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Constraining the form of the galactic halo with deep star counts Davis, David Sau 2004

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Constraining the Form of the Galactic Halo With Deep Star Counts by David Saul Davis B.Sc.(Physics), McGil l University, 2002 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E OF M A S T E R OF SCIENCE in The Faculty of Graduate Studies (Department of Physics and Astronomy) We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y OF BRITISH C O L U M B I A -October 2004 ' © David Saul Davis, 2004 THE UNIVERSITY OF BRITISH COLUMBIA FACULTY OF G R A D U A T E STUDIES Library Authorization 9 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Name of Author (please print) Date (dd/mm/yyyy) Title of Thesis: Co^^tMtKjifO 6f Trf£T FofcM OF Tt/tf HtCtcY I^^Y Ufru^> Degree Year: Department of "PHY^CS £ h&r£c4*&!*y The University of British Columbia Vancouver, BC Canada grad.ubc.ca/forms/?formlD=THS page 1 of 1 last updated: 20-Jul-04 11 Abstract The Canada-France-Hawaii Telescope Legacy Survey - Deep ( C F H T L S - D e e p ) represents a truly unique data set in terms of angular size and depth. A l though the survey is only ~ 40% complete, it is already complete to a magnitude of 25 in u*,g',r ' , i ' , and z' . We use this data for an ambitious star-count project. B y comparing the observed colour-magnitude diagrams (CMDs) wi th simulated C M D s , one can place many constraints on the form of the Galaxy. Th is thesis represent the first stage of this project - the study of the stellar halo of the Galaxy. We find the half-light radius of the de Vaucouleurs profile to be slightly higher than the literature value. We find the slope of the ini t ial-mass function of stars wi th masses between 0 . 4 M Q < M < 0.8M© is slightly lower than Salpeter. We find that the halo is oblate wi th a ratio of minor axis to major axis of ~ 0.9. Final ly, we find that the best value of the stellar binary fraction is 0.35. The other elements of the C F H T L S ( C F H T L S - W i d e and C F H T L S - V e r y Wide) provide an immense number of observations that can be compared wi th theoretical models. There is plenty of data wi th which to constrain the form of the other components of the Ga laxy - the thick and th in disks. Th is study, along wi th the subsequent studies, wi l l give us the most detailed picture of the stars in our home Galaxy ever. i i i Contents A b s t r a c t i i C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i 1 I n t r o d u c t i o n 1 1.1 The History of Star Count ing 1 1.2 Thesis Goals 4 1.3 Present Status 5 1.4 Layout 7 2 D a t a . . 9 2.1 C F H T Open Star Cluster Survey 9 2.1.1 D a t a Reduct ion , 10 2.1.2 Cal ibrat ion H 2.1.3 Combinat ion of deep and shallow exposures 13 2.2 C F H T Legacy Survey 14 2.2.1 Reduct ion 15 2.2.2 Star-galaxy separation 18 3 T h e m o d e l s 33 3.1 Monte Car lo Simulations 34 Contents iv 3.2 Probabi l i ty M a p 37 4 Data - Model comparison 43 4.1 Statistics 43 4.1.1 Gaussian Likel ihood Rat io 43 4.1.2 Poisson Likel ihood Rat io 44 4.1.3 Bayesian Inference 45 4.2 Parameter Search and Error Est imat ion 46 4.2.1 Searching Parameter Space 46 4.2.2 Error Est imat ion 50 5 Constraints on Parameters 51 5.1 Init ial Mass Funct ion 51 5.2 Physical size 59 5.3 A x i a l ratio 59 5.4 B inary fraction 62 6 Conclusions and Discussion 65 6.1 Conclusions 65 6.2 Discussion 66 6.3 Future Work , • • • • 66 Bibliography 68 List of Tables 2.1 Coordinates of the centers of the CFHTLS-Deep fields v i List of Figures 2.1 The locations of the C F H T L S - D e e p fields 16 2.2 C M D of D4 20 2.3 Fan's colour-magnitude space selection 21 2.4 Half-l ight radius vs. magnitude 23 2.5 C M D wi th H L R cut 24 2.6 Fan colour-colour space selection 25 2.7 Colour-Colour diagram 26 2.8 C M D with colour-colour selection 27 2.9 Colour-colour selection 28 2.10 C M D wi th colour-colour selection 29 2.11 Stellarity vs. magnitude 30 2.12 C M D of D l , D2, D3 & D4 32 3.1 Schematic of simulations 35 3.2 Monte Car lo C M D 38 3.3 Probabi l i ty maps 41 3.4 Number densities vs. distance modulus 42 5.1 Contour plot - D l 52 5.2 Contour plot - D2 53 5.3 Contour plot - D3 54 5.4 Contour plot - D4.. -. •. > . 55 5.5 I M F vs. B F 57 List of Figures vii 5.6 IMF vs. axial ratio 58 5.7 H L R vs. binary fraction 60 5.8 H L R vs. A R 61 5.9 Binary fraction vs. A R 63 List of Figures A c k n o w l e d g m e n t s vi i i I would like to thank my supervisor, Dr . Harvey Richer. Th is thesis would not have been possible without his ideas, encouragement and support. I am indebted to h im for giving me the opportunity to conduct my research in many places around the world, and to collaborate wi th experts in the field. I would like to thank Jasper Wa l l , who led me down the correct path in terms of statistical analysis. I would like to thank Michele Cignoni . His three years of hard work gave us a framework wi th in which to conduct the project. Furthermore, his stellar evolutionary tracks were indispensable. I would like to thank Giuseppe Bono for being such an gracious and attentive host for the durat ion of my stay in Monte Porzio. I would like to thank Jason Ka l i ra i for being so patient while explaining the basic concepts of stellar photometry. I would like to thank A n n a Saj ina for helping me conceptualize Markov Cha in Monte Carlos. I would like to thank Robert Ferdman, Jason Rowe, and Mark Huber for dealing wi th my endless barrage of Fortran, L inux, and awk questions. Final ly, I 'd like to give a shout out to Celine for making sure that I ate and slept everyday for the duration of the project. 1 C h a p t e r 1 I n t r o d u c t i o n 1.1 The History of Star Counting Galaxy formation is one of the most actively pursued fields in modern astrophysics. The most direct approach it to detect high redshift galaxies as they are forming. Th is ap-proach has proved to be successful at answering a large range of questions. However, there is an alternate approach. The galaxies in the Local Group can be studied in de-tai l unimaginable for high redshift galaxies. The absolute magnitudes, metallicities and velocities of individual stars may be measured. If the record of galaxy formation is imprinted in some observable of these stars, we have an alternate and independent ap-proach wi th which to answer fundamental questions regarding galaxy formation. Indeed the manner in which a particular galaxy formed is imprinted in its stars. The stellar age, metallicity, and distr ibution in phase space al l give clues to the way in which a Ga laxy formed. Because of our proximity to stars in the M i l ky Way, the Ga laxy is one of the best places to conduct a study of this nature. On ly in the M i l k y Way are large numbers of low mass stars easily resolvable. If one can reconstruct a detailed model of the Galaxy, then it becomes possible to perform 'stellar archeology', and gain insight into processes that occured when the Universe was a fraction of its current age. In this thesis I conduct a study of the M i l ky Way halo v ia comparison of observations wi th predictive star-count models. In the early 1600s Gali leo resolved the band of light known as the M i l k y Way into separate stars. Th is fundamental result paved the way for following generations of sci-entists to understand the nature of our home Galaxy. In the mid-1700s, Emanuel Kant and Thomas Wright proposed that the circular nature of the M i l k y Way could be ex-Chapter 1. Introduction 2 plained if the Galaxy was a vast stellar disk and our solar system was merely a small component [5]. Th is postulate represented the first recognition that we live in a galaxy that was not spherical. Count ing stars was one of the first quantitative studies under-taken by astronomers, yet it remains a fruitful endeavor to this day. In the 1780s W i l l i am Herschel produced the first map of the M i l k y Way. His map was based on counting the number of stars in 683 separate regions of the sky. B y today's standard, his understanding of stars was extremely crude. In order to interpret his results he simplif ied his analysis by assuming that the intrinsic luminosity of al l stars was equal, the stellar density throughout the Galaxy was constant, there was no obscuring medium between the stars, and the end of the stellar distr ibution was visible. From his data, Herschel concluded that the sun was at the centre of a distr ibution wi th a vertical extent one fifth that of its horizontal extent [5]. Star-count models took a leap forward wi th the work of Jacobus Kapteyn. His work, during the early 20th Century, confirmed Herschel's conclusions, albeit in a much more quantitative manner. Kapteyn was able to associate a physical distance and scale lengths wi th his model. The Kapteyn "Universe" consisted of a flattened spheroidal system of stars. The stellar density dropped steadily, and decreased to 50% of its peak value 800 pc and 150 pc from the center along the major axis and minor axis respectively, and to 1% by 8500 pc and 1700 pc along the same axes. He concluded that the Sun lay at a distance of 650 parsecs from the centre, 38 pc above the Galact ic mid-plane[5]. Star-count models became very popular after Kapteyn. In the 1920's and 30's these models were used by the likes of Bok[9], Basinski[2], Seares[32], V a n Rh i j n [33], and Oort[23] in a effort to determine the geometrical form of the Ga laxy in terms of its stars and dust. These efforts concentrated on directly inverting the equations of stellar statistics, and met w i th l imited success for several reasons. F i rs t , dust is distr ibuted in a stochastic manner throughout the disk of the Galaxy. A l though there are general trends to the distr ibution, such as decreasing levels of dust at higher Galact ic latitudes, there exist patches of dust that buck the general trend. Secondly, the relatively shallow l imi t ing magnitude of the data then available led to a relatively smal l number of stars Chapter 1. Introduction 3 by today's standards. The inversion of the integral equation for the projected number of stars on the sky is an unstable mathematical function, and produces unreliable results for small numbers of stars. For these reasons, interest in star-count models of the Ga laxy waned during the middle of the 20th century. Star-count models went through a renaissance in the 1980's wi th the publ icat ion of the Bachal l & Soneira's modelf l ] . Bahcal l credited the renewed interest in star counts to three factors: 'the use of automatic measuring devices, a change of theoretical tactics, and these developments, of a detailed.model that can be used wi th modern computers.'[1] A l l three of the developments occurred, more or less concurrently in the early 1980's, and each is essential to the modern field of star counting. -The use of automatic measuring devices for photographic plates, and the subsequent development of C C D s , led to a revolution in terms of the number, and accuracy, of stellar photometric measurements available. The change in theoretical tactics was to abandon the attempt to map the dust distr ibut ion in the Galaxy. Rather, the focus was placed on patches of the sky that had very low extinction. Final ly, the geometrical form of the Galaxy was assumed to be that of other galaxies of similar Hubble type. These so called 'copy cat' models do not attempt to yield a unique solution to the form of the Galaxy, but rather attempt to restrict the possible parameter space that accurately describes the Galact ic form by iteratively comparing models wi th observations. W i t h the advent of large mosaiced charge coupled devices ( C C D s ) , and powerful computers, using the technique of star-counts to place meaningful constraints on the form on the M i l k y Way has become feasible. These developments open a door to exploring the M i l k y Way in a manner that was never before possible. In 1996 Ried et a l . conducted a major study of the M i l ky Way halo attempting to determine the luminosity function of the halo at very small masses (O .5M 0 ) . Taking advantage of the extremely good seeing (0.5") during his observing run, he was able to distinguish stars from galaxies to a magnitude of R — 24.5. However, he was l imited in colour to (R-I)< 1 by confusion w i th th in disk M-dwarfs. Unfortunately, the lowest mass main-sequence stars have redder (V-I) colours than (V- I )= l . He was unable to put constraints on the luminosity function Chapter 1. Introduction 4 at very low masses. [26] The advent of the Sloan Dig i ta l Sky Survey (SDSS) led to an increased interest in star-count models. In 1999 Fan conducted an important theoretical study predicting where different contaminating populations (eg. QSOs) would lie in colour-colour space[8]. A l though the SDSS had extremely large areal coverage (~ 1000 square degrees), the l imit ing magnitude of the study was relatively bright (g' ~ 22). A study by Chen et al.[6] attempted to determine the degree of flattening of the halo, and to fit the density of the halo to a power law. A l though the combination of the two were highly degenerate the best f i tt ing combination was determined to be a very flattened halo (c/a ~ 0.55) and a relatively flat power law index (2.5 ± 0.3) [6]. There have also been several attempts to constrain the star formation history of both the disk of the Galaxy, and local dwarf galaxies. Some examples of these include papers by Val lenari et al.[37], Rob in et al.[28], and Dolphin[7]. A l though the paper by Dolph in was concerning dwarf spheroidal galaxies, it was part icularly useful for this project because of the rigorous statistical analysis it contained. Recently there have been several more studies of the Galact ic Halo. The most recent of these is a study by Lemon et al . [21]. This study was based on SDSS and the Mi l lenn ium Galaxy catalogue, and again found a flattened halo (c /a = 0.56). These projects are always plagued by similar problems. First, there is only a small colour-magnitude space that is dominated by the Galact ic halo. Second, the derived parameters are often quite co-variant wi th one another. There has been no study that attempts to constrain more that two parameters simultaneously. This thesis wi l l focus on the author's recent efforts toward this end. 1.2 Thesis Goals The goal of this thesis is to be able to put tight constraints on al l parameters necessary to describe the form of the Galact ic Halo. It is important to realize that the method of predictive star-count models can never yield a unique solution due to the fact that a Chapter 1. Introduction 5 small perturbation may be added to any well f i tt ing model without significantly changing the value of the fit. Recognizing this, in accordance wi th Occam's razor, one attempts to construct a model w i th the fewest number of parameters that can model a l l the essential observational features wi th an acceptable accuracy. Furthermore, one must ensure that none of the assumptions that are used in the model (i.e. geometrical form of the population) is not overly restrictive. As the data improve, the number of observable stars, and the l imit ing magnitude of those stars increase. W i t h an increased number of stars, signal to noise is increased, and one may add more elements to the model. The correct number of elements to in include in a model can be determined by experimenting wi th statistics such as the run test. For the purposes of this thesis we restrict ourselves to the study of a Galact ic Halo hypothesized to be well described by five parameters. These are a normalizing number density of halo stars, slope of the initial-mass function for stars wi th masses between O . 4 M 0 and 0 . 8 M Q , half-light radius of the halo, binary fraction, and ratio between the major and minor axes of the halo. The potential of this study extends far beyond the scope of this thesis. The high density of stars in the disk allows for a more in depth study. The more complicated form of the disk requires a larger number of free parameters than a model of the halo. W h e n attempting to model the Galact ic disk a typical Megacam field wi l l contain tens of thousands of stars, compared w i th several hundred halo stars. W i t h numbers like these, one can imagine fitting a model that includes: scale length, scale height, star-formation history, metall icity gradients, binary fraction, the slope of the init ial-mass function over several mass ranges, and the local stellar density of disk stars. 1.3 Present Status A large port ion of this project was made up of developing an approach wi th which to attack the problem. This approach had several components. The first of these was to discover the correct question to ask. The beginning of this project focused on the spiral Chapter 1. Introduction 6 structure of the Galaxy. Whi le this is a very interesting question, it is an extremely chal-lenging one. The clumpy distr ibution of dust throughout the Galaxy makes it impossible to suppose an analyt ical model that can predict the true dust distr ibution. Wi thout an independent measure of reddening as a function of distance from the sun, predictive star count models have a very small chance at success. It was realized that it would be far more fruitful to concentrate on fields wi th high Galact ic latitude. Th is changed the focus of the study from attempting to constrain the spiral structure of M i l k y Way to a focus on the form of the M i l ky Way halo. The second challenge was the choice of data. Dur ing the earlier phases of the project we attempted to use data from the Canada-France-Hawaii Telescope Open Star Cluster Survey ( C F H T - O S C S ) . Whi le there is certainly a large amount of information contained in these rfields, they are not appropriate for a study of this scope. It was only realized relatively recently that the C F H T Legacy Survey ( C F H T L S ) is an ideal source of data for this project. Th is data has now been reduced. Furthermore, scripts have been developed which means that any further reductions can be performed quickly and easily. When attempting to use predictive star-count models to constrain Galact ic structure, it goes without saying that one must make predictions. These predictions typical ly take the form of 'art if icial data' . The third challenge was to determine the method to produce the artif icial data, and indeed in what form it was most useful. It was first assumed that the most useful form of the simulated data would be the same for that the observed data was in - a photometry list. To create data in this form, the best technique is a Monte Car lo simulation. A Monte Car lo simulation code, M I L K Y W A Y M C , was wri t ten by Michele Cignoni . Later, it was realized that it was more efficient to compare the observed photometry wi th probabilit ies directly. Th is necessitated the development of a code that was able to produce a map of probabilit ies in colour-magnitude space, M I L K Y W A Y P M . Both these codes are now functioning, and promise to be very useful in future studies. The fourth challenge was the determination of the technique used to compare the sim-ulated and observed data. Several different methods were used. A t first, Hess diagrams were employed to compare the quality of fits of different models. For this technique to Chapter 1. Introduction 7 be useful, it must be used in a regime where points are distributed wi th Gaussian errors. Th is project is not in this regime. Compar ing luminosity functions, and colour functions were also experimented with, but found to be difficult to interpret. Eventually, we came upon a useful statistic wi th which to determine maximum likelihoods in the Poisson regime, namely the Poisson l ikelihood ratio. The fifth challenge was determining a manner in which to search a high-dimensional parameter space. We experimented wi th several different approaches from the simple grid search to more complicated Markov Cha in Monte Carlos, gradient calculations, and simplex methods. A l l the aforementioned challenges have been met, and we now posses a unique data set and a powerful algorithm wi th which to analyse it. Fields that are dominated by the halo of the Galaxy, taken from the C F H T L S - D e e p , have been analysed. However, as the Legacy Survey wi l l continue to. accumulate images of the aforementioned fields, the photometry wi l l continue to improve. As the analysis of the fields has been automated, it takes litt le effort to re-analyse the fields as more data become available. A t the moment we are l imited not by the depth- of the photometry, but by bur inabi l i ty to distinguish stars from galaxies at faint magnitudes. If this problem can be overcome we wi l l be able to take full advantage of the data we have in hand. If indeed we can access the data fainter than g' = 25 we have a marvelous opportunity to study the lowest mass stars in the halo - stars that are generally inaccessible to other star count studies. 1.4 Layout This thesis wi l l follow the following form: Chapter 2 wi l l be devoted to discussing the particular data that has been, and wi l l be used for this project. There are two distinct sources of data used in this project, and both wi l l be discussed in turn. The first source of data were f rom the C F H T L S -Deep. These data were the pr imary source of information in the analysis of the Chapter 1. Introduction 8 Galact ic Halo. The second source of data is the C F H T Open Star-Cluster Survey ( C F H T - O S C S ) . These data wi l l be used in further studies of the Galact ic disk. Chapter 3 wi l l discuss the details of the model used to simulate the Galaxy. The model, coauthored by myself and Michele Cignoni from the University of P isa , was developed in the spirit of Bachal l & Soneira. The Galaxy is modeled as consisting of three independent components: the halo, the thick disk, and the th in disk. Whereas most preceding models used an empirically determined luminosity function to predict stellar counts, these models use a theoretical init ial-mass funct ion ( IMF) and stellar evolutionary tracks to predict stellar counts. Th is yields the enviable abil i ty to place observational constraints on the stellar I M F , and star-formation history of the Galaxy. Chapter 4 wi l l discuss the various statistics used to compare the observed C M D s wi th the simulated one. Th is was a crit ical aspect of the project. Furthermore, it wi l l discuss the methods used to determine the best f i tt ing parameters. The methods discussed wi l l highlight the advantages and disadvantages of various parameter search techniques in multi-dimensional parameter spaces - specifically the downhil l simplex method, the Markov Cha in Monte Car lo, and the grid search. Final ly, we wi l l discuss the method for determining the error on the fitted parameters. Chapter 5 wi l l discuss the results we obtained from our analysis. We present results for the halo; these results demonstrate the power of the technique. Chapter 6 wi l l conclude and discuss the direction of future work on this project. Chapter 2 9 Data The data used for this study came primari ly from two sources: The Canada-France-Hawaii Telescope Open Star Cluster Survey ( C F H T - O S C S ) and the Canada-France-Hawai i Telescope Legacy Survey ( C F H T - L S ) . The two data sets required different levels of processing and therefore wi l l be discussed in the separate sections. 2.1 C F H T Open Star Cluster Survey The Canada-France-Hawai i Telescope Open Star Cluster Survey was an ambitious study originally intended to obtain deep (V ~ 24.5) multi-colour photometry of 19 open star clusters in the disk of the M i l ky Way. The C F H T - O S C S is a data set of unprecedented depth and area compared wi th previous open star cluster studies. The study had the ambitious goals of the comparison of observational results and theoretical models, the study of star-cluster dynamics, and the determination of the distance to and age of the clusters. To date five papers have been published directly on the study, and many collaborations have used the data in some other way. The data for the 19 open clusters was taken during a three-night observing run wi th the Canada-France-Hawai i Telescope ( C F H T ) from October 15 to October 18, 1999. The images were taken wi th C F H - 1 2 K , a high-resolution, wide-field mosaic camera, mounted at the f /4 prime focus of C F H T . The camera is made up of twelve 2048 x 4096 pixel C C D s wi th an angular size of 0.206 arcseconds/pixel. The wide field of the camera was necessary for imaging open clusters wi th a large angular extent. Fortuitously, the clusters often only filled the inner four chips of the camera, leaving 2/3 of the camera available to obtain uncontaminated images of the background populations. Chapter 2. Data 10 Images of the clusters were taken in three filters: Johnson B , V , and R. The R images were shallow, and were used only to get an estimate of the reddening to the cluster. The choice of B and V filters rendered the survey rather sensitive to dust extinction - not an ideal choice for this study. However, the primary science goals of the project was the study of white dwarfs, and it was this consideration that drove the filter selection. Furthermore, it was this goal that determined the limiting magnitudes of the various fields. In most cases the observations had a limiting magnitude of V ~ 24.5 at a colour of (B — V) = 0, but unfortunately significantly brighter at redder colours. For the majority of the clusters, it was determined that a single 300s second exposure was sufficient to meet the goals. For several of the clusters, deeper images were obtained by stacking several 300s exposures. It is these fields that are of the most use for this spin off study. Because of the very low Galactic latitude of the majority of these fields, the star counts are dominated by..the Galactic thin disk. This thesis will be.concentrating on the Galactic halo, and therefore this data will not be used directly in this thesis. However, as this project continues, and the analysis expands to include the Galactic disk, this data will become extremely useful. Furthermore, many of the principles involved in this reduction were also used to reduce the C F H T L S data, and hence were necessary for the ultimate completion of the project. 2.1.1 Data Reduction The first stage of the reduction involves the preprocessing. The preprocessing of this data were performed by Jason Kalirai as part of his M.Sc. thesis. The preprocessing can be divided into several steps: 1. Bias frames (exposure time = 0 s) and dark current frames are taken, and subtracted from the science frame. This accounts for the non-zero readout from a C C D even if the C C D is not exposed to any light. 2. After the images are de-biased, the data is flat-fielded. This involves taking images of an even background such as the twilight sky, or a screen inside the dome. The Chapter 2. Data 11 differing responses of the individual pixels is calibrated in this way. 3. These corrections are applied using the FITS Large Images Processing Software (FLIPS). Furthermore, one may normalize the sky background to the chip with the highest sky value (lowest gain), CCD4. This step provides a scaled data set with a smooth background on all chips, and consequently the zero points for all the chips will be nearly equal. For the CFHT-OSCS the flat field was good to better than 1% in all filters. 4. In order to reach the fainest magnitudes, multiple deep exposures were taken of several of the clusters. These multiple deep exposures were stacked and averaged using FLIPS. 5. The photometry was performed using Emanuel Bertin's SExtractor and PSFex software packages. These are highly automated, very fast programs that quickly produce accurate photometry for moderately crowded fields. These programs are ideal for extracting photometry for the huge numbers objects present in the modern astronomical surveys. 2.1.2 Calibration Calibration is a difficult but critical step in the creation of a useful photometric catalog. In order to convert the instrumental magnitudes to real magnitudes we use the following equations: vinstr = V + avX + pv(B-V) + Zv • (2.1) bin8tr = V + abX + p\{B-V) + Zb (2.2) In the above equations vinstr and binstr are instrumental magnitudes given by PSFex, a is the coefficient of the air-mass term X, (3 is the coefficient of the colour correction term (B — V), and Z is the zero-point shift. A total of 23 calibration frames of different exposure times, air-masses, and filters were used to calculate the various coefficients, and solve for the calibrated apparent magnitudes. There are subtle systematic differences Chapter 2. Data 12 between each chip, so individual calibrations are required for every chip, and further for every individual cluster. The value of /3 remains constant for al l nights of observation. The vast major i ty of the difference between the different calibrations is accounted for by the zero-point term, Z. When one desires to compare their photometric measurements wi th another study, that may have used a slightly different set of broadband filters, it is of utmost impor-tance that the magnitude systems used are consistent wi th one another. In 1983, Landolt published a catalogue of bright stars (11.5 < V < 16.0) around the celestial equator [15]. Th is allowed for people using slightly different filter sets to compare their photometric measurements directly. The magnitudes of the standard stars used by Landolt are based on non-linear transformations of the magnitudes and colours of the stars in several dif-ferent filters. The final magnitudes derived for the stars are based on the total amount of flux emitted in a filter wi th in a large aperture designed to ensure min imal light is lost. In contrast, PSFex fits the point spread functions of the stars such that signal-to-noise ratio is maximized. Th is means that some of the flux from the star is sacrificed in order to eliminate flux from the sky. In order to calibrate the data, total flux must be compared wi th total flux. Therefore, it is required that aperture photometry is used in conjunction wi th P S F photometry during the calibration phase. To solve for the calibration coefficients one must find many unsaturated stars, wi th high signal to noise, that appear both in Landolt 's catalogue[16][17] and the cal ibrat ion images taken of the Landolt standard fields: SA-92 and SA-95. D A O P H O T was used to calculate the magnitudes of stars wi th various apertures, ranging from 8 - 2 0 pixels. From this data, a curve of growth was constructed. It was found that an aperture of 16 pixels, the measured magnitude was within 0.01 M a g (99% of the total flux) of its final value, and the flux from the sky was minimized. It was these magnitudes that were compared wi th the standard magnitudes taken from Landolt . The cal ibration coefficients were calculated by using a least squares method, in which a large number of stars that satisfy the above criteria (i.e. high signal to noise, unsaturated, and appearing on both Landolt 's catalogue and calibration images) are fit simultaneously Chapter 2. Data 13 for al l three coefficients in both the B and V filters. The specific algorithm was based on a method designed by Harris, Fi tzgerald & Reed [11]. Unfortunately, not a l l chips contain enough stars that fit these criteria to obtain a low-sigma fit. Including fits that do not have an acceptable sigma do not aid wi th the analysis, and are therefore disregarded. In this case, only chips 02, 04, 07, and 08 gave results that had an acceptable sigma. However, these four independent calibrations were all very close to each other, making it reasonable to extend these results to the other chips. From this stage we take away the following coefficients: Zv, Zb,av,ab, fiv, and /3b. We then create on-frame science standards. The stars selected to be on-frame stan-dards are bright, yet unsaturated. Using the following equations: we determine Z'v and Z'b where Z'v and Z'b represent the final shifts from the instru-mental magnitudes to the apparent magnitudes. We then solve equations 2.1 and 2.2 for the terms V + f3v(B — V) and B + (3b{B — V) which are then substituted back in equations 2.3 and 2.4 along wi th the P S F magnitude values to solve Z'v and Z'b for each chip. 2.1.3 Combination of deep and shallow exposures After the final zero-points are calculated and applied to the photometry lists, the final stage of the data reduction is to merge photometry lists of different exposure time. Th is task was easily accomplished using a program writ ten by Pat Durel l . Th is program reads in the photometry list to be merged, and for al l stars that appear in both lists selects only one to appear in the final list. The decision on which star to put in the final list is based on several criteria. The default choice is to take the star from the longer exposure, as it wi l l typical ly have higher signal to noise. However, near the saturation l imit of the deeper exposure, it is often unclear which star wi l l have been more accurately measured. In this case the program examines the star on the deeper image for indications that it V = vPSF -0v(B-V)-Z'v- 2.5log(t) (2.3) B = bPSF - pb(B - V ) - Z ' b - 2.5 log(t) (2.4) Chapter 2. Data 14 was improperly measured, such as a unusually high error or bad x 2-value. If the star appears to be well measured in the deeper exposure, the values from the deeper exposure wi l l be chosen. Conversely, if there appears to be a problem wi th the measurement on the deeper frame, the measurement from the shallower frame wi l l be selected. Final ly, if the star on the deeper exposure is clearly saturated, the program chooses the star from the shallower exposure. 2.2 C F H T Legacy Survey The C F H T L S is an extremely ambitious project that wi l l take over 500 nights of dark and grey time at the C F H T over the next five years (~ 50% of the total dark and grey time available). The survey is divided into three subsurveys: the deep survey, the wide survey, and the very-wide survey. Whi le, both; the wide and very wide surveys are fascinating projects, wi th many diverse science goals, the images obtained as part of these surveys wi l l not be used in the thesis. Therefore, al l remaining references to the C F H T L S wi l l be specifically referring to the deep survey. The C F H T L S - D e e p survey has been allotted 202 nights over the durat ion of the survey. The pr imary science goal of C F H T L S - D e e p is to map the distr ibut ion of galaxies out to high redshift and to detect thousands of type 1 supernovae. The survey consists of four single independent megacam fields of 1 square degree. Each field is imaged in the full set of C F H T ' s filters: u*,g',r',i' and z'. In order of obtain the same approximate l imi t ing magnitude in each band, the exposure time must vary from filter to filter. Th is is to compensate for the varying sensitivity of the camera, Strehl ratio, and sky brightness in the various bands. The final exposure times are intended to be: u* - 33 hours, g' - 33 hours, r' - 66 hours, i' - 132 hours, and z' - 66 hours. The survey wi l l reach a projected l imit ing magnitude of ~ 28 in al l five bands. The fields were selected to be at high Galact ic latitude to avoid dust and disk stars, making them perfectly suited to the needs of this project. The coordinates and reddening values of the four fields, named D1,D2,D3 and D4, are listed in the table 2.1. The positions of the fields are shown schematically Chapter 2. Data 15 Name R A (hr:min:sec) Dec (deg:min:sec) 1 b E ( B - V ) D l 02:26:00 -04:30:00 172° - 5 8 ° 0.02 D2 10:00:29 +02:12:21 237° +42° 0.01 D3 14:19:28 +52:40:41 96° +60° 0.01 D4 22:15:31 -17:44:06 39° - 5 2 ° 0.03 Table 2.1: Coordinates of the center of the C F H T L S - D e e p fields. The excess colour is not constant over the field. The value quoted is a typical value for each field, but there are variations of ~ 0.01 mags. These value are derived from Schlegel's dust map [35]. in figure 2.1. 2.2.1 Reduction A n integral part of the C F H T L S is the pipeline to reduce the data, making it easily used by a wide cross section of the astronomical community, from astronomer that work pr imari ly at other wavelength to theorists. Every level of the data is (or wi l l be) available to the community, from raw images, through preprocessed images, to stacked images and 'data products' such as photometry lists. Th is is made possible through several facilities, preprocessing is performed at C F H T with the El ix i r pipeline. The images are stacked, and reduced at T E R A P I X (Traitement Elementaire, Reduct ion et Analyse des PIXe ls de megacam). Final ly, images and data products are distributed through the Canadian Astronomical Da ta center ( C A D C ) . Elixir W i t h i n several weeks of the end of an observing run raw images and preprocessed images are available to the astronomical community on-line through the C A D C . The prepro-cessed images are pipeline processed wi th El ix i r . E l ix i r applies dark images, flat fields, and bias images to every image. As more and more data accumulates, the flat field Chapter 2. Data 16 A) D l : 1-172, b=-58 D2:1=237, b= 42 D3:1= 96, b= 60 Figure 2.1: The locations of the fields D l , D2, D3, and D4. In frame A) the view is from the Galactic pole, with 1=0 to the right. In frame B) the view is from the Galactic plane again with 1=0 to the right Chapter 2. Data 17 images may be averaged to construct a 'superflat'. Th is allows the zero-points for each image to be determined very accurately for every chip in Megacam. Zero-point are now known to < 0.01 magnitudes. However, E l ix i r does not release any combined images, or data products. Terapix T E R A P I X is an astronomical data reduction center dedicated to the processing of ex-tremely large data flows from digital sky surveys. T E R A P I X is located at I A P (Institut d'Astrophysique de Paris). The Terapix organization is responsible for making the data available and useful for people without the knowledge, time, or computing power neces-sary to perform their own reductions. Terapix wi l l stack al l images using the Bert in 's SWarp program. This wi l l give the images their advertised depth. Furthermore, it wi l l reduce the stacked images, creating photometry lists for each of the images. The im-age reduction wi l l be performed by a pipeline based on Bert in 's SExtractor and PSFex . Terapix has yet to release data, but is due for its first data release very soon. University of Victoria Megacam images are very large, and the process of stacking images is very time con-suming. Terapix has been plagued by delays, and has not yet released stacked images. Fortunately, the Department of Physics and Astronomy at the University of V ic to r ia has a group working on galaxies in the C F H T L S - D e e p fields. The data reduction at this group is led by Stephen G w y n . Stephen G w y n generously provided me a wi th copy of his stacked, E l ix i r processed, images. High photometric precision is not required for this project. As long as the photometry is accurate to wi th in the size of a typical b in we are unaffected by photometric errors. Because of this, aperture photometry was sufficient, and point-spread photometry was not necessary. The images were reduced using the fast and highly automatic software package SExtractor. The images were obtained as multi-extension F I T S ( M E F ) files. Th is means that one file contained the images for a l l 36 chips. W i t h most software packages, M E F files have to be decompressed, and each Chapter 2. Data 18 extension analysed individually. SExtractor can analyse M E F s directly, without decom-pressing the file. After the process was scripted, the processing of al l 36 chips in a field in al l filters took approximately 90 minutes on a desktop P C . Th is is an impressive ac-complishment considering that that entails approximately 2 x 10 6 accurate photometric measurements. 2.2.2 Star-galaxy separation Al though the main goal of the C F H T L S - d e e p is to detect galaxies, the survey wi l l in-evitably image a fascinating set of stars. The ut i l i ty of this data set in star-count modeling is l imited not only by the l imit ing magnitude of the survey, but also by our abil i ty to distinguish between stars and galaxies. Because of the difficulty of distinguishing be-tween stars and galaxies at faint magnitudes, the data used for this study wi l l be several magnitudes brighter than the detection l imit. Th is means that the completeness wi l l be close to 100%. There are several methods used to distinguish stars from galaxies. Each method has its advantages and disadvantages. The simplest methods use a distinctive feature to distinguish stars from galaxies in some 2-d parameter space. These methods include magnitude-isophotal area[25], magnitude-peak intensity[13], and magnitude-surface brightness[10]. These methods typ-ically work well at bright magnitudes. However, these methods do not use al l the infor-mat ion contained in the brightness profile, and hence are l imited at fainter magnitudes. More sophisticated methods such as the Sebok classifier[34], the r~2 moment [14], the Q classifier[19], and the T/> parameter[20] , use more of the information in the intensity profile, but are not found to be robust when confronted wi th merged objects or crowded fields. B y viewing objects as vectors of parameters, the separation of star and galaxies can be thought of as the definition of a hypersurface that separates the populations effectively in a mult i-dimensional parameter space. This is the philosophy used in Emanuel Bert in 's stellarity index [3], which is incorporated directly into SExtractor. The hypersurface Chapter 2. Data 19 dividing stars from galaxies is determined using a neural network consisting of one input layer, one hidden layer, and one output layer. A neural network is only as good as its training. The neural network used to calculate stellarity in SExtractor was trained by Ber t in in a magnitude regime where star counts are approximately equal to galaxy counts. Th is means that the stellarity index is most accurate from 18 > V > 22 [3]. For objects brighter than the V ~ 18 l imit, it is quite simple to distinguish stars from galaxies. The more challenging regime is for objects fainter than V ~ 22. The next several paragraphs I describe the steps in the separation of stars from galaxies in the C F H T L S - D 4 field (chosen because it has the highest number of stars). Shown in figure 2.2 is a C M D of al l the objects the C F H T L S - D 4 field. There are 2.7 x 10 5 objects in the diagram. For the purposes of this study we were concerned only wi th halo stars. In order to distinguish halo stars from disk stars we take advantage of their different metallicities. Stars wi th a significant metal content, i.e. disk stars, have far more absorption lines in their atmospheres than metal-poor stars, i.e. halo stars. Th is has the impl icat ion that the total f lux emitted at short wavelengths by disk stars is less than that emitted by halo stars. Th is results in redder colours for disk stars. A diagram of the theoretical distr ibut ion of disk stars and halo stars in shown in figure 2.3. In al l colour-colour spaces and the colour-magnitude space denoted in this figure, halo stars and disk stars separate. For our study we chose the boundaries g' < 24, g' > 18, (g' — r') < 0.8 and (g' — r') > 0.1. Th is space was chosen as the largest area dominated by halo stars w i th small contamination by thick disk stars. Stars are point sources, and therefore follow a tight magnitude-half light radius relation. If the stars are faint enough that saturat ion effects, such as potential well overflow and bleeding, are not important, their half-light radi i wi l l be independent of the magnitudes, and should be close the telescope P S F . Th is creates the tight sequence that can be seen for half-light radi i smaller that 0.9". For magnitudes brighter than g ~ 22 we can easily separate stars from galaxies using this sequence. A magnitude - half light radius diagram is shown in figure 2.4. Figure 2.5 shows a smaller area of colour - magnitude space wi th only objects that satisfy the Chapter 2. Data 20 Chapter 2. Data 21 Figure 2.3: A figure taken from Fan 1999. In al l four frames, disk stars are shown as '+ ' while halo stars are shown as 'o'. In frame d, notice that at magnitudes brighter than g ~ 18 the space is dominated by disk stars. A t magnitude g > 18 the disk and halo populations separate in colour. The halo stars are of lower metall icity and are therefore bluer.. Chapter 2. Data 22 half-light radius < 0.9" cut. Many quasars and compact galaxies satisfy the colour-magnitude space selection and have small half-light radii. Fortunately, quasars and compact galaxies typically occupy a different region of colour-colour space. A theoretical plot showing the positions of stars, compact emission line galaxies, quasars, and white dwarfs is shown in figure 2.6. A n observed colour-colour plot is shown in figure 2.7. Figure 2.8 shows a colour magnitude diagram after quasars and galaxies have been removed with a colour-colour cut. Note that there are almost no objects bluer than the main sequence turnoff. If analysis is restricted to the magnitude brighter than g' = 24 then these cuts are sufficient. We can do slightly better with aggressive pruning in colour-colour space. Halo stars occupy a strict region of parameter space. We can safely reject any object that does not lie close to the sequence. Figure 2.9 shows the selections in colour-colour space. Figure 2.10 show the C M D of the resulting selections in colour-colour space. One can-now take advantage of the stellarity index. Up to this point we can be quite sure that we have not cut any stars from out data set. A n examination of where our objects lie on the magnitude-stellarity plane shows that the vast majority of objects brighter than g' = 24 have a stellarity, s, greater than s = 0.97. This is interesting for the following reason: when trying to separate stars from galaxies the stellarity index is often the only figure of merit examined. A much less stringent stellarity cut is typically employed. This diagram shows that in doing this one may be selecting compact "star like" galaxies. We can now choose to set our limiting magnitude at the elbow of the stellar sequence in the stellarity-magnitude diagram. This occurs at approximately g' = 24. This is shown in figure 2.11. A small number of objects have very high stellarity and small half-light radius in one band, but not in others. We take advantage of the multi-colour aspect of the CFHTLS-Deep by requiring that the above criteria are satisfied in every filter. Every magnitude gained deeper than g' = 24 will yield a windfall of stars. Unfortu-nately in this regime galaxies and stars both appear as unresolved point sources in ground based images. The extremely small PSF of HST would allow one to probe deeper, but Chapter 2. Data 23 Figure 2.4: A diagram of half-light radius versus magnitude. There is a clear stellar sequence at half-light radii smaller that 0.9". There are ~ 4 x 103 objects in the stellar sequence brighter than i' = 22 and ~ 8 x 103 objects with half-light radius smaller that 0.9" in total. Chapter 2. Data 24 Figure 2.5: A C M D of al l objects satisfying the half-light radius < 0.9" cut. There are ~ 8 x 10 3 objects in this diagram. Chapter 2. Data 25 u ' - g ' g ' - r * • • • • l i • • • I • i i • i i i i i I r*—r Figure 2.6: Figure taken from Fan 1999. A colour-colour plot showing the theoretical positions of compact emission line galaxies (CELG) , quasars (QSO), stars, blue horizontal branch stars (BHB), and white dwarfs (WD) Note that al-though the quasar region is very large, the number of quasars with blue u* -g' colours is much higher than the number of red u* — g'. Elliptical galaxies and quiescent spiral galaxies are not shown on this plot, but are cleanly separated from the stellar sequence. Chapter 2. Data 26 Figure 2.7: A colour-colour plot of objects wi th a half light radius less than 0.9". The stellar sequence is clearly seen on the right of the diagram. Extragalact ic objects occupy the region blue-ward of u* — g' = 0.6. Chapter 2. Data 27 D4 I i i i I i i i i i i i I i i i I i I 0 0.2 0.4 0.6 0.8 g'-r1 Figure 2.8: A C M D of objects that satisfy the colour cut of u* — g' > 0.6 and the half-light radius cut. There are almost no objects blue-ward of the main sequence turnoff at magnitudes brighter than g' = 24. Th is indicates that we have removed the vast majority of extragalactic objects brighter than g' = 24 Chapter'2. Data 28 Figure 2.9: A colour-colour diagram showing the selection criteria used in this study. Th is is designed to follow the obvious stellar sequence as closely as possible without removing stars that may have been poorly measured in one filter. Chapter 2. Data 29 Figure 2.10: A colour-magnitude diagram like figure 2.8, but showing the set of objects satisfying the more stringent colour-colour selection criteria shown in figure 2.9. Chapter 2. Data 30 D4 0.6 h CO CD M 0.4 0.2 h ~i 1 r T 1 r 0.8 14 _ J i i i i • i i 16 18 20 —r 1 i_ 22 24 26 Figure 2.11: Stellarity as a function of magnitude. A l l objects are shown in black, and points satisfying the aforementioned criteria are shown in blue. Note the elbow in the distr ibution of stars at i' ~ 24 and s ~ 0.95. Th is marks the end of the range where stellarity is a useful measurement. Chapter 2. Data 31 the field of view of H S T is small. Furthermore, the increase in the number of galaxies overwhelms the increase in the number of stars. F ind ing the one star in thousands of galaxies becomes a search for a needle in a haystack, and contamination is inevitable. Unt i l there are space-based telescopes equipped wi th cameras that have large fields of view, magnitudes of g ~ 25 may be the l imit at which star-count studies are pract ical. C M D s of the four fields (with objects in the desired colour-magnitude space subjected to the above selection criteria) are shown in the figure 2.12. Note the differences between the C M D s from the different lines of sight in the parameter space denoted wi th a box. Because of the different Galact ic coordinates of the fields these lines of sight pass through different sections of the various Galact ic components.' Chapter 2. Data 32 Figure 2.12: A C M D of al l the objects present in the field C F H T L S - D 1 , C F H T L S - D 2 , C F H T L S - D 3 , C F H T L S - D 4 . The colour-magnitude space analysed in the project is denoted wi th a box. Th is area was selected to be the region of colour-magnitude space dominated by halo stars. Objects in this area are subjected to. a stellarity cut, half-light radius cut, and a colour-colour cut. There are 2.7 x 10 5 objects in each panel. There are 2.2 x 10 3 , 2.8 x 10 3 , 3.9 x 10 3 , and 1.5 x 10 3 objects contained wi th in the boxes in D l , D2 , D3 , and D4 respectively. C h a p t e r 3 33 T h e m o d e l s The degeneracy between the distance to a star and its intrinsic luminosity is at the heart of the challenge of determining the form of the Galaxy. The"- fact that it is impossible to distinguish a local M-dwarf from a distant red giant from photometry alone creates a degeneracy.between the physical shape of the Galaxy and the types of stars that inhabit the Galaxy. Given a hypothetical form of the Galaxy, it is a relatively straightforward task to simulate an artif icial colour-magnitude diagram. Unfortunately, the reverse is not true. There are many different combinations of parameters that can create vir tual ly identical C M D s , making the direct mapping from C M D to Galact ic form impossible. Th is fact necessitates a more statistical approach to the problem. The most natural approach to simulating C M D s is to use Monte Car lo simulations. A Monte Car lo simulation uses a sequence of random numbers to sample stochastically a given probabil i ty distr ibution. Using a Monte Car lo simulation it is possible to create a photometry list - the same form of data that one obtains from analysing C C D images. Monte Car lo simulations are simple to use, and can incorporate subtle features, such as photometric errors and incompleteness. Monte Car lo simulations have several drawbacks. First , the results they give are stochastic - the output of the model is dependent on the ini t ia l random seed. Th is is the very quality needed for some applications. For others, especially those dealing wi th relatively small numbers of simulated objects, the stochastic nature of the simulations is just a source of noise. For our project, we are counting the number of stars observed in bins. In order to resolve features distributed across colour - magnitude space we must make those bins small. Due to the number of large number of bins, we have a small number of stars in any one bin, and are therefore dominated by Poisson noise. Chapter 3. The models 34 This means that if we expect several stars in a bin, variations on the order of 100% for different random realizations are not uncommon. The most common methods to overcome the inherent noise involved in Monte Car lo simulations is to run the simulations many times. Th is is done by either casting several simulations wi th different random seeds, or by generating many more points than are needed and then statistically scaling the results back. The second drawback is l inked to the first - Monte Car lo simulations are computationally very expensive, and for some applications, prohibit ively slow. The negative features of the Monte Car lo technique are very apparent for our ap-pl icat ion. It takes minutes for the simulation of one C M D using modern desktop P C s . Th is begs the development of an alternate method. The method used in this project was to create a map of probabil i ty contours from the probabilit ies directly, rather than to stochastically sample probabilit ies, as in a Monte Car lo approach. The main drawback of this method is that it is impossible to create a photometry list. One must directly simulate a probabil i ty map. Furthermore, the inclusion of stochastic effects, such as photometric error, were far more difficult to make part of this simulation. The implementation of these two techniques wi l l be discussed in the next sections. A flow diagram for both the aforementioned methods is shown in figure 3.1. 3.1 Monte Carlo Simulations The Monte Car lo code used for this project, MlLKYWAYMC, was inspired by the pioneer-ing work of Bachal l and Soneira in the 1980's. In this model[l], the Ga laxy is composed of three components: a young th in disk that contains most of the stellar mass of the Galaxy, an intermediate thick disk, and an old spherical halo. Whereas the Bachal l & Soneira code made use of an empirical luminosity function, MlLKYWAYMC uses theo-retical stellar evolutionary tracks and star formation histories to generate the luminosity distr ibution. MlLKYWAYMC gives us the abil i ty to constrain the star formation history of the Galaxy, as well as the I M F of the stars wi th in the Galaxy. The following section is a description of the steps the code goes through to create an artif icial photometry list. Chapter 3. The models 35 a) Monte Carlo Simulation input parameters increase distance determine N 0 assign a mass assign an age assign a metallicity determine Mg> determine g' print to list isN<N 0? i are we far enough?; <ye; simulation is done <8> b) Probability M a p input parameters increase distance increase mass -determine M g-determine g' increase probability in proper pixel is mass < mj? i are we far enough?; <^ej simulation is done <8> Figure 3.1: A flow diagram for the Monte Car lo and probabil i ty map methods. The process for dealing wi th binaries is not shown, but occurs concurrently wi th the main loop for both procedures. For the Monte Car lo method, a random number is drawn, and if this number is lower than the binary fraction, a second mass is assigned. The fluxes of the stars are added, and the new magnitude and colour calculated. For the probabil i ty map, a sequence of diminishing probabil i ty is added above the main sequence up to Amag = 0.75 - the maximum deviation from, the main sequence for a binary star (equal mass binary). Chapter 3. The models 36 1. The first step in the Monte Car lo simulation is the input of parameters. Parameters of the observations (field of view and l imit ing magnitude), specifics of the spatial distr ibution (Galactic coordinates, local density and scale length) and specifics for each thick disk, th in disk, and halo (star formation history, metall icity distr ibut ion and binary fraction) must a l l be input to the code. For our project, only the halo was simulated. Th is decreases the number of input parameters substantially. We can reasonably model the halo wi th one metall icity and one burst of star formation. Due to the long main-sequence lifetime of the low mass stars in the halo, the results are relatively insensitive to the age of the halo populat ion. Ages from 10 G y r to 14 G y r produce similar results. We choose the metall icity to match that of metal-poor globular clusters. 2. The main section of the code involves stepping through distance modul i unt i l the edge of the Galaxy has been reached. Given the location of the Sun in the Galaxy, the supposed spatial distr ibution of the halo, the local density of halo stars, and a heliocentric distance, we calculate the corresponding volume and the number of stars, No, of each populat ion brighter than a l imit ing absolute magnitude, M 0 , expected. Th is l imit ing magnitude is chosen to agree wi th the l imi t ing magnitude of the Hipparcos mission to facilitate comparison wi th the Hipparcos data set. A Monte Car lo is then employed to choose the masses of stars stochastically. The mass selection works in the following manner: The value of C is calculated such that C / rrradm = 1. (3.1) 3. A random number generator then picks a number, R, and the mass of the star, m 0 , is calculated such that C / m-adm = R. (3.2) 4. A random number is drawn to decide if the star has a binary companion. Aga in , TUQ is calculated for the companion star. Chapter 3. The models 37 5. The metall icity and age of the star(s) are determined in a similar manner. For this thesis, the metall icity and age were held constant, so this step could be omitted. We adopted an halo age of 12 G y r and a halo metall icity similar to metal poor globular clusters, [Fe/H] = - 2 . 0 . 6. Using a star's mass, age, and metallicity, its magnitude in various filters is calculated from a set of homogeneous evolutionary tracks. These tracks were provided by the astronomy group at the University of P isa. These tracks have evolutionary stages from Zero Age M a i n Sequence ( Z A M S ) , through post main sequence evolution, to supernova and white dwarf stages. 7. Photometr ic errors are added to the star's magnitude as a function of magnitude. 8. Stars are picked as described in steps 3 to 6 unt i l there are N0 stars w i th an absolute magnitude brighter than M 0 . 9. Steps 2 to 8 are then repeated for each distance modulus unt i l the entire Ga laxy is simulated. A sample of a C M D created from a simulated photometry list is shown in figure 3.2. 3.2 Probability Map The code used to make the probabil i ty maps had the same basic form as the Monte Car lo , but w i th some important differences - the most important being that it is unnecessary to calculate integrals to determine specific values of mass thereby dramatical ly reducing the computational cost. The following section is a description of the steps the code goes through to create a colour-magnitude probabil i ty map. 1. A s wi th the Monte Car lo simulation, the first step of this code is to input various parameters. However, in this case, the probabil i ty map remains unnormalized unt i l the last stage of the code. Th is means that the probabil i ty map is independent of Chapter 3. The models 38 18 Monte Carlo Simulation: D1 20 22 24 4 A " * • i * . A A A * * 4 ^ A ^ A A ^ A ^ V A / ' ' \ ^ A A A A ^ SJ'* ^ ^ ^ A V * t ' A " ' ' / * V f A . V A A A A ^ ^ "A ^ A ^ - A " ^ i > ' A A 1 ' » ' A A A A * A . A A A < A * F A A | » * A A A A A A A A A * , A ^ A * A * * ' / A i 4 i * A —C 1 1 1^ A f A , | ^ A * l I t L» I* * I A A A j A + 0.2 0.4 0.6 0.8 Figure 3.2: A C M D created from a Monte Car lo simulation of the Galact ic Halo. Chapter 3. The models 39 local density and field of view of the camera. When comparing a simulated prob-abil i ty map wi th observed data, the map is normalized to the number of observed stars. Th is means that it is impossible to constrain the local density of stars v ia this method. 2. Aga in , the bulk of the code involved stepping through distance modul i unt i l the edge of the Galaxy was reached. Like step 2 in the previous list, we calculate a number, N, based on the Sun's Galactocentric distance, the supposed spatial distr ibution, and the calculated volume element. Because the probabil it ies are unnormalized, this number does not have a physical meaning. It simply serves to scale the probabilit ies wi th distance and hypothesized stellar density. Because we cannot discover the local density of halo stars directly using this code, the best fit for only the four other parameters wi l l be determined wi th this method. The normalizat ion wi l l be found using the Mote Car lo technique. 3. The next step differs the most from the Monte Car lo approach . Whereas in the Monte Car lo approach masses are stochastically assigned, in this approach probabilit ies, P(m), are calculated for every mass according to the equation P(m) = m-a. (3.3) 4. A s wi th Monte Car lo, colours and magnitudes are calculated v ia stellar evolutionary tracks. 5. The product of N x P[m) is stored in the appropriate pixel for each mass. 6. Steps 2 through 5 are repeated over the appropriate range of distance modul i . 7. Final ly, the entire array of pixels is mult ipl ied by a constant to ensure the sum over al l the pixels is equal to the number of stars in the observed field. Probabi l i ty maps of the expected stellar distr ibution created in this manner are shown in figure 3.3. The differences in the shape of the contours is accounted for by the different Chapter 3. The models 40 number densities at various distance modul i . The number density as a function of density for the four lines of sight is shown in figure 3.4. Chapter 3. The models 41 D1 D2 D3 D4 Figure 3.3: A probabil i ty map created for the fields D l ( l=172°,b=-58°), D2 ( l=237°,b=42°),D3 ( l=96°,b=60°), and D4 ( l=39 0 ,b=-52°) . The peaks for D1,D2,D3, and D4 are in different locations due to the different number of stars expected at various distance modul i for the various lines of sight as shown in figure 3.4. Chapter 3. The models 42 F i gure 3.4: Relative number, relative number density, and relative volume as a function of distance modulus. These quantities are scaled so they wi l l a l l fit in the same dynamic range, and hence the numbers do not have a physical meaning. The strong peak in the D4 field shows that the bulk of stars in this field are expected to be close to a distance modulus of 14.5. Th is results in the probabil i ty map for this field looking more like an isochrone that for the other fields. C h a p t e r 4 43 D a t a — M o d e l c o m p a r i s o n 4.1 Statistics After the construction of a simulated data set, be it a Monte Car lo simulation or a probabil i ty map, the next step must be the comparison of the simulated data wi th the observed data. There were three methods that were used for this study, the x 2 test, the Poisson l ikel ihood ratio, and Bayesian inference. For al l these methods it was necessary to bin the observed data into pixels. The pros and cons of each method wi l l be discussed in the following sections which are based on an analysis performed by Dolph in (2002). 4.1.1 Gaussian Likelihood Ratio Generally, when comparing a model wi th data the obvious choice of statistic to use is the X 2 test. A x 2 test measures the difference between a model and the data and, as such, is intuit ive to understand, and comfortable to use. Indeed, this was the first method used for the analysis of our models. Furthermore, if the data has known uncertainties at each point and Gaussian error, minimizing x 2 is a maximum-l ikel ihood calculat ion. Th is is demonstrated in the following manner: Let P denote the probabil i ty that the observation d is drawn form model TO. Pi is the l ikel ihood that a value of di is observed in bin i considering the model predicts a value of rrii in that pixel, and a* is the uncertainty of the bin. We can then write that p. = J—e-oMdi-mtW (4.1) We define a "Gaussian l ikelihood rat io", G L R , as the probabil i ty that the observed data point di was drawn from a model equal to TO* divided by the probabi l i ty that it was Chapter 4- Data - Model comparison 44 drawn from a model equal to di. GLR = Y[ J^f-e-0-5^-™*?^. (4.2) j y °~mi B y taking the logarithm of both sides, we obtain: - 2 \n(GLR) = M^f) + Y, (di~2mi)\ (4.3) or - 2 ln(GLR) = x2 + ]T In (?f) . (4.4) j \adi J If we can assume that the errors of the observations have a smooth Gaussian distr ibu-t ion and the Gi values do not change during the fit then ln(o-^/<T^) = 0 and minimizing X 2 w i l l determine the model most likely to have produced the observations. Unfortu-nately, neither of these conditions are true in the comparison in our situation. For the pixels to be small enough to resolve al l the interesting features they wi l l contain only a smal l number of stars. Th is means that we are dominated by Poisson errors rather than Gaussian errors. Second, h i^cr^/cr^) does not vanish. Mighel l (1999) showed that the inappropriate use of x 2 in a Poisson regime can lead to an incorrect value of the min imum (up to 42% in his example) [22]. 4.1.2 Poisson Likelihood Ratio The Poisson probabil i ty is defined as: Pi = - ^ T T (4-5) with rrii and di defined as in the previous section. When dealing wi th smal l numbers it is natural to define the "Poisson Likel ihood Ra t io " , PLR. Th is is defined in a manner analogous to the definition Gaussian Likel ihood Rat io . We obtain: PLR = TT = TT ( ! ? ) ( V * - m \ (4.6) 11 ddiedi l i v d.J v ; Chapter 4- Data - Model comparison 45 Analogously to the previous section, we can take the logari thm of the above expression and obtain the small number equivalent of x2, namely This statistic shares many features wi th the G L R , it vanishes for a part icular pixel when model and data predict the same number, and has the same value of variance and expectation value for larger values of m^. 4.1.3 Bayesiah Inference Both the above mentioned methods can compare observational data wi th both Monte Car lo simulations and probabil i ty maps. The "bin-free" method of Bayesian inference has the obvious advantage that it does not require the data to be binned, ensuring al l the information in the data set is retained. However, this means that there is no natural way to use this statistic to compare observational data wi th a Monte Car lo simulation. Furthermore, to take full advantage of unbinned data one would have to construct a probabi l i ty map such is a smoothly changing function of colour and magnitude. A method to do this remains unknown to me as of present. Supposing one could construct a smoothly varying probabil i ty map, the probabi l-ity of finding a star, j, at a certain posit ion in colour-magnitude space is wri t ten as Pj(cj,m,j)/ J^jTrij, or just Pj(cj,rrij) assuming the sum of al l probabilit ies over the entire colour-magnitude space is normalized to 1. Assuming that al l the points are inde-pendent, the total probabil i ty is simply: If the binning of colour-magnitude space is sufficiently fine (the b in size is much smaller than C M D features) then the probabil i ty wi l l be close to constant over a b in. In this case, we can write the cumulative probabil i ty of al l the points in a b in as (4.7) P = l[Pr (4.8) j (4.9) Chapter 4- Data - Model comparison 46 Thus, the total probabil i ty is (4.10) A s wi th GLR and PLR we now take the logarithm to obtain (4.11) If this binning is fine enough, equations 4.11 and 4.7 should find identical min ima. P L R , the Bayesian inference method is does not assist further to search for a min imum. It does not find a more secure value for a minimum, and is not normalized. However, it is useful as an indicator of the correct choice of pixel size. Whi le we are not in danger of finding an erroneous min imum if the data is binned too finely, the signal to noise in a given pixel wi l l decrease. Therefore, the error found on our solution wi l l be greater if the binning is too fine. It is important to b in the data as coarsely as possible while st i l l resolving al l the features of the C M D , and obtaining the correct answer. 4.2 Parameter Search and Error Estimation Efficiently searching a high-dimensional parameter space is a very challenging task. Over the course of this study three different methods were used. The first is the downhil l simplex method, the second is the Monte Car lo Markov Cha in and the th i rd is the simple, but computationally expensive grid search. Each of these methods has its pros and cons. Each wi l l be described in the following sections. 4.2.1 Searching Parameter Space Downhill Simplex The downhil l simplex method was invented by Nelder and Mead in 1965. The appealing attr ibute of this method is that it only requires the abil i ty to evaluate a function at a For the purposes of this study, it was practical to.bin.qui te finely. If one can use the Chapter 4- Data - Model comparison 47 part icular location, and not the derivatives of the function. A simplex is a geometrical figure wi th N +1 dimensions in an Af-dimensional space. To be clear, in a 2-dimensional space, a simplex is a triangle. For this project we used an algorithm described in Numerical Recipes called AMOEBA [ 24 ] . A M O E B A works by moving the simplex toward the minimum through a series of four possible types of steps. In the case of N = 2 a simplex is a triangle. Unfortunately, i n mult i-dimensional space it is impossible to bracket the minimum. One must init ial ize the simplex to some posit ion. If the surface of the parameter space is smooth, the algori thm works regardless of the ini t ial posit ion of the simplex. However, if there are local min ima the choice of the ini t ial simplex can influence the minimum that A M O E B A wi l l f ind. The algorithm begins by evaluating the figure of merit of the fit (for this study P L R was used) at each vertex. The most basic move is the reflection of the highest point through the opposite side - in the N = 2 case the triangle flips over its downhil l edge. The second type of move is related to the first. If A M O E B A can find a lower posit ion by stretching in the direction of the vertex being moved it wi l l do so - flip over the downhil l edge and stretch. In this manner, A M O E B A can move quickly over flat smooth surfaces, and on steep descents. If everything works as it should, and indeed the function does minimize, eventually A M O E B A wi l l approach a minimum. In this case, f l ipping the highest vertex over the downhil l edge could move the vertex over the minimum, and back uphi l l . When A M O E B A finds itself in this situation there are two types of moves it may perform. Bo th involve shrinking the size of the simplex, allowing it to find a precise value for the minimum. A M O E B A may move the posit ion of the highest vertex toward the downhil l edge wi th no flip. Alternately, A M O E B A may move the highest two verticies toward the downhil l vertex. Th is is a very simple, but powerful algorithm. Unfortunately, our space was ful l of local min ima. I postulate that these minima are not real, but are a product of the discreet nature of our dataset and of our simulated probabil i ty maps. Regardless of the nature of our local minima, they prevent the use of A M O E B A to search the global min imum. Chapter 4- Data - Model comparison 48 Markov Chain Monte Carlo The Markov Cha in Monte Car lo ( M C M C ) is an interesting technique for f inding a min-imum in mult i-dimensional parameter space. Like the downhil l simplex method it does not require the calculation of derivatives of a function. The core of the technique is very simple. Start ing at a certain locat ion x 0 , one calculates the value of the fit (again P L R in our case), denoted f ( x o ) • Then a value for a location, x i , randomly located within a n-dimensional box centered on xo, f is calculated. If the value of A / (x*i) (defined as / (x i) — / (xn)) is negative, the process repeats start ing now from x\. If the value of A / (x*i) is positive then the process repeats from xo, and a new model is calculated from another random point replacing x \ . In this way the function wi l l slowly be minimized. The collection of locations, x*o ,Xi, . . . , x n is in fact the Markov Cha in . The process of getting reasonably close to the min imum is called the "burn- in" period. Whi le each step during the burn- in can have any direction, the average of al l the directions of the steps wi l l be in some direction. Once the general area of the min imum is found, the algorithm continues as before. However, instead of progressing in a relatively straight line toward the minimum (as happens during the burn-in) the average direction of the steps wi l l now be zero. Movement is guaranteed because the value of the "function" wi l l not be the same for two realizations of the simulations wi th different random seeds. The simulation finishes after a certain number of iterations. The chain is then 'pruned' by removing the locations that are part of the burn-in. After the burn in , the number of steps at a given location wi l l be proport ional to the quality of fit at that point. Therefore, to estimate the value of the min imum one examines the distr ibut ion of the points. The beauty of the Markov Cha in Monte Car lo is that the errors are determined, in the course .of finding a minimum. The' distr ibut ion of points around the error gives the error estimate directly. Simulated Anneal ing is a simple modification to the Markov Cha in Monte Car lo designed to ensure that the entire parameter space is well sampled. Th is means that the start ing point has litt le bearing on the subsequent behaviour of the Markov Cha in . Chapter 4- Data - Model comparison 49 To achieve this, the algorithm takes the counter-intuitive step of sometimes accepting a posit ion x n + i even if / (xn+i) is greater than / (xn). Th is means that the chain can cross between widely separated peaks of probability. Th is is achieved by giving the algori thm a "temperature", T. The probabil i ty of the chain accepting a point w i th a poorer fit is proport ional to e'A^Xn^T. The temperature is high during the ini t ia l stages of the simulation. Th is ensures that the the location of xn wi l l move relatively quickly away from the point of init ial ization, and wi l l sample a good fraction of parameter space. A s the simulation continues, T decreases slowly, eventually making it v i r tual ly impossible for the Markov Cha in to accept steps that involve a positive A / (xn). The number of projects that use M C M C s is growing steadily, but the process remains a 'black art ' , i:e. a rigorous prescription of how to apply the M C M C to a particular problem does not exist. The choice of ini t ia l temperature, cooling rate, the size of the box in which to search for a new point, the length of the burn-in, and the number of iterations to include in the simulation are left to the designer of each part icular M C M C . Very vague guidelines, such as the M C M C should accept approximately 50% of steps during the burn-in, can be found scattered throughout the literature, but a definitive guide does not exist. Grid Search The simplest type of search for a minimum in parameter space is just to search the entire parameter space. If the defined boundaries bracket the minimum, then one is guaranteed to find the global min imum. The drawback of the grid search is that it is extremely computat ional expensive. This problem is part icularly pronounced if one is try ing to find a min imum in a high-dimensional space. Efficiency is maximized wi th a judicious choice of search boundaries and spacing. For this thesis, we never searched in a space of dimension higher than four. Further-more, the values of the parameters we were searching were al l confined to a relatively small range. We found a grid search to be acceptable for these purposes. A l l results shown in this thesis were obtained using a grid search. Chapter 4- Data - Model comparison 50 However, when this project is extended to include the thick and th in disks, this search technique wi l l cease to be a viable method for finding the minimum. 4.2.2 Error Estimation One attractive aspect of the Markov Cha in Monte Car lo is the fact that error estimation is an integral part of the parameter estimation. For both the grid search and the downhil l simplex method, uncertainties are not obtained so naturally. For our purposes the Boot-strap method, was the simplest method of estimating our uncertainties. The bootstrap method, invented by Efron in 1979, allows one to obtain many 'sample' datasets from a single observed data set. It seems like one is getting something for nothing. The method is named after the image of achieving goals wi th no external help - pul l ing oneself up by one's bootstraps. Th is method creates a simulated dataset by randomly sampling the observed data wi th replacement. Each star is labeled. iV random numbers are drawn from a range of 1 to N, where TV is the number of stars in the original dataset. For each random number drawn, the corresponding star is copied to a list. In this manner a new dataset is composed of the same length of the original observed dataset. The fit is then calculate w i th the new data set. In this way one can obtain a distr ibution of values of the fit, and hence an estimate of the errors. C h a p t e r 5 51 C o n s t r a i n t s o n P a r a m e t e r s At the time of writ ing, the model definition remains incomplete. The derived error on the best estimates of our parameters are larger than we expect. A l l results stated here are preliminary. The goal of this project was to constrain the value of the four parameters necessary to describe completely the Galact ic halo. The constraints put on the parameters wi l l be discussed in the following section. The dependence of two parameters at a time on contour plots wi l l be shown. Note that in al l the plots the contours are exponentially spaced. The one-sigma errors, determined from bootstrapped datasets, are shown as dashed lines. These errors were determined at the best-fitting model, and were assumed to be somewhat insensitive to small displacements in parameter space. We were most interested in determining the initial-mass function and half-light radius. The plots 5.1, 5.2, 5.3, and 5.4 show how the fit varies wi th initial-mass function and half-light radius for the four fields, D l , D2 , D3 , and D4. Following the aforementioned plots, al l subsequent plots wi l l be for the 'representative' field D l unless otherwise stated. D l was chosen because, like D2 , it has a monotonically decreasing stellar density along the line of sight. B o t h D 3 and D4 peak at some point, making the true determination of the fit, especially half-light radius, problematic. 5.1 Initial Mass Function One of the most important parameters to understand when studying stellar populations is the number of stars of various masses of a freshly formed populat ion. Th is quantity is called the ini t ia l mass function, £ ( M ) . Specifically, the number of stars of a mass Chapter 5. Constraints on Parameters 52 D1 1 1.5 2 2.5 3 Slope of the IMF Figure 5.1: Contours showing the goodness of fit varying wi th I M F and H L R for the field D l . B inary fraction is fixed to 35% and axial ratio is set to 1. Chapter 5. Constraints on Parameters 53 F i gure 5.2: Contours showing the goodness of fit varying with IMF and H L R for the field D2. Binary fraction is fixed to 35% and axial ratio is set to 1. Chapter 5. Constraints on Parameters 54 D3 Slope of the IMF Figure 5.3: Contours showing the goodness of fit varying with IMF and H L R for the field D3. Binary fraction is fixed to 35% and axial ratio is set to 1. Note the unphysically high value of the half-light radius. It is postulated that this is due the the fact that the density profile along this line of sight is not monotonically decreasing, but rather has a peak. Chapter 5. Constraints on Parameters 55 1 1.5 2 2.5 3 Slope of the IMF Figure 5.4: Contours showing the goodness of fit varying wi th I M F and H L R for the field D4. B inary fraction is fixed to 35% and axial ratio is set to 1. Note the unphysically high value of the half-light radius. It is postulated that this is due the the fact that the density profile along this line of sight is not monotonically decreasing, but rather has a peak. Chapter 5. Constraints on Parameters 56 between M and M + dM can be writ ten in the form: dN = NQ£(M)dM (5.1) where N0 is some normalizing constant. If we require that / N0£(M)dM = 1 M 0 (5.2) JM then N0 represents the number of solar mass stars in a freshly formed populat ion of stars. Th is is a difficult quantity to determine directly for a general populat ion of stars. If a populat ion of stars is coeval, as in a cluster, one can measure a related quantity, the luminosity function, allowing the derivation of the I M F . In 1955 Salpeter [30] postulated that the I M F could be well represented by a power law of the form £ oc M~a. (5.3) Salpeter found that £ oc M - 2 ' 3 5 . (5.4) Th is form of the I M F is known as the Salpeter I M F . Whi le equation 5.3 may be val id for a large range of masses, it can not be true for al l masses. Th is is because if a < 2 the total mass involved in a starburst diverges at high masses, while if a > 2 total mass diverges for low masses. Th is implies that a changes as a function of mass. In 1986 Scalo published a study advocating an I M F described by a three part broken power law wi th a < 2 at low mass and a > 2 at high masses [31]. A broken power law is generally accepted now, but the location of the breaks, and the slopes of the indiv idual sections remains somewhat indeterminate. The effects of metall icity on I M F are poorly understood. Theory suggests that without metals to aid in the cooling process, much larger masses of gas wi l l fragment out of giant molecular clouds. Th is would imply a flatter slope at the high end of the I M F for low metallicity. populations. In this thesis we attempt to constrain the slope of the I M F of stars from a mass of ~ 0 . 4 M Q to the halo turn off (~ 0 .8M o ) .We found the init ial-mass function to be slightly flatter than Salpeter, i.e. a < 2.35. The plots 5.5 and 5.6 show the dependence of the fit on the slope of the I M F on binary fraction and axial ratio respectively. Chapter 5. Constraints on Parameters 57 Slope of the IMF Figure 5.5: Contours showing the goodness of fit varying wi th I M F and binary fraction. Half-l ight radius is fixed to 7 kpc and axial ratio is set to 1. Chapter 5. Constraints on Parameters 58 D1 Slope of the IMF Figure 5.6: Contours showing the goodness of fit varying wi th I M F and axial ratio. Half-light radius is fixed to 7 kpc and binary fraction is set to 0.35. Chapter 5. Constraints on Parameters 5.2 Physical size 59 The stellar halo of the M i l k y Way is an important, but poorly studied, component of our Galaxy. Fundamental to our understanding of the halo is knowledge of i ts physical dimension. The form of the density profile of the Galaxy is uncertain. It is typical ly modeled as a de-projected de Vaucouleurs profile or a power law. If the halo density profile is a de Vaucouleurs profile, the physical size of the halo is described by a characteristic radius, RQ. If the density profile is a power law, the physical size is measured by the power-law index, and a characteristic scale. For this study we used a de-projected de Vaucouleurs profile. The density as a function of Galactocentric distance, p(R) is wri t ten as where RQ is the half-light radius of the Galaxy. The best values for half-light radi i were found to be higher than Bahcal l 's estimate [1]. Furthermore, there is a degeneracy between half-light radius and axial ratio. In fact, for the D3 and D4 fields, extremely large values of half-light radius coupled wi th a small value of axial ratio are favoured over half-light values more in accordance wi th the literature value [1]. The dependence of the fit on half-light radius and binary fraction (fraction of stars found in binary systems) and axial ratio are shown in figures 5.7 and 5.8. 5.3 Axial ratio Halo stars formed when the I S M was very metal poor - before star formation in the Galaxy, and subsequent evolution, had time to enrich the I S M . This means that they formed when the Ga laxy was very young, and therefore contain information about the environment in which the M i l ky Way formed. One of the most interesting unknowns is the dynamical environment from which the Galaxy was formed. The axial ratio gives (5.5) Chapter 5. Constraints on Parameters 60 Half-Light Radius (kpc) Figure 5.7: Contours showing the goodness of fit varying with binary fraction and H L R . Slope of the IMF is held at 2, and binary fraction is set to 35%. Chapter 5. Constraints on Parameters 61 D1 5000 1 0 4 1.5xl04 2xl0 4 2.5xl0 4 Half-Light Radius (kpc) Figure 5.8: Contours showing the goodness of fit varying with H L R and axial ratio. Slope of IMF is fixed to 2 and binary fraction is set to 35%. Chapter 5. Constraints on Parameters 62 information on this. The flattening of the halo is described by the ratio of its major to minor axes - c/a. The literature values have quite a range for this parameter, from as high as 1 [28] to as low as 0.5 [6]. Th is parameter has interesting implicat ions for theories seeking to explain the origin and dynamics of the halo[4]. For instance, numerical simulations of A - C D M galaxy formation predict that dark matter halos should have axial ratios between 0.6 — 0.8 [12]. If there is coupling between the dark matter halo and the stellar halo, or if they have a common dynamical origin, one would expect the stellar halo to be flattened in a similar manner [28]. For our best estimates of half-light radius, the axial ratio minimized to ~ 0.9. The number of stars at a particular distance is a product of the half-light radius and the axial ratio. Because of this, for any one direction there is a degeneracy between axial ratio and half-light radius. Th is is shown in figure 5.8. We had hoped that we would be able to break the degeneracy by combining information from fields in different directions. Th is possibil i ty is st i l l being investigated. 5.4 Binary fraction Because the solar neighbourhood is dominated by stars from the th in disk, one can discover the binary fraction for this component relatively easily. The disk binary fraction is approximately 50% [5]. The paucity of halo stars in the solar neighbourhood makes the binary fraction of the halo much harder to determine. Most constraints come from spectroscopic studies of high-velocity stars. The results of these studies are somewhat inconclusive. Some suggest that the binary fraction is independent of metall icity [18], while other studies favour a smaller binary fraction for halo stars (~ 20%) [29]. A recently published study of Messier 4 that I was involved in found that the observed C M D was best fit by a binary fraction of several percent [27]. The significance of the M 4 result is unclear as the binary fraction is subject to dynamical effects. It is important to determine the halo binary fraction for star-count studies because it is degenerate wi th the half-light radius. Many models assume of 50% binary fraction. In 2002, Siegel[36] showed that this Chapter 5. Constraints on Parameters 63 Binary Fraction Figure 5.9: Contours showing the goodness of fit varying with binary fraction and axial ratio. Half-light radius is fixed to 7 kpc and slope of the IMF is set to 2. Chapter 5. Constraints on Parameters 64 assumption can lead to significant (~ 20%) errors on derived scale lengths and heights. The binary fraction was the most tightly constrained parameter in our study. It mini-mized to a physically reasonable value for almost every other combination of parameters. The value was typically ~ 35%, with the one sigma error bars constraining the value between 20% — 50%. The dependence of the fit on binary fraction and axial ratio is shown in figure 5.9. Chapter 6 65 Conclusions and Discussion 6.1 Conclusions This project has developed a framework wi th in which which to conduct star-count studies, and has set the stage for a more iri depth study. Some of the key findings related to the development of the project are listed below. • Th is study suggests, that because of the difficulty in distinguishing stars from galaxies at very faint magnitudes, the practical l imit to depth for a ground based star-counting study may be close to g = 25 no matter how much telescope time one can get. • Monte Car lo simulations are a working method wi th which to conduct star-count studies. However, they are computationally expensive, and are therefore only viable if one has extensive computational resources. • Because of its low computational expense, probabil i ty mapping is the preferred method wi th which to conduct star-count studies. However, it is impossible to determine the local density of halo stars v ia this method. Monte Car lo simulations remain the only way to determine the local density of halo stars. • If colour - magnitude space is correctly binned, minimizing Poisson Likel ihood Rat io ( P L R ) and maximizing probabil i ty v ia Bayesian inference wi l l lead to the same conclusions. Th is means that we have an independent measure wi th which to determine the correct b in size. Chapter 6. Conclusions and Discussion 66 • Markov Cha in Monte Car lo simulations, downhil l simplex methods, and grid searches are al l val id methods wi th which to determine a maximum likel ihood. However, M C M C simulations are computationally expensive, and downhil l simplex methods are sensitive to local maxima. For a low dimensional (N ~ 5) the grid search is not too computationally expensive, and gives a sense of the overall shape of the f i t t ing surface. It was the preferred choice for studying the halo. However, for studies of the disk, this method would be prohibit ively slow. 6.2 Discussion The principle of analyzing only one component of the Galaxy at a time is good in theory, but has some pract ical difficulties. There are two obvious, and related, points. F i rs t , one has to worry about contamination from the components not being modeled. The results obtained were somewhat sensitive to our (arbitrary) choice of where the thick disk contamination was 'low enough'. Because we cannot identify the populat ion that a part icular star is drawn from, a quantitative choice is difficult to make. We simply had to examine colour-magnitude diagrams by eye, and estimate. Second, in order to examine the halo exclusively, severe restrictions in colour-magnitude space must be implemented. Th is in turn restricts the physical range of stars that may be studied. For our choice of colour cut, we were l imited to stars more massive than 0 . 4 M Q . A s the range of masses examined shrinks, the range of the slope of the I M F , and al l its associated degeneracies, that can acceptable reproduce the observations grows. 6.3 Future Work There is a vast amount of fascinating work to be performed in this field. The C F H T L S represents a t ruly unique dataset for this type of project. The penultimate stage of this project wi l l focus on fields that are comprised pr imari ly of thick disk and halo stars in a large section of colour-magnitude space. Fields wi th Chapter 6. Conclusions and Discussion 67 moderate Galact ic latitudes are ideal for this purpose. 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