 • ; - 0 - • .— r .e t a „ ,n .Oo, ft, .m9f&* | - 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 — 1—|—1—1—1—|—1—1—1—1—1—1—1—|—1—1— " WF4, / o V o -9 ,? macro q t f i t f i a S & ^ — i i i i i i i i i i i i i i i i i — 18 20 22 24 18 20 22 24 b . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . r PC, V '•. ~ o • • ~ j " , . . o r ° . . . . ^ . i . . ^ ' 1 ~l 1 1 1 1 1 1 '1 1 1 1 1 1 1 t _ _ i i | i i i | i i i | i i i | i i _ - °o ••: -- WF2, v 0<*>" -- <-> . -r o ~ - . * — M ^ c u , , ^ » r t i c & a ? . B u g & £ £ ** : "*i i 1 i i i i i i i i i i i i i i _ 1 1 I I 1 1 I I 1 1 I I 1 1 I I 1 _ r WF3, V- 0 r o . r — : <*•»'.- = " o ~ ; .•^».. ,^«..-aaA- r -_Sa-~a» < i .o• ( ' - f f , • , -— i i i i i i i i i i i i i i i i i — i I I  i   I I   I I - WF4. V °@ :' -- o -. • ~ 0 •• : ° ° .C : — i i i i i i i i i i i i i i i i i — 20 22 24 26 20 22 24 26 magnitude Figure 3.2: Shown again are the RMS magnitude distributions for each chip and filter. Star chosen as candidate variables are labelled as open circles. Chapter 3. Data Analysis 31 Table 3.1: RMS outliers chosen as candidates for variability search. Shown for each star are the WFPC2 chip on which it was found, its identification number and coordinates, the filters for which it was found as an outlier, its V magnitude and V — I colour, their associated errors, and comments about the star. Chip ID RA (J2000) (hh:mm:ss.ss) Dec (J2000) (dd:mm:ss.s) Filters V <jv V - I <Tv-I Comments PC 1 16:23:56.26 -26:32:27.0 /, v 22 349 0.024 0 831 0.042 isolated; very near to edge of image PC 2 16:23:55.28 -26:32:35.8 i,v 23 383 0.017 2 360 0.020 isolated; close to edge of image PC 3 16:23:54.59 -26:32:42.2 i,v 24 798 0.011 2 862 0.014 isolated; close to edge of image PC 5 16:23:55.56 -26:32:32.1 V 23 615 0.007 1 690 0.014 isolated PC 132 16:23:55.10 -26:32:04.2 I, v 25 969 0.033 3 295 0.036 isolated; next to saturated star WF2 21 16:23:53.41 -26:31:29.1 V 24 041 0.034 2 468 0.036 next to saturated star; very near to edge of image WF2 24 16:23:53.56 -26:31:29.8 V 24 952 0.021 2 806 0.026 next to saturated star WF2 73 16:23:53.87 -26:31:24.1 V 24 040 0.021 2 552 0.027 next to saturated star WF2 120 16:23:54.35 -26:31:24.6 I 26 096 0.029 2 418 0.036 next to saturated star WF2 277 16:23:56.44 -26:31:31.1 I, v 24 097 0.036 2 839 0.046 next to saturated star WF2 395 16:23:59.37 -26:31:57.2 V 24 923 0.033 2 218 0.035 close to edge of image WF2 511 16:24:00.26 -26:31:48.3 V 25 243 0.014 3 173 0.018 near to saturated star; close to edge of image WF2 515 16:23:58.25 -26:31:10.3 I 25 377 0.029 3 046 0.036 isolated; appears to be two overlapping PSFs WF2 585 16:24:00.68 -26:31:37.1 V 21 197 0.024 1 368 0.031 isolated; close to edge of image WF2 661 16:23:56.63 -26:32:21.7 V 25 530 0.027 3 006 0.029 close to bright star WF3 58 16:23:59.89 -26:33:21.0 V 20 853 0.011 1 865 0.012 isolated WF3 392 16:24:02.42 -26:32:59.9 r,v 24 117 0.036 0 565 0.039 isolated; very near to edge of image WF3 554 16:24:00.53 -26:31:49.8 V 21 152 0.015 1 203 0.020 isolated WF3 611 16:24:01.13 -26:31:49.2 V 22 406 0.025 1 349 0.026 isolated; close to edge of image WF3 613 16:24:01.34 -26:31:52.8 V 25 322 0.023 3 122 0.025 isolated; close to edge of image WF3 619 16:24:01.02 -26:31:45.7 I 19 978 0.053 1 411 0.059 isolated; close to edge of image WF4 11 16:23:53.74 -26:32:53.0 V 24 614 0.033 2 711 0.036 near another star - PSFs may be overlapping WF4 53 16:23:56.65 -26:32:32.6 i,v 24 965 0.021 2 781 0.022 isolated; very near to edge of image continued on next page Chapter 3. Data Analysis 32 Table 3.1: continued Chip ID RA (J2000) (hh:mm:ss.ss) Dec (J2000) (dd:mm:ss.s) Filters V av V - J crv-l Comments WF4 55 16:23:56.58 -26:32:33.7 V 21.333 0.005 1 960 0.006 isolated WF4 58 16:23:52.32 -26:33:13.8 V 21.965 0.024 2 070 0.027 next to saturated star WF4 63 16:23:56.58 -26:32:34.8 I, v 22.648 0.009 1 880 0.010 isolated WF4 190 16:23:53.79 -26:33:16.9 V 24.564 0.020 2 638 0.023 next to saturated star WF4 345 16:23:53.61 -26:33:42.4 V 24.964 0.022 1 436 0.023 isolated; very near to edge of image WF4 427 16:23:56.32 -26:33:32.0 I, v 25.599 0.016 2 735 0.020 near another star - PSFs may be overlapping WF4 550 16:23:56.72 -26:33:48.3 I, v 24.235 0.030 0 580 0.032 isolated; near to saturated star WF4 551 16:23:59.03 -26:33:26.9 I, v 24.550 0.013 2 742 0.016 isolated; near to saturated star WF4 555 16:23:59.14 -26:33:26.1 i, v 24.877 0.024 2 794 0.028 isolated WF4 563 16:23:59.19 -26:33:26.5 I,V 22.291 0.031 1 431 0.035 isolated; near to saturated star WF4 579 16:23:55.60 -26:34:02.3 V 22.006 0.020 2 118 0.021 isolated WF4 581 16:23:55.30 -26:34:05.6 V 23.336 0.022 2 413 0.029 isolated; close to edge of image Chapter 3. Data Analysis 33 Variability Index While the above method of choosing variability candidates based simply on the scatter of the light curves of the stars in the sample is a good first pass in the candidate search, is not terribly robust against data corrupted by cosmic rays and warm pixels, which are quite common in HST images. A more robust candidate selection criterion that attempts to account for these effects is the variability index statistic formulated by Stetson [32]. Another attractive feature of this method is that it makes use of data in all filters employed in the dataset in determining the variability index statistic. What follows is summary of the method; details can be found in the original paper. The variability index J is given by: j = E L i wk sgn(Pfc)y / |^| Here, the user has defined n pairs of observations, each with an associated weight wu-A pair is usually defined as two observations taken within some maximum separation in time, which is chosen by the user: It is important to note here that equation 3.1 also allows single observations to be used in the instances where a data point is not within the chosen date threshold of any other data point. Pk is the product of the normalized residuals of the two observations (denoted by the subscripts i and j) that form the A;th pair, and is given by: 0 J $i(k)$j(k) if i(k) / j(k) (observation pair); Pk = < (3.2) I $i(k) ~ 1 if i(k) = j(k) (single observation), where 8^k) (or $j(k)) is * n e magnitude residual of a given observation from the weighted average of all observations in the same bandpass, scaled by the standard error: / n rrn-m = V ^ ^ T ' ( } where n is the total number of observations contributing to the mean, i.e., the number of observations taken in the same filter as the ith. data point. Instead of using cr-clipping techniques in an effort to reject bad data points from the light curve of a star, which can potentially remove a measurement that reflects a real Chapter 3. Data Analysis 34 variation in the brightness of an object, the stars are reweighted according to the size of their residuals as calculated in equation 3.3. To do this, the average magnitude m in equation 3.3 is redefined as follows: where mnew is the new reweighted mean, and fninit is the initial arithmetic weighted mean, which is multiplied by a weighting factor. This procedure is iterated until the mean and the individual weights stabilize. According to Stetson, experimentation with artificial corrupt datasets show that the effectiveness of this method is not very sensitive to the values of a and b above. Following values used by Stetson for HST data, I have adopted the values a = b = 2. For my purposes, I have considered two consecutive points as a pair if they are separated by no more than 0.03 days, which is typical of the spacing in time between data points within a given orbit of observations in this dataset. If this is not the case for any one data point, it is treated as a single observation in equation 3.2. Pairs of observations were assigned a weight Wk = 1 and single observations were assigned a weight of u>k — 0.25 in equation 3.1, following work by Mochejska et al. [33], in which 47 low-amplitude variable stars were discovered in the open cluster N G C 6791. Values of J were computed for every star, and those with values of J that were obvious ouliers in the variability index distribution as a function of magnitude were considered as variable star candidates. These distributions are shown in Figure 3.3, where stars selected as candidates are plotted as open circles. A final list of candidates arising from this process, excluding any stars satisfying the additional criteria described in §3.2, is given in Table 3.2. Both candidates listed in this table are common to this method and the R M S outlier method discussed in the previous subsection. 3.2.2 Main Sequence Outliers As explained in Section 1.1.1, binary stars are expected to form a sequence on the H-R Diagram (and CMD) that is parallel to the main sequence (MS). Looking at the colour-(3.4) Chapter 3. Data Analysis 35 " i — | — i — i — i — | — i — i — i — | — i — i — i — | — i — r PC 6 0 4 0 20 "•a 0 —I I I I I 6 0 4 0 2 0 o o o \' ]'• 'Y'r'l- "|' i WF3 J I i i i ] i a ' • ••«*"•< Y T I I I I I I I I I I I I I I I T" WF2 i | i i i | ' i i i [• i - - f - j i - | i WF4 o o ± _l I I I I V • -1 • i '.i • -J | 18 2 0 2 2 2 4 18 2 0 2 2 2 4 Figure 3.3: Variability index J as a function of I magnitude for stars found in each chip of W F P C 2 in the M4 field. Outliers of this distribution are plotted as open circles. Chapter 3. Data Analysis 36 Table 3.2: Variability index outliers chosen as search candidates. Shown for each star are the WFPC2 chip on which it was found, its identification number and coordinates, its V magnitude and V — I colour, their associated errors, the variability index calculated for the star, and comments. Chip ID RA (J2000) (hh:mm:ss.ss) Dec (J2000) (dd:mm:ss.s) V °V V-I av-l J Comments PC 1 16:23:56.26 -26:32:27.0 22.349 0.024 0.831 0.042 27.39 isolated; very near to edge of image WF2 21 16:23:53.41 -26:31:29.1 24.041 0.034 2.468 0.036 11.63 very near to edge of image; near to saturated star magnitude diagram of M4 (Figure 3.4, left; [27]), one can see no obvious second sequence in the magnitude range observed, other than the expected faint white dwarf sequence. Nevertheless, there are stars that appear to be outliers to the main sequence, lying above it, or between it and the white dwarf sequence. The C M D on the right hand side of Figure 3.4 is the same as that on the left, but stars which obviously do not lie on the MS are plotted as open circles. Out of these stars, period searches were performed on those which were not excluded according to the criteria listed in §3.2. These remaining candidates are listed in Table 3.3. 3.2.3 ZZ Ceti Candidates Section 1.1.2 describes the properties of ZZ Ceti stars, as well as their location on the H-R and colour-magnitude diagrams. The C M D on the left hand side of Figure 3.5 shows which stars lie within the instability strip in the observed white dwarf sequence of M4 [34]. This is where it is expected that the ZZ Ceti variable stars would be found. To be conservative, the stars considered for period searches as possible ZZ Ceti variables were chosen to lie within a broader magnitude and colour range ( A V = 2 magnitudes and A(V — I) = 1 magnitude) than the region actually thought to contain these stars. The C M D shown on the right hand side of Figure 3.5 highlights the stars chosen as ZZ Chapter 3. Data Analysis 37 1 20 > 25 30 1 [ I 1 1 1 1— ' 1 • -V 0 \ 0 -- 0 -«0 o %: 0 '•- o o ° \ 0 o-h O T- 0 0 0 • . \ 0 •• 0 o . 0 0 0 0 0 . .v*;:;!,.-';.- 0 0 -0 1 1 ' I . ' 1 V - I V - I Figure 3.4: Left: Proper-motion cleaned colour-magnitude diagram of M4 [27]. There are no obvious signs of a binary star sequence running parallel to the main sequence of the cluster. Right: Main sequence outliers chosen as variability candidates are plotted as open circles on the same C M D . Chapter 3. Data Analysis 38 Table 3.3: M4 main sequence outliers chosen as candidates for variability search. Shown for each star are the WFPC2 chip on which it was found, its identification number and coordinates, its apparent V magnitude and V — I colour, their associated errors, and comments about the star. ID RA (J2000) Dec (J2000) (hh:mm:ss.ss) (dd:mm:ss.s) V-I av-i Comments 16:23:56.26 -26:32:27.0 22.349 0.024 0.831 0.042 PC 97 16 23 54 33 -26 32 20 6 22 639 0 009 0 706 0.015 WF2 404 16 23 56 79 -26 31 07 1 24 480 0 027 2 882 0.029 WF3 78 16 23 58 95 -26 32 58 1 25 262 0 023 3 554 0.027 WF3 532 16 24 02 37 -26 32 29 9 24 014 0 023 2 786 0.027 625 16:24:02.30 284 16:23:54.03 -26:32:08.4 -26:33:29.6 27.190 0.027 2.821 0.029 23.985 0.018 2.044 0.019 isolated; very near to edge of image isolated isolated isolated; near to saturated star isolated; next to diffraction spike of saturated star isolated; next to diffraction spike of saturated star; very near to edge of image isolated; near two saturated stars WF4 327 16 23 57 94 -26 32 59 9 23 409 0 021 2 572 0 025 isolated WF4 335 16 23 56 22 -26 33 16 9 22 937 0 016 1 884 0 018 isolated WF4 473 16 23 54 87 -26 33 53 3 24 443 0 010 3 007 0 016 next to diffraction spike of saturated star WF4 519 16 23 57 81 -26 33 31 4 22 926 0 013 2 153 0 018 isolated; near to diffraction spike of saturated star Chapter 3. Data Analysis 39 Ceti candidates as open circles, and the range of parameter space within which these candidates were chosen is bordered in the diagram by a dashed line. Period searches were then performed on these stars, provided they are not excluded further, as outlined in §3.2. The remaining candidates are listed in Table 3.4. 22 24 h 26 28 V •V* S m k&l HB-1 — 0 - 1 HHH > 25 V - I Figure 3.5: Left: Stars lying within the white dwarf instability region on the M4 C M D are bordered by a black box [34]. Right: Stars chosen as candidates to search for ZZ Ceti variables are plotted as open circles, and the ranges of V and V — I enclosing these candidates is bordered by a dashed line. This is a more conservative magnitude and colour range than that shown in the C M D on the left. Chapter 3. Data Analysis 40 Table 3.4: M4 white dwarf sequence stars chosen as ZZ Ceti variable star candidates. Shown for each star are the WFPC2 chip on which it was found, its identification number and coordinates, its V magnitude and V — I colour, their associated errors, and comments about the star. Chip ID RA (J2000) (hh:mm:ss.ss) Dec (J2000) (dd:mm:ss.s) V av V - / °~V-1 Comments WF2 150 16:23:54.72 -26:31:26.3 23.659 0.018 0 501 0.021 isolated; near diffraction spike of saturated star WF2 259 16:23:56.97 -26:31:44.0 24.978 0.031 0 720 0.034 isolated; near diffraction spike of saturated star WF2 273 16:23:58.12 -26:32:03.1 24.648 0.027 0 752 0.028 isolated WF2 318 16:23:58.16 -26:31:50.8 24.817 0.017 0 749 0.019 isolated; near diffraction spike of saturated star WF2 340 16:23:58.88 -26:32:00.0 24.515 0.025 0 649 0.028 isolated WF2 485 16:23:58.90 -26:31:28.3 23.951 0.032 0 552 0.036 isolated WF2 623 16:23:56.88 -26:31:47.0 25.431 0.028 0 943 0.032 isolated WF2 654 16:23:57.50 -26:30:52.8 25.053 0.008 0 868 0.011 isolated WF3 146 16:23:58.75 -26:32:40.1 25.218 0.023 0 835 0.025 isolated WF3 186 16:24:00.10 -26:32:58.7 24.821 0.023 0 776 0.025 isolated WF3 392 16:24:02.42 -26:32:59.9 24.117 0.036 0 565 0.039 isolated; very near to edge of image WF3 431 16:23:59.94 -26:32:04.6 25.330 0.054 1 017 0.068 isolated WF3 469 16:24:00.59 -26:32:09.6. 25.315 0.022 0 841 0.025 next to diffraction spike of saturated star WF4 14 16:23:55.54 -26:32:36.6 24.184 0.010 0 591 0.017 isolated WF4 35 16:23:53.91 -26:32:54.8 24.742 0.013 0 721 0.018 isolated WF4 81 16:23:52.76 -26:33:13.0 24.601 0.026 0 296 0.028 isolated WF4 165 16:23:54.11 -26:33:11.3 24.142 0.021 0 559 0.024 isolated WF4 223 16:23:54.87 -26:33:12.0 24.095 0.018 0 524 0.022 isolated WF4 286 16:23:54.13 -26:33:29.0 24.621 0.026 0 700 0.028 isolated; near to diffraction spikes of two saturated stars WF4 550 16:23:56.72 -26:33:48.3 24.235 0.030 0 580 0.032 isolated; near to saturated star 3.3 Periodicity Analysis Ever since Stellingwerf's landmark paper [35] describing the technique, phase dispersion minimization (PDM) has been used extensively and successfully in the discovery of count-less numbers of variable stars and binary systems. It is one of the favourite methods used Chapter 3. Data Analysis 41 by astronomers in determining variable star periods. As with most techniques, it has its advantages and disadvantages. Unlike Fourier transform analysis, which is particularly well-suited for sinusoidal-type pulsations, it is not sensitive to the shape of a star's light curve. In addition, its implementation is very straightforward. However, determination of the statistical significance of results obtained from use of P D M is not so simple, as is the case for Fourier techniques. It has been shown that the P D M probability distribu-tion does not follow an F distribution [36], as originally thought. Methods used here to estimate the reliability of findings made by P D M analysis will be discussed in the next section. The P D M Method A set of observations for a given star can be characterized by two vectors: the magnitudes rrii and the observation times fy. Supposing that there are a total of N observations in all, the variance of the magnitude for a star is given by * = S £ f f ^ W ) ' . (3-5) where m is the mean magnitude. We next define a phase corresponding to each magnitude rrii, given by *i = ^ T - mt [ ^ ] , (3.6) where int[...] refers to the integer part of the term inside the brackets. P is a trial period, and ti is the earliest date in the light curve data. The full phase interval is (0,1) and the phase of a particular data point indicates when it occurs during a given cycle of the trial period. If the star is variable, and the period P is its true period, then the scatter around the phase-folded light curve will be minimized. To investigate a given trial period, the following steps are taken. (1) Using notation employed by Stellingwerf, the phase vector is divided into Nb bins, each of length 1/Nb. (2) Each bin is given Nc "cover" bins, also of length l/Nb, which are offset in phase by l/(NbNc) from the previous cover, using periodic boundary conditions on the full Chapter 3. Data Analysis 42 phase interval to obtain uniform coverage. Thus, in total there are M = NbNc bins, each of length l/Nb and whose midpoints are separated by l/(NbNc) uniformly in the unit interval. Each data point will fall into exactly Nc bins. A given bin structure is denoted (3) The variance for each bin is then calculated using equation 3.5, and the overall variance for all M bins is given by where s2 is given by equation 3.7 and a2 is given by equation 3.5. If s 2 corresponds to a phase-folded light curve that does not represent the true period for the data, then 0 ~ 1. However, if s2 does correspond to a correct period, 0 will reach a local minimum with respect to neighbouring trial periods, and will hopefully be near zero. For this study, I have chosen a bin structure (Nb, Nc) = (5,2). This was chosen to ensure that a significant number of data points would reside in each bin, especially in the case of trial periods for which phase coverage is poor. The distribution of 0 as a function of trial periods is a type of periodogram, and is similar to those encountered when utilizing Fourier methods. As an example, the light curve of a simulated data set with known period is shown in Figure 3.6. This is a sine curve with identical time sampling to the M4 /-band data (148 data points), with a period of 0.42 days. The magnitude errors on each data point are typical of the noise levels found in the dataset. The top half of Figure 3.7 shows the 0 distribution obtained by performing P D M analysis on the data with the above-mentioned bin structure. The lower half of the figure shows the phase-folded light curve for the period found to be associated with the lowest calculated value of 0 . There is no question that the analysis was successful in finding the correct period for this light curve. The other minima in the periodogram correspond to aliases of the input period. by (Nb,Nc). is its variance. (4) The (3.7) (3.8) Chapter 3. Data Analysis n i i r i i p r 22.5 23 h 23.5 \ i ± -I I L. 2150 2155 2160 - i 1 1 r - | 1 r 22.5 h 23 23.5 h _l I I l_ _i i i_ 2210 2215 MJD - 49791 2220 Figure 3.6: Example light curve of a sinusoid with the same time sampling as the / -M4 data, having a period of 0.42 days. Chapter 3. Data Analysis 44 1 h ® 0.8 0 .6 0 .4 0 .2 2 2 . 5 2 3 h 2 3 . 5 0 .5 period (days) T i r _i i i _ _i i i_ 0 .2 0 .4 0 .6 phase 0.8 Figure 3.7: P D M analysis results of the artificial light curve shown in Figure 3.6. The periodogram (above) shows that the analysis finds the correct period for that light curve. Below is the light curve folded into the period asscoiated with the lowest value of 9 , 0.42 days. Chapter 3. Data Analysis 45 P D M analysis was performed on all candidate variable stars, and the results of this analysis will be discussed in the following chapter. 3.4 Reliability Tests In an effort to understand the reliability of the P D M analysis performed on candidate variable stars, a number of artificial stars were created by sampling different types of variable star light curves at the same observation times as the M4 dataset. These stars were then added to the WF2 images, and P D M analysis was performed on the light curves extracted from the resulting images. Two types of eclipsing binary star light curves and two sinusoidal light curves of differing periods were simulated. The latter simply involved sampling a sine curve of specific amplitude and period at the observation times. The binary star curves were simulated by employing the widely-used Wilson-Devinney (W-D) code for eclipsing binary star light curve analysis [37]. A short description of this code is given in what follows, but for details the reader is referred to the original paper. The Wilson-Devinney code is based on a physical model of close binary stars, whose effects on the light curves of these systems were first treated in detail by Kopal [3]. This physical model describes effects such as Roche equipotential surfaces, tidal and rotational distorion, limb and gravity darkening, and mutual irradiation of the stars. Before the inception of the W-D code, modelling of binary star parameters was performed using the geometric modelling approach, popularized by Russell and Merrill [4], which assumes that the two members comprising the binary system are similar ellipsoids, representing only an approximation to the true shapes of the stars. This model handled only some of the parameters in the physical model described above, and even then only indirectly. The advantage to this model, however, was that it could be treated analytically. The W-D code was an enormous improvement to the Russell and Merrill model, and can handle cases such as eccentric orbits, non-sychronous rotation, and all types of close binary systems. It can even be used to analyse some types of x-ray binary systems. For the purposes of this study, however, the full capabilities of the W-D code were Chapter 3. Data Analysis 46 not needed, and it simply served as a light curve generator. Using the code, two types of eclipsing binary star light curve were generated: an Algol and a W U M a system (see §1.1.1), each of amplitude approximately 0.2 magnitudes, and with period 0.42 days. The simulated Algol system has an orbital inclination (i) of 67°, and has a relative orbital ellipse semimajor axis length (a) of 351.6 RQ, and the simulated W U M a system has i = 47.5° and a = 1.88 RQ. A period of 0.42 days was chosen because it exhibits a case of full phase coverage for this dataset, and thus represents a best-case scenario for diagnostics performed for the purpose of understanding the reliability of the P D M analysis of this dataset. For the sinusoidal light curves, two types were simulated. One was similar to the above eclipsing binary curves, with an amplitude of 0.2 magnitudes and period of 0.42 days. The other was created in an effort to simulate ZZ Ceti variable light curve parameters, with an amplitude of 0.1 magnitudes and a period of 1000 seconds, or about 0.01 days. These are values typical of ZZ Ceti stars. Although it is believed that the 1300 s exposures taken of M4 prohibit the discovery of ZZ Ceti stars, these simulations would test the possiblity of finding these objects simply based on the time sampling of the data and the noise levels associated with stars of these magnitudes. The light curve data was used in conjuction with the A D D S T A R program in D A O P H O T [28] to add artificial stars to the actual M4 images at different magnitudes. This ensured that the output light curves would have noise levels typical of real stars in the image with similar magnitudes. In all, 961 stars were simulated and added for each of the Algol, W UMa, and longer period sinusoid light curves. 900 sinusoids typical of a ZZ Ceti variable star amplitude and period were added to the images, within the magnitude ranges for which these stars were searched (see §3.2.3). Examples of these artificial star light curves, phased to their corresponding input periods, are shown in Figures 3.8-3.10. Examples of the artificial ZZ Ceti-type sinusoids would appear identical to the sinusoids with 0.42 day periods (Figure 3.10), so they are not included. Photometry was performed on these artificial stars in the same way as on the original images, as described in section 2.2.2. In order to verify the consistency between the Chapter 3. Data Analysis 47 16 .8 17 17 .2 17 .4 17 .6 17 .8 h 18 1 8 . 2 18 .4 11111111111111111 i _i i i i I i _ 0 0 .5 18 .8 19 19 .2 19 .4 19 .6 19 .8 2 0 2 0 . 2 2 0 . 4 2 0 . 6 2 0 . 8 21 2 1 . 2 2 1 . 4 1111111111111111111 x' * -I 1 I I I I I I 1_ 2 1 . 8 2 2 2 2 . 2 2 2 . 4 2 2 . 6 2 2 . 8 2 3 2 3 . 2 i i i I i i i | i i i | i i i |»i i i 2 3 . 4 2 3 . 6 2 3 . 8 2 4 2 4 . 2 2 4 . 4 Im _ 1 • -• * • -0.5 phase 0 .5 Figure 3.8: Artificial Algol-type light curves. The curves are phase folded to their input period of 0.42 days. The curve on the top left is without noise, and is plotted on the same scale as the rest shown. Note the increase in noise levels and scatter of the curves at fainter magnitudes. Chapter 3. Data Analysis 48 1 1 1 | 1 1 1 | 1 1 1 | 1 1 1 | 1 1 1 18.8 i i 11 rrrTT 1 | 1 1 1 | 1 1 1 21.8 11 VJ I T T J i i i | i i i | i i i • 7 V / -19 19.2 22 22.2 • - i i i 1 i i i 1 i i i 1 i i i 1 i i i - 19.4 22.4 19.6 22.6 -16.8 i i i | i i i | i i 1 j 1 1 1 | 1 1 1 19.8 22.8 -17 17.2 20 , 20.2 • I ? 23 23.2 17.4 20.4 - 23.4 -17.6 20.6 - 23.6 -17.8 _ 20.8 23.8 -18 18.2 r <0 -21 21.2 7 * " 24 24.2 •a • * • 18.4 > i i i 1 i i i i 21.4 i i i i 1 i I i I 24.4 • • 1 1 1 1 1 1 1 1 1 0 0.5 1 0 0.5 1 0 0.5 1 phase Figure 3.9: Artificial W UMa-type light curves with period Figure 3.8. 0.42 days, presented as in Chapter 3. Data Analysis Figure 3.10: Artificial sinusoidal light curves, presented light curves with a period of 0.42 days. as in Figure 3.8. Shown are Chapter 3. Data Analysis 50 photometry performed on the original images and the artificial star images, the output magnitudes of several stars common to both images were compared for each observation time. A n example of this comparison is shown in Figure 3.11. It can be seen that the magnitude difference between images is very small. In fact, the mean value of this difference is 0.0008 magnitudes, with a a of 0.003 magnitudes. This check was performed on a large sample of stars on all chips, and demonstrated consistent photometry between the original and artificial star images. 3.5 Searches for Non-Periodic Variable Stars In addition to periodic variables, there may exist objects in the M4 field that vary in magnitude, but not periodically. There is also the possible presence of objects whose period is too long compared with the time frame in which the observations were taken (or too short relative to the time sampling), so that their periodicity may not be evident. Clearly, tools such as P D M are not useful in searches for these objects. Non-periodic objects include, but are not limited to: transits of stars by planetary-sized or other objects, cataclysmic variables, supernovae, and flare stars. The variability candidates chosen by some of the methods outlined so far in this chapter (MS outliers and variablity index, to be specific) are valid for these cases and those candidates chosen by these methods were used to search for the existence of non-periodic variability in the M4 field. 3.5.1 Median-Smoothed Light Curves The primary method used to search for variable star candidates with non-periodic vari-ability involved smoothing the light curve of each star found in the dataset. This was done by replacing each data point in the light curve with the median of that point and its two nearest neighbouring points. This has the effect of reducing the scatter in the light curve, and real variations in the light curve would be pronounced. For each smoothed light curve, the standard deviation was calculated, and the number Chapter 3. Data Analysis 51 0.05 h CD CO 6 -0.05 n i 1 r ~i 1 1 r 0.05 - 0 . 0 5 h 2150 2210 40 4$ _L 2155 2215 MJD - 49791 J _ 2160 2220 Figure 3.11: Comparison between photometry performed on real and artificial star im-ages. Shown is the difference in output magnitude between an artificial star image and the original image for a sample star (WF2, ID #211) for each image in the M4 dataset. V'-band data are represented by triangles, and /-band data by open circles. The deviations from zero difference are very small. Chapter 3. Data Analysis 52 of standard deviations away from the mean was calculated for the highest and lowest points in each smoothed light curve (which will be called the a-value for simplicity). Thus, for any data point i in a given smoothed light curve, Imj — ml , a -value = — ,^ (3.9) a where rrii is the magnitude of the ith data point, m is the mean magnitude of the median smoothed light curve, and a is the standard deviation. Candidates were chosen by finding outliers in the distribution of a-values obtained. These outliers represent data points that deviate significantly more than is expected from the scatter of the smoothed light curve, and may indicate that the light curve demonstrates a real variation in the star's magnitude. Figure 3.12 shows this distribution for each filter. The triangles represent the brightest points in the light curve, and the squares represent the faintest data points. There was one obvious outlier in the distribution for the V filter, and is marked by a solid circle. The outliers in the I filter distribution were not as obvious. In order to allow for the consideration of several stars showing a high a-value, a somewhat conservative cut was made at a cr-value of 6.85. This seemed to be a value that more or less clearly separated the highest data points in the /-band distribution from the rest in that plot. Above this threshold, all points were considered as candidate stars. This cut is shown by a dashed line in the figure. The remaining candidates are marked by dotted circles. It turns out that the most obvious outlier in the distribution (solid circle) is a star that was near the edge of the image and the variations in its light curve were due to the star disappearing from and re-appearing in the frame over time. The rest of the candidates from this search are listed in Table 3.5. These candidates are not very promising, as none of these outliers were found to have high cr-values in both the I and V-band data. 3.5.2 Searching for Supernovae In addition to the median smoothing of light curves, additional techniques were employed in the search for possible supernovae (SNe) occuring in the M4 field. Within the dataset, Chapter 3. Data Analysis 53 Figure 3.12: cr-value as a function of / and V magnitude for stars found on all W F P C 2 chips. Candidates were chosen if they were above a cr-value threshold of 6.85 (dotted line). These candidates are circled. The most obvious candidate is marked by a solid circle. er 3. Data Analysis 54 bo a T 3 (L> J 3 * a O O a =3 a cS T3 § O J 3 .a .3 a t I CO " g 9 I S .2 !> 1 I a 3 Co O c o o § 1 3 1 a no J3 bo •c T 3 CD a 0) CO O •s 3 O •a .s? a SI Q 3 « a a o T P oo in co C N . H T P 1-4 Ss '3. a c o •a -.3 o § c« 3 CO I G ht ht 5 bp "S 2? bp bp "G •3 *n 'C 'C C4H J 3 J 3 to a> 00 CN T P CN CN CN T P CN O O o O o O d d d d o> OS T P CN T-l CO CO CN p CO i—! P CN r H CN CO CN CN CN CN CO CN O O O o O d d d d d 00 m T P m 00 t ~ 00 CO CO CN CN CO 00 CO Tt< CO d CN CN CN CN CN CN >-< *-< OS O p CO CN ai a> CO CO T P «—I T P I- l CO in CN i-i ,—( ,—( CN co CO CO CO CO CO CO CO CO CO CN i CN CN CN CN CO CN CO O ) CO p 00 o CO 00 od d in in "S in in CO CO CO CO CO CN CN CN CN CN CO CO CO CO CO i - l i-) CO 00 o> CO T-l T P CO CO T P CO CO T P in CN CN CN CN CN T P fa fa fa fa fa Chapter 3. Data Analysis 55 there is a ~ 45 day gap in the data. For this section, and further discussions pertaining to searchng for supernova with this dataset, epoch 1 will refer to the data taken before this gap, and epoch 2 to the data taken after the gap. The data from epoch 1 and epoch 2 were combined separately for each bandpass using M O N T A G E 2 , and PSF photometry was performed on each resulting image using D A O P H O T and A L L S T A R in exactly the same way as described in §2.2.2. The combined epoch 1 images used 100 images in I and 66 in V, taken over 14 days. The combined image for epoch 2 used 48 images in I and 32 in V, taken over 9 days. Three methods employing these combined images were used to search for supernova candidates. The first of these involved searching for stars which were found on the combined image of one epoch, but not on the other. A l l of the stars that were chosen as candidates as a result of this search turned out to be either saturated, or too faint to be found on the epoch 2 image with enough signal-to-noise to be considered a real detection, or else were stars that were near the edge of the image. By contrast, the second method used stars that were common to both frames. The difference in magnitude between epoch 1 to epoch 2 was calculated for each star, forming a tight distribution about a zero magnitude difference, with scatter increasing toward fainter magnitudes, as expected. Outlier stars in this distribution were chosen, since they deviate significantly from the expected magnitude difference compared with other stars of similar magnitude. The distribution for each chip and filter, as well as candidates selected, are shown in Figure 3.13. A l l but one of the candidates derived from this search met at least one of the exclusion criteria described in §3.2. That remaining star and corresponding information related to this searchare given in Table 3.6. This remaining candidate is not an encouraging prospect for a supernova event, since the difference in magnitude between epochs was not found to be significant in both 7. and V. A supernova would, however, show a large magnitude difference in all filters, especially two so close in wavelength range. The third and final method employed in searching for supernovae in the M4 data involved examining the background galaxies found in the images and comparing them Chapter 3. Data Analysis 56 J 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _ 1 p c . J J ~i i i l i i i l i i i I i i i I i i i I i i j i i | 1 1  | 1 1 1 | 1 1 1 | 1 1 1 | 1 1 _ j . WF2, / . _= r ° c a > •= "l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 j i i i i i i I I i i I I i i I I i i I I i -_ WF3, / J "i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i i J 1 1 | 1 1 1 | 1 1 1 | 1 1 I | 1 | 1 1 -_ WF4, / 2 - .f.^V • -~l 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I 1 1 1 1 -18 2 0 2 2 2 4 2 6 18 2 0 2 2 2 4 2 6 | i i i | i i i | i i i | r i i • - [ - I - T - I — | — r : PC, V : r Oi L :6J " i i i i i i 1 i i i 1 i i i 1 i i i 1 i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 l_ - WF2, V ^ © '-r o i L §>' ' : l - Q : 1 1 i 1 i i i 1 i i i 1 i i i 1 i \r i 1 r i i I I 1 i i I I i i i i i i i i i i I I i l_ WF3, V - . " -. . . . -.''IK^';., .' • -r o •= 1 i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 r I I 1 1 I I 1 1 I I 1 1 I I 1 I I I 1 1 [ -_ WF4, V J L o • i 1 i i i 1 i i i 1 i i i 1 i i i 1 W i i 1 i 18 2 0 2 2 2 4 2 6 2 8 18 2 0 2 2 2 4 2 6 2 8 magnitude Figure 3.13: Distribution of magnitude differences between the epoch 1 and epoch 2 combined images. Outliers to this distribution were chosen as candidate supernovae and are marked by open circles. er 3. Data Analysis 57 ! a CD a 3 c3 a 1 3 3 O bo 3 •a •a a CD CO •s 3 is •8 o .5 £ 3 * S3 m T3 IS =3 ta H3 CD T3 3 3 SP a T3 CD 05 5 o- a E3 £ 5 "S )mrai ilatec O CO 1 O OS CO o d j lO T f CO i-t d i 1 b o d |-i 1 CO o d b S9 0.009 22.6; Filt< 2000) i:ss.s) CO d Dec (J: (dd:mn -26:32:: )0) ss.ss) CO <¥* RA (J20( (hh:mm:E 16:23:54.; Q t~ os Chip PC Chapter 3. Data Analysis 58 from one epoch to the next. Galaxies were identified using two methods. The first method involved using the SExtractor program [38],[39], which uses neural network software to identify galaxies separately from stars in an image. The second method was simply to look for "fuzzy" objects that SExtractor was not able to identify on the images. The comparison between epochs was performed by subtracting the combined epoch 2 image from the combined epoch 1 image, in the hopes of discovering residual flux in the subtracted image, possibly indicating the presence of a supernova. Before subtracting the two images, the signal due to the sky background was removed from each image, and the total flux for each combined image was normalized. This is necessary since the amount of images that make up the combined epoch 1 image, and hence its total flux, is greater than for the epoch 2 image. Fortunately, the M O N T A G E 2 software described in the previous chapter performs this sky subtraction and flux normalization for each combined image it creates. The subtracted image produced was not very good, and it is thought that this is due to the well-documented poor sampling of the point spread function of stars in HST images. In an attempt to improve the quality of the subtracted images, the combined images were created again as described above, but this time after resampling the pixel grids of the single images to increase the resolution of the point spread function. This was accomplished again using M O N T A G E 2 , which allows the user to expand the images in this way. this process is similar in concept to the popular "drizzling" technique [40]. The P C chip was expanded by a factor of two, and the WF2, WF3, and WF4 chips were expanded by a factor of three. The resulting subtracted images were improved, but not by a significant amount. However, it seemed that the subtraction did work well for faint objects, which included most of the galaxies found in the field. Several of the galaxies contained bright cores that did not subtract very well, but they were in the minority. The results of subtracting the epoch 2 image from the epoch 1 image are shown for /-band data of each chip in Figures 3.14 - 3.17. Shown in these figures are three colums of images, which are subsets of the original epoch 1, epoch 2, and (epoch 2 — epoch 1) images, and are centred on the coordinates of each galaxy identified, with each row Chapter 3. Data Analysis 59 corresponding to a different galaxy. Each image is a 20 x 20 pixel subset for the P C chip, and a 30 x 30 pixel subset for the WF2, WF3, and WF4 chips. There is no evidence for any residual flux greater than about \a above (or below) the noise levels for the image, except in cases where the nucleus of the galaxy was too bright, and the quality of the subraction was poor. 60 1 2 S U B 1 2 S U B 1 2 S U B 1 2 S U B Figure 3.14: Results of image subtraction performed on the P C chip /-band data. Shown are 20x20 pixel subsets of the full image, each one centred on the coordinates of the galaxies found on the image. These images are organized into three columns: the epoch 1 combined image, the epoch 2 image, and the (epoch 1 — epoch 2) image. The resulting subtracted image showed no signs of residual flux > la above or below the mean noise level. Chapter 3. Data Analysis 61 1 2 S U B 1 2 S U B 1 2 S U B 1 2 S U B Figure 3.15: Results of image subtraction performed on the WF2 chip /-band data. Shown are 30 x 30 pixel subsets of the full image, presented as in Figure 3.14. Chapter 3. Data Analysis 62 1 2 S U B 1 2 S U B 1 2 S U B 1 2 S U B » * • OH E H • * • • • • BB  U B • • * " M: \ mmm* m • • m * • D B B • H i • a Figure 3.16: Results of image subtraction performed on the WF3 chip /-band data. Shown are 30 x 30 pixel subsets of the full image, presented as in Figure 3.14. Chapter 3. Data Analysis 63 1 2 S U B 1 2 S U B 1 2 S U B 1 2 S U B Figure 3.17: Results of image subtraction performed on the WF4 chip /-band data. Shown are 30 x 30 pixel subsets of the full image, presented as in Figure 3.14. 64 Chapter 4 Results After candidates were selected as described in the previous chapter, their light curves were examined more closely and were analysed using P D M to search for periodicity. Of all the variable star candidates listed in the tables in the previous chapter, only one of these stars (PC, ID #97) showed signs of variability in its light curve. Even for the remaining candidate star, indicators of periodicity were not significant enough to be conclusive. It is therefore possible that this remaining candidate is non-periodic, or else there is not enough data in the light curve of that star to decisively find a period for it through P D M analysis. For the next three sections, the results obtained from analysis of the remaining candidate star will be discussed. 4.1 Overview of Remaining Candidate The only star remaining from the entire list of candidates compiled in the previous chapter was found on the P C chip as both a main sequence outlier candidate (Table 3.3), and as an outburst candidate (Table 3.6; it has been identified as a cluster member by its proper motion, and so is excluded as a supernova candidate, but it may be a flare star, which could explain the magnitude difference seen between epochs). The star is relatively isolated (it is not close to any saturated star or diffraction spikes), and is near the centre of the chip. Figure 4.1 shows a combined /-band image of the P C chip observations, on which the star in question is circled. It has a magnitude V = 22.6 and V — I — 0.71, which places it at the bright end and just red of the white dwarf sequence in the C M D of M4, as shown in Figure 4.2. There is another star next to this candidate on the C M D . Chapter 4- Results 65 I M ^ & * ' Mi Figure 4.1: The combined /-band P C chip image of the M4 field taken with HST. The remaining candidate from the variable star search is enclosed by a dashed circle. It is a relatively isolated star, in that it is not close to a saturated star or a diffraction spike. It is also very far from the edge of the image. Chapter 4- Results 66 This star (ID #1), also found on the P C chip image, was very close to the edge of the frame. Its partial appearance and disappearance from the field over time was responsible for the observed variations in its light curve, causing it to be flagged as a possible variable star candidate. 4.2 Light Curve of Candidate Star The light curve of the star is shown on Figure 4.3, and is displayed in two sets of rows, which show the light curve data for each filter in which the observations were taken. The first set shows the data before the 45 day gap in the observations, and the second set of rows shows the data after this gap. The reason for presenting the data in this way is to highlight what is perhaps its most desirable feature in being a candidate star: the light curve seems to show the same behaviour in both bandpasses. This was not seen in the the light curves of any other candidate star. The notable feature in the light curve is the apparent rise in amplitude by ~ 0.1 magnitudes, seen within the first few days the data were collected. If the light curve were periodic, it would seem to have a period of approximately 7 or 8 days, but would have to be modulated in amplitude over time. However, it is hard to tell if this is indeed the case. Since the star is isolated on the image, an effort was made to improve the measure-ments of the star's magnitudes in each individual image by performing simple aperture photometry on it using D A O P H O T . A n aperture of 1.5 pixels was found to yield mea-surements with the smallest corresponding errors. Magnitude offsets were then added as described in §2.2.3. The light curve extracted for the star as a result of this process is shown in Figure 4.4. It can be seen that the photometry has indeed improved, and that the same trends observed in Figure 4.3 are still present in the light curve of the star. One note should be made; it is believed that the error bars on the magnitude measurements in Figure 4.4 are underestimated, especially in the V-band data. As shown, they would imply the existence of short term variability. Inspection of the light curves of a few other isolated stars resulting from the same aperture photometry also show measurements with Chapter 4- Results 67 o •Y V 7 - / Figure 4.2: The same C M D of M4 shown in Figure 3.4, with the remaining candidate star circled. It is at the bright end and slightly to the red of the white dwarf sequence of the cluster. Chapter 4- Results 68 Figure 4.3: Light curve of the remaining candidate variable star. The first set of rows shows the light curve before the 45 day gap in the observations, and the second set shows the light curve after that gap. It can be seen that the light curve shows a ~ 0.1 magnitude rise in its amplitude a few days into the observations in both the I and V filter data, which is a highly desirable characteristic in the selection of this star as a candidate. Chapter 4- Results 69 2 1 . 8 2 1 . 9 2 2 22 .1 2 2 . 9 2 3 2 3 . 1 2 3 . 2 ~i 1 I r -i 1 1 r + I i i i i 2 1 5 0 2 1 5 5 2 1 6 0 2 2 1 0 2 2 1 5 MJD - 49791 2 2 2 0 Figure 4.4: Light curve of the remaining variable star candidate, after performing aper-ture photometry in an attempt to improve the measurements of the single-image photometry of the star. It can be seen that the errors on the magnitude measurements are smaller as hoped. The main features in the light curve seen in Figure 4.3 are still present in this light curve. Chapter 4- Results 70 error bars significantly smaller than the scatter of the data. In addition, these stars are found in significantly different positions of the C M D of the cluster. In fact, two of these stars are not cluster members. This seems to indicate that this effect is not due to short term variability. It may be that there are uncertainties that have not been taken into account in performing the photometry of these stars. This issue is currently being examined. 4.3 Phase Dispersion Minimization Analysis Period searches were performed on all stars selected as candidate variables using the phase dispersion minimization technique described in §3.3. Light curves of all candidate stars were searched for periods ranging from 0.01 to 15 days. The lower limit to this range was chosen to correspond to slighty less than the Nyquist frequency calculated for this dataset. The Nyquist frequency vn is the highest frequency of variability detectable, given the time sampling of the data. For evenly spaced data, it is given by vn — 1/2Ai , where A i is the spacing in time of the data. For cases such as this one, where the data is unevenly spaced, a pseudo-Nyquist frequency can be calculated with A i being the minimum spacing between any two data points in the data. For this dataset, this minimum spacing is 843 seconds, yielding a Nyquist frequency of 5.93 x 10~4 s _ 1 , which corresponds to a period of 1687.5 seconds, or 0.02 days. The upper limit to the period search was chosen to match the length of time for which the data was taken before the 45 day gap in the observations. The results obtained were not encouraging. None of the periodograms derived from this process showed any of the candidate stars to have a period with a corresponding value of 6 that was significantly lower than one, compared to other trial periods. The results from P D M analysis of two candidate stars in the M4 field are shown in Figures 4.5 (WF3, ID #532) and 4.6 (WF4, ID #327) for different period ranges. The values of 6 obtained do not deviate significantly from unity, and the analysis of these stars shows no correlation between the periodograms of I and V-band data. Shown in Figure 4.7 Chapter 4- Results 71 1 0.8 0 .6 0 .4 0 .2 ® 0 1 0 .8 0 .6 0 .4 0 .2 0 2 1 . 5 2 1 . 4 2 1 . 3 2 1 . 2 2 4 . 1 2 4 2 3 . 9 2 3 . 8 1 1 1 r _i i i_ _i i i i_ _i i i i_ 0 .5 1 1.5 period (days) 1 1 1 1 1 1 • | i • i i i i 1 i i i 1 i i i 1 i i 1 1 1 1 1 i i i | i i i | i i i I I i • 1 1 1 1 1 • • ! • • .-• i i i i i i i i i i i i i i i i i i i 0 .2 0 .4 0 .6 phase 0 .8 Figure 4.5: P D M analysis performed on a star in WF3 (ID #532). The periodograms and phased light curves for each filter show no signs of periodicity. Chapter 4- Results 72 0 - i 1 i i 5 1 1 1 1 1 1 1 1 1 1 6 7 i i i i i i i 8 1 i i i i 1 r 9 10 period (days) 21 .1 l 1 i | i i i j i i i 1 i i i 2 1 ^ 2 0 . 9 » * • • • • • • • 2 0 . 8 1 1 i 1 i i i 1 i i 1 1 1 1 1 1 1 | 1 1 1 2 3 . 6 1 1 i i i i i i 1 1 1 1 1 1 : ^ 2 3 . 5 2 3 . 4 • : • * tit' ' 2 3 . 3 ~ 1 1 , i , , , i , , , 1 , , , 1 , , , -0 .2 0 .4 0 .6 phase 0 .8 Figure 4.6: P D M analysis performed on a star in WF4 (ID #327). The periodograms and phased light curves for each filter show no signs of periodicity. Chapter 4- Results 73 are the results of the P D M analysis performed on the remaining candidate variable star, searching trial periods of 4-10 days, which is the range of possible periods that one might expect by examining the light curves in Figures 4.3 and 4.4. On the top half of Figure 4.7 the O periodograms obtained for this trial period range are shown, and indicated are the periods that are associated with the lowest value of © for data taken in each filter. The bottom half of the figure shows the light curve of the star in each bandpass, phase-folded into the period corresponding to the minimum value of O found. As with the rest of the stars probed, the 0 distribution does not seem to venture far below a value of one. However, as with the light curves shown for this star in the previous section, the periodograms show a very similar structure for the light curves obtained from both the / and V-band data. This is encouraging, even though no definite period was found, because it seems to indicate that the light curves corresponding to observations in both filters follow similar trends in their progression over time, which is a quality that was not seen in the other candidate stars. After performing aperture photometry on the remaining candidate, P D M analysis was carried out on its light curve, and the results are shown in Figure 4.8. Interestingly, the period found to correspond to the lowest value of 0 was the same for both the I and V-band observations, with a value of 9.8 days. This value as a possible period is not evident by inspecting the light curves for this star, and its corresponding 0 value is still not very much lower than unity. If one were to remove the points that define the rise in the amplitude of the phase-folded light curves in Figure 4.8, which occur at a phase of about 0.2, the remaining points would not indicate variations in the light curve of this object. In fact, this amplitude peak in the phase-folded light curve is defined only by those points (6 points in / , 2 points in V) which define the peak present a few days into the light curves of Figure 4.4. This points to the fact that there is no other data in the light curves of this star which show this peak to be repeating. This could be because, as argued at the beginning of the chapter, the candidate is perhaps a non-periodic variable star, or has a period that is long compared to the duration of the observations. Another reason may be that a second (or third) rise in amplitude may have occurred during the Chapter 4- Results 74 6 8 period (days) F 1 1 r 1 1 1 r • 10 ~ i r i i r 2 1 . 8 2 1 . 9 2 2 2 2 . 1 2 2 . 9 2 3 2 3 . 1 2 3 . 2 H 1 h H h H 1 1-_i i i _ *.4 .j i i_ _i i i_ 0 .2 0 .4 0 .6 phase 0 .8 Figure 4.7: Results of phase dispersion minimization analysis of the remaining candidate variable star. Top: Q distribution periodograms for observations in each bandpass. Although there is no significant deviation of 6 from a value of 1, the features of the periodogram are very similar between the two sets of data. Bottom: Phased light curves, folded into the period associated with the lowest found values of ©. Chapter 4- Results 75 Figure 4.8: Results of P D M analysis of the remaining candidate, after performing aper-ture photometry on the star. The periods found in the data taken in each filter agree with other. Chapter 4- Results 76 45 day gap in the observations. In any case, the observations of this star leave a great deal of room for speculation and interpretation, which will be discussed. further in the following chapter. 4.4 Reliability Tests Tests of the reliability of the P D M analysis performed on this dataset were outlined in §3.4. The results of these tests are found on Tables 4.1-4.6. These tables show the number of stars added at each magnitude, the number recovered, and the percentage of those stars recovered with 0 values corresponding to four different groups of periods. The first group is simply the correct input period (0.42 days), the second group is half the input period, and the third group is § of the input period. It was found that these are the three major groups within which the periods of the artificial stars were recovered. The fourth group is for periods that do not fall into any of the three previous categories. The reasons for the P D M analysis finding periods at half or § the input period is evident once one observes the results of the analysis for a couple of cases. As an example, Figure 4.9 shows the results of P D M analysis performed on a W UMa-type light curve added to the V-band images. This artificial star was found by P D M analysis to have a period of ~ 0.21 days (= | x 0.42 days). It is expected that a strong signal would be found at half the input period, especially in the case of W U M a light curves, which typically have eclipse minima that, while unequal in depth, are very similar in width. However, inspection of the periodogram shows that there is a significant drop in 0 at the correct period of 0.42 days, and is comparable to the value of 0 corresponding to a 0.21 day period. The bottom two plots show the phase-folded light curves for each of these periods. One can see that while the curve corresponding to a 0.42 day period seems to have less scatter, the curve corresponding to the 0.21 day period seems to show less variation over the unit interval. The reason the P D M analysis chooses a lower value of 0 for the 0.21 day period instead of the 0.42 day period is due to the bin structure used in the analysis (see §3.3). One can see that for a bin length of 0.2 in phase (the bin length used), the variance of Chapter 4- Results 77 1 0 .8 © 0 .6 0 .4 0 .2 0 _1 I I L_ _ l I 1_ 0 .5 period (days) -i 1 r ~i I 1 1 1 1 1 P 1 1 1 1 r 2 0 . 8 h } ' ' V . • 2 1 h 2 1 . 2 h 2 0 . 8 — • • • Pt = 0.21 days H 1 h H 1 h H 1 h -I 1 h •• • 2 1 2 1 . 2 _i i i i i ' i _i i i_ • * » P2 = 0.42 days J I I I I I 1 L 0 .2 0 .4 0 .6 phase 0 .8 Figure 4.9: Results of P D M analysis on an artificial W UMa-type light curve. Top: The period corresponding to the lowest value of 0 is 0.21 days (1). This is half the input period, which has a corresponding © value which is only slightly greater (2). Bottom: Phase-folded light curves corresponding to 0.21 and 0.42 day periods. The lower © value for the 0.21 day period is probably due to the size of bins used in evaluating the total variance of the phase-folded light curve. Chapter 4- Results 78 each bin, calculated using equation 3.5, will be larger on average for the 0.42 day folded light curve. Using a small bin length would no doubt lower the variance of each phase bin for the 0.42 day curve, but due to the amount of data available from these observations there would be very little data within each phase bin upon which to base a result. Thus, although the lowest value of © did not correspond to the input period, the bin structure that was used does seem to work, since a quick inspection of the periodogram obtained for the light curve demonstrates that the input period was indeed recovered; it does correspond to a significantly low value of 0 compared to neighbouring trial periods. This was generally not an issue for the /-band images, since the data taken in this filter has greater time sampling than in V. It was also not a problem for the sinusoidal light curve in both bandpasses, and this is probably due to their shape. At periods \ or § the input period, the periodogram would exhibit a large 0 value, since these periods would cause the peaks and troughs of the sinusoid to coincide in the phase-folded light curves. Thus, a period recovered that was equal to, one half of, or three-halves of the input period was considered a successful recovery. These results suggest that for all types of stars added to the /-band images, down to a certain input magnitude, the P D M analysis is highly successful in recovering their input periods. For the Algol-type light curves, this limiting magnitude is at / ~ 24, at which point only half of stars are recovered with the input period. For the W UMa-type light curves, this limit is / ~ 24 as well, since at this magnitude about three quarters of the input curves are found at either half or three-halves the input period. For the sinusoidal light curves, the recovery of the input period was highly successful for the data added to images in both filters, right down to / ~ 25 and V ~ 27. For the Algol curves in the V-band images, successful recovery of input periods is present down to V ~ 25, at which magnitude about half the input stars are found with correct periods. For W UMa-type curves, this limit is at V ~ 25 as well, with a success rate of about 97%. Below these / and V limiting magnitudes, almost none of the stars were recovered with light curves having more than 6 data points (see §2.2.2), let alone with the correct input period as a result of P D M analysis. The limiting magnitudes found for the Algol Chapter 4- Results 79 and W UMa-type light curves are shown on the M4 C M D in Figure 4.10. The region above the two intersecting dashed lines represents the C M D parameter space within which period recovery was successful. No variable stars resembling a binary star light curve were actually found, so it is thought that within this portion of parameter space, there were no binary systems observed with periods on the order of half a day (which have good phase coverage given the time sampling of this dataset), since the simulations show that these types of stars can be found. Binaries with periods of this length are typical of contact binary stars, which include the W UMa-type systems. In addition, the successful recovery of sinusoidal light curves down to even fainter magnitudes seems to indicate that variable stars with sinusoidal-type pulsations with periods on the order of half a day are are virtually nonexistent within the same region of the colour-magnitude diagram. Of course, there is the possibile presence of variable stars in the M4 field with periods exhibiting poor phase coverage in their phase folded light curves. In these cases, P D M analysis would not be very useful without more data, which would be needed to prove or disprove the existence of such stars. ZZ Ceti Variable Star Candidates It is not surprising that ZZ Ceti variable stars were not identified in this dataset. As discussed in §1.1.2, this is due to the exposure times being 1300 seconds for each single observation, which is on the order of a typical period of the ZZ Ceti variables. So any variability in these stars, if they are indeed present in the sample, would be "washed out" since the integration time of each observation is too long to capture the star at various stages of its pulsation cycle. Independent of the integration time is the consideration of the Nyquist frequency for this dataset, which corresponds to a period of 1687.5 seconds, as shown in §4.3. This is longer than a typical ZZ Ceti period, providing further proof that this particular dataset is not suited for searches of these stars. er 4- Results Table 4.1: Results from artificial star tests using Algol-type light curves with the same time sampling as /-band observations of M4. Shown are input magni-tude, amount of input curves, amount recovered, and the percentage recovered at notable factors of the input period. / Input Output % Recovered P 2r 3p 2r other P 17.0 74 72 100.0 0.0 0.0 0.0 18.0 74 68 100.0 0.0 0.0 0.0 19.0 74 73 100.0 0.0 0.0 0.0 20.0 74 71 100.0 0.0 0.0 0.0 21.0 74 70 100.0 0.0 0.0 0.0 22.0 74 68 100.0 0.0 0.0 0.0 23.0 74 69 98.6 1.4 0.0 0.0 24.0 74 65 41.5 4.6 0.0 53.8 25.0 74 3 33.3 33.3 0.0 33.3 Table 4.2: Results from artificial star tests using Algol-type light curves with the same time sampling as V-band observations of M4. Shown are input magni-tude, amount of input curves, amount recovered, and the percentage recovered at notable factors of the input period. V Input Output % Recovered P ±P 2r 3p 2r other P 19.0 74 72 0.0 0.0 100.0 0.0 20.0 74 68 0.0 0.0 100.0 0.0 21.0 74 73 2.7 1.4 94.5 1.4 22.0 74 71 9.9 8.5 76.1 5.6 23.0 74 70 20.0 12.9 42.9 24.3 24.0 74 68 25.0 10.3 26.5 38.2 25.0 74 69 24.6 15.9 8.7 50.7 26.0 74 65 9.2 4.6 1.5 84.6 27.0 74 3 0.0 0.0 0.0 100.0 er 4- Results Table 4.3: Results from artificial star tests using W UMa-type light curves with the same time sampling as /-band observations of M4. Shown are input magnitude, amount of input curves, amount recovered, and the percentage recovered at notable factors of the input period. Input Output % Recovered P i p 2r 2r other P 17.0 74 71 98.6 0.0 0.0 1.4 18.0 74 68 100.0 0.0 0.0 0.0 19.0 74 73 98.6 1.4 0.0 0.0 20.0 74 72 98.6 1.4 0.0 0.0 21.0 74 71 93.0 7.0 0.0 0.0 22.0 74 69 82.6 17.4 0.0 0.0 23.0 74 68 66.2 32.4 0.0 1.5 24.0 74 64 21.9 53.1 0.0 25.0 25.0 74 2 0.0 50.0 0.0 50.0 Table 4.4: Results from artificial star tests using W UMa-type light curves with the same time sampling as V-band observations of M4. Shown are input magnitude, amount of input curves, amount recovered, and the percentage recovered at notable factors of the input period. V Input Output % Recovered P 2r 2r other P 19.0 74 71 0.0 100.0 0.0 0.0 20.0 74 68 1.5 98.5 0.0 0.0 21.0 74 73 2.7 97.3 0.0 0.0 22.0 74 72 5.6 94.4 0.0 0.0 23.0 74 71 12.7 84.5 0.0 2.8 24.0 74 69 30.4 66.7 0.0 2.9 25.0 74 68 35.3 61.8 0.0 2.9 26.0 74 64 9.4 26.6 1.6 62.5 27.0 74 2 0.0 100.0 0.0 0.0 Chapter 4- Results 82 Table 4.5: Results from artificial star tests using sinusoidal light curves with the same time sampling as /-band observations of M4. Shown are input magnitude, amount of input curves, amount recovered, and the percentage recovered at notable factors of the input period. / Input Output % Recovered P i P 3r 1 r other P 17.0 74 68 100.0 0.0 0.0 0.0 18.0 74 69 100.0 0.0 0.0 0.0 19.0 74 73 100.0 0.0 0.0 0.0 20.0 74 72 98.6 0.0 0.0 1.4 21.0 74 71 100.0 0.0 0.0 0.0 22.0 74 69 100.0 0.0 0.0 0.0 23.0 74 69 100.0 0.0 0.0 0.0 24.0 74 65 96.9 0.0 0.0 3.1 25.0 74 12 100.0 0.0 0.0 0.0 Table 4.6: Results from artificial star tests using sinusoidal light curves with the same time sampling as V-band observations of M4. Shown are input magni-tude, amount of input curves, amount recovered, and the percentage recovered at notable factors of the input period. V Input Output % Recovered P ± P 2r other P 19.0 74 68 100.0 0.0 0.0 0.0 20.0 74 69 100.0 0.0 0.0 0.0 21.0 74 73 100.0 0.0 0.0 0.0 22.0 74 72 100.0 0.0 0.0 0.0 23.0 74 71 100.0 0.0 0.0 0.0 24.0 74 69 100.0 0.0 0.0 0.0 25.0 74 69 98.6 0.0 0.0 1.4 26.0 74 65 95.4 0.0 0.0 4.6 27.0 74 12 91.7 0.0 0.0 8.3 83 20 h ^ 25 30 0 V-I Figure 4.10: The same colour-magnitude diagram as in Figure 3.4, with the limiting magnitudes shown as intersecting dashed lines, above which P D M analysis was successful in recovering artificial binary star input periods. 84 Chapter 5 Discussion The previous chapters discuss efforts made to search for variable objects in a field six core radii from the centre of the globular cluster M4, observed with the Hubble Space Telescope. Various criteria were employed to narrow the sample of stars found in the field to a smaller group of candidate stars. The existence of periodic variable stars in the field was investigated using the technique of phase dispersion minimization. The reliability of this technique was tested for sensitivity to various light curve types using synthetic data. The field was also examined for the presence of non-periodic variability, as could arise from transiting planets, cataclysmic variables, flare stars, and supernovae. 5.1 A Lack of Variability Results from these searches indicate that there are most probably no eclipsing binary stars in the field observed with periods ~ 0.5 days, down to limiting magnitudes of V ~ 25 and I ~ 24, based on simulations of Algol and W UMa-type light curves. This is consistent with a lack of contact binary systems, which have periods on the order of half a day. Searches for other types of periodic variable star have yielded no candidates. It can also be concluded that there are most likely no periodic variables in the field with periods on the order of half a day, to limiting magnitudes of V ~ 25 and I ~ 25, based on analysis of simulated sinusoidal light curves. In addition, a search for variable stars with non-periodic light curves found only one possible candidate. Chapter 5. Discussion 85 Interpretation and Potential Explanations The lack of eclipsing binary stars in the field can be explained in a number of ways. One may be due to the fact that the observations were taken in a field that is considerably distant from the core of M4. It is expected that mass segregration in the cluster over time will preferentially cause binary systems to sink toward the cluster centre, since binary stars are typically more massive on average than single stars. However, a considerable number of ~ 0.6 M© white dwarf stars are seen in the field, so it is not unreasonable to expect, for example, binary star systems comprised of 2 x 0.3 M© stars in that same field. Another possibility is that due to the high-density environment of globular clusters, low-mass binaries are more prone to disruption before they have the opportunity to harden over time through interaction with third bodies. Conversely, the relatively large numbers of blue straggler stars in globular clusters (30 have been found in M4 [9]), may represent the coalescence of many low-mass binary systems which have acted as a heat source to the cluster in preventing core collapse. This could reduce the number of binaries expected in this magnitude range. However, the real explanation for the lack of eclipsing binaries in this sample may simply be small-number statistics. As speculated in §1.2, based on the low fraction of stars in the study of 47 Tuc found to be binary systems, the predicted number of objects recoverable in these data was low to begin with. The small number of background galaxies in the M4 images is the most probable reason that supernovae were not found in the field observed. Previous studies of the Hubble Deep Field, whose observation was comparable in depth to the M4 field, yielded a discovery of two supernovae. Due to the substantially larger background flux in the M4 observations compared to that of the H D F (see §1.2), considerably fewer galaxies were observable, and so it was not believed that any should be found in these observations. Chapter 5. Discussion 86 5.2 Speculations on the Remaining Candidate In the above summary of results, an otherwise firm declaration of a null result for the presence of variability in this field has been modified by phrasing such as most probably and most likely. This is due to the still unresolved issue of the remaining candidate star found in the P C field of the observations. There is no clear reason that this star should not be considered as variable: it is relatively bright, it is not close to any image defects (such as cosmic rays or diffraction spikes from saturated stars) and is far from the edge of the image (see Figure 4.1). A n apparent peak in magnitude is seen in its light curve (Figures 4.3 and 4.4) in both bandpasses, further supporting the claim that this object is indeed variable. A clear second peak in brightness is not seen. This could mean that the star is not periodic, however, it may very well be that this object has a period such that a second peak occured during a gap in the observations. Inspection of its light curve seems to indicate a period in the range of 5 - 9 days, but periodicity analysis performed on this star was inconclusive. The location of this star on the colour-magnitude diagram, as shown in Figure 4.2, indicates that it may be a binary star, composed of a low-mass main sequence star and a white dwarf. In order to investigate this possibility further, a simple decomposition of this candidate into two stars was performed, by investigating the magnitudes and colours of two separate stars that would be needed to produce a single observed star of magnitude and colour similar to this candidate (V = 22.639, V — I = 0.706). This would give further insight as to the types of stars that might exist in a binary system found in the position of the candidate star on the C M D of M4. The composite magnitudes of such a binary system were calculated as follows: Vcomp = -2 .51og[ l0^/ - 2 - 5 ) + 10^/- 2- 5)] (5.1) IcomP = -2.5 log [ lO^/- 2 - 5 ) + 10(/2/-2-5>] , (5.2) where the subscripts 1 and 2 refer to each star in the system. It turns out that a binary system consisting of a white dwarf with magnitude and colour V ~ 22.7, V — I ~ 0.38 Chapter 5. Discussion 87 and a star very low on the main sequence, with magnitude V ~ 28.3, V — I ~ 4.5, result in an object with a position on the C M D that is very close to the candidate star. This is illustrated in Figure 5.1. The positions of the two speculated binary star constituents on the C M D are shown as concentric circles, and the resulting binary is shown as a star shape. A n open circle is centred on the position of the candidate star. As one can see, the composite star is very close to the position of the candidate. It was expected that such an object would be dominated in luminosity by a white dwarf of similar magnitude. The position of the lower main sequence constituent would indicate that this is an object that is fainter than any known field subdwarf at the metallicity of M4, implying a mass near 0.09 M© [18], which is very near to the hydrogen-burning limit. These stars are difficult to observe due to their faintness, but if such a star is found in a binary system, it could be exploited to uncover many of its properties, such as its mass, which would help constrain low-mass MS models. It may also be possible that this object is a cataclysmic variable. It is thought that such objects can be found in the colour-magnitude diagrams of globular clusters, between the main sequence and the white dwarf sequence [20]. In addition, those that are found close to the white dwarf sequence will be dominated in magnitude by the white dwarf component. This scenario is very similar to what has been found through the decomposition of the candidate object as described above. The light curve of the object does not seem to show the characteristics of an eclipsing binary system. The possibility does exist that it is a face-on binary system that includes a B Y Draconis variable. The period range of 0.4 - 10 days of the B Y Dra stars found in the 47 Tuc study does not discount this as a possibility. In addition, that study reported amplitudes for these stars ranging from V ~ 0.001 - 0.061. This is not much smaller than the ~ 0.1 magnitude increase seen in the candidate object's light curve. It is difficult to say with any certainty whether this object is a B Y Dra variable. Further simulations were performed on 625 artificial sinusoidal light curves with an amplitude of 0.9 magnitudes with V = 22.0 - 25.0 and J = 20.0 - 23.0; these are magnitude ranges within which the candidate star is observed. They were given an input period of 7.0 days, as this is within the range of periods for the star that its light curve would Chapter 5. Discussion 88 2 0 0, „ . • 2 5 3 0 V i V V-I Figure 5.1: Colour-magnitude diagram of M4 showing the positions of the two stars (con-centric circles) needed to produce a binary system that would be observed with similar magnitude and colour as the candidate star (star shape). The resulting object is very close in position to the candidate, which is located at the centre of the open circle. Chapter 5. Discussion 89 seem to indicate. These stars were added to the frame and analysed as described in §3.4. Results of the P D M analysis performed on these light curves show that all of the input light curves were recovered with the correct input period. This indicates that in all probability, this candidate is not periodic, or else it has a period that is too long to be detected with P D M analysis. In this case, more data would be needed to obtain full phase coverage in order to determine the nature of this object's variability. Yet another possibility is that the observed increase in brightness is due to flaring on the star's surface, which would not necessarily be periodic, thus explaining the null result of the P D M analysis. However, this is unlikely, since the timescale typical of stellar flares rising to maximum brightness and diminishing is on the order of hours, whereas the rise in amplitude in the light curve of the candidate star seems to last for days. 5.3 The Future While this research did not produce any confirmed discoveries of variability in the mag-nitude range V ~ 20 — 26 for M4, it is far from having been fruitless. Firstly, there is one interesting candidate variable star. However, it seems that more observations will be needed to identify the nature of this object. Since B Y Draconis stars are believed to be binary systems, their investigation is crucial for the study of globular cluster evolution, as discussed in §1.1.1. Also, because they are magnetically active objects, they are also important to research concerning highly magnetic stars and their behaviour. The identification of the white dwarf stars in the M4 field suspected of being ZZ Ceti variables may prove to be of considerable value. Even though their identification as these rapid pulsators was not confirmed, it is quite possible that some of them are ZZ Ceti stars, and future observations with adequate time sampling can be performed to verify their identity. Table 5.1 lists the candidate ZZ Ceti stars from Table 3.4, and displays information useful for such observations. Also, it identifies those stars that are believed to be in close proximity to, or even within the ZZ Ceti instability strip. The discovery of these stars, as discussed in §1.1.2, is of the utmost importance to the understanding of Chapter 5. Discussion 90 the inner structure and behaviour of white dwarfs, as well as the star formation history of the Galaxy. Table 5.1: Table listing the same ZZ Ceti variable star candidates as those in Table 3.4, with extra information useful for future observations, including the (X,Y) position on the chip, and whether it is inside or very close to the ZZ Ceti instability region. Chip ID X Y Inst. Strip? Chip ID X Y Inst. Strip? WF2 150 678.6 197.0 yes WF3 392 505.4 796.4 yes WF2 259 354.2 340.4 WF3 431 555.9 144.6 WF2 273 105.4 354.2 WF3 469 598.2 236.6 WF2 318 203.9 430.8 WF4 14 62.2 200.2 yes WF2 340 70.9 456.9 WF4 35 86.9 487.8 WF2 485 330.5 646.8 yes WF4 81 149.4 721.4 WF2 623 336.2 312.8 WF4 165 240.1 560.9 yes WF2 654 734.0 702.3 WF4 223 304.7 480.0 yes WF3 146 215.0 344.7 WF4 286 388.4 660.9 WF3 186 255.2 604.6 WF4 550 750.2 483.1 yes 5.4 Closing Remarks As evidenced by this study of M4 and previous studies of 47 Tucanae, the Hubble Space Telescope can be used to search globular clusters for variability to unprecedentedly faint magnitudes (V" ~ 25 for both programs). In particular, the study of binary stars is, as stated before, of critical importance to the understanding of the dynamical evolution of globular clusters. It is a well known fact that the observing time required to conduct these searches is very difficult to obtain on HST, but hopefully it is time that will continue to be granted in the hopes of gaining the answers to the fundamental questions discussed here; answers that these types of studies can provide. And by the way, it was exciting to analyse Hubble Space Telescope images. 91 Bibliography [1] P. N . Kholopov et al. Combined General Catalogue of Variable Stars, 4.1 Ed., 1998. [2] R. W. Hilditch. An Introduction to Close Binary Stars. Cambridge, U K : Cambridge University Press, 2001. [3] Z. Kopal. Close Binary Systems. London: Chapman & Hall, 1959. [4] H. N . Russell and J . E . Merrill. The Determination of the Elements of Eclipsing Binaries. Contributions from the Princeton University Observatory, Princeton, 1952. [5] B . W. Carroll and D. A . Ostlie. An Introduction to Modern Astrophysics. Reading, Mass. : Addison-Wesley Pub., 1996. [6] P. Hut et al. Binaries in Globular Clusters. PASP, 104:981-1034, November 1992. [7] J . Binney and M . Merrifield. Galactic Astronomy. Princeton, N J : Princeton Uni-versity Press, 1998. [8] R. W. Romani and M . D. Weinberg. Limits on Cluster Binaries. ApJ, 372:487-493, May 1991. [9] S. M . Rucinski. W UMA-Type Binary Stars in Globular Clusters. AJ, 120:319-332, July 2000. [10] D. E . Winget. Seismological Investigations of Compact Stars. In IAU Symp. 123: Advances in Helio- and Asteroseismology, volume 123, page 305, 1988. [11] G . Fontaine. White Dwarf Stars as Cosmochronometers. In ASP Conf. Ser. 245: Astrophysical Ages and Times Scales, page 173, 2001. Chapter 5. Discussion 92 [12] N . A . Bahcall, J . P. Ostriker, S. Perlmutter, and P. J . Steinhardt. The Cosmic Triangle: Revealing the State of the Universe. Science, 284:1481, May 1999. [13] R. L. Gilliland, P. E . 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