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Decomposition of the Globular Cluster NGC 6397 Tsui, Hong 2008

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        Decomposition of the Globular Cluster NGC 6397  for Physics or Astronomy 449  by  Hong Tsui     A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Bachelor of Science (Hons)  in  The Faculty of Graduate Studies  (Physics & Astronomy)    The University Of British Columbia  (Vancouver, Canada)  April, 2008   Hong Tsui 2008         ii    Abstract   The kinematics and white dwarf distribution have been studied for the Globular Cluster NGC 6397. The data was obtained from NASAs Hubble Space Telescope in 2005. In particular, we used the images of a field 5 Southeast of the core of NGC 6397 from Advanc ed Camera for Surveys to conduct our analyses. The first part of the study is about the kinematics of the globular cluster. Isotropy of velocity distribution and cluster rotation have been considered. As anticipated, this relaxed cluster exhibited no stron  signs of anisotropy. However, there appears to be some level of rotation. The rotational motion turns out to be ?? ?? cos(?? ) = 3.88  1.41 ??????  ?? ?? ? 1  and  ?? ?? = ? 14.83  0.58 ??????  ?? ?? ? 1.  This  result  is  not  entirely expected and deserves further investigation in future studies.   The second of the thesis is based on white dwarf populations in the globular cluster and the Galactic Bulge.  As  a  first  glance,  there  appears  to  be  a  lacking  of  white  dwarfs  at  the  age  of  approximately 0.6 ?????? . Further investigation reveals this to be statistically insignificant. Through this analysis, another pattern of white dwarf abundance is discovered. There appeared to be much more stars at the age between 0.9 ? 2.0 ?????? . This could be a manifestation of modeling error. As the f inal consideration of this thesis, white dwarf candidates in the Galactic Bulge are illustrated. Approximately 10 candidates are found at the most probable location of stars in the Bulge.    The analyses conducted in this thesis set stage for further development in understanding of globular clusters. In particular, the rotation analysis raises curiosity about the dynamics of NGC 6397 in the plane of the sky. Moreover, the velocity distribution analysis confirms properties and theories pertaining to globular clusters.      iii    Table of Contents  Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v Acknowledgement . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . .  vi 1   Introduction and Background   1 1.1   Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . .   1 1.2   Globular Cluster NGC 6397 . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    1 1.3   Thesis Overview and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . 2 2   Isotropy of NGC 6397  3 2.1   Formation of Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . 3 2.2   Relaxation of Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . 4   2.2.1  Strong Closer Encounters . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . .  4   2.2.2  Weak Distant Encounters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . 5   2.2.3  Relaxation of NGC 6397 . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  6 2.3  Methods of Observing Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . .  7 2.4   Hubble Data . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7 2.5   Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  9 2.6   Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . 12 3   Rotation of NGC 6397   13 3.1   Rotation in Globular Clusters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . 13 3.2   Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . .  13 3.3   Conclusion . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   15 4  Missing White Dwarfs in NGC 6397   16   4.1   White Dwarfs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16     4.1.1  Degenerate Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  17 4.1.2 Chandrasekhar Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17   4.2  Reddening and Extinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   18   4.3   Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  19   4.4  Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 5  White Dwarfs in Galactic Bulge  25   5.1  Distribution of Starlight in Milky Way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  25   5.2  Analysis and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26   5.3   Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6  Conclusion  29   6.1  Summary and Implications of Results . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  29    6.2   Direction for Further Research . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30    iv    List of Tables  3.1   Cluster Centre Proper Motions . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  15  4.1   Coefficients for the Transformation from WFC to BVRI . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . .   21  4.2   Synthetic Zeropoints for WFC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   22  4.3  Extinction Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . 22   v    List of Figures  2.1   Orientation of ACS and WFPC 2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . 8  2.2   CMD of NGC 6397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . .  8  2.3   PM Cleaned CMD of NGC 6397 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   9  2.4   Geometry of Proper Motion Normalization . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . .   10  2.5  Real and Uniform Cumulative Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  10  2.6  Real and Uniform Collapsed Cumulative Distribution . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . .  11  3.1  Proper Motion of ACS Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . .  14  3.2  WFPC 2 Cluster Centre CMD . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . .   14  4.1  Distribution of Star Types in CMD . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . .   18  4.2  Cluster White Dwarf CMD . . . . . . . . . . . . . . .  . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . .  20  4.3  Johnson/Bessell UBVRI Filter Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . .  21  4.4   WFC Filter Characteristics . . . . .  . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . .  21  4.5   Cluster White Dwarf CMD with 0.5 Solar Mass Bergeron Model . . . . . . . . . . . . . . .  . . . . . . . .   22  4.6  Bergeron Model: Age versus Magnitude . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . .  23  4.7   White Dwarf Age Distribution in NGC 6397 . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . .  23  4.8  Real and Uniform Age Cumulative Distribution . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . .  24  5.1  COBE View of the Milky Way . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  26  5.2  Most Probable Location of White Dwarfs in Galactic Bulge . . . . . . . . . . . . . . .  . . . . . . . . . . . . .  27  5.3  CMD of NGC 6397 with White Dwarfs Distinguished . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . .  . .  27  5.4  Possible White Dwarf Candidates in the Galactic Bulge . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . .   28     vi    Acknowledgement  I would like to thank Dr. Harvey Richer and Dr. Jeremy Heyl  for guiding the way through my research. I appreciate the extra time that they have to spend showing me basic analytical tools. I have learned a great deal about astronomical research and methods from them. Moreover, I would like to thank Dr. Heyl for meticulously inserting comments throughout the Monte Carlo simulation program; I regret not being able to make that component a useful part of this thesis.  Chapter 1. Introduction and Background _____________________________________________________________________________  1  Chapter 1  Introduction and Background  1.1 Globular Clusters   There are many interesting astronomical objects that would yield insight into the formation, kinematics, and age of our own Milky Way. Star clusters are of particular interest because they are readily observable and consist of hundreds of thousands of star s. Two general classes of clusters exist, essentially differing in age, metal abundance, size, and location within the galaxy.     For this thesis, we are interested in the class of cluster known as globular cluster. The other type of cluster,  open  cluster,  will  be  briefly  described  for  completeness.  Open  clusters  contain  up  to  several hundred gravitationally bounded stars. The core radius (radius at which the surface brightness is half of its central value) is generally a few parsecs. The stars have small pe uliar velocities and a mass-to-light ratio close  to  unity.  These  clusters  are  typically  located  within  50  pc  of  the  plane  of  the  galaxy,  where extinction is the most severe. The average open cluster has an age less than 300 Myr. Their metallicities are very scattered and do not follow the general trend that younger clusters are more metal abundant [25].    Globular  clusters  (GC)  are  very  different  than  open  clusters  in  terms  of  size,  location,  age  and metallicity. These are spherically distributed clusters with much greater stellar density. Their luminosities range from 104?? ??????  ????  106?? ?????? . The core radius is usually less than 1 pc. GCs are more interesting than open  clusters  because  they  are  amongst  the  oldest  objects  in  our  galaxy.  Their  metalliciies  can  range from 1/3 to 1/300 of the solar abundance of metals, indicating very old stellar populations. In fact, an accurate modeling of their age would provide a lower limit for the age of the Milky Way. A benefit for studying globular clusters over open clusters is provided by their high galactic latitudes. Their features can be thoroughly observed in optical without too much worry about the dust extinction in the galactic plane. They are roughly spherically distributed around the galaxy and we observ e more globular clusters towards the centre of the galaxy than away from it. This means we can get an independent estimate of our present location from the galactic centre just from globular cluster distribution [25].     Trigonometric parallax cannot be used to provide information for the distances to the globular clusters because  they  are  typically  a  few  kilo  parsecs  away.  The  best  standardized  method  is  to  conduct  an isochrone  fitting  on  the  colour-magnitude  diagram  (CMD)  of  the  cluster  while  assuming  cert in parameters like extinction and metallicitiy.   1.2 Globular Cluster NGC 6397   NGC 6397, located in the constellation Ara, was discovered about 250 years ago by Abbe Nicholas Louis de  la  Caille.  Its  right  ascension  and  declination  are 17? 40?? 41??  and  ? 5340? 25"  [22].  In  terms  of galactic coordinates, its longitude and latitude are 338.17 and ? 11.96, respectively [22].   Chapter 1. Introduction and Background _____________________________________________________________________________  2    The effect of reddening caused by interstellar dust in the direction of NGC 6397 is fairly consistent among  various  observations.  The  colour  excess  values  are  typically  ?? (?? ? ?? ) = 0.18.  Through  a comparison between field subdwarfs and cluster stars, an ?? (?? ? ?? ) = 0.183  0.005 was determined in 2003 [10].  Another  source determined the colour excess to be 0.18  [11].  The  most  recent  value  was determined using cluster white dwarfs, and it turned out to be  ?? (?? 606?? ? ?? 814?? ) = 0.20  0.3 [28].   The  distance  of  NGC  6397  has  been  estimated  by  fitting  field  subdwarfs  to  the  clusters  main sequence. A true distance modulus of (?? ? ?? )0 = 12.13  0.15 was determined using subdwarfs from HIPPARCOS in 1998 [21]. A few years later, a similar technique was used to obtain a value of 12.01 0.08 [10]. The most recent estimate was done using cluster white dwarfs and a value 12.03  0.06 was determined [28].    Another important parameter that is used to quantify star clusters is their metal content. NGC 6397 is a metal-poor cluster, but unfortunately there is no clear agreement for its metallicity. A typical value for [Fe/H]  is  in  the vicinity of  -2. Two independent  estimates  suggest  a  value  of ? 2.03  0.05  [10]  and ? 2.02  0.07 [17].    Finally, the latest estimate of age of NGC 6397 has been conducted using the cluster white dwarfs [28]. In that particular study, a Monte-Carlo simulation of the white dwarf population was compared to the observed white dwarf cooling sequence. The result i  an age of 11.47  0.47 ??????  at 95% confidence. If we take NGC 6397 to be amongst the oldest objects in the Milky Way, it would place the epoch of original assembly of the our galaxy at  ?? = 3.1  0.6 [28] .   1.3 Thesis Overview and Objectives   There  are  four  primary  objectives  for  this  thesis.  First  objective  is  to  extract  information  about  the velocity distribution in the cluster from proper motion of the stars (Chapter 2). Second objective is to determine the possible rotation in NGC 6397 (Chapter 3). Third  objective is to determine the significance of  missing  the  white  dwarfs  in  the  cluster.  Moreover,  the  masses  of  the  cluster  white  dwarfs  will  be established through fitting of cooling models (Chapter 4). Fourth objective is to determine the possible white dwarf candidates in the galactic Bulge (Chapter 5).    The kinematics (Chapters 2 and 3) will be determined through a series of proper motion analyses using images from the Hubble Space Telescope (HST). It is interesting to note that rotation of globular cl sters have only been measured for the line of sight from radial velocities up until 2000 [26]. The unique abilities and resolution of the HST allows astronomers to readily measure the motion in the plane of the sky. For our data, the period between observ ations is about 10 years with the first epoch data taken by the Wide Field and Planetary Camera 2 (WFPC 2) in 1995. The second epoch was captured by the Advanced Camera for Surveys (ACS) in 2005.     The masses of the white dwarfs in NGC 6397 and the Bulge o f the galaxy are determined through white dwarf cooling model fitting. The detail will be provided in Chapters 4 and 5, but the outline will be briefly  mentioned.  ACS data  is  used  to generate  a colour -magnitude  diagram  of  the  cluster  stars.  In particular,  we  focused  our  attention  to  the  fainter  stars  in  the  lower  left  region  of  the  CMD,  which represent  the  white  dwarfs.  Bergeron  models,  along  with  correct  distant  moduli  and  reddening  and extinction coefficients, are used to fit the cluster and Bulge white dwarf branches.    Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   3  Chapter 2  Isotropy of NGC 6397  2.1 Formation of Globular Clusters One  of  the  first  dynamical  models  of  clusters  was  introduced  by  King  in  1966.  This  simple  model assumed  an  isotropic distribution  of  velocities  for  all  members.  Needless  to  say,  it  is  a  good  first approximation to understanding stellar kinematics in clusters; however, this oversimplification does not reflect  the  observed  behavior  of  typical  GCs.  To  understand  the  origin of  anisotropy,  it  would  be necessary to consider the formation process and relaxation of GCs.  At present, the formation of galactic globular clusters is still not completely understood. For a number of years, astronomers believed that GCs are formed durin one massive collapse event. However, recent observations from HST on the cluster NGC 2808 provide evidence for multiple collapses, as there are three distinct generations of stars [27]. The current explanation for this observation is that NGC 2808 is not a typical cluster because of its enormous mass. It has more than one million stars in the most recent estimates,  whereas  typical  clusters,  like  NGC  6397,  have  a  fraction  (~1/10   2/10)  of  that.  Average clusters only have one stellar generation because theadiation from the first batch of stars would push out the residual gas after the initial collapse. Contrarily, a massive cluster would have a stronger gravitational field  to  retain  the  gas  and  allow  for  second  or  even  third  generation  collapses.  Another ossible explanation is that NGC 2808 is not a genuine globular cluster, but a dwarf galaxy that has been stripped of most of its mass during capture by the Milky Way [27]. This multiple ollapse concept is still open to question and much research needs tobe done. For our discussion, we will use the traditional approach of single-collapse to describe the formation of GCs.         Globular clusters are formed during the collapse of a molecular cloud which exceeds the critical mass Beyond this mass, the gas sure can no longer overcome the gravitational attraction amongst the gas particles. tential energy of a sphere of radius with gas of density ven by [6] ?? ?? =? 12? ?? (?? )?? (?? )?? 3??  ?  ? 16?? 215 ?? ?? 2?? 5.                                  1.1 )  The kinetic or thermal energy in the same sphere with speed of sound  ?? ??  is [6]   ?? ?? ? ?? ?? 22 4?? ?? 3 ??3  .                                                               If ? ?? ?? ? > ?? ?? , then the clouds diameter  2r must exceed the lower limit 2?? ? ? 52?? ? ?? ?? 2????   ?  ?? ??  ? ?? ?? ? ??????                                                       where  ?? ??  is known as Jeans length. The mass enclosed within  ?? ??  is simply Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   4   ?? ?? ? 43 ?? ? ?? ??2 ?3?? = ?? 6 ?? ?? 3?? ,                                                         where  ?? ??  is known as Jeans mass. This is the criterion that must be satisfied in order for the molecular cloud  to  collapse.  Once  initiated,  a  process  of  cascading  fragmentation  develops   due  to  irregular overdensities in real molecular clouds . From this model, the ages of the cluster members would be very similar because they are formed during the same collapse. An accepted belief is that globular clusters are generally formed when the galaxy is still very young, making them the oldest objects and th etal abundant.    One can also generalize the Jeans length and mass equations for collapse of stars into clusters. The same concept is applied, where the  gravitational potential  energy  must  exceed the  kinetic  energy. The only parameter that needs t be adjusted is the speed of sound. Instead of ?? , one would use the average random velocity of stars, . For example, Jeans length would become ?? ?? = ?? ? ?????? = 27 ????   ? ??1 ????  ?? ? 1? ? ??1 ????????  ?? ?? ? 3? ?12                                     (2.1.5)   in typical astronomical units. Jeans mass would take the form,   ?? ?? = 1.0 ? 104 ?? ?????? ? ??1 ????  ?? ? 1?3 ? ??1 ?? ??????  ?? ?? ? 3?? 12.                                   1.6)   Regardless of how the cluster is formed, either by collapse of a larger structure or expansio n of multiple smaller structures, the initial motions of the particles or stars should be mostly radial [15]. Assuming this to  be  true,  conservation  of  angular  momentum  would  continue  to  preserve  these  radial  orbits.  This anisotropy due to formation should  be slowly erased as gravitational interaction amongst the members smoothes the velocity distribution towards isotropy. The gradual evolution towards isotropy is known as relaxation and will be the focus of the next section.   2.2 Relaxation of Globular Clu sters  Relaxation is an important process that occurs to many systems modeled by particles. Particles of air or scent will eventually spread around the room because of random collisions that happen about  1011  times per second per particle at room temperature and atmospheric pressure at sea level [25]. These collisions allow the particles to  exchange  energy and  momentum  with  each  other, smoothing  out the anisotropic distribution. Similar to this concept, stars within a globular cluster also undergo avitational interactions that allow the cluster to gradually approach isotropy. 2.2.1 Strong Close Encounters Strong  encounters are  defined such that a star, A, approaches  very close to another star, B, insofar it completely alters the velocity of st . To simplify the derivation, we will assume all stars in the cluster to  have  mass ??   and  random  velocity  of  equal  magnitude   We  will  also  assume  other  gravitational sources to be negligible compared to the forces involved in the Ateraction. If  and B are initially far Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   5  away, the kinetic energy dominates their potential energy with each other. However, as their separation  decreases, the kinetic and potential energies become comparable to each other. In fact, strong encounter is defined such that their potential energy is greater than or equal to their kinetic energy, ?? ?? 2?? ??? ?? 22 .                                                             (2.2.1)   This means   ??   ? ?? ?? ? 2???? ?? 2 ,                                                             where  ?? ??  is called the strong encounter radius  [25] . The frequency of such encounters can be calculated by determining the number of stars that would be enclosed by the circle drawn out by the strong encounter radius  as  the  star  travels  around  the  cluster.  For  a  star  travelling  at  velocity  ??   for  a  time ?? ,  a  strong encounter will occur every time another star is within the volume  ?? ?? ?? 2????  centered on its path. If there are ??  stars per unit volume and one strong encounter within a time  ?? ?? , suc h that ?? ?? ?? 2 ???? = 1, then  ?? ?? = ?? 34?? ?? 2?? 2??  ? 4 ? 1012  ????   ? ??10???? ?? ? 1 ?3 ? ???? ?????? ?? 2 ? ??1?? ?? ? 3?? 1,                     .2.3)   where  ?? ??  is the mean time between strong encounters.  In general, the time for such strong encounters in typical  regions  of  galaxies  is  even  longer  than  the  age  of  the  universe.  These  interactions  are  only important when discussing the dense cores of globular clusters and galactic nuclei [25].    2.2.2 Weak Di stant Encounters   For distant encounters, we assume the forces involved are weak such that the stars only slightly deviate from their original paths after each interaction. This allows for the usage of impulse approximation to calculate the forces that the  stars would feel had they not been perturbed. A star of mass  ?? 1 is moving with velocity  ??  approaches a star of mass  ?? 2 within a distance of  ??  (impact parameter) . ?? 1 is originally travelling in a straight line, but  ?? 2 gravitationally attracts  ?? 1 and gives it a slight perpendicular velocity ?? ? . If we start measuring the time at closest approach, the perpendicular force is     ?? ? = ?? ?? 1?? 2 ??(?? 2 +?? 2?? 2)32= ?? 1 ?? ?? ????? .                                                       Integrating over time, we obtain    ? ?? ? = 1??1? ?? ? (?? )?????? ? = 2?? ?? 2???? .                                                5)   The parallel component of the velocity is not changed through the interaction because the duration of the force accelerating  ?? 1 as it approaches ?? 2 is the same as it moves away and decelerates from 2. The deflection angle is thus   Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   6  ?? = ? ?? ??? = 2?? ?? 2?? ?? 2 .                                                               Because  ?? 1is  deflected  towards  ?? 2 and  moment um  in  the  perpendicular  direction  is  conserved,  the distance  between  the  stars  will  decrease.  However,  in  order  for  the  impulse  approximation  to  remain valid, this distance cannot change significantly  [25] . In particular, the induced perpendicular velocity  ? ?? ?  must  be small compared to the initial approach velocity  ?? . This gives rise to the condition   ??   ? 2?? (?? 1+?? 2 )?? 2 .                                                                During the passage of the cluster, a star will have many distant weak encount ers. Each interaction will change the velocity by an amount  ? ?? ?  in random directions. Provided  the  directions  are random , it is not informative to find ? ?  as it will turn out to be zero. We will need to add the squares of ?  and find ? ? ?? ?2?   instead.  Following similar procedures as strong encounters, we want to find out the frequency of weak encounters. In time  ?? , the number of stars  ?? 2  passing  ?? 1  with separation between  ??  and  ?? +  ? ??  is the  product  of  number  density  ??   and  volume  ???? ? 2???? ? ?? .  Next  we  multiply  by  ? ?? ?  in  the  form  of Equation 2.2.5  and integrate over  ??  in order to get  ? ? ?? ? 2 ?  [25] ,  ? ? ?? ? 2 ?  = ? ?????? ? 2?? ?? 2???? ? 2 2???????? = 8?? ?? 2 ?? 22??????  ln(?? ???????????????? ???????? ??????  ).                           (2.2.8)   Aft er  a  time ?? ?????????? , as ? ? ?? ? 2? = ?? 2, the expected perpendicular velocity of ?? 1 has the same magnitude as its original velocity. This means the star has completely lost its original trajectory and is smoothed out by random gravitational interactions. If we define ?? ? ? ?? ??????????????? , we can rewrite the relaxation time   ?? ?????????? = ?? 38?? ?? 2?? 2???????? ? 2 ? 109?????????? ? ??10 ????  ?? ? 1?3 ? ?? 2?? ?????? ?? 2 ? ??103 ?? ?? ? 3?? 1.                       .2.9)   Now one must define the parameters  ?? ??????  and ?? ?????? . The exact values are open to question, but in our derivation,  ?? ??????  should not be smaller than  ?? ??  and ?? ??????  should be the maximum size of the system of interest.  However,  ?? ??????????   does  not  heavily  depend  on  these  parameters  as  the  ratio  will  go  through  a logarithmic operator. Essentially we will arrive at a correct order of magnitude for the relaxation time of a cluster even with compromised values of  ?? ??????  (?? ??????  is usually taken to be equal to  ?? ?? ).     2.2.3 Relaxation of NGC 6397 Besides  strong  and  weak  encounters,  there  are  additional  interactions  that  must  be  considered  for  the relaxation of a globular cluster. One particular interaction that must be considered is the changes in the Galactic  potential  in  which  NGC  6397  exists.  Thbit  of  NGC  6397  has  a  disk  crossing  time  of approximately  100  Myr  [12,  18].  This  leads  to  the  possibility  of  disk  shocking  to  interfere  with  the relaxation process and cluster dynamics. For typical relaxed clusters, like the NGC 6397, disk shocking is the most important close to the tidal radius, and less significant towards the half mass radius [9].    In regards to isotropic velocity distribution, the essence of the previous discussion lies in the fact that NGC 6397 is a relaxed cluster. The age of thuster is 11.5  0.5 Gyr [22], whereas the core and halfmass radius relaxation times are 0.079 Myr and 0.29 Gyr, respectively [11]. One can see that relaxation Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   7  time  decreases  towards  the  centre  of  the  cluster  because  stellar  masses  and  densities  increa . This dependence can also be seen in the functional form of Equation 2.2.3 and 2.2.9. It is worthy to mention that mass segregation has also been observed in NGC 6397 [5, 14]. Mass segregation contributes to faster evolution towards isotropy for the clunter because it tends to move massive stars towards the core.       2.3 Methods of Observing Anisotropy Until  the  dawn  of  the  21st  century,  anisotropy  of  stellar  motion  in  globular  clusters  has  been  studied through  very  crude  and  inaccurate  method s.  It  is  accomplished  by  comparing  radial  drop -off  that  is observed  with the theoretical  models  of varying anisotropy. This technique  has a serious flaw insofar there  is  no  guarantee  for  the  models  to  be  correct.  The  combination  of  observational  error  and  more important,  inadequate  dynamical  modeling  would  lead  to  fairly  unreliable  results  [15].  However,  the proper  motions  provided  by  Hubbles  ACS  and  WFPC  2  allow  astronomers  to  measure  anisotropy directly, without the need for complicated dynamical models.     Previous analysis of anisotropy required radial velocity observations. This means the result is heavily dependent on various geometrical perspectives at different radii of the cluster. The velocities observed for the center of the cluster is almost pure ly ?? ?? . But as we move out towards the edge of the cluster, a larger component of ?? ??  will contribute to the observed velocity. Moreover, due to the escape velocity of clusters, there is a natural dropoff  in ? ?? 2 ? ?????????????????? ? ??  even in clusters wit h isotropic velocities [15]. Another flaw with radial velocity studies is that they are very sensitive to the model used to calculate the distribution. All these weaknesses make radial velocity a bad choice for analyzing anisotropy.    Proper  motions  allow  a stronomers  to  compare  velocity  dispersion  in  the  radial  and  transverse direction. Although there will be projection problems along the line of sight, this method still provides a much  more  direct  measurement  of  anisotropy  [15].  One  possible  source  of  error   with  proper  motion analysis  is  due  to  cluster  rotation.  As  one  can  imagine,  a  rotating  cluster  will  distort  the  radial  and transverse velocities in the plane of the sky. Along any line of sight, the components of rotation will be added or subtracted to pr oper motions that are parallel to the cluster equator. In another words, rotation would most severely distort the proper motions near the center of the cluster and less towards the edge. A possible correction for this is to find a dynamical model that woul d account for the rotation. However, this leads to the same modeling problem as radial velocity studies, but with much less dependence on the particular model.   2.4 Hubble Data   A single field located 5 Southeast of the cluster core is imaged with the ACS  for 126 orbits (179.7 ksec) in  2005  [22].  This  field  is  chosen  because  it  has  a  60%  overlap  with  pre -existing  WFPC  2  images. However, due to the short exposure time of the WFPC 2 images (3.96 and 7.44 ksec in 1994 and 1997), the level of detail is fairly  limited compared to the ACS image. The location of the ACS field is depicted in Figure 2.1.  The regions traced out by bold lines represent the most recent images. The deep outer field taken by the ACS is represented by the rectangular box. A glance at Fig ure 2.1 would actually reveal two bolded  regions.  This  is  because  WFPC  2  was  simultaneously  exposed  during  the  ACS  observations. Coincidentally, these WFPC 2 exposures nicely fall on the core of the cluster with the roll angle of the telescope adjusted [22 ]. This extra set of data will be useful for determining cluster rotation in Chapter 3.  Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   8      Figure 2.1: Orientation of ACS and WFPC 2 Fields. The horizontal and vertical axes are right ascension and declination, respectively.     The  ACS  data  were  reduced by two independent groups at Rice University by J. Anderson and UBC by J. Brewer and H. Richer. We will not go through the details, which can be found in [2, 22] but the selection criteria will be briefly mentioned. In order to eliminate all objects but th e stars, all sources with the following properties were removed: (1) noticeably boarder or narrower than a Point Spread Function (PSF), (2) so close in proximity to brighter stars as to be likely PSF artifacts, or (3) on linear features so they were likely  to be bumps on diffraction spikes. This reduced the number of objects in the ACS field from 48785 to 8357 true stars [22]. The CMD generated with all 8357 stars is shown in Figure 2.2.    Figure 2.2: CMD of NGC 6397. Distinct main sequence and white dwarf  branches can be seen.   Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   9    2.5 Analysis and Results  We  are  interested  in  the  velocity  distribution  of  NGC  6397.  This  equates  to  finding  a  method  to systematically eliminate all the stars which are not part of the cluster. One solution is to make use of theWFPC 2 images taken in 1994 and 1997. The positions of the stars between the 2005 ACS image and the WFPC 2 images were compared, generating proper motions for the stars within the 60% overlap in areas. Using these proper motions, we can distinguish cluste stars from non-cluster stars because NGC 6397 as a whole has large space motion [22]. The caveat for adapting this method is that we will lose about 40% of the cluster stars due to incomplete overlap of images.    The proper motions were determined such t at the zero point is moving with the cluster. Therefore we can isolate the cluster stars by imposing the proper motion condition ?? ?? 2 = ?? ?? 2 + ?? ?? 2 < 1. Within the ACS field, 2395 stars meet the proper motion requirement. The proper motion cleaned CMD is shon in Figure  2.3.  The  result  from  this  reduction  procedure  is  very  nice  as  one  can  see  a  very  tight  main sequence branch and a very distinct white dwarf branch.  Figure 2.3: Proper Motion Cleaned CMD of NGC 6397. The main sequence and white dwarf branches are much more distinct.   Using the 2395 stars, a distribution of the proper motion directions has been generated. In particular, we have normalized the distribution such that towards the cluster centre is represented by an angle of 0, clockwise rotation is positive, directly away from centre is ? , counter clockwise rotation is negative. The geometry is illustrated in Figure 2.4.  Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   10              Figure 2.4: Geometry of Proper Motion Normalization.   In  order  to  readily  determine  the  isotropy  of  the  observed  distribution,  we  have  plotted  the  angles against a uniform distribution on a cumulative scale. This is depicted in Figure 2.5. The result is fairly transparent as the two distributions are very similar. To quantify the degree of  dissimilarity between the real and uniform distributions, t he  Kolmogorov -Smirnov test (KS -t est) was used. The KS -test, like other statistical  tests  tries  to  determine  if  two  dataset differ  significantly. However,  the  KStest  has  the advantage of making no  underlying  assumption about the distribution of data.  The level of significance is summarized by the KS value. A large value means the two distributions are alike and a small value means the distributions are distinct. For our analysis, this is the most important property to keep in mind. For a good introduct ion to the workings of the KS -test, please refer to [16].     Figure 2.5: Real and Uniform Cumulative Distribution. The KS value between the two distribution is 0.36, indicating a high level of similarity.   Proper motion direction Cluster Centre Angle 0 ?/ 2  -?/ 2  Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   11    From this preliminary analysis, the difference between two distributions is not statistically significant. A KS value of 0.36 is not low enough to draw upon any conclusive statement (for  anisotropy). Therefore, in hopes to amplify the signal, we have combi ned the angles and collapsed them into a distribution from 0 to ? /2. First, the clockwise and counterclockwise proper motions are combined. This means we no longer distinguish  between  the  directions  of  proper  motion  around  the  centre.  Second,  the  proper  mo tions directly towards and away from centre are combined. After the two collapses, we can only determine if the proper motions are: (1) moving radially with respect to the cluster centre, or (2) rotating around the cluster centre in the plane of the sky. T he idea here is to sacrifice a portion of the directional information in order to amplify the signal. The collapsed distribution is illustrated in Figure 2.6.    The  KS  value  increased  as  expected.  It  is  now  closer  to  a  2 -?  result,  indicating  s ome  level  of distinction  between  radial  and  rotational  proper  motion.  If  we  qualitatively  analyze  Figure  2.6,  three trends can be seen: (1) the two distributions are almost identical between angles 0 to 0.3 radians, (2) there is an overabundance of proper motion between  angles of 0.3 to 0.85 radians, and (3) there is an under -abundance  or  proper  motion  between  angles  of  0.85  to  1.4.  According  to  our  collapsing  scheme,  this translates to isotropy in radial orbits but slightly anisotropic rotational orbits around the clust er centre on the plane of the sky. As a side note, due to the nature of cumulative distributions, it is the slopes of lines which determine the relative increase or decrease in abundances, not the absolute position of the lines.     Figure 2.6:  Real and Uniform Collapsed Cumulative Distribution. There is observable anisotropy towards the centre of the distributions.   Chapter 2. Isotropy of NGC 6397 _____________________________________________________________________________   12   2.6 Conclusion   Observing isotropy in an old, relaxed globular cluster, such as NGC 6397, is an anticipated result. The measured distribution  of velocities is very similar to that of a uniform distribution; the resulting KS value is 0.36. Even if the cumulative distributions are collapsed to amplify the signal, the KS value is still only 0.037 (~2 ? ). The caveat  is that some  directional  informat ion  is sacrificed  during the collapse, but the signal is still not high enough to determine anisotropy in the cluster as a whole. However, the collapsed distribution yields insight into possible cluster rotation. There are two reasons for this: (1) there s eems to be  more  stars  in  orbits  around  the  cluster  centre  in  the  plane  of  the  sky,  and  (2)  the  number  of  stars travelling  towards  and  away  from  the  cluster  centre  appears  to  be  fairly  similar  to  the  uniform distribution. In other words, there is a good cha nce for the cluster to be rotating in our line of sight. This peculiarity is the main investigation of the next chapter.   Chapter 3. Rotation of NGC 6397 _____________________________________________________________________________  13  Chapter 3  Rotation of NGC 6397  3.1 Rotation in Globular Clusters Globular clustes are known to exhibit rotational behavior. The first method for the detection of rotation in GCs dates back in the first half of the 20th century. The main evidence spawned from the observations that showed flattening of the spherical structurestation which compresses the cluster along its  spin  axishese  ellipticity  measurements  were done  in  the  1920s [23].  Another  method  for determining  rotation  in  GCs  depends  on  radial  velocity  measurements  of  individual  cluster  stars. Comparing t  radial velocity with the stars location within the cluster could provide a lower limit for the  degree  of rotation of the  cluster . The latest  method for  measuring rotation  of  globular clusters relies  on  proper  motion  of  individual  stars. h  a  stud was  conducted  for  t   rotation  of  Globular Cluster 47 Tucanae23] It was studied using images  taken by  WFPC 2,  fairly similar to the quality  of our images   in  terms  of  proper  motion.  The  conclusion  was  a  rotational  proper  motion  of  0. 233 0.055 ??????  ?? ?? ? 1. Similar  concepts will  be used  in order to  extract rotational information pertaining to  NGC 6397.   3.2  Analysis and Results   The  space  motion  of  NGC  6397  was  studied  by  [13]   very  recently  using  the  same  set  of  ACS  data obtained in 2005.  For details of the data reduction and analysis, please refer to . However, we will mention the most important step in determining an accuratece motion of the clustert is to find  a  s et  of suitable reference objects. Fortunately,  398 suitable galaxi es were found  within the ACS field which can be used as references.  The potent information from their study to our analysis is  the proper motion in the ACS field.  The motion  is ?? ?? cos(?? ) = 3.56 0.04 ??????  ?? ?? ? 1 and ?? ?? = ? 17.34 0.04 ??????  ?? ?? ? 1.  Figure 3.1 illustrates the distribution of the proper motion.    To determine the rotation of the cluster as a whole, we need to find the absolute motion of the cluster core. This data is provided by the WFPC 2 exposures taken simultaneously with the ACS in 2005. Our method of isolating cluster stars and finding reference objects is not as sophisticated as  [13]; nevertheless, it should be provide a good estimate of the rota tion NGC 6397 in the plane of the sky.    The first step is to adapt a robust method to isolate cluster stars and field stars using the CMD. There are basically two techniques. One requires the use of proper motion and the other requires the use of the CMD. Isolation of cluster stars in colour space (CMD) turns out to be superior because it does not bias the selected stars. After all, we want to extract proper motion information; therefore it is unreasonable to select stars based on their proper motions in th e first place. Figure 3.2 illustrates the groups of stars used to determine the proper motions of the cluster and the field.    Chapter 3. Rotation of NGC 6397 _____________________________________________________________________________  14   Figure 3.1 : Proper Motion of ACS Field: Red dots represent reliable reference objects, in particular, soild red dots and heavil y weighted. The clump near the bottom of the plot is the cluster  [13] .    Figure  3.2 : WFPC 2 Cluster Centre CMD. The red crosses are all the observed stars. The green and blue crosses are the stars used for calculating proper motions for the cluster and  field, respectively. The green crosses represent a large portion of the observed main sequence stars.Chapter 3. Rotation of NGC 6397 _____________________________________________________________________________  15     It should also be noted that the proper motions are given in units of pixels, so a transformation matrix is used to convert the change in pixel (dx and  dy) into  ?? ?? cos(?? ) and ?? ?? . The transformation has been tested and agrees with the right ascension and declination parameters recorded in the WFPC 2 data set. The values are summarized in Table 3.1.             Table 3.1 : Cluster Centre Proper Motions     Using the values from Table 3.1, the proper motion of the cluster  stars in the centre of NGC 6397 can be  readily  determined.  It  turns  out  to  be  ?? ?? cos(?? ) = ? 0.32  1.37 ??????  ?? ?? ? 1  and  ?? ?? = ? 2. 51 0.54 ??????  ?? ?? ? 1. The rotation of the cluster can then be estimated by taking the difference between the values in the centre  of the cluster  and the  values determined in the  outer field by  [13] . The rotation al motion  is    ?? ?? cos(?? ) = 3.88 1.41 ??????  ?? ?? ? 1  and  ?? ?? = ? 14.83  0.58 ??????  ?? ?? ? 1. This  result  is  not entirely expected because the position  of the ACS field is almost directly Southe ast of the center. A nave prediction would presume the  direction of the proper motion  to be closer to 45  or approximately equal  in magnitude for  ?? ?? cos(?? ) and ?? ?? . Moreover, the direction of ?? cos(?? ) and ?? ??  do not correspond to either clockwise  or  counterclockwise  rotation  of  the  c luster  in  the  plane  of  the  sky.  A  positive  ?? ?? cos(?? ) represents motion to the left in Figure 2.1, whereas a negative  ?? ??  represents motion towards  the bottom. This means the estimated proper motion corresponds to stars moving away from the cluster centre.  This peculiar  proper  motion is probably a direct result of not having suitable reference objects.    3.3 Conclusion   Rotation in globular clusters i s known to exist. Combining the ACS data  (outer field ) and WFPC 2 data (inner field ), we have generated a rough estimate of the rotation of NGC 6397  in the plane of the sky. The rotational motion at the location of the ACS field (5 SE of cluster centre) i s approximately  ?? ?? cos(?? ) =3.88 1.41 ??????  ?? ?? ? 1 and ?? ?? = ? 14.83  0.58 ??????  ?? ?? ? 1. This result deserves further analysis as we believe the  estimate of the  proper motion determined from the WFPC 2 data is not very accurate. As a first  step,  there  needs  to  be  a  more  sophisticated  method  for  identifying  reference  objects.  The  two methods  we  considered,  proper  motion isolation and  CMD  isolation,  appear  to  be  insufficient  for  the WFPC 2 data. The first method of proper motion isolation, as discussed in Section 3.2, is biased. The second method of CMD isolation, which we used, does not very accurately distinguish between field stars and cluster stars because the main sequence is not very distinct (Figure 3.2). Cluster Centre Proper Motions (  ?? ?? ? 1) Cluster Stars ?? ?? cos(?? ) ?? ??  2.23  0.13  -1.38  0.06  Field Stars ?? ?? cos(?? ) ?? ??  2.55  1.24  1.13  0.48  Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  16  Chapter 4  Missing White Dwarfs in NGC 6397  4.1 White Dwarfs White dwarfs (WD), categorized as class D  stars, are a family of stars with mass comparable to the Sun and radius comparable to the Earth. They occupy a region that is below and approximately parallel to main  sequence  stars on  the CMD.  The  name,  white  dwarf,  is  somewhat  misleading  as  it  implies  the surface colour of the stars to be white. In fact, WDs exist in a large range of colours with temperatures varying from less than 5000 K to more than 80 000 K[6]. Ther e are t hree s divisions to WDs according to their spectral type, DA, DB, and DC. The most common form is known as DA white dwarfs which only have pressure -broadened Hydrogen absorption lines. This is the type of WDs that  pertains to  our analysis.    White dwarfs are bu rnt out cores of sun -like stars with no nuclear energy source. In particular, we can crudely  estimate the pressure at the centre of a WD  using  [6]   ?? (?? ) = 23 ???? ?? 2 (?? 2 ? ?? 2 )                                                        which is derived from hydrostatic equilibrium. Setting  ?  and R to typical WD values, we arrive at the pressure at the core of the star to be   ?? (?? = 0) ? 3.8 ? 1022 ??  ?? ? 2                                                 2)   We can proceed  on to  estimating the core temperature assuming this pressure to be true. Using  principles of radiative transfer, the change in  pressure  over  the change in radius is rel ated by   ?? ?? ?????????? = ??? ? ????   ?? ??????                                                            (4.1.3)   where  ??  is the average opacity and  ?  is the density.  Using   ?? ?????? = ?? ??4?? ?? 2                                                             1.4)   and   ?? ?????? = 43 ?? ?? ?? 4                                                                 which  is  derived  from  the  Planck  function  and  integrating  over  energy  density,  we  arrive  at  the temperature gradient for radiative transport equation,   ???????? = ?34?????? ? ???? 3?? ??4?? ?? 2                                                           (4.1.6)  Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  17   where  ?? ? 4???? = 7.56591 ? 10? 16??  ?? ? 3?? ? 4 is the radiation constant. If we assume the  core temperature to be much greater than the surface temper ature  and set typical WD value s for each parameter, Equation 4.1.6  simplifies to  ?? ???????? ? ? 3??? ??4???? ?? ????4?? ???????14 ? 7.6 ? 107??                                               (4.1.7)   From this crude approximation, one can learn that WDs cannot  be composed mainly of hydrogen or else nuclear  energy  generation  would  be  possible,  increasing  their  luminosities  by  several  orders  of magnitude. In other words, the particles  in  the core of WDs  must not undergo nuclear fusion at these densities and temperatures.    4.1.1 Degenerate Matter    Given the extreme densities of WDs, it is possible to show that normal gas and radiation pressure  are not sufficient  for  supporting  the  structurom  gravitational  collapse.  The  mechanism  which  provides  the force required to counteract gravity  direct manifestation of the Pauli Exclusion Principle). More specifically, the stellar structure is held up against collapse by electron degenerssure. Combining PEP with Heisenbergs Uncertainty Principle, we arrive at thet   ?? = ? 3 ?? 2 ?235? 2?? ??   ? ????? ????? ?? ?53   ,                                                       (4.1.8)   where Z and A are  proton and nucleon numbers, respectively. For ful l derivation, please refer to  [6].    4.1.2 Chandraskhar Limit Having derived the electron degeneracy pressure, we  combine it with Equation 4.1.6nd obtain  23 ???? ??2?? ???? 2 = ? 3?? 2?235? 2?? ??   ? ????? ????? ?? ?53  .                                              (4.1.9)   Assuming constant density,  ?? = ?? / 43 ?? ?? 3, Equation 4.1.9  becomes   ?? ???? ? (18?? )2310? 2?? ?? ?? ?? ????13? ? ?? ?? ? ???????53 .                                            (4.1.10)   This result is amazing becau se it implies ?? ???? ?? ????13  is constant or ?? ???? ?? ????3  is constant. In more familiar terms, more massive WDs are actually smaller than less massive WDs!   However, one can imagine adding more and more mass to an existing WD. What woulappen then? Essentially the increasing pressure would increase the velocity of the electrons  eventually they would approach the speedof light. The pressure is now described by Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  18  ?? = ? 3?? 2?134  ? ?? ? ????? ????? ?? ?43 .                                                      (4.1.11)   If  one  assumes  Z/A  =  0.5  for  carbon -oxygen  WD  along  with  constant  density,  ?? = ?? / 43 ?? ?? 3,  then combining Equation 4.1 .1 with 4.111 will provide an estimate of the maximum mass of WDs, ?? ?? ? ? 3? 2??8   ? ? ???? ?32 ? ? ???? ?1?? ?? ?2? 0.44 ?? ??????                                    1.12)   A precise derivation of the Chandrasekhar Mass,  ?? ?? ? , would yield a result of  ?? ?? ? = 1.44 ?? ?????? .  Once again, for full derivation , please refer to  [6]. Being able to theoretically derive characteristics of WDs allow s us to more confidently establish the mechanisms governing their  activities. In particular, no WDs with  mass  greater  than ?? ?? ?  have ever been found, which supports t he concept that electron degeneracy pressure is responsible for maintaining hydrostatic equilibrium in a WD.     Figure 4.1 : Distribution of Star Types in CMD  [20] .   4.2 Reddening and Extinction   When we look into the sky on a clear night, we see little m ore than the bright stars luminous enough to transmit their light through all the obstacles that stand between us. In particular, there  is aluminous matter which obscure and block out light in our galaxy. This  ma t t er  is composed of dust and gas particles,  which is collectively known as  the  interstellar medium (ISM).  This obscuration due to scattering and absorption of light is referred to as interstellar extinction.       Due tothe presence of obscuration, one must modify the distance modulus equation by addextra absorption factor, ?? ?? ? ?? ?? = 5(log10 ?? ? 1) + ?? ??                                                 1)  Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  19  where  d  is  measure  in  parsec ,  ?? ??  and  ?? ??   are  the  apparent  and  absolute  magnitudes  respectively .  In addition, this absorption factor should somehow be  related to the optical depth. To quantify this, we must combine the simplest form of the radiative transfer equation  [6] ,  ?? ???? ?? , 0 = ??? ?? ??                                                                  (4.2.2)   where  ?? ??  is the optic al depth,  with the flux -magnitude equation,   ?? 1 ? ?? 2 = ? 2.5 log10 ? ?? 1??2?                                                        and arrive at   ?? ?? ? ?? ?? ,0 = ? 2.5 log10 (?? ? ?? ?? ) ? 1.086?? ??                                      (4.2.4)   where  ?? ?? ? ?? ?? ,0 is just the change in apparent magnitude due to the obscuration. But we know that is equivalent to the absorption factor  according to Equation 4.2.1 , ther efore  ?? ?? ? 1. 086?? ??                                                           2.5)   The optical depth through a cloud of dust is simply,   ?? ?? = ? ?? (?? )?? ?? ??????0                                                             where  ?? (?? ) and  ?? ??  are the number density and scattering  cross section of the dust particles along the line of sight.  Intuitively speaking, absorption should be more severe if there is more dust particles obscuring the light. This can be seen by combining Equations 4.2.5 and 4.2.6.    We can try to qualitatively u nderstand the wavelength dependence of absorption.  Dust particles come in a variety of sizes and compositions.  One can make a simplifying assumption that each particle is a sphere with radius  ??  and has a scattering cross section of  ?? ???????? = ?? ?? 2. If the wavelength of light  is much larger than  the  radius of the dust particles, the light passes through the cloud without much extinction. However, if the wavelength is small relative to  ?? , then the light is either completely blocked or severely obscured.  A  useful  analogy  to  understand  this  idea  would  be  water  waves  passing  through  obstacles. Large objects simply block out the waves whereas small obstacles minutely perturbs the passing wave s. This cross sectional  analysis was first done by G. von Mie in 1908  [6] .   Now that we have a grasp on the wavelength dependence of dust extinction, we can discuss a related concept  known  as  interstellar  reddening.  As  longer  wavelengths  can  more  easily  pa ss  through  a  dust cloud, blue light is generally more obscured than red light. For this reason, stars commonly appear redder than their effective temperatures should imply under the influence of dust. Conveniently for astronomers, this is usually not a sig nificant problem because absorption and emission lines in the stars spectrums can be used to detect this change  [6] .   4.3 Analysis and Results Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  20  Isochrone fitting is commonly used to determine properties of observed stars in  a CMD. The model is theoretically developed with certain fixed parameters, for example, masses, ages, and metallicities of the stars. Essentially one would use a variety of models to f it the pattern observed on the CMD . At the end, the characteristics of the stars would be assumed to b e most similar to the matching model.     The outline of the procedures taken for our analysis will be summarized in the next few paragraphs. As a first step,  a CMD of the WDs in NGC 6397 is plotted using the ACS data, Figure 4.2. Immediately one would see a  noticeable  gap between 0.35 and 0.40 in the colour axis. The objective of this section is to determine if the gap is statistically significant. In particular, we will convert those magnitudes into ages using  the cooling models developed by  Bergeron et al.  (2006).      Figure 4.2 : Cluster White Dwarf CMD. There is a noticeable gap near the 25 th magnitude.     In  order to conduct the  fitting procedure, the band pass filters for the  model and  observation  must correspond to each other. The ACS data is provided  in F606W and F814W filters , whereas the Bergeron models are provided in Johnson UBVRI filters.  Figure 4.3 illustrates typical wavelength dependence of the Johnson filters. Figure 4.4 illustrates the F60 6W and F814W filters specific to  the Hubble ACS.     2 3 2 4 2 5 2 6 2 7 2 8 2 9  0   0 . 2   0 . 4   0 . 6   0 . 8   1   1 . 2   1 . 4F606WF 6 0 6 W - F 8 1 4 WC M D  ( F 6 0 6 W  v s  F 6 0 6 W - F 8 1 4 W )Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  21   Figure 4.3 : Johnson/Bessel UBVRI Filter Characteristics  [4].     Figure 4.4 : WFC Filter Characteristics [1] .   The conversion coefficients are summarized in Table 4.1  and 4.2 . They are provided by  [24]   SOURCE S M AG TARGET  Observed   Synthetic  T M AG  TCOL  c0  c1  c2  c0  c1  c2  TCOL range*  F606W   F814W  V  I V  I  V  I 26.325   25.495  0.236   -0.002  0.000   0.000  26.394  26.331  25.489  25.496  0.153  0.340  0.041  -0.014  0.096  -0.038  -0.930  0.015  < 0.4  >0.4  <0.1  >0.1  *TCOL range applies only to synthetic transformations .  Table 4.1:  Coefficients for the Transformation from WFC to BVRI  [24] .     Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  22  FILTER  VE GA  M AG F606W  F814W  26.398  25.501   Table 4.2 : Synthetic Zeropoints for WFC  [24] .      As  discussed  in  Section  4.2,  interstellar  extinction  and  reddening  interfere  with  the   apparent magnitudes that are observed . Fortunately, the extinction and reddening have been previou sly determined for NGC 6397  [22] . The extinction coefficients are summarized in Table 4.3.   Extinction Coefficient   Value ?? ??  ?? ??  ?? (?? ? ?? ) 0.56  0.31  0.25   Table 4.3:  Extinction Coefficients.     Once the  extinction corrections have been applied, the  models  are converted to F606W and F814Wbands. Then the distance modulus of 12.03 06, determined by 22] is added to the F606W magnitude.The  best  fiting Bergeron model is the 0.5 solar mass WDs with pure hydrogen atmospheres (Figure 4.5).  Now  that  we  have  a  working  model,  a  plot  of  the  its  age  versus  magnitude  is  plotted  to  observe  the dependence (Figure 4.6).     Figure 4.5 : Cluster  White Dwarf CMD with 0.5 Solar Mass Bergeron Model. The kink near the horizontal position of 0.4 is a result of transformation using the numbers from Table 4.1 and 4.2.   2 3 2 4 2 5 2 6 2 7 2 8 2 9  0   0 . 2   0 . 4   0 . 6   0 . 8   1   1 . 2   1 . 4F606WF 6 0 6 W - F 8 1 4 WC M D  ( 0 . 5  s o l a r  m a s s ,  p u r e  H  m o d e l )Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  23   Figure 4.6 : Bergeron Model: Age versus Magnitude.     Using the model, we have converted all the sta rs magnitudes, down to F606W = 28.5, to age in Gigayears. The distribution is generated and is illustrated in Figure 4.7. One feature that is quickly observable is the over -abundance of white dwarfs between the ages of 1 and 2  ?????? .  Moreover, the gap that originally motivated the study turns out to be fairly insignificant. It is located in the 0.6  ??????  bin of the histogram in Figure 4.7.     Figure 4.7 : White Dwarf Age Distribution in NGC 6397.     Once again, in order to de termine the statistical significance, we have conducted a KS test of this distribution against a uniform distribution. The KS value is 0.24, indicating a fair level of similarity between the distributions. The  plot of the cumulative distributions are shown  in Figure 4.8     1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 e + 0 0 6 1 e + 0 0 7 1 e + 0 0 8 1 e + 0 0 9 1 e + 0 1 0 1 e + 0 1 1 1 8  2 0  2 2  2 4  2 6  2 8  3 0  3 2Age (Years)F 6 0 6 WA g e  v s .  F 6 0 6 W  ( M o d e l )0.00%10.00%20.00%30.00%40.00%50.00%60.00%70.00%80.00%90.00%100.00%01234567891011120.2 0.611.4 1.8 2.2 2.633.4 3.8 4.2 4.655.4 5.8 6.2 6.677.4 7.8 8.2 8.6FrequencyAge (Gigayear)WD Age Distribution in NGC 6397 (F606W < 28.5)Chapter 4. Missing White Dwarfs in NGC 6397_____________________________________________________________________________  24    Figure 4.8 : Real and Uniform Age Cumulative Distribution.   4.4 Conclusion   The  missing  white  dwarfs, implied by the  gap in the CMD (Figure 4.2), in NGC 6397 turn out to be statistically  insignificant.  In  other  words,  this  gap  probably  d oes  not  represent  any  unusual  event  that happened  in  the  past  leading  to   an  under -abundance  of   white  dwarfs  just  under  the  age  of  1 ?????? . However, from the cumulative distribution (Figure 4.8), one can see a large deviation from uniformity at the age of 0.9 to 2 ?????? . At 2  ?????? , it slowly settles down to uniformity e gigayears again. There are two explanations for this observation: (1) approximately 0.9 to 2 number  of  white  dwarfs  were  born,  or  (2)  there  apto  be  some  inaccuracy  in  the te  dwarf modeling. he second explanation is more likely as there is no evidence to suggest why there would be a white  dwarf  population  burst  about  1 ago.  However,  before  stating  any  conclusions,  one  should obseve that the KS value is merely 0.24. This means the deviation could very well have just happened by chance.  Chapter 5. White Dwarfs in Galactic Bulge _____________________________________________________________________________ 25  Chapter 5  White Dwarfs in Galactic Bulge As the final section of this thesis, we will consider the possible white dwarf candidates in the Bulge of the Milky  Way.  This  is  possible  because  the  stellar  density  of  NGC  6397  is  not  high  enough  to  obstruct distant sources in the ACS field. 5.1 Distribution of Starlight in the Milky Way From  observations  of  other  spiral  galaxies  and  data  from  star  counts,  distance  indicators,  abundance analyses and other measurements, astronomers have been able to reconstru a model of our own galaxy, the Milky Way [6]. The light from spiral galaxies is generally strongest in the near frared region of the electromagnetic spectrum, close to 1 ? m. This wavelength is commonly emitted by older stars, such as K giants. The reason for this bias in wavelength is because blue light is absorbed by interstellar dust and reradiated  in  far-infrared, beyond ~10  ? m. However, the  optical spectrum is still the  most important for determining galactic structure.     The Suns  distance from t he center of the  galaxy, also known as the Solar Galactocentric Distance (?? 0 ), is recently determined to be  7. 62   0.32 ??????  through S2 orbit fits [7]. It is interesting to note that the original estimate by Shapley was closer to  15 ?????? . The full diam eter of the galactic disk is believed to be in the range of  40 to 50 ??????  [6].     The galactic disk appears to be composed of three separate components. The young disk probably has a  scale  height  of  50 ???? . The  scale  height  is  defined  to  be  the  distance  over  which  the  stellar  number density drops by 1/e.  The young disk is the region where Galactic dust and gas distribution is the most abundant. Another component, the old thin disk, has a larger vertical scale height  (?? ?? ? ???? ), approximately 325 ???? . The third component, the thick disk, has a scale height  (?? ?? ? ?????? ) of 1. 4 ?????? .  This disk has much lower stellar density compared to the thin disks. The number density of stars determined from star -count data follows [6],   ?? (?? , ?? ) = ?? 0 ? ?? ????? ?? ? ???? + 0. 02?? ????? ?? ? ?????? ? ?? ???? ?? ,                                     (5.1.1)   where  R  is  the  distance  from  the  center  of  MW,  ? ?? = 3.5 ??????  is  the  disk  scale  length,  and  ?? 0 =0.02 stars  ?? ?? ? 3 for  the  absolute  magnitude  range    4.5 ? ?? ?? ? 9.5.  The  Sun  is  approximatel30 ????  above the midplane of the young thin disk. Another important relation to consider is the luminosity density for the old thin disk is,   ?? (?? , ?? ) = ?? 0?? ???? ?? sech2( ???? 0)                                                   (5.1.2)   Chapter 5. White Dwarfs in Galactic Bulge _____________________________________________________________________________ 26  where  ?? 0 = 2?? ?? ? ????  and ?? 0 ? 0.05 ?? ??????  ?? ?? ? 3. Besides distinguishable stellar densities, the thin and thick disks are also different in terms of their chemical composition and stellar kinematics.    The  galactic  Bulge  was  originally  thought  as  a  roughly  spherical  st ructure  near  the  center  of  our galaxy. However, observations of Mira variable stars reveal the Bulge to be a bar that is inclined at a particular angle from our line of sight. In another words, it closer resembles an ovular structure from the Suns point o f view. The radius of the Bulge is approximately  1 ??????  and has a scale height of  400 ????  along the minor axis. The minor axis to major axis ratio is about 0.6.       Figure 5.1: COBE View of the Milky Way [19].      5.2 Analysis and Results   Using the  CMD generated from the ACS data taken 5 Southeast of NGC 6397, we can see a number of faint field stars dimmer than the cluster white dwarfs. More specifically, we see a branch approximately 2 magnitudes below and parallel to the cluster white dwarfs. Th is pattern deserves investigation because it can possibly represent white dwarfs in the Galactic Bulge.     If we calculate the most probable distance of the Bulge white dwarfs through basic trigonometry, we arrive  at  a  value  of  approximately  7.45 ?????? .  Recall  from  the  previous  section,  the  stellar  distribution decreases exponentially with radius, so the most probable location is the least distance from the center of the Milky Way. The geometrical setup is illustrated in Figure 5.2. Moreover, the central di ion of gas and dust in our galaxy is located in the midplane (b = 0) of the young thin disk. As we head away from the plane, extinction and reddening effects both decrease. At the location of NGC 6397, the vertical distance from the plane is approxiately 0.52 ?????? . Due to the large distance from the plane, we can make a simplifying assumption. The dust extinction and reddening effects between the Earth and NGC 6397 are approximately equal to the effects between the Earth and the Galactic Bulge.  Chapter 5. White Dwarfs in Galactic Bulge _____________________________________________________________________________ 27    The  distance  modulus  of  NGC  6397  is  previously  determined  to  be  12.03  0.03  and  the  most probable  location  of  the  Bulge  white  dwarfs  is  approximately  7.45 ??????   from  our  calculation.  Using Equation 4.2.1, the distance can be converted to a distance modulus  and is approximately 14.36. Taking the difference between these two moduli (Earth -to-6397 and Earth-Bulge) would give us an estimate of the decrease in magnitude. The parallel branch approximately 2 magnitudes lower than the cluster WDs is very visible in Figure 5.3.   Figure 5.2: Most Probable Location of White Dwarfs in Galactic Bulge   Figure 5.3: CMD of NGC 6397 with White Dwarfs Distinguished.  Chapter 5. White Dwarfs in Galactic Bulge _____________________________________________________________________________ 28   Figure 5.4: Possible White Dwarf Candidates in the Galactic Bulge. The red crosses represent  original observations from the ACS. The green crosses are the cluster white dwarfs with an adjustment to the distance modulus corrected for Earth -to-Bulge distance. There is clear overlap in the circled region. 5.3 Conclusion Through  the  unparallel  capablities  of  the  ACS  onboard  the  Hubble  Space  Telescope,  we  are  able  to reliably  observe  the  faint  stars  down  beyond  ~26  magnitude.  This  provided  the  opportunity  to  study potential white dwarf candidates behind the location of NGC 6397. Making reasonable ass ions about distance  modulus and  extinction,  it is  likely that ~10 white  dwarfs of 0.5 solar  mass are found in the galactic Bulge. This is probably one of the first observations of Bulge white dwarfs, let alone determining their possible masses.  Chapter 6. Conclusions _____________________________________________________________________________  29  Chapter 6  Conclusions  6.1 Summary and Implication of Results Using  the  observations  from  the  Hubble  Space  Telescope,  we  have  conducted  a  number  of nalyses regarding the  Globular Cluster NGC 6397. The kinematics of the cluster, including isotropy of velocity distribution  and rotation, have been examined. The velocity distribution of roughly 2400 stars in the ACS deep field appears to be virtually uniform . This result is anticipated as relaxed globular clusters are known to evolve towards isotropy. Further analysis of thisribution reveals signs of probable cluster rotation.   Consequently, a rotational study is conducted by comparing motion of the cluster in the outer field with the inner field.he  motion at the location of the ACS field (5 SE of cluster centre) is a pproximately ?? ?? cos(?? ) = 3.88 1.41 ??????  ?? ?? ? 1  and  ?? ?? = ? 14.83  0.58 ??????  ?? ?? ? 1.  The  result  is  not  entirely expected as the motion does not correspond to clockwise or counterclockwise rotation in the plane of the sky. This result is prob ably manifested in the method which is used to isolate cluster stars from field stars. In order to arrive at a conclusive result, a more reliable method is necessary for determining reference sources.    The second half of the thesis is pertained to white dw arfs in NGC 6397 and the galactic Bulge. The missing  white  dwarfs,  or  gap,  observed  in  the  CMD  of  the  outer  field  turns  out  to  be  statistically insignificant. The Bergeron cooling models are used to fit the cluster white dwarfs. The best fitting model is the 0.5 solar mass with pure hydrogen atmosphere white dwarfs. After converting the stars observed magnitudes to ages, a substantial overpopulation of 0.9 to 2.0  ??????  white dwarfs are found. Although the statistical analysis reveal this pattern to be f airly common, there is still reason to believe the cause to be manifested in modeling error.    Finally,  potential  white  dwarf  candidates  in  the  Galactic  Bulge  are  studied.  After  correcting  for distance modulus and extinction, there are approximately 10 star s in the Bulge which resembles 0.5 solar mass white dwarfs. This is a pleasant result as few studies have been able to detect w hite dwarfs in the Bulge.    6.2 Direction for Further Research    The rotation of NGC 6397 in the plane of the sky demands more at tention as our preliminary estimates definitely raises curiosity. Better images of the cluster core and more sophisticated methods of identify reference  objects  would  increase  the  reliability  of  the  rotational  proper  motion  estimates.  Another interesting t opic would be to study white dwarfs in the Galactic Bulge. If one can accurately determine a sequence of such stars in the centre of the Milky Way, one can learn a phenomenal amount of information about the age and  formation of our  galaxy.     30  REFERENCES    [1]   Advanced Camera for Surveys Instrument Handbook for Cycle 17, Version 8.0. December 2007. Space Telescope Science Institute. Retrieved from http://www.stsci.edu/hst/acs on March 2008.   [2]   Anderson, J., et al. 2008, in preparation.  [3]   Anderson, J., King, I. (2003) The Astronomical Journal, 126: 722-777.   [4]   Andover Corporation. Retrieved from http://www.andovercorp.com/Web_store/UBVRI/Johnson.php on March 2008.   [5]   Andreuzzi, G., Testa, V., Marconi, G., Alcaino, G., Alvarado, F., & Buonanno, R. 2004, A&A, 425, 509.   [6]   Carroll B.W., Ostlie D.A. (1996) An Introduction to Modern Astrophysics. AddisonWesley Publishing Company, Inc.   [7]  Eisenhauer et al. (2005). The Astrophysical Journal, 628:246-259.   [8]   Ernst, A., Glaschke, P., Fiestas, J., Just, A., Spurzem, R. (2008). N -body Models of Rotating Globular Clusters. Accepted for Publication in the Astronomical Journal. arXiv:0702206v2.   [9]   Gnedin, O. Y., Lee, H. M., & Ostriker, J. P. 1999, ApJ, 522, 935   [10]  Gratton, R.G., Bragaglia, A., Carretta, E., Clemintini, G., Desidera, S., Grundahl, F., &  Lucatello, S. 2003, A&A, 408, 529   [11]  Harris, W.E. 1996, AJ, 112, 1487.   [12]  Kalirai, J. S. et al. 2007, astro-ph/0701781.   [13]  Kalirai et al. (2008). The Space Motion of the Globular Cluster NGC 6397. arXiv: astroph/07017181v2.   [14]  King, I. R., Sosin, C., & Cool, A. M. 1995, ApJ, 452, L33   [15]  King, I., Anderson, J. (2000) Dynamics of Star Clusters and the Milky Way. ASP Conference Series, Vol. 000.   [16] Kolmogorov-Smirnov Test. Retrieved from http://www.physics.csbsju.edu/stats/KS-test.html on March 2008.   [17]  Kraft, R.P., & Ivans, I.I. 2003, PASP 115, 143.   [18]  Milone, A. P., Villanova, S., Bedin, L. R., Piotto, G., Carraro, G., Anderson, J., King, I. R.,& Zaggia, S. 2006, A&A, 456, 517.   [19]  NASA. Retrieved from http://map.gsfc.nasa.gov/media/ContentMedia/dirbe123_2p6dec.jpg on March 2008.   [20]  Northern Arizona University. Retrieved from http://www4.nau.edu/meteorite/Meteorite/Images/bt2lf1509_a.jpg on March 2008.   [21]  Reid, I.N., & Gizis, J.E. 1998, AJ, 116, 2929 .   [22]  Richer, H., Dotter, A., Hurley, J., Anderson, J., King, I., Davis, S., Fahlman, G., Hansen , B., Kalirai, J., Paust, N., Rich. M., Shara., M, (2008) Deep ACS Imaging in the Globular Cluster NGC 6397: The Cluster Color Magnitude Diagram and Luminosity Function. Accepted for Publication in the Astronomical Journal. arXiv:0708.4030v1.   [23]  Shapley H., Star Clusters, McGraw-Hill (1930).   [24]  Sirianni et al. (2005). Accepted for publication in PASP. arXiv: astroph/0507614v1.   [25]  Sparke L.S., Gallagher J.S. (2000) Galaxies in the Universe: An Introduction. Cambridge University Press.   [26]  van Leeuwen, F., Le Poole, R. S., Reijns, R. A., Freeman, K. C., & de Zeeuw, P. T. 2000, A&A, 360, 472.   [27]  Weaver, D., Villard, R. (2007). Hubble Finds Multiple Stellar Baby Booms in a Globular Cluster. Retrieved from http://hubblesite.org/newscenter/archive/releases/2007/18/full/ March 2008.   [28] Hansen et al. 2007, The White Dwarf Cooling Sequence of NGC 6397. arXiv:astro-ph/0701738v2. 

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