An Analysis of Stellar Populations in Globular Clusters by Ryan Goldsbury B.Sc., Clemson University, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE STUDIES (Astronomy) The University Of British Columbia (Vancouver) October 2011 c© Ryan Goldsbury, 2011 Abstract This thesis is composed of three chapters, as well as an introduction, which de- scribe three distinct projects. In Chapter 2 we present new measurements of the centers for 65 Milky Way globular clusters. Centers were determined by fitting ellipses to the density distri- bution as well as the symmetry of the clusters. All of the determinations were done with stellar positions derived from a combination of two single-orbit Advanced Camera for Surveys images of the core of the cluster. We find that the ellipse- fitting method provides remarkable accuracy over a wide range of core sizes and density distributions, while the symmetry method is difficult to use on clusters with very large cores, or low density, requiring a larger field, or a very sharply peaked density distribution. Chapter 3 deals with a re-analysis of previous work on white dwarf natal kicks, and expands on this to analyze the radial distributions of stellar populations in globular clusters at earlier stages of stellar evolution (earlier referring to pre-white dwarf). The effects of stellar incompleteness, and a method to account for this are discussed. Finally, the results of a statistical analysis of completeness corrected radial distributions in 56 globular clusters are presented. No significant evidence of kicks is found, however multiple clusters show evidence that stars along the horizontal branch have not relaxed since undergoing mass loss after leaving the main sequence. In Chapter 4, we present a novel method for determining the distance to a star cluster by fitting spectral energy models to the spectral energy distributions of cluster white dwarfs in multiple filters. The statistics of our fitting method are discussed in detail. This approach results in a true distance modulus of (m−M)0 = ii 13.35±0.02±0.06, which corresponds to a physical distance of 4.67±0.04±0.13 kpc. The first error given is random, and the second is systematic. iii Preface Chapter 2 is a reformatted version of previously published work (Goldsbury et al. [12]). I was the primary author of this work. In addition, I did all of the required work for the paper after the initial data reduction. The data reduction itself is described in Sarajedini et al. [22]. Section 3.1 builds on work from Davis et al. [6]. The code to generate Fig- ure 3.2 was written by Saul Davis, and modified by me to include the corrections described in that section. Chapter 4 is based on work that went into a forthcoming paper by Woodley et. al. For this work, I wrote the code that did all of the computing of likelihood distributions, as well as the relevant statistics. However, I did not write the ma- jority of the paper itself. The text and figures in this chapter are entirely from my contribution to that work. iv Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Galactic Globular Clusters . . . . . . . . . . . . . . . . . . . . . 1 1.2 Determining Cluster Centres . . . . . . . . . . . . . . . . . . . . 2 1.3 The ACS Survey of Galactic Globular Clusters . . . . . . . . . . 2 1.3.1 The Survey . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3.2 The Data and the Resulting Catalog . . . . . . . . . . . . 3 1.4 White Dwarf Kicks . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.5 The Distance to 47 Tuc . . . . . . . . . . . . . . . . . . . . . . . 4 2 Determining Centres for Resolved Globular Clusters . . . . . . . . . 5 2.1 A Brief History of Approaches . . . . . . . . . . . . . . . . . . . 5 2.2 Methods for Determining The Center . . . . . . . . . . . . . . . . 6 2.3 From Iso-density Contours . . . . . . . . . . . . . . . . . . . . . 7 v 2.3.1 Avoiding Incompleteness . . . . . . . . . . . . . . . . . . 7 2.3.2 Calculating the Projected Density Profile . . . . . . . . . 8 2.3.3 Fitting Ellipses . . . . . . . . . . . . . . . . . . . . . . . 9 2.3.4 Determining Uncertainties . . . . . . . . . . . . . . . . . 12 2.4 From Cluster Symmetry . . . . . . . . . . . . . . . . . . . . . . 13 2.4.1 Generating an Incompleteness Mask . . . . . . . . . . . . 16 2.5 Comparison to Previous Works . . . . . . . . . . . . . . . . . . . 16 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3 Distributions of Various Stellar Populations . . . . . . . . . . . . . . 20 3.1 White Dwarf Kicks: Possible Evidence in NGC 6397 . . . . . . . 20 3.1.1 New Data . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.1.2 Issues of Coordinates and Centers . . . . . . . . . . . . . 21 3.2 Calculating Completeness Corrected Radial Distributions . . . . . 23 3.2.1 Isolating Regions of the CMD . . . . . . . . . . . . . . . 25 3.2.2 Bias Introduced by Incompleteness . . . . . . . . . . . . 26 3.2.3 Correcting for Incompleteness Using Artificial Star Tests . 27 3.2.4 Calculating the Distributions . . . . . . . . . . . . . . . . 29 3.2.5 Comparing the Distributions . . . . . . . . . . . . . . . . 31 3.3 The Distributions of Horizontal Branch Stars . . . . . . . . . . . 31 3.3.1 Selecting the Sample Populations . . . . . . . . . . . . . 31 3.3.2 Results of the Analysis . . . . . . . . . . . . . . . . . . . 37 3.3.3 Distributions and Relaxation Time . . . . . . . . . . . . . 39 3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4 Measuring the Distance to 47 Tucanae (NGC 104) . . . . . . . . . . 44 4.1 The Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.2 Using Models to Determine a Distance . . . . . . . . . . . . . . . 44 4.3 Converting Spectral Models to Magnitudes in Hubble WFC3 Filters 46 4.4 A Summary of the Model-Fitting Procedure . . . . . . . . . . . . 47 4.4.1 Calculating Statistical Errors . . . . . . . . . . . . . . . . 50 4.4.2 Calculating Systematic Errors . . . . . . . . . . . . . . . 51 4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 vi 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 vii List of Tables Table 2.1 Centers of Milky Way globular clusters . . . . . . . . . . . . . 19 Table 3.1 Results from the analysis of HB distributions. The first column is the cluster NGC name. The second column is the number of stars in the entire ACS field. The third column is the core ve- locity disperion in km/s. The fourth column contains the mean relaxation time of the HB population in log10(years). The last two columns contain the results of the comparison between the HB, MS, and TO stars. CON means that the HB population was more concentrated. N.S. means that the difference between the distributions is not significant. . . . . . . . . . . . . . . . . . . 43 viii List of Figures Figure 2.1 Faint star incompletness in NGC 1851 . . . . . . . . . . . . . 8 Figure 2.2 An illustration of radial symmetry in incompletness . . . . . . 9 Figure 2.3 Ellipse fitting summary . . . . . . . . . . . . . . . . . . . . . 10 Figure 2.4 Resulting ellipse fits for NGC 1261 . . . . . . . . . . . . . . 11 Figure 2.5 Uncertainty estimation for the iso-density contour method . . 12 Figure 2.6 Symmetry method pie-slice orientation . . . . . . . . . . . . 14 Figure 2.7 Pie slice orientation and symmetry sensitivity . . . . . . . . . 15 Figure 2.8 A comparison of center coordinates to other sources . . . . . 17 Figure 3.1 Incorrect group selection in the NGC 6397 kick result. . . . . 23 Figure 3.2 Original and corrected radial distributions (NGC 6397 WD kicks). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 Figure 3.3 The autoselect.pro interface . . . . . . . . . . . . . . . . . . 25 Figure 3.4 Completeness fraction in 47 Tuc . . . . . . . . . . . . . . . . 27 Figure 3.5 Bins for completeness corrections . . . . . . . . . . . . . . . 29 Figure 3.6 The completeness correction as a function of radius for 2 dif- ferent magnitude bins in 47 Tuc. Data is shown as black trian- gles. The red curve is the best fit Gaussian. . . . . . . . . . . 30 Figure 3.7 HB mass-magnitude relation . . . . . . . . . . . . . . . . . . 32 Figure 3.8 HB masses for NGC 5272 . . . . . . . . . . . . . . . . . . . 33 Figure 3.9 MS mass-magnitude relation . . . . . . . . . . . . . . . . . . 34 Figure 3.10 MS masses for NGC 5272 . . . . . . . . . . . . . . . . . . . 35 Figure 3.11 HB and MS mass functions for NGC 5272 . . . . . . . . . . . 36 Figure 3.12 Rejection sampling of MS objects . . . . . . . . . . . . . . . 37 Figure 3.13 Example radial distributions. . . . . . . . . . . . . . . . . . . 38 ix Figure 3.14 Mean HB relaxation time vs. HB concentration significance. . 40 Figure 3.15 Mean HB relaxation time vs. Number of stars in the field. . . 41 Figure 3.16 HB concentration significance vs. Number of stars in the field. 42 Figure 4.1 47 Tuc Cycle 17 observation map . . . . . . . . . . . . . . . 45 Figure 4.2 Combining the spectrum with throughput and reddening . . . 48 Figure 4.3 Bergeron’s model grid. . . . . . . . . . . . . . . . . . . . . . 49 Figure 4.4 An example fit to the photometry . . . . . . . . . . . . . . . . 51 Figure 4.5 Systematic error in the distance to 47 Tuc. . . . . . . . . . . . 52 Figure 4.6 47 Tuc distance modulus likelihood . . . . . . . . . . . . . . 54 Figure 4.7 CMD turn-off magnitude as a function of [Fe/H] and cluster age. 55 Figure 4.8 Cluster age vs. CMD turn-off magnitude for [Fe/H]=-0.74. . . 55 x Glossary ACS Advanced Camera for Surveys PI Principle Investigator MCT Multi-Cycle Treasury WFC Wide Field Camera HST Hubble Space Telescope PSF Point Spread Function 2MASS 2 Micron All Sky Survey RA Right Ascension IMBH Intermediate Mass Black Hole CCD Charge-Coupled Device AGB Asymptotic Giant Granch HB Horizontal Branch RGB Red Giant Branch TO Turn Off MS Main Sequence WD White Dwarf xi CMD Colour-Magnitude Diagram RS Rank Sum KS Kolmogorov-Smirnoff STMAG Space Telescope MAGnitude UVIS Ultraviolet-Visible IR Infrared DM Distance Modulus SED Spectral Energy Distribution xii Acknowledgments I would like to thank Harvey Richer and Kristin Woodley for their guidance and numerous helpful discussions on almost every aspect of this thesis. Specifically, the fourth chapter benefited greatly from our weekly meetings. I would also like to thank Jasper Wall for his fantastic course on statistics, and for multiple one on one discussions about various statistical approaches, and how to use them to improve my analysis. I would not have been able to do much of the work presented in this thesis without the knowledge gained from this course. xiii Chapter 1 Introduction 1.1 Galactic Globular Clusters Globular star clusters are abundant in our own galaxy, and also appear in every galaxy close enough that we should expect to be able to observe them. These clusters form over relatively short periods of time, resulting in fairly homogeneous stellar populations. Although there is strong evidence for multiple populations within individual globular clusters, these variations are very small when compared with inter-cluster variations. Globular clusters can also spend long periods of time orbiting far from the disk of the galaxy and so their structure is dominated by the internal dynamics of the stars within each cluster. These two facts make globular clusters perfect laboratories for studying both stellar evolution, and internal cluster dynamics Kalirai and Richer [16]. Because the formation of globular clusters is tied directly to the formation history of our galaxy, analysis of globular clusters can also lead to insights about our galaxy’s history, and galaxy formation in general. Recent advances in instrumentation have given us the deepest and highest res- olution images of globular clusters ever taken. When these images are combined with advanced reduction techniques, the result is high quality data catalogs that are almost 100% complete down 22nd magnitude. “The ACS Survey of Galactic Glob- ular Clusters” Sarajedini et al. [22], undertaken in 2007, provides what is likely the most comprehensive catalog of stars in Milky Way globular clusters ever con- structed. This thesis covers a number of topics that can be addressed utilizing high 1 quality images of globular clusters from the Hubble Space Telescope (HST). 1.2 Determining Cluster Centres Before beginning any work on the [22] data set, it quickly became apparent that the precision and accuracy of the center coordinates was paramount to generating reasonable results for the studies we wished to undertake, and the centers from Harris [14] were simply not sufficient to pursue this analysis. Precise centers are key to analyzing the dynamics in the central regions of clusters, and in particular, searching for evidence of IMBHs and analyzing the distributions of separate stellar populations within a cluster. Because of this, new centers needed to be calculated with much better precision than what was currently available. Additionally, these centres would be quite useful for other astronomers doing similar work with glob- ular clusters, as many of the centres had not been re-determined since the advent of modern CCDs. This data set provided a perfect opportunity to do this. The analysis and results are discussed in Chapter 2. 1.3 The ACS Survey of Galactic Globular Clusters 1.3.1 The Survey ACSacsACS Survey of Galactic Globular Clusters” (PI: Sarajedini, Ata). This survey was a part of Hubble’s Multi-Cycle Treasury (MCT) programs in 2009. The MCT is a program intended “to provide astronomers with the opportunity to tackle key scientific questions that cannot be fully addressed through the standard time allocation process” [25]. The survey used a total of 134 orbits, which would be difficult to obtain in a standard proposal cycle. A total of 65 globular clusters were observed in the ACS filters F814W and F606W with one orbit per cluster per filter. All of the data used for the work in Chapter 2 and Chapter 3 (with the exception of Section 3.1) are from the aforementioned survey, described in more detail by Sarajedini et al. [22]. A thorough discussion of the reduction methods used for the data can be found in Anderson et al. [2]. A brief summary follows. 2 1.3.2 The Data and the Resulting Catalog Each cluster was observed with the Advanced Camera for Surveys (ACS) / Wide Field Camera (WFC) on HST for one orbit through each of F606W and F814W . Each orbit contained one short exposure (to fill in the saturated stars) and four to five deep exposures, which were stepped over the ACS chip gap for more uniform spatial coverage. The catalog was constructed by analyzing the field patch by patch in all the individual exposures simultaneously. In each patch, the brightest stars were found first and fitted with a Point Spread Function (PSF) that was tailored to each particular exposure, then subtracted to allow fainter neighbors to be found. Only stars that clearly stood out above the known subtraction errors and known PSF artifacts in multiple exposures in both filters were included in the catalog. The very few stars that were saturated even in the short exposures were measured by fitting the PSF to the surrounding unsaturated pixels. The 2 Micron All Sky Survey (2MASS) catalog was to used convert the positions measured in the image- based frame into absolute Right Ascension (RA) and Declination. The operations that follow in this paper were based on the RA and Dec. coordinates, as well as the F814W magnitude for each star. This entire data catalog is now public on Dr. Sarajedini’s personal webpage [21]. 1.4 White Dwarf Kicks In 2008, a result was discovered by Davis et al. [6] that suggested the possible existence of some unexpected dynamical effects during the birth of White Dwarf (WD) stars in the globular cluster NGC 6397. This phenomenon was described as a white dwarf birth “kick” that would impart some extra bit of kinetic energy to these objects. This paper was preceeded by a theory paper, Heyl [15], that discussed the initial asymmetry in mass-loss along the asymptotic giant branch, and constraints that could be put on this mass-loss through observation. Fregeau et al. [11] later discussed the dynamical effects of such a kick for star clusters as a whole. With the recent availability of the treasury data [22], it seemed worth pursuing evidence for this kick in earlier stages of stellar evolution (post turn-off but pre- white dwarf). This would be done by analysing the radial distributions various post-turn off groups and comparing them to multiple pre-turn off “control” groups. 3 This was the original motivation for the analysis of this data set. This early work is discussed in more detail in Section 3.1. 1.5 The Distance to 47 Tuc The final section of this thesis (Chapter 4) presents a new technique for determin- ing the distance to a globular cluster. HST cycle 17 the globular cluster 47 Tuc was imaged for 121 orbits using WFC3 and ACS ( GO-11677 , PI - Harvey Richer). The cluster was observed in F390W , F606W , F110W , and F160W , giving good coverage of visible and near Infrared (IR) wavelengths. The observation program and data reduction is discussed in great detail in Kalirai et al. 2011 (in preparation). The method we use relies on white dwarf spectral models, and a few relatively con- servative assumptions about the white dwarfs themselves. This results in the most precise distance to 47 Tuc ever determined. Distance is a very important parameter when studying globular clusters, as any uncertainty in an age determination from isochrone fitting is dominated by the uncertainty in the distance. The distance pre- sented in Chapter 4 has decreased the error bars by a factor of two from previous determinations. This leads directly to a factor of two increase in the precision of ages determined using the isochrone fitting method. 4 Chapter 2 Determining Centres for Resolved Globular Clusters 2.1 A Brief History of Approaches The need for precise centres of Galactic globular star clusters is more critical to- day than ever before. For example, there is much work being done searching for Intermediate Mass Black Holes (IMBHS) in globular cluster cores. The masses of these IMBHs are expected to be quite small compared to their galactic counter- parts, a few hundred to a few thousand M, and so the distance from the center of the black hole within which the stars would be observably influenced is also quite small. As shown in Anderson and van der Marel [1], a center that is off by only 12 ′′ can greatly affect conclusions about the presence of an IMBH. Additionally, recent work analyzing the radial distributions of various stellar populations within globular clusters, such as Bellini et al. [3] and Ferraro et al. [9], also depends on the precision of the centers used. An incorrect center will dilute the differences between any two radial distributions centered on the same erroneous location. This is complicated by the fact that, the “centre” of a cluster can refer to many things, i.e. the centre of light, the centre of mass, the centre of the star count, or even the centre of symmetry. However, assuming that the cluster is dynamically relaxed, and the amount of dark matter is negligible, these should all refer to the same lo- cation. The methods discussed below will discuss how the centre of the star count 5 distribution was determined. Historically, coordinates for the centers of globular clusters have been deter- mined by a variety methods, and have been compiled in a catalog made available online by Harris [14]. The parameters in this catalog are still widely used today. The majority of the centers in this catalog come from a list compiled by Djorgovsky and Meylan [7]. Of the 143 globular clusters contained in this 1993 list, 109 of the center coordinates come from Shawl and White [24]. The centers from Shawl and White were taken from scanned photographic plates. These scans were smoothed and the center was found with an algorithm known as the SWIRL technique, de- veloped by C.R. Lynds (described in Shawl and White [23]). This method involves analyzing the density over a number of regions symmetrically selected in the X and Y directions, and manually adjusting the center point until the densities in the corresponding X and Y regions are as similar as possible. Uncertainties were given as the standard error of repeated determinations with this method. These center coordinates have stood the test of time quite well, but with the advancement of modern Charge-Coupled Devices (CCDS) and better reduction techniques, more precise centers can be determined. Whereas previous determinations were made on photographic plates and fo- cused on the outskirts of clusters, our measurements here will focus on the central regions. If there are any asymmetries in the stellar distribution, the two methods will not arrive at the same center. The SWIRL method analyzes the symmetry of light from the entire cluster, but has no way to deal with asymmetries in the distribution caused by small numbers of giant stars. We caution that the method discussed here can only be used to determine the center of the starcount distribu- tion. The center of mass must be determined through analysis of the dynamics of the cluster. 2.2 Methods for Determining The Center Two methods for determining the center of the starcount distribution were em- ployed. Both methods begin with the positions and magnitudes from the catalog, constructed as described in Chapter 1. The density method searches for the average center of a number of isodensity contours by fitting ellipses to these contours. The 6 pie-slice method searches for the most symmetric point of the cluster by analyzing the distributions of stars as a function of radius on opposite sides of the cluster. All of the centers reported in Table 1 were derived with the density contour method, as it was found to be more reliable over a wide range of density distribu- tions than the pie-slice method. 2.3 From Iso-density Contours 2.3.1 Avoiding Incompleteness We began with the catalog as constructed above and converted the apparent F814W magnitudes into an absolute magnitude scale using the distance moduli for each cluster from Harris [14]. All stars with an absolute F814W magnitude fainter than 8 were immediately excluded. This has the effect of smoothing out the distribution, as stars fainter than this are not likely to be found near much brighter stars, and they are very incomplete in the cores of denser clusters. When stars of all magnitudes are included in the density distribution, many underdense regions around bright stars appear. These holes do not reflect the true underlying density distribution, and so an arbitrary cutoff was implemented to limit these effects (Figure 2.1). For denser clusters, stricter magnitude cutoffs were used. In general, the cutoff varied from cluster to cluster, and was selected so that the stars in the inner most region of the analysis were at least 50% complete prior to the application of com- pleteness corrections. In most cases, this meant a cut at an absolute magnitude of 8 was necessary. However, the densest clusters required that only stars brighter than an absolute F814W magnitude 2 be used, in order to remove the appearance of these holes. These regions of incompleteness begin to significantly overlap as you approach the center of the cluster. The number density of very bright stars in the cores of clusters is so large, that very faint stars are almost impossible to reliably detect. As a result, fainter stars are almost entirely incomplete in the inner region of the cluster. Because density is a function of radius, and the ability to reliably detect a star is dependent on the density of the bright stars in the region surrounding that star, the incompleteness of fainter stars is also a function of radius. So, although incompleteness is also present to a lesser extent in stars above the implemented 7 Figure 2.1: Cluster NGC 1851. The density contours (left) of the stars that were rejected from the center determination method due to the imposed limiting F814W magnitude cutoff, and positions of the stars in the field (right) after the initial data reduction (star positions are in black). Large voids are seen near brighter stars, as these make it difficult to find fainter stars nearby. The center is almost entirely empty, as it is far too dense to reliably find stars fainter than absolute magnitude 8. The coordinates are in projected distance with respect to the center in Harris [14]. cutoff, the radial dependence of the incompleteness means that there will be no directional bias introduced, as evidenced in Figure 2.2, and center determinations should not be affected. 2.3.2 Calculating the Projected Density Profile The coordinate system used for the determination is oriented to right ascension and declination, but with the necessary cosine term to generate a projection onto the plane tangent to the sky at the center of the field. The density of the cluster was then constructed over a grid of points. This grid was centered on the RA and Dec. from Harris [14]. The grid extends 100 ′′ in each direction, sampling every 2 ′′. As mentioned in Anderson & van der Marel 2009, significant overbinning is required to generate a smooth distribution. Circular bins centred at each grid point were used with a radius of 25 ′′ in most cases. However, clusters with very flat 8 Figure 2.2: On the left, density contours for NGC 1851 from the method out- lined in Section 2.3.2 are shown in black (thick). Overplotted in red (thin) are the inner contours from the valley shown in Figure 2.1. El- lipses fit to each of these sets of contours find average centers that agree to within 1.6′′ (within the estimated uncertainty of the two centers). On the right, slices of the density surface are shown through ∆Y = 0 for the faint stars excluded, and the remaining stars used in the center deter- mination. Gaussians fit to these distributions find centers that agree to within 0.12 ′′. This agreement supports the conclusion that, due to the radial dependence of incompleteness, this effect will not bias the center determinations. The coordinates are in projected distance with respect to the center in Harris [14]. distributions required larger bins (up to a radius of 40′′) to smooth the distribution. To remove the effects of the edge of the field on the density distribution, the density values for points on the sample grid that were within a bin radius of the edge of the field were set to 0, so as not to be included in any of the analysis. That is to say, if the circular bin at that point included area outside of the field, that point was excluded from the ellipse fitting portion of this method. 2.3.3 Fitting Ellipses After the distribution was generated and the edge effects were accounted for, the distribution was broken up into eight contours. The contours were spaced evenly between the minimum non-zero density and the maximum density. The outer three 9 Figure 2.3: Cluster NGC 1851. A visual representation of the method de- scribed in Section 2.3. Each row shows the points on the sample grid that fall within a specified density bin, as well as the ellipse fit to these points, and the center for this ellipse. The top row corresponds to a den- sity bin of ∼2.5-3.3 stars per square arcsecond, while the bottom row corresponds to a bin of ∼4.0-4.8 stars per square arcsecond. The coor- dinates are in projected distance with respect to the center in the catalog [14]. contours were not used due to the fact that they were often quite azimuthally in- complete, and this could potentially bias the ellipse fits. The innermost contour was also ignored as it is actually a solid two dimensional region of points and could not be fit well with an ellipse. Each of the four remaining contours was then fit with an ellipse. This step is outlined in Figure 2.3. For most of the denser clusters, it is possible to fit many more than four ellipses (in some cases up to 20). However, larger numbers of smooth contours cannot be generated for the clusters with low numbers and/or flat distributions. In order to maintain consistency, eight contours and four ellipses were adopted as the maximum numbers that could be used for every cluster. The final center value was determined as the average center of these 10 Figure 2.4: Cluster NGC 1261. Density contours are shown in black (thin). Fit ellipses are shown in red (thick) with the centers as red dots. The calculated center is shown as a blue circle with the radius equal to the estimated uncertainty. Distances are given in seconds of arc in projected distance from the catalog center. Relevant information is contained in the table to the upper right including the center in Right Ascension and Declination, the difference from the catalog center in seconds of arc in the RA and Dec. directions, and the uncertainty of the center determined as the standard deviation from all of the ellipse centers. The smaller plot in the lower right shows a close-up of the determined center as well as the Harris center [14], which is [0,0] in this coordinate system. four ellipses. The uncertainty was estimated as the standard deviation of the ellipse centers. A plot of the density contours as well as the ellipses fit to these contours and their centers for the cluster NGC 1261 is shown in Figure 2.4. Plots such as this one are available for all 65 clusters in Table 1 as part of the supplementary material of Goldsbury et al. [12]. 11 Figure 2.5: The standard deviation (used as the estimated uncertainty) and actual error as a function of the number of stars in the field. Values were determined through simulation. The standard deviation of the ellipse centers appears to be a good estimate for the uncertainty in the center. 2.3.4 Determining Uncertainties The consistency of the method discussed above as well as the error estimates for these measurements were analyzed through simulation. Synthetic cluster distribu- tions were generated and analyzed with the density contours method. The number of stars in the synthetic distributions ranged from 25,000 to 250,000 in increments of 25,000. These values cover roughly the range of the 65 clusters presented in this paper after the initial faint magnitude exclusion. For each increment between 25,000 and 250,000, one hundred distributions were generated, and their centers were measured through the density contours method. The standard deviation of the centers used in each determination, as well as the actual errors from the true center of the generated distributions were averaged over these 100 samples. The estimated uncertainty and actual error are plotted against the number of stars in Figure 2.5. This figure suggests that the standard deviation of the ellipse centers 12 provides a very good estimate of the actual uncertainty in the determination. 2.4 From Cluster Symmetry To begin, a grid of sample points was constructed around the center value from the Harris Catalog. The grid extends 40 ′′ in each direction, sampling every 2 ′′. At each point on the grid, the stars within a radius of 1.2′ were divided into eight different pie slices corresponding to the cardinal and semi-cardinal directions in RA and Dec. (see Figure 2.6). A cumulative radial distribution was then generated for each of the eight pie-slices. The four pairs of opposing distributions were then compared. Figure 2.7 shows two pie-slices and the corresponding cumulative dis- tributions of the stars within these regions. To compare the distributions at a single point, the integrated area between the curves was used. Four pairs of distributions were compared, giving four values for each point on the grid. These values were then added together to yield a measure of the symmetry of the cluster at that point on the sample grid. When determined in this way, higher values correspond to less symmetry, as this indicates a larger difference in the opposing distributions at that point. This was done for every point on the sample grid. The result is a two- dimensional array of values that describe the symmetry of the cluster as a function of position in projected space. It is important to note that a minimum of two pairs of orthogonal pie slices must be used to get a center determination with this method, as each pair of pie slices is only sensitive to changes along one axis. This is demonstrated in Figure 2.7. Mov- ing a pair of pie slices perpendicular to their orientation keeps them in a position that is still symmetric along their axis. For this reason, a second pair of pie slices must be used to sample the symmetry in the direction orthogonal to the first. For our sample of clusters, it was found that eight pie slices worked better than four, however switching to sixteen pie slices resulted in sample sizes that were too small to be useful. Determining the size of pie slices to use also takes some consideration. The radius of the pie slices must be determined based on the size of the field, and the size of the sample grid. It is important to constrain the radius so that no point on the grid is affected by the edge of the field. It is also important that the radii of the 13 Figure 2.6: The orientation of the eight pie slices used in the symmetry method are shown here. Each pair is color coded. The cumulative dis- tributions of like colors are compared at each point on a sample grid. The coordinates are in projected distance with respect to the center in the catalog [14]. pie slices remain constant across the grid, so as not to bias the determination. This creates a problem, as larger radii create larger samples and thus better distributions, however the sample grid must then be smaller so that the pie slices at the outer edges of the grid do not run out of the field. For this same reason, sampling over a larger area requires the radii of the pie slices to be smaller. After the symmetry value is calculated at each point in the sample grid, con- tours can be created and a center determined by finding the most symmetric point on the grid. Due to the orientations of the pie slices, these contours will be eight sided figures (or 2N-sided figures for N pairs of pie slices), and so it is not appro- 14 Figure 2.7: Cluster NGC 1261. The North and South pie slices as well as their cumulative distributions are shown at three different locations in the cluster. The top row corresponds to the center found by the density contours method. The middle row corresponds to a point 30 ′′ West of this center. The bottom row corresponds to a point 30 ′′ South of this center. The distributions remain almost identical as the pie slices are moved perpendicular to their orientation. The area between the oppos- ing radial distributions begins to grow as the pie slices are moved away from the true center of the cluster parallel to their orientation. The coor- dinates are in projected distance with respect to the center in the catalog [14]. 15 priate to fit them with ellipses as discussed in the density contours method. This method can only show the most symmetric point on the sample grid, and so the uncertainty of this center is limited by the spacing of the grid points. Even for cen- trally concentrated clusters with good numbers across the field, the symmetry did not appear to change significantly on scales smaller than 2′′, and so a grid spacing of less than this was not used. This makes the centers determined by this method considerably less precise than those determined by the density contours method. 2.4.1 Generating an Incompleteness Mask As discussed in the previous section and shown in Figure 2.1, bright giant stars create large holes in the apparent stellar distribution. These holes are often asym- metrically distributed due to their small numbers, and as a result large differences appear in opposing radial distributions at locations that are otherwise quite sym- metric. Because of this effect, many of the denser clusters required that an incom- pleteness mask be applied before using the pie-slice method. In this case, simply imposing a strict magnitude cutoff does not work, as this lowers the number of stars in each pie slice to the point where the sample sizes are no longer useful. This is discussed in more detail in Anderson and van der Marel [1] (see Fig. 12 of that paper). 2.5 Comparison to Previous Works A distribution of the differences in X and Y from the catalog centers is shown in Figure 2.8. X and Y are oriented to RA and Dec., and represent seconds of arc on the sky. The center coordinates were also compared to those presented in Noyola and Gebhardt [19]. There appears to be no systematic difference between the centers determined in this paper and those determined by Noyola & Gebhardt. The means of the differences in RA and Dec. are consistent within the standard errors with a distribution about the origin. With respect to the centers in the Harris Catalog, the distribution of differences in X is consistent with a random normal distribution centered around zero, suggesting that there are no systematics biasing the centers in RA. However, the average difference in the Y direction is greater than the standard error of the differences in that direction, indicating that the distribution 16 Figure 2.8: The scatter of the differences between the determined centers and the center values given in Harris [14]) and Noyola and Gebhardt [19]. The X and Y axes correspond to seconds of arc in the RA and Dec. directions. The cluster NGC 7089 is not included in the left plot as it differs by greater than 20′′, however it is still included in the average and standard deviation. Of the 65 centers presented in this paper, 24 were also determined by Noyola & Gebhardt. The differences between these 24 clusters are shown on the right. The uncertainties given in each case are the standard error of the mean. With respect to the Harris Catalog, the standard deviation of the scatter in RA is 6.65 ′′ and the standard deviation of the scatter in Dec. is 7.03′′. is just barely inconsistent with a center of zero. This suggests a possible systematic difference of about 1.7 ′′ in the Dec. direction between the centers provided in Table 1, and the centers given in Harris [14]. The centers presented by Noyola & Gebhardt also exhibit this systematic difference with a distribution of differences from the Harris Catalog centered approximately 1′′ South of the origin. 2.6 Conclusions In this chapter we have presented new centers for 61 Milky Way globular clusters and outlined the methods used to determine them. These centers are significantly more precise than what is currently available in the literature with an average un- certainty of less than 1′′. In addition, many of these centers differ significantly from the values in Harris [14]. Of the 61 clusters analyzed, 24 differ by more than 5 ′′, 17 and 7 clusters differ by more than 10′′. We have described the general outline of the density contours method and shown that incompleteness in faint stars should not bias this determination due to the dependence of incompleteness on the distance from the cluster center. We have also discussed a separate method that relies on analyzing the symmetry of the clusters to find the center, however the dependence of this method on sample size and cluster size relative to the imaging field make it difficult to apply in most cases. Denser clusters that require masking bright stars can make the symmetry method even more time consuming, and so we suggest that the density contours method is the preferred method of determing globular cluster centers. 18 Table 2.1: Centers of Milky Way globular clusters Cluster ID Alternate ID RA J2000 Dec J2000 Estimated ` b ( h : m : s ) ( ◦ : ′ : ′′ ) Uncertainty (′′) ( ◦) ( ◦) NGC 104 47 Tuc 00:24:05.71 -72:04:52.7 0.2 282.7334 -29.5475 NGC 288 00:52:45.24 -26:34:57.4 1.8 308.4303 36.0769 NGC 362 01:03:14.26 -70:50:55.6 0.1 316.5174 -12.8101 NGC 1261 03:12:16.21 -55:12:58.4 0.1 316.7442 -60.6231 Palomar 1 03:33:20.04 +79:34:51.8 0.8 133.3390 22.7027 Palomar 2 04:46:05.91 +31:22:53.4 0.1 103.5433 -28.4620 NGC 1851 05:14:06.76 -40:02:47.6 0.1 248.3607 -20.6108 NGC 2298 06:48:59.41 -36:00:19.1 0.1 243.1166 -22.4883 NGC 2808 09:12:03.10 -64:51:48.6 0.1 328.9732 -19.2632 E 3 09:20:57.07 -77:16:54.8 1.5 316.8235 -30.9358 NGC 3201 10:17:36.82 -46:24:44.9 1.0 278.1893 9.2654 NGC 4147 12:10:06.30 +18:32:33.5 0.1 17.7755 65.4396 NGC 4590 M 68 12:39:27.98 -26:44:38.6 0.2 22.8622 -50.2242 NGC 4833 12:59:33.92 -70:52:35.4 0.3 289.7864 -43.0012 NGC 5024 M 53 13:12:55.25 +18:10:05.4 0.1 192.5475 0.2276 NGC 5053 13:16:27.09 +17:42:00.9 1.4 198.6739 11.2637 NGC 5139 ω Cen 13:26:47.28 -47:28:46.1 0.1 273.0120 4.0733 NGC 5272 M 3 13:42:11.62 +28:22:38.2 0.2 43.0822 80.7444 NGC 5286 13:46:26.81 -51:22:27.3 0.1 319.1657 8.1490 NGC 5466 14:05:27.29 +28:32:04.0 0.6 61.6991 6.5728 NGC 5904 M 5 15:18:33.22 +02:04:51.7 0.0 329.0530 62.5516 NGC 5927 15:28:00.69 -50:40:22.9 0.2 330.6481 1.7546 NGC 5986 15:46:03.00 -37:47:11.1 0.2 4.1513 -35.0802 Lyngå 7 16:11:03.65 -55:19:04.0 0.9 274.5328 -55.0366 NGC 6093 M 80 16:17:02.41 -22:58:33.9 0.2 219.5801 -43.8833 NGC 6101 16:25:48.12 -72:12:07.9 0.5 282.8777 -27.5859 NGC 6121 M 4 16:23:35.22 -26:31:32.7 0.4 231.7876 -23.6896 NGC 6144 16:27:13.86 -26:01:24.6 0.6 236.1344 -12.1475 NGC 6171 M 107 16:32:31.86 -13:03:13.6 0.1 233.5681 10.4371 NGC 6205 M 13 16:41:41.24 +36:27:35.5 0.1 186.8649 57.7933 NGC 6218 M 12 16:47:14.18 -01:56:54.7 0.8 272.9232 57.2127 NGC 6254 M 10 16:57:09.05 -04:06:01.1 0.1 339.7755 52.5219 NGC 6304 17:14:32.25 -29:27:43.3 0.2 4.9970 -10.3158 NGC 6341 M 92 17:17:07.39 +43:08:09.4 0.3 74.5945 13.8815 NGC 6352 17:25:29.11 -48:25:19.8 0.6 350.4569 -44.6173 NGC 6362 17:31:54.99 -67:02:54.0 0.5 318.4562 -46.5374 NGC 6366 17:27:44.24 -05:04:47.5 1.4 52.7583 -42.8224 NGC 6388 17:36:17.23 -44:44:07.8 0.3 328.0561 -70.0233 NGC 6397 17:40:42.09 -53:40:27.6 0.3 296.6536 -63.2206 NGC 6441 17:50:13.06 -37:03:05.2 0.2 239.5334 -54.6078 NGC 6496 17:59:03.68 -44:15:57.4 1.3 250.5333 -29.8676 NGC 6535 18:03:50.51 -00:17:51.5 0.4 213.9027 1.2416 NGC 6541 18:08:02.36 -43:42:53.6 0.1 258.8651 -7.1202 NGC 6584 18:18:37.60 -52:12:56.8 0.2 283.3375 5.5601 NGC 6624 18:23:40.51 -30:21:39.7 0.1 288.7654 30.9569 NGC 6637 M 69 18:31:23.10 -32:20:53.1 0.1 317.2313 28.8707 NGC 6652 18:35:45.63 -32:59:26.6 0.1 331.1239 22.9512 NGC 6656 M 22 18:36:23.94 -23:54:17.1 0.8 338.5181 29.5142 NGC 6681 M 70 18:43:12.76 -32:17:31.6 0.1 350.1062 8.1499 NGC 6715 M 54 18:55:03.33 -30:28:47.5 0.1 9.7305 -24.3068 NGC 6717 Pal 9 18:55:06.04 -22:42:05.3 0.2 17.7368 -21.8003 NGC 6723 18:59:33.15 -36:37:56.1 0.3 6.3887 -39.3572 NGC 6752 19:10:52.11 -59:59:04.4 0.1 317.8388 -55.1195 NGC 6779 M 56 19:16:35.57 +30:11:00.5 0.2 127.4063 -32.5365 Terzan 7 19:17:43.92 -34:39:27.8 0.3 260.5393 -79.4448 Arp 2 19:28:44.11 -30:21:20.3 0.9 229.4959 -46.4803 NGC 6809 M 55 19:39:59.71 -30:57:53.1 0.8 241.7989 -11.9084 Terzan 8 19:41:44.41 -33:59:58.1 1.4 247.0191 -8.2681 NGC 6838 M 71 19:53:46.49 +18:46:45.1 0.5 219.2432 55.7621 NGC 6934 20:34:11.37 +07:24:16.1 0.1 51.9165 -18.6097 NGC 6981 M 72 20:53:27.70 -12:32:14.3 0.1 150.5005 -73.7548 NGC 7078 M 15 21:29:58.33 +12:10:01.2 0.2 230.1238 53.5815 NGC 7089 M 2 21:33:27.02 -00:49:23.7 0.1 261.6423 54.6937 NGC 7099 M 30 21:40:22.12 -23:10:47.5 0.1 307.1349 39.5768 Palomar 12 21:46:38.84 -21:15:09.4 0.4 334.1046 34.9822 19 Chapter 3 Distributions of Various Stellar Populations 3.1 White Dwarf Kicks: Possible Evidence in NGC 6397 As discussed in Section 1.4, the original goal of this project was to investigate the radial distributions of the post Turn Off (TO) stars in the 65 globular clusters probed by the ACS treasury survey. The motivation for this analysis was the apparent evidence presented by Davis et al. [6] that the young white dwarfs in NGC 6397 appear to be radially extended compared to their older counterparts. This would indicate that, at some point during the evolution of the stars, after the stars leave the Main Sequence (MS), but before becoming white dwarfs, these stars could be getting a “kick”. This would input energy into this population, while the rest of the cluster would relax (or stay relaxed as the case may be) as normal. The result would be a distribution for the young white dwarfs that looks far too extended or “fluffy” for the mass of these objects. One would expect that this increase in kinetic energy of the young white dwarfs would dissipate on timescales similar to the relaxation time of the cluster. And so, as the white dwarfs cool and age this effect will no longer be present. Through repeated interactions with other cluster members, the aging white dwarfs will exchange this excess of kinetic energy, and sink back to a more concentrated distribution that is consistent with their mass. This is exactly the result presented in Davis et al. [6]. 20 With the apparent evidence of this kick in NGC 6397, we were motivated to use the newly acquired treasury data to probe for an indication of a kick at earlier stages of stellar evolution. Perhaps this kick could even be seen by analysing the radial distributions of the Asymptotic Giant Granch (AGB), Horizontal Branch (HB), or even the Red Giant Branch (RGB) stars. Mass-loss occurs to some degree during all of these stages and so measuring how the distributions of stars in each cluster change during these various phases could provide insight into when, and over what period of time, a “kick” occurs. This analysis is presented in Section 3.3. 3.1.1 New Data In July of 2010, new Cycle 17 data became available for NGC 6397 that covered the same field as the original study by Davis et al. [6]. This new epoch provided an opportunity to generate much better proper motions for our field, and produce a very clean Colour-Magnitude Diagram (CMD). A natural next step was to use these data to attempt to reproduce the original result [6]. This appeared to be a very straight forward task. However, when the distributions of the young and old white dwarfs were calculated using the new data, they did not seem to be at all consistent with the original results. The first discrepancy was that there appeared to be no difference between the young and old white dwarf groups in radial extension. The original result was not apparent. The second, was that the new distributions covered a smaller range in radial distance, and appeared to be closer to the center than the original. 3.1.2 Issues of Coordinates and Centers After some careful analysis, it was determined that the calculation of the original radial distributions by Davis et al. [6] involved three mistakes. The first was the lack of correction for the distortions in a spherical coordinate system that occur for large values of declination. While 1 ′′ of dec. is 1 ′′ of dec. anywhere on the sphere, this is not the case for right ascension. Because NGC 6397 is almost 54 ◦ below the equator, 1′′ of RA actually subtends a much smaller angle on the sky than 1′′ of dec. (by a factor of 0.59). This needs to be corrected with the inclusion of a cos(δ ) term (where δ=declination) that transforms the coordinate 21 system of the field to a projection onto a flat plane tangent to the sphere at the center of the field. This transformation was not done before calculating the original distributions. This is part of the explanation for the discrepancy in the radial range of the distributions. The second mistake, which also contributes to the different range of distances, was the use of incorrect coordinates for the center of the cluster. It appears that the center used was around [17:40:42.96 , -53:40:30.6] , but the actual center is in fact much closer to [17:40:42.09 , -53:40:27.6] Goldsbury et al. [12]. This, along with the lack of cos(δ ) transformation, entirely accounts for the distance discrepancy. The final mistake was the inclusion of some questionable group members in the young white dwarf selection. The original groups can be seen in the left hand side of Figure 3.1. A few stars that do not appear to be associated with the young white dwarfs can be seen as blue triangles near the main sequence. These objects were included as young white dwarfs in the calculation of the original distributions, but were excluded in the corrected groups. New WD groups were chosen by proper-motion cleaning the CMD and select- ing only those stars which fell tightly along the WD sequence in colour-magnitude space. This is shown in the right panel of Figure 3.1. The magnitude range was chosen in the same way as the original distributions, with models giving a mag- nitude/age relation. The coordinates were then corrected for large δ distortions by transforming to a projection onto a two-dimensional flat space tangent at the center of the field. Using the newly selected groups, the new center coordinates, and working in projected space, corrected radial distributions were generated in an identical way to the originals. These are shown in Figure 3.2. When calculated in the correct way, the young and old white dwarf distribu- tions show no statistically significant difference. This invalidates the original result and indicates either that the white dwarfs are not receiving a kick during their for- mation, or that the distribution has already relaxed. 22 Figure 3.1: On the left: The original groups used for the comparison of the young and old white dwarfs to the 0.5 M and 0.2 M main-sequence groups. On the right: The groups selected for the more recent analysis. These are more tightly bound to the sequence, and utilize the new data. 3.2 Calculating Completeness Corrected Radial Distributions The original goal of this entire project was to test the hypothesis that the radial dis- tribution of objects on the horizontal branch (HB) is consistent with their expected distribution, given their range of masses, and assuming that the cluster is relaxed. This study is motivated by the evidence provided by Davis et al. [6] that the young white dwarfs in NGC 6397 appear to be significantly more extended than the older white dwarfs. However, given that the results from that study appear to be incor- rect, as shown in Section 3.1, this analysis seems less motivated. This analysis was actually begun prior to the discovery of the errors made in the original white dwarf kick result, and so the results found by analyzing the radial distributions of these 23 Figure 3.2: Top: The original distributions using the incorrect center and no cos(δ ) correction. Bottom: The corrected distributions calculated in tangent projected coordinates with the new center. 24 Figure 3.3: The basic user interface of autoselect.pro. The user runs the rou- tine from the IDL command-line (or as a part of a larger script) and selects the region with a few mouse clicks. post turn-off groups will be presented below. 3.2.1 Isolating Regions of the CMD The first step of this process is to use the color magnitude diagram produced for each cluster in F606W and F814W to determine which stars would be considered group members for each population to be analyzed. What is essentially being done is creating some binding region in a two dimensional space (color and magnitude), and finding all of the objects that fall within that region. Two procedures were writ- ten in IDL that will easily accomplish this task. They are called bind2d.pro and autoselect.pro. Both procedures will allow the user to select some two dimensional region in color-magnitude space with a few clicks on the CMD. bind2d.pro requires the user to select upper and lower magnitude bounds, and 25 then to bind the left and right sides of the sequence with N clicks, on each side of the region to be bound, that generate N− 1 line segments. autoselect.pro only requires the user to select upper and lower magnitude bounds. The sequence between these is then broken up into N bins (as specified by the user). The mean and standard deviation in color of each bin is then found. Objects are included if they fall within M (also specified by the user) standard deviations of the mean in each bin. Keeping all of data for a given cluster organized in a single two-dimensional array (or a series of one-dimensional arrays ordered in the same way) makes this group selection considerably easier. Once the indices in colour-magnitude space are known for some group of stars, the position arrays can then be evaluated with these same indices to analyze the spatial distribution of these objects. 3.2.2 Bias Introduced by Incompleteness Analyzing the radial distributions of populations within globular clusters is consid- erably complicated by incompleteness. In general, one is interested in comparing how the distribution of objects changes with distance to the center and with the mass of the objects in question. Unfortunately, the incompleteness of some popu- lation depends on exactly these two parameters. The incompleteness of a field centered on the core of a globular cluster can be thought of, and most easily shown, as a surface (or contours) in two dimensional parameter space (see Figure 3.4). Completeness fraction decreases with decreasing luminosity, and also decreases with projected stellar density, and thus decreases towards the core of the cluster. A naturally mass segregated cluster will show less massive, and therefore less luminous, objects in more radially extended distributions than the more massive, more luminous cluster members. Not accounting for incompleteness has the effect of biasing such a result to higher significance. Because the low mass members are less likely to be found near the center of the cluster due to crowding and the presence of many bright objects, the distribution of these low mass members can appear very radially extended entirely due to incompleteness. These effects must be taken into account before any statistical analysis of the radial distributions can 26 Figure 3.4: Contour plot showing the completeness fraction in 47 Tucanae as a function of absolute magnitude and distance from the cluster center. These values were determined from the results of artificial star tests. be undertaken. 3.2.3 Correcting for Incompleteness Using Artificial Star Tests Generating completeness corrections for a given group of stars requires us to quan- titatively analyze how easily those stars can be found, and how that varies across the field. The first step is to insert a large number (∼ 105 in this case) of arti- ficial stars into the observed field for each cluster. To “input” an artificial star, a point source of the appropriate magnitude is added, convolved with the point spread function, and given appropriate noise. These stars are input one at a time so as not to interfere with each other. The region around the artificial star is then reduced following exactly the same process used to find the real stars (described in Anderson et al. [2]). This results in an output list of found artificial stars. The input 27 artificial star list can then be compared to the output found list to determine where stars of a given magnitude can be reliably found. My analysis of incompleteness began with these generated input and output lists for each cluster. Because this work is concerned with analyzing the radial distributions of popu- lations of varying brightness, the most straight forward approach is to characterize the incompleteness of a cluster as a function of those two parameters (distance from the centre and magnitude). To do this, six parameters from the artificial star tests are needed: input and output magnitude, and input and output projected po- sitions on the sky (two coordinates each). The x,y positions of the stars were then converted to radial distances using centres determined as described in Chapter 2. The cluster was then broken up into ten radial bins, and ten bins in magnitude. The range of these bins can be seen graphically in Figure 3.5. The radial bins ranged from 0 to 2500 pixels (0 to 125 arc-seconds) with an annular width of 250 pixels (12.5 arc-seconds). The magnitude bins ranged from 0 to 11 absolute F606W with a bin width of 1 magnitude. This range was chosen to cover the main sequence below the turn off for the majority of clusters. Incompleteness above the turn-off is not an issue except for the very centres of the largest clusters. These very central regions were excluded during the analysis of the radial distributions for this very reason, and so the analysis of the incompletness of stars above the turn-off is not needed. There is no correlation between these two parameters for the input stars and they can therefore be treated as orthogonal. This gives us a two-dimensional space with 100 grid points. In each of these 100 2-d bins, the completeness fraction is calculated as: C f rac = N f ound Ninput (3.1) The completeness correction for any bin is actually the inverse of this com- pleteness fraction. i.e. if only 50% of stars are found, then there would be a correction factor of 2 (each star that is found should be weighted as 2 stars). The shape of this completeness correction factor as a function of radius, for some given magnitude bin, is well approximated by a Gaussian (see Figure 3.6). For the pur- pose of smoothing out the completeness surface, Gaussians were fit to these slices in magnitude. The values in the grid were redefined as the fit Gaussians at these 28 Figure 3.5: The cluster shown is NGC 104 (47 Tuc). These bins are those used to determine the incompleteness of a field as a function of magni- tude and position. The left plot shows the distribution of the 10 radial bins. The right plot shows the 10 bins in magnitude. points. The result across the entire 2-d grid is then what is shown in Figure 3.4. With a completeness grid now defined for each cluster. An IDL routine was written that would interpolate between the grid points in two dimensions. So, for any given cluster, magnitude, and distance from the centre, a completeness cor- rection value was returned. How this is worked into the radial distribution of the objects is discussed in the following section. 3.2.4 Calculating the Distributions To calculate a cumulative radial distribution, a group must first be selected as de- scribed in Section 3.2.1. Once the appropriate indices for these group members are found using a routine similar to those described in Section 3.2.1, the distances of all of these objects must be determined. Using the positions in R.A. and Dec. from the catalog (Anderson et al. [2]), and the determined center values (Chapter 2), the projected distance from the center could be calculated for each object in the group. These distances must then be sorted into ascending order, which gives the x-values for the distribution. With the distance and magnitude for all of the objects, a com- 29 Figure 3.6: The completeness correction as a function of radius for 2 different magnitude bins in 47 Tuc. Data is shown as black triangles. The red curve is the best fit Gaussian. pleteness correction value can then be assigned to each group member. Once these correction values are sorted in the same order as their corresponding distances, the y-values of the distribution are then the cumulative total of the completeness cor- rection values, normalized by the total of all completeness correction values in the group. y(n) = n ∑ i=0 Ccorr(i) T ∑ i=0 Ccorr(i) (3.2) Here y(n) are the y-values of the cumulative radial distribution, Ccorr are the completeness correction values for each group member sorted in order of increas- ing distance, and T is the total number of group members. If the completeness correction for every group member is 1, then this reduces to the case of the non- completeness corrected cumulative distribution. 30 3.2.5 Comparing the Distributions Once the distributions are generated, a statistic is needed to determine the signif- icance of any apparent differences. The two obvious choices for this would be the Wilcoxon Rank Sum (RS) test [27], or the Kolmogorov-Smirnoff (KS) Statis- tic [18]. Unfortunately, the Rank-Sum test is no longer suitable after applying completeness corrections to the distributions. This would require interpolating the distances along the distribution to evenly weighted points so that the arrays could be ranked and compared. The KS statistic is still easily applied to such distri- butions, as it only requires calculating the largest vertical separation between the cumulative distributions of the two samples. The significance of this separation can then be assessed using the KS distribu- tion and the appropriate sample sizes. 3.3 The Distributions of Horizontal Branch Stars We would like to compare the radial distribution of the horizontal branch stars to two separate control groups. One of the comparison groups should have the same mass distribution as the HB stars do now, the other should have a mass distribution that is consistent with their original masses, prior to any mass-loss that occured along the giant branch. The current turn-off stars are the obvious choice for the latter, but choosing a MS group that matches the current mass distribution of the HB stars is not so simple. First, the mass distribution of the HB stars in each cluster must be determined. Then, masses must be assigned to the MS stars, and a sample of MS stars must be drawn that reproduces the HB mass distribution. 3.3.1 Selecting the Sample Populations Two sets of models were used to assign masses to the HB and MS stars respectively. Horizontal branch models from the Pisa evolutionary group (Cariulo et al. [4]) were used, as described by Gratton et al. [13], to determine the masses of stars on the HB for each cluster. Two functions are provided in that paper; one which relates object mass to colour (V-I), and another which relates mass and magnitude (V). Both of these functions also depend on the metallicity of the cluster. These two relations are shown in Figure 3.7 for [Fe/H] = −1.57 (appropriate for NGC 5272). This 31 Figure 3.7: The mass-magnitude and mass-colour relations from Gratton et al. [13], shown over the colour and magnitude range of the HB in NGC 5272. allows us to make two separate estimates of the mass of each object on the HB, one from the colour and one from the magnitude. The final mass estimate was then determined as the weighted mean of these two determinations in the same way as Gratton et al. [13] (the mass determined from V-I weighted four times as much as the mass from V). The horizontal branch and the determined masses are shown for NGC 5272 in Figure 3.8. After establishing the masses of objects along the HB of each cluster, isochrones from Dotter et al. [8] were used to determine a mass-magnitude relation for the main sequence of each cluster (this also depends on metallicity). This relation is shown in Figure 3.9, again for [Fe/H] =−1.57. The masses of these objects on the CMD of NGC 5272 is shown in Figure 3.10. After assigning masses to the objects along the HB, as well as the MS, the mass function can be determined within some mass range. This is shown for the HB and the MS in Figure 3.11, between 0.55 and 0.75 Solar masses. There are two ways in which samples could be pulled from each population 32 Figure 3.8: Masses determined for the objects on the horizontal branch in NGC 5272. 33 Figure 3.9: The mass-magnitude relation from Dotter et al. [8], shown for the main sequence of NGC 5272. The mass window is selected to match the mass range of the HB objects. 34 Figure 3.10: Masses determined for the objects on the main sequence in NGC 5272. 35 Figure 3.11: The mass functions of the HB and MS stars in NGC 5272. that have matching distributions. One could use all of the MS stars and pull a smaller sample from the HB to match the distribution of MS stars, or one could use all of the HB stars and pull a sample from the MS to match the distribution of HB stars. Because the significance of any stastical comparison will obviously be limited by the sample size of the HB, it makes much more sense to chop our MS group down to match the HB. This can be done by rejection sampling our MS stars under the HB distribution. However, because our MS stars are not uniformly distributed within this mass range, a correction for the original MS distribution needs to be included as well. Our rejection distribution will be: MDre j(M) = MDHB(M) MDMS(M) (3.3) Now MDre j is our mass distribution that we will use for accepting or reject- ing MS objects, and MDHB and MDMS are the original mass distributions for the horizontal branch and main sequence stars respectively. However, this rejection distribution is only defined at the center of the bins from our original HB mass histogram. The distribution is next interpolated to all mass values from our MS 36 Figure 3.12: The rejection sampling mass distribution used to sample MS ob- jects. Accepted objects are shown in green, rejected in red. group, using quadratic splines fit to each four point neighborhood. We then assign a random number to each MS object, uniformly distributed between 0 and the max- imum value of MDre j. Because the random values are assigned over this range, any sort of normalization for the original distributions is entirely unneccessary. If this random value is less than MDre j(M) for the given mass M, then the object is ac- cepted into our sample. If it is above this value, then it is rejected. This process is illustrated in Figure 3.12. This leaves us with a sample of MS objects that matches the mass distribution of the HB. 3.3.2 Results of the Analysis Of the 65 clusters that were a part of the Legacy Survey, 56 were found to be suitable for analysis with the previously discussed methods. The remaining 9 were rejected after some preliminary analysis on the grounds of small numbers of stars in the cluster, poorly populated horizontal branches, or poorly measured incomplete- ness. Of the 56 clusters analysized, 43 show a statistically significant difference 37 Figure 3.13: Completeness corrected radial distributions for the MS, HB, and TO stars in the core of NGC 5272. Values in the upper right of the plot are the probability of the two respective distributions differing that much by chance alone, as determined by a KS test. (less than 5% probability of the difference arising by chance alone) between the cumulative radial distribution of the turn-off stars, and the distribution of the MS stars selected from the mass distribution of the HB objects. Of these 43 clusters, the TO group is more centrally concentrated than the MS stars in every single case. This is the result expected from a relaxed, mass segregated cluster. The TO stars are more massive and thus sink to slower, smaller orbits through the exchange of kinetic energy from repeated interactions with less massive cluster members. The clear evidence of a kick here would be HB distributions that are signifi- cantly more extended than their MS counterparts. However, there are no clusters that exhibit such a result. Of the 56 clusters studied, 28 show a significant differ- ence between the radial distributions of the HB and MS stars. In all 28 of these clusters, the HB stars have a more centrally concentrated distribution. In fact, there 38 are no clusters that even show the HB stars to be significantly more extended than the TO stars, indicating that, in all cases, the HB has not even fully relaxed since undergoing mass-loss along the red giant branch. An example of these distributions is shown for NGC 5272 in Figure 3.13. The relevant results from this analysis of the distributions are shown in Table 3.1. The core velocity dispersions are taken from Harris [14]. The mean relaxation times are calculated by finding the relax- ation time from the radial distance of each object in our HB group. These are calculated based on the half-mass radius and half-mass relaxation time from Har- ris. It is important to note that this mean relaxation time is not a meaningful value except in reference to our specific group of HB objects in this field. 3.3.3 Distributions and Relaxation Time Using models from Dotter et al. [8], we can find that, assuming a time of 1 Gyr on the RGB, the current HB stars are, at most, a few hundredths of a solar mass heavier than the current TO stars. Although the difference is small, perhaps this could explain the fact that many of the HB distributions seem to be significantly more concentrated than the TO. If this were true, one would expect that this group would relax from the more concentrated distribution on a timescale proportional to the relaxation time in our field. Therefore, there should be a correlation between the mean relaxation time of the HB objects, and the significance of the difference between the HB and TO groups. A plot of these two parameters for our 56 objects is shown in Figure 3.14. The significance of the correlation between these was determined by calculat- ing the Spearman rank correlation coefficient from 100,000 bootstrapped iterations. This results in a mean correlation coefficient of -0.39. The bootstrapped distribu- tion indicates that this coefficient is consistent with zero at less than 0.1%, indi- cating that the correlation is highly significant. However, this correlation clearly goes the opposite way from what one would expect, given the previous explana- tion. As the mean relaxation time of the field decreases, the difference between the distributions becomes more significant. The explanation for this is quite trivial. The mean relaxation time in our field is a strong function of the physical stellar density, which is strongly correlated with 39 Figure 3.14: Mean HB relaxation time vs. HB concentration significance. the number of stars in our field (the hidden parameter being the distance to the cluster). So, the mean relaxation time correlates strongly with the number of stars in our field, shown in Figure 3.15. The rank-sum statistic, used to determine the significance of the HB concentration, is also strongly dependent on the number of objects in the sample Figure 3.16. So, because HB concentration significance correlates with number of objects, and mean relaxation time correlates with number of objects, it is also the case that HB concentration significance correlates with mean relaxation time to a lesser degree. 3.4 Conclusions Given the results presented here and the corrections made to the original result from Davis et al. [6], there is no indication that a kick occurs sometime along the giant or horizontal branch, or that white dwarfs receive birth kicks at all. Addition- ally, trying to correlate the concentration of the horizontal branch objects with the relaxation time in the field is complicated by the fact that the significance of the 40 Figure 3.15: Mean HB relaxation time vs. Number of stars in the field. Boot- strapping in the same way as before, the mean Spearman rank coeffi- cient is 0.68, and it is consistent with zero at less than 0.001%. difference between two radial distributions is highly dependent on the number of stars in the field which also correlates with the relaxation time. A more thorough statistical analysis would be necessary to further investigate this. 41 Figure 3.16: HB concentration significance vs. Number of stars in the field. Spearman rank coefficient of -0.58. Consistent with zero at less than 0.001%. 42 Table 3.1: Results from the analysis of HB distributions. The first column is the cluster NGC name. The second column is the number of stars in the entire ACS field. The third column is the core velocity disperion in km/s. The fourth column contains the mean relaxation time of the HB population in log10(years). The last two columns contain the results of the comparison between the HB, MS, and TO stars. CON means that the HB population was more concentrated. N.S. means that the difference between the distributions is not significant. Cluster ID Num. Stars V. Disp. (km/s) Mean RT (log yrs) HB vs. MS HB vs. TO NGC0104 158154 11.0 8.49 CON CON NGC0288 30170 2.9 8.57 N.S. N.S. NGC1261 108577 n/a 9.26 N.S. N.S. NGC1851 144355 10.4 9.27 CON CON NGC2298 22372 n/a 8.89 N.S. N.S. NGC2808 308733 13.4 9.29 CON CON NGC3201 35578 5.0 8.36 N.S. N.S. NGC4147 21879 2.6 9.13 CON CON NGC4590 66950 2.5 8.79 N.S. N.S. NGC4833 67219 n/a 8.60 N.S. N.S. NGC5024 249264 4.4 9.56 N.S. N.S. NGC5053 26878 1.4 8.80 N.S. N.S. NGC5139 341424 16.8 8.80 N.S. N.S. NGC5272 179330 5.5 9.52 CON CON NGC5286 210729 8.1 9.29 CON CON NGC5466 33245 1.7 8.90 CON CON NGC5904 118902 5.5 8.58 CON CON NGC5927 106704 n/a 8.73 CON N.S. NGC5986 165883 n/a 9.05 CON N.S. NGC6093 136854 12.4 8.98 CON N.S. NGC6101 74586 n/a 8.74 N.S. N.S. NGC6121 13714 4.0 7.61 CON N.S. NGC6144 24768 n/a 8.66 N.S. N.S. NGC6171 21003 4.1 8.03 N.S. N.S. NGC6205 152392 7.1 8.94 CON N.S. NGC6218 32589 4.5 8.16 N.S. N.S. NGC6254 60069 6.6 8.31 CON N.S. NGC6304 112828 n/a 8.30 CON CON NGC6341 142970 6.0 8.81 CON N.S. NGC6352 29025 n/a 8.24 N.S. N.S. NGC6362 34791 2.8 8.53 N.S. N.S. NGC6366 12200 1.3 7.87 N.S. N.S. NGC6388 351435 18.9 9.10 CON CON NGC6397 16100 4.5 7.71 N.S. N.S. NGC6441 384591 18.0 9.34 CON CON NGC6496 25716 n/a 8.36 N.S. N.S. NGC6535 11103 2.4 8.17 N.S. N.S. NGC6541 121587 8.2 8.75 CON N.S. NGC6584 70632 n/a 9.24 N.S. N.S. NGC6624 70851 5.4 8.62 CON N.S. NGC6637 68437 n/a 8.89 CON N.S. NGC6652 34114 n/a 8.56 N.S. N.S. NGC6656 100958 7.8 8.12 N.S. N.S. NGC6681 53856 5.2 8.56 CON N.S. NGC6715 389646 10.5 10.43 CON CON NGC6717 17341 n/a 8.61 N.S. N.S. NGC6723 66386 n/a 8.75 CON N.S. NGC6752 52817 4.9 8.04 N.S. N.S. NGC6779 87488 4.0 8.80 N.S. N.S. NGC6809 47161 4.0 8.41 N.S. N.S. NGC6838 16464 2.3 7.82 CON N.S. NGC6934 90542 5.1 9.29 CON N.S. NGC6981 49589 n/a 9.23 CON N.S. NGC7078 270678 13.5 9.14 CON N.S. NGC7089 250705 8.2 9.32 CON CON NGC7099 73710 5.5 8.66 N.S. N.S. 43 Chapter 4 Measuring the Distance to 47 Tucanae (NGC 104) 4.1 The Data The globular cluster 47 Tuc was imaged for 121 orbits in Hubble Space Telescope (HST) cycle 17 using Wide Field Camera 3 (WFC3) and The Advanced Camera for Surveys (ACS) ( GO-11677 , PI - Harvey Richer). The cluster was imaged in Ultraviolet-Visible (UVIS) and IR channels, and the program obtained a total of 7.5×105 seconds of exposure. Two regions of the cluster were observed. A map of these regions can be seen in Figure 4.1. The work below deals with data only from the “swath field” of this program. Over this region, the cluster was observed in F390W , F606W , F110W , and F160W , giving good coverage of visible and near IR wavelengths. The observation program and data reduction is discussed in great detail in Kalirai et al. 2011 (in preparation). 4.2 Using Models to Determine a Distance This novel approach to measuring the distance to a globular cluster requires well calibrated photometry of a number of white dwarfs in a cluster, in multiple fil- ters. Using white dwarf spectral models, and making some assumptions about the cluster white dwarfs allows us to predict an expected absolute magnitude for 44 !" #" $%"&'()*+" Figure 4.1: An observation map of the program in 47 Tuc. Two distinct re- gions are apparent. Directly to the West of the core is the primary ACS field. The large horseshoe shaped region surrounding this is called the “swath field”, which was used in this analysis. these objects, and then fit the distance modulus by comparing the expected abso- lute magnitude to the measured apparent magnitude. Statistical uncertainties can be determined by analysing the distribution of distance measurements among all of the white dwarfs in the cluster. Properly considering the uncertainties in our parameter assumptions and propagating this through to the final distance estimate gives a reliable indication of the range of distances that are consistent with these intial assumptions. Combining these errors gives us an overall uncertainty for our measurement. 45 4.3 Converting Spectral Models to Magnitudes in Hubble WFC3 Filters We began with a grid of pure Hydrogen atmosphere white dwarf spectral models from Tremblay et al. [26]. Spectra were defined for objects with surface gravities (log(g)) ranging from 6.0 to 10.0 in steps of 0.5, and for temperatures ranging from 6,000 K to 120,000 K. Spectra on the grid were defined in steps of 500 K between 6,000 K and 17,000 K, steps of 5,000 K between 20,000 K and 90,000 K, and steps of 10,000 K between 90,000 K and 120,000 K. Each spectrum was in units of Fλ (ergs cm−2 s−1 Å−1) as a function of λ (Å). Each individual spectrum in the grid was then converted to four Space Tele- scope MAGnitude (STMAG) through our filters in order to fit to the photometry of our white dwarfs. To derive an expected magnitude through each filter for a given spectrum, the total system throughput curves and STMAG zeropoints from the Space Telescope Science Institute (STScI) website were used. The STMAG ze- ropoints are conveniently defined so that a flat spectrum of 3.63×10−9 ergs cm−2 s−1 Å−1 (F0) through any filter will give a magnitude of 0. The magnitude in some filter (Mmod) for any given model spectrum can then be calculated as shown below. Mmod =−2.5log10 ( r2 d2 ∫ ∞ 0 λFmodEλSλ dλ∫ ∞ 0 λF0Sλ dλ ) (4.1) Here Fmod is the energy flux per unit wavelength for the spectral model, Sλ is the total system throughput as a function of wavelength for the filter, and Eλ is the standard interstellar extinction curve for fraction of light scattered/absorbed as a function of wavelength, given by Fitzpatrick [10] with some input E(B−V ) and Rv = 3.1. The inclusion of λ here is necessary due to the way that CCDs measure fluxes. The CCD does not know the individual energy of each photon that it receives, only whether it has received a photon or not. Because photon energy is inversely proportional to wavelength, for any given energy density, the number of photons received will increase linearly with wavelength. Ephot = ch λ (4.2) The result of this is that, for a relatively flat spectrum, fluxes measured on a 46 CCD through some filter will be more affected by the energy flux at longer wave- lengths, as this region of the spectrum will necessarily have a higher number of photons per unit energy. Because the models give the energy flux per unit wave- length at the surface of the object, the factor of r 2 d2 is necessary to scale the model flux to the flux observed for an object of radius r at a distance d. This is left as a free parameter, and the equation can be rewritten as shown below. Mmod =−2.5log10 (∫ ∞ 0 λFmodEλSλ dλ∫ ∞ 0 λF0Sλ dλ ) +5log10 ( d r ) (4.3) The model magnitude in any given filter is a function of only log(g), tempera- ture, reddening (E(B−V )), d, and r. Fmod is dependant on log(g) and temperature. Eλ depends only on E(B−V ) (with Rv held as 3.1), and the dependance on d and r is explicit in the equation. The terms from the integrand in Equation 4.3 are shown visually in Figure 4.2. 4.4 A Summary of the Model-Fitting Procedure Before fitting our real photometry, we needed to select a high quality sample from all of the objects that fall along our WD sequence. The goal of this selection was to have a very clean sample of WDs that are free of photometric contamination. The first test was that candidates needed to have a FWHM that differed by less than 1 standard deviation from the mean FWHM of the image in F160W. All of the images were also visually inspected for obvious contamination of possible ellipticity in the objects. Those candidates were thrown out as well. These two criteria chopped our original group of 171 WDs down to a high quality sample of 59 objects. The selection is described in more detail in Woodley et al. 2011 (in preparation). With the entire model grid converted to magnitudes for our four filters (F390W , F606W , F110W , and F160W ), we can now fit to our photometry while varying the various parameters that go into the model. From the models of Tremblay et al. [26], we can derive a relationship between log(g), temperature, and mass. White dwarfs are only stable on some surface in this 3-dimensional parameter space. This means that we only need to fit two of these parameters, while the third can then be derived from the other two. We have chosen to parametrize this by treating mass and temperature, as our independent variables, and determining the log(g) from 47 Figure 4.2: The figures show various components of Equation 4.3. The top figure shows the unreddened model spectrum over the range of all of our filters, as well as the filter throughputs. The bottom left figure shows the interstellar extinction curve from Fitzpatrick [10] for E(B−V ) = 0.04 and R = 3.1. The bottom right figure shows the combined model spectrum (Fmod), extinction curve (Eλ ), and filter throughput (Sλ ) for F606W . these. Additionally, because we fit a mass and temperature, and can subsequently determing log(g), we also know the radius from: r = √ GM g (4.4) Lines of constant log(g) and radius in Teff, mass parameter space are shown in Figure 4.3. This means that although there are six parameters that really control the spec- trum (temperature, mass, object distance, surface gravity, reddening, and object radius), due to the physical relations between these parameters, we only need to fit 48 Figure 4.3: These two figures show appropriately labeled lines of constant log(g) and radius in the parameter space of Teff and mass. four of them. We can now generate a model Spectral Energy Distribution (SED) in these four filters given a mass (M), temperature (Teff), reddening (E(B−V )), and a distance (d). The first three parameters control the shape of the SED, while the distance controls the overall scaling. The likelihood of the data D, for a single ob- ject j out of our 59 objects, given any temperature (Teff), Distance Modulus (DM), reddening (E), and mass (M) is given by: L j(D | (Teff,DM,M,E)) = 4 ∏ i=1 exp [ −(Di−mi) 2 2σ2i ] (4.5) Here Di are the measured magnitudes in our 4 filters (i running from 1 to 4) and mi are the the magnitudes from the model calculated as described in the previ- ous section. σi is the photometric uncertainty for each data point. The likelihood function for each object was then calculated over a grid in this four dimensional parameter space. Although our method relies on making some assumptions about the mass and reddening of the cluster white dwarfs, we wished to take proper un- certainties in these assumptions into account. For this reason, the likelihood was calculated while varying these parameters as well. An example of the likelihood distribution for a single WD with mass and reddening constrained is shown in Fig- ure 4.4. Next, the likelihood grids for each individual object were marginalised over 49 mass and effective temperature, while including a Gaussian prior on mass with a mean of 0.53 M and a standard deviation of 0.01 M, and a uniform prior on Te f f . This follows from the recent direct spectroscopic measurements from [17] of 0.53±0.01 M for WDs in M4. Further support for our mass selection comes from predictions by Renzini and Fusi Pecci [20] that the WDs forming today in globular clusters should have a mass of 0.53±0.02 M. This step is described by Equation 4.6. L j((DM,E) | D) = ∫ ∞ 0 ∫ ∞ 0 L j(D | (Teff,DM,M,E)) exp [ − (0.53−M) 2 2(0.01)2 ] dM dTeff (4.6) L((DM,E) | D) = 59 ∏ j=1 L j((DM,E) | D) (4.7) After combining the distance modulus-reddening likelihood of all 59 objects (described in Equation 4.7), we marginalise over the reddening parameter with a Gaussian prior with a mean of 0.04 and a σ = 0.02 (from Harris [14]) to obtain the distance modulus likelihood for the cluster as a whole (Equation 4.8). L((DM) | D) = ∫ ∞ 0 L((DM,E) | D) exp [ −(0.04−E) 2 2(0.02)2 ] dE (4.8) 4.4.1 Calculating Statistical Errors Statistical errors for our measurement of distance modulus were calculated di- rectly from the resulting distribution. After marginalizing out the other parameters, the value which maximized the likelihood was found. The normalized likelihood distribution was then integrated from this maximum value (Xmax) to some value (Xmax+∆X), which gave a total area of 0.3413 (for a standard 1σ error bar). 0.3413 = ∫ Xmax+∆X Xmax L(X)dX (4.9) Here L is the likelihood distribution, marginalized over all parameters except for one, and Xmax is the value of the parameter that maximizes that distribution. Equation 4.9 can be solved for ∆X to give the value for an upper 1σ error bar. 50 Figure 4.4: An example object fit for temperature and distance modulus. Here mass is constrained to 0.53 M, and reddening is held as 0.04. There is an obvious degeneracy between temperature and distance. As the temperature of the model object increases, it also becomes brighter. This requires the object to be farther away in order to have the apparent brightness of our real photometry. Switching the integral to go from Xmax−∆X to Xmax gives the lower error bar. If the distribution is symmetric, these will obviously be identical. 4.4.2 Calculating Systematic Errors In addition to the statistical errors of this measurement, we also need to consider the effects of systematic errors caused by the overall calibration of our photometry. We began with the values for the uncertainties in the absolute photometric cali- bration in F390W , F606W , F110W , and F160W (errors of 0.023, 0.023, 0.013, and 0.011 magnitudes respectively) from Kalirai et al. 2011 (in preparation). We then generated 10 synthetic objects from the model grid. Next, we calculated the likelihood over our parameter space and marginalized down to just distance mod- ulus for each object, using the systematic errors as our error when calculating the likelihood distribution. From this, we could calculate the error in distance mod- ulus as a function of temperature. The width of this distribution comes entirely from the calibration uncertainties, as the magnitudes themselves are drawn straight from the model grid that is being fit. This dependence on temperature is due to 51 Figure 4.5: Triangles show the final distance modulus uncertainty for each of the 10 synthetic objects generated from our model grid. A cubic spline was then interpolated between these points to find the systematic error values corresponding to the temperatures fit for our 59 objects. The histogram of these 59 values can be seen in the inset. the shifting of the SED peak with temperature. As the temperature gets higher, the largest amount of energy from the object is given off outside the range of our filters. Changing the temperature further only changes the slope of the line going through our filters, rather than drastically altering the shape by moving the peak of the spectrum around within the range of our filters. This results in a wider range of temperatures fitting reasonably well at higher temperatures. The mean of the 59 values in the histogram from Figure 4.5 is 0.055. As all 59 objects contribute equally to the likelihood, this mean is a reasonable way to characterize the overall systematic errors in our distance modulus estimation. 4.5 Conclusion Our likelihood analysis results in best fit distance modulus of (m−M)0 = 13.35± 0.02± 0.06, which corresponds to a physical distance of 4.67± 0.04± 0.13 kpc. 52 The first error given is random, and the second is systematic. This measure- ment is entirely consistent with previous distance measurements of 47 Tucanae, (m−M)0 = 13.30±0.15 given by Woodley et al. 2011 (in preparation). Our final likelihood distribution after marginalizing over all other parameters is shown in Figure 4.6. Additionally, we may wish to consider how our distance estimate de- pends on our assumptions about mass and reddening. To do this, we can consider our original 5 parameter likelihood grid. After marginalizing out the temperature for each object individually, and combining the likelihood of all objects as in Equa- tion 4.7, we are left with likelihood values on a grid over reddening, mass, and DM. For each combination of reddening and mass on the grid, there is a corresponding 1-dimensional DM likelihood distribution. The most likely DM can then be cal- culated for each mass and reddening pair. The result is a surface of most likely DM values that is well approximated by a plane. This plane is described by Equa- tion 4.10. DM = 14.634−2.573M+2.552E (4.10) Here, M must be in units of solar masses, and E is magnitudes of E(B−V ). This approximation to our likelihood grid is good to within 0.003 magnitudes and is valid within the ranges of M = 0.5−0.56 M and E = 0.0−0.08. Our analysis of both random and systematic errors suggest that this is the most precise distance measurement to 47 Tucanae ever presented. As the distance modu- lus uncertainty is one of the largest contributing factors to cluster age uncertainty as determined by fitting isochrones, a measurement this precise will allow the turn-off age to be better constrained than was previously possible. To determine a simple relation between turn-off magnitude and age, isochrones from Dotter et al. [8] were used. The turn-off magnitude was calculated from each isochrone, defined as mag- nitude at which the most negative value of the slope ∆colour∆mag occurs. If we look at the value of the turn-off magnitude as a function of metalicity and age, we get the plane-like surface shown in Figure 4.7. A slice of this surface at [Fe/H]=-0.74, appropriate for 47 Tucanae (Carreta and Gratton [5]), is shown in Figure 4.8. To first order, the uncertainty in the age of the cluster is linearly proportional 53 Figure 4.6: The final likelihood distribution for the unreddened distance modulus of 47 Tucanae. The values shown are the mean and 1σ er- ror of this distribution to the uncertainty in the turn-off magnitude. This means that a factor of two im- provement in the DM uncertainty, as we have presented here, leads to a factor of two improvement in age measurements using the turn-off fitting technique. More specifically, an overall uncertainty of 0.07 in the DM leads to an uncertainty of 0.7 Gyr in the age. 54 Figure 4.7: CMD turn-off magnitude as a function of [Fe/H] and cluster age. Figure 4.8: Cluster age vs. CMD turn-off magnitude for [Fe/H]=-0.74. 55 Chapter 5 Conclusions The work in this thesis has covered a number of varied topics that all deal with globular clusters and, more specifically, stellar populations within globular clusters. Chapter 2 describes a robust method for finding the centre of the projected stellar distribution of a globular cluster. This method was then applied to the 65 clusters from the ACS legacy survey to determine the most precise centres for these clusters ever presented. These centre coordinates are essential for any radial distribution work, and for probing the existence of IMBHs in globular clusters. In fact, the work in Chapter 3 would not have been possible without these precise centres. This work is also presented in Goldsbury et al. [12]. Chapter 3 summarizes the analysis of the radial distributions of horizontal branch, turn-off, and main sequence stars in 56 globular clusters from the ACS legacy survey. The first part of this chapter shows a re-analysis of the work from Davis et al. [6]. This re-analysis concludes that, due to the mistakes made in the original work, the result itself is not valid. Following up on this with the horizontal branch analysis, we find no evidence of extended radial distributions that would indicate the existence of an asymmetric velocity kick. Chapter 4 details my contributions to Woodley et al. 2011 (in preparation). The work is an application of a novel method for measuring the distance to globu- lar clusters by fitting real WD SEDs with WD atmosphere models to determine a distance. This method is applied to 47 Tucanae to derive the most precise distance to that cluster ever presented. 56 Bibliography [1] J. Anderson and R. P. van der Marel. New limits on an intermediate-mass black hole in omega centauri. i. hubble space telescope photometry and proper motions. The Astrophysical Journal, 710(2):1032–1062, 2010. → pages 5, 16 [2] J. Anderson, A. Sarajedini, L. R. Bedin, I. R. King, G. Piotto, I. N. Reid, M. Siegel, S. R. Majewski, N. E. Q. Paust, A. Aparicio, A. P. Milone, B. Chaboyer, and A. Rosenberg. 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An analysis of stellar populations in globular clusters Goldsbury, Ryan 2011
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Title | An analysis of stellar populations in globular clusters |
Creator |
Goldsbury, Ryan |
Publisher | University of British Columbia |
Date Issued | 2011 |
Description | This thesis is composed of three chapters, as well as an introduction, which describe three distinct projects. In Chapter 2 we present new measurements of the centers for 65 Milky Way globular clusters. Centers were determined by fitting ellipses to the density distribution as well as the symmetry of the clusters. All of the determinations were done with stellar positions derived from a combination of two single-orbit Advanced Camera for Surveys images of the core of the cluster. We find that the ellipse-fitting method provides remarkable accuracy over a wide range of core sizes and density distributions, while the symmetry method is difficult to use on clusters with very large cores, or low density, requiring a larger field, or a very sharply peaked density distribution. Chapter 3 deals with a re-analysis of previous work on white dwarf natal kicks, and expands on this to analyze the radial distributions of stellar populations in globular clusters at earlier stages of stellar evolution (earlier referring to pre-white dwarf). The effects of stellar incompleteness, and a method to account for this are discussed. Finally, the results of a statistical analysis of completeness corrected radial distributions in 56 globular clusters are presented. No significant evidence of kicks is found, however multiple clusters show evidence that stars along the horizontal branch have not relaxed since undergoing mass loss after leaving the main sequence. In Chapter 4, we present a novel method for determining the distance to a star cluster by fitting spectral energy models to the spectral energy distributions of cluster white dwarfs in multiple filters. The statistics of our fitting method are discussed in detail. This approach results in a true distance modulus of (m-M)₀ = 13.35 ± 0.02 ± 0.06, which corresponds to a physical distance of 4.67 ± 0.04 ± 0.13 kpc. The first error given is random, and the second is systematic. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-10-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivs 3.0 Unported |
DOI | 10.14288/1.0072295 |
URI | http://hdl.handle.net/2429/37991 |
Degree |
Master of Science - MSc |
Program |
Astronomy |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2011-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/3.0/ |
AggregatedSourceRepository | DSpace |
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