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Exploring the Milky Way Halo with SDSS-ll SN Survey RR
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5/08 KIProj/AocaPresIClRCIDateDueindd

EXPLORING THE MILKY WAY- HALO WITH SDSS-II SN SURVEY RR LYRAE
STARS

By

Nathan De Lee

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

2008

ABSTRACT

EXPLORING THE MILKY WAY HALO WITH SDSS-II SN SURVEY RR LYRAE
STARS

By

Nathan De Lee

This thesis details the creation of a large catalog of RR Lyrae stars, their lightcurves,
and their associated photometric and kinematic parameters. This catalog contains
421 RR Lyrae stars with 305 RRab and 116 RRc. Of these, 241 stars have stellar
spectra taken with either the Blanco 4m RC spectrograph or the SDSS/SEGUE sur-
vey, and in some cases taken by both. From these spectra and photometric methods
derived from them, an analysis is conducted of the RR lyrae’s distribution, metallicity,

kinematics, and photometric properties within the halo.

All of these RR Lyrae originate from the SDSS-II Supernova Survey. The SDSS-II
SN Survey covers a 2.5 degree equatorial stripe ranging from -60 to +60 degrees in
RA. This corresponds to relatively high southern galactic latitudes in the anti-center
direction. The full catalog ranges from go magnitude 13 to 20 which covers a distance

of 3 to 95 kpc from the sun.

Using this sample, we explore the Oosterhofl dichotomy through the AlogP
method as a function of |Z| distance from the plane. This results in a clear divi-
sion of the RRab stars into 001 and 0011 groups at lower |Z I, but the population

becomes dominated by 001 stars at higher |Z|.

The idea of a dual halo is explored primarily in the context of radial velocity
distributions as a function of |Z I In particular, V937», the radial velocity in the
galactic standard of rest, is used as a proxy for V¢, the cylindrical rotational velocity.

This is then compared against a single halo model galaxy, which results in very similar

V937. histograms for both at low to medium |Z|. However, at high |Z| there is a clear
separation into two distinct velocity groups for the data without a corresponding
separation in the model, suggesting that at least a twoocomponent model for the halo
is necessary.

The final part of the analysis involves [Fe/H] measurements from both spectra
and photometric relations cut in both |Z| and radial velocity. In this case, there
is less of a clear change as a function of these cuts, although that may be due to
metallicity effects on the shape of the horizontal branch. The metallicity groups may
be truncated at both the metal-rich and metal-poor end of the histograms because
at those metallicities the horizontal branch stars may occur primarily out of the

instability strip, removing them from our sample.

Copyright by
NATHAN DE LEE
2008

To Mom and Dad

ACKNOWLEDGMENTS

There are many people I would like to thank for helping make this possible. Of course
my advisers Horace Smith and Timothy Beers, whose invaluable guidance was key to
the success of this project. I would also like to thank all those who did work at the
24inch telescope for providing a basis to test some of the relationships used in this
work. I am deeply indebted to Young-Sun Lee for all of his help in processing SDSS
data as well as helping me understand the field of galactic structure generally. Ronald
Wilhelm for his help with the RR Lyrae metallicities. Suzanne Hawley and Andrew
West for providing the single-epoch spectra so critical to this work. My collaborators
at Cambridge University and the Univ. of Washington, without their photometric
catalogs, there would be no RRL survey. Barbara Anthony-Twarog and Bruce Twarog
for starting me on the path of Astronomy in the first place. Chris Waters and Adam
Kraus for their insight into statistics. Thanks to Cluze for listening to my endless
talking about this, and finally, Shannon Bell for her patience and keeping me on task
till the end. Thanks to Christopher Waters for the MIEX class used to format this

thesis.

vi

TABLE OF CONTENTS

List of Tables ........................................................ x
List of Figures ....................................................... xi
List of Symbols ...................................................... xiii
1 Introduction ...................................................... 1
2 Galactic Formation and Structure ................................ 3
2.1 The canonical Structure of the Milky Way ............ 3
2.1.1 Bulge .............................. 3

2.1.2 Disk ............................... 5

2.1.3 Halo ............................... 5

2.2 Formation Models of the Milky Way Halo ............ 6
2.2.1 ELS ............................... 6

2.2.2 SZ ................................ 8

2.2.3 Fr0m the Ground up Hierarchical Halo Formation ...... 9

2.2.4 Stellar Streams ......................... 11

2.3 The Dual Halo and Beyond .................... 12
RR Lyrae Stars .................................................. 14
3.1 Basic Properties .......................... 14
3.2 Methods of Analysis ........................ 20
3.3 Methods for Period Determination ................ 22
3.4 Pulsation Theory .......................... 24
3.5 Oosterhoff Dichotomy ....................... 26
3.6 Usefulness as Kinematic Tracers of the Halo ........... 29
The Sloan Digital Sky Survey and Other Surveys ................. 32
4.1 Overview .............................. 32
4.2 SDSS-II SN Survey ......................... 35
4.3 Variable Star Catalogs ....................... 36
4.3.1 Cambridge Catalog ....................... 36

4.3.2 Washington Catalog ...................... 37

4.4 Other RRL Surveys ........................ 37
4.4.] N SVS .............................. 38

4.4.2 QUEST-I ............................ 39

4.4.3 LONEOS ............................ 39

vii

Photometric Analysis ............................................. 40

5.] Variable Star Selection ....................... 40
5.1.1 Welch-Stetson Index ...................... 41
5.1.2 Finding the Mean ....................... 45
5.1.3 Color Cuts ........................... 45
5.1.4 Final Candidate List ...................... 47

5.2 Variable Identification ....................... 49
5.2.1 Finding the period: Supersmoother .............. 49
5.2.2 Finding the period: Template Fitting ............. 52
5.2.3 Period-Amplitude Diagram .................. 53

SpectroscoPic Observation and Data Reduction ................... 59

6.1 CTIO Blanco 4m Data ....................... 60
6. 1. 1 Observations .......................... 60
6.1.2 Image reduction and Spectral Extraction ........... 60
6.1.3 Radial Velocities ........................ 61
6.1.4 Determining Metallicity .................... 64

6.2 SDSS Data ............................. 66

Production of Derived Parameters ................................ 70

7.1 Determining Metallicities & Distances .............. 70
7.1.1 Fourier Metallicities ...................... 71
7.1.2 Period-Amplitude-Metallicity ................. 73
7.1.3 Distance ............................. 75

7.2 Center of Mass Calculations .................... 76

7.3 Frame of Reference ......................... 79

Tracing the Halo ................................................. 83

8.1 Photometric Properties ...................... 83
8.1.1 Spacial Distribution ...................... 83
8.1.2 Oosterhoff Properties ..................... 85

8.2 Kinematic Properties ....................... 88
8.2.1 Radial Velocities ........................ 88
8.2.2 A Question of Bias ....................... 91
8.2.3 The Thick Disk ......................... 95

8.3 Composition Properties ...................... 97

Final Analysis and Conclusions ................................... 101

Configuration files ................................................ 105

A.1 Example phasepar ......................... 105

A2 Parameter Files for Besacon Model ................ 106
A21 Parameters for Part A ..................... 106
A.2.2 Parameters for Part B ..................... 108

Example Lightcurves ............................................. 111

viii

C Data Tables ...................................................... 113

References ....................................................... 170

ix

2.1

3.1

3.2

5.1
5.2

5.3

6.1

7.1

7.2

8.1
8.2

8.3

CI
C2
C3

LIST OF TABLES

Properties of the Halos ............................ 12
Properties of RR Lyrae stars. ........................ 15
Properties of the Oosterhoff Groups ..................... 27
RRL Candidate Search Parameters ...................... 48
Period Search Parameters .......................... 50
Variable Typing Metric ............................ 51
Lines used in Line by Line Radial Velocity Determination ........ 63
(1531 Conversion Chart ............................. 71
Methods of Metallicity Determination .................... 76
Gaussian Parameters to Vgsr Fits ...................... 89
Gaussian Parameters to Vgsr Fits with D < 30 kpc ............ 90
The KS-Test of the Null Hypothesis ..................... 94
Parameters in Data Tables .......................... 113
Photometric Properties of SDSS RRL .................... 115
Kinematic Properties of SDSS RRL ..................... 142

2.1

3.1

3.2

3.3

3.4

3.5

3.6

4.1

4.2

4.3

5.1

5.2

5.3

5.4

5.5

5.6

6.1

6.2

6.3

6.4

LIST OF FIGURES

Schematic Diagram of the Milky Way .................... 4
Schematic HR Diagram ........................... 16
RRL Bailey Types .............................. 18
Multicolor Photometry of RS Boo ...................... 19
Radial Velocity Curve of RS Boo ...................... 20
Example of HJD Versus Phase ........................ 22
The Oosterhoff Groups in the Milky Way and Local Group ........ 28
Comparison of ug'rz'z and UBVRI Filters .................. 33
Schematic Layout of the SDSS Camera ................... 34
Variable Star Surveys Locations ....................... 38
Variability Index versus 9 Magnitude .................... 44
RRL with Color Cuts ............................. 55
Histogram of Color Cuts as a Function of L ................ 56
Templates Used for Fitting Lightcurves ................... 57
Period-Amplitude Diagram for Final Typing ................ 58
Final Period-Amplitude Diagram ...................... 58
CTIO Spectra at Different Stages of Reduction. .............. 62
Spectra Changing as a Function of Phase. ................. 65
Recombination of SDSS Spectra. ...................... 67
SDSS versus CTIO Metallicities ....................... 68

xi

7.1 Fourier Parameter Metallicity Fit ...................... 73

7.2 Period-Amplitude Metallicity Fit ...................... 74
7.3 Radial Velocity Templates .......................... 77
7.4 Radial Velocity Comparison ......................... 78
7.5 Astronomical Coordinate Systems ...................... 82
8.1 An Extinction Map as a Function of RA .................. 84
8.2 Spacial Map of RRL Stars .......................... 85
8.3 Alog P with |Z| Cuts ............................ 86
8.4 A log P with Metallicity ........................... 88
8.5 V93,- Histograms with IZ I Cuts ........................ 90
8.6 Vgsr Histograms with | Z I Cuts with D < 30 kpc .............. 92
8.7 Vgsr versus V¢ ................................. 93
8.8 Model V93,» Histograms with |Z| Cuts with D < 30 kpc .......... 95
8.9 Thick Disk Contamination .......................... 96
8.10 Metallicity Histograms with |Z I Cuts .................... 98
8.11 Metallicity Histograms with IZ I Cuts and Vgsr cut ............ 99
8.12 Metallicity Histograms with V220 cuts .................... 100
B1 Example Lightcurves in ugrz’z ........................ 111
B2 Example Lightcurves in ugrz’z ........................ 112

Images in this dissertation are presented in color.

 

LIST OF SYMBOLS

pc A parsec is 3.086 x 1016 meters. ...................... 5
MV Absolute magnitude in the Johnson V filter ................. 15
Mg Mass of the Sun ................................ 15
Re Radius of the Sun ............................... 15
A A unit of length used for measuring wavelength (1 x 10—10 meters) . . . 18
HJD Heliocentric Julian Date (in days) ..................... 21
mg Apparent magnitude in the SDSS 9 filter ................. 75
Mg Absolute magnitude in the SDSS 9 filter. ................. 75
|Z| The vertical distance away from the galactic plane (in kpc) ....... 87
V¢ The velocity in the qb galactic coordinate (in km /s) ............ 88
Vgsr The radial velocity in the galactic center of rest (in km / s) ........ 89

xiii

 

CHAPTER 1:
INTRODUCTION

 

The Milky Way galaxy has been an eternal backdrop to our stories, our myths, and
our lives since time immemorial. Yet, it was not until relatively recently, only since
the early decades of the last century, that we’ve begun to understand what it is and
as a result, understand our place in the universe. It has not been because of a lack
of interest or intellect that the questions surrounding the nature of our galaxy have
been so hard to answer. Rather it is because the Earth is embedded in the Milky
Way’s spiral disk. The problem with being in the disk of the Milky Way is two-fold.
The first issue is that there is a large amount of dust that absorbs light in the disk,
so there is a visual limit to what we can observe. This can be avoided to some extent
by going to redder wavelengths, but it is still a problem. The second issue, and by far
the most problematic, is that we have only one vantage point. The farthest human
technology has ever made it is to the very edge of our solar system, which from a
galactic stand-point, is an insignificant distance. This leads to the great irony that
we know quite a lot about the structure of other galaxies, millions of parsecs (1 pc
~ 1016 meters) away, but have only in the last few years begun to understand the
structure of our home.

One major component of the Milky Way is the galactic halo. It is a spherically
distributed collection of stars that engulfs the more familiar disk of the Milky Way. It
is composed primarily of older stars that are on a random collection of orbits, although

there are definite kinematically defined structures. These structures, or streams are

likely the result of the our galaxy cannibalising smaller, dwarf galaxies. Beyond this,
there is still rigorous debate about the particulars of the populations of stars that the
halo contains, how it was formed, and what it’s exact structure is.

One method of tracing the structure of the halo is to use some sort of object as a
test particle. One kind of object that is useful for this purpose is an RR Lyrae star.
RR Lyrae stars are radially pulsating stars, that have a known intrinsic brightness.
As they pulsate their brightness changes, tracing out a lightcurve in less than a day.
These properties are important because it means that not only are they identifiable
by their lightcurve shape, but since their intrinsic brightness is known, the distance
to them can be calculated. Distances are a precious commodity in astronomy since
the sky is basically a two—dimensional projection of three-dimensional space. One of
the fundamental questions in mapping any object is knowing the distance from one
place to another, and RR Lyrae are one of the few tools that can serve that purpose
well.

The purpose of this work is two-fold. The first is to create a catalog of RR
Lyrae stars with complete lightcurves in the five SDSS filters ugrz’z. This catalog will
also include photometric parameters such as the period of pulsation for each star, as
well as observed brightness, and thus distance. For a substantial part of the catalog
there will be additional information about the kinematics of the RR Lyrae and their
chemical composition. The second purpose of this work is to use this catalog to
investigate the structure of the halo. The main question is the following: “Is the halo
a single coherent object, or does it contain two or more discrete parts?” The following
chapters will discuss the previous work on this subject, how the catalog was made,

and, ultimately, what the analysis shows.

 

CHAPTER 2:
GALACTIO FORMATION AND
STRUCTURE

In this chapter, the current picture of the Milky Way galaxy and our understanding
on how it was formed will be explored. This is meant as an overview, and as such
will not worry too much about the details of the current work in these fields. In the
case of the halo, however, particular attention will be paid to its structure and even

more so to formation models.

2.1 THE CANONICAL STRUCTURE OF THE MILKY WAY

The Milky Way is a barred-spiral galaxy composed of three distinct luminous parts,
each representing very different populations of stars, and each with it’s own structure
and formation history. Figure 2.1 gives a schematic break down of the Milky Way.
Each of these parts: the bulge, the disk, and the halo will be considered in turn in

the following sections.

2.1.1 BULGE

The bulge of the Milky Way is located at the very center of the galaxy. It is a
bar-shaped region with an axial ratio of 1:0.35:0.26 (Rattenbury et al., 2007). This
region contains a population of stars the covers a wide range in metallicity —1.5 S

[Fe/ H] _<_ 0.75 (Zoccali et al., 2008). The metallicity, [Fe/ H], is a shorthand method

8.5 kpc
Halo Bulge

 

  
  

  

'
7

Sun Thin Disk

Thick Disk

 

Galactic
Center

Figure 2.1 A schematic version of the Milky Way labeling the many constituents
including the bulge, the disk, and the halo.

of describing the amount of non-primordial elements in a star. It is usually defined
as the logarithm of the ratio of number density of Fe to H in a star compared to
that value in the Sun, or log(NFe/NH)3tar — log(NFe/NH)@. In general, most of
the other elements not generated in the big bang track the iron to hydrogen ratio.
The mean metallicity of the Bulge is around +0.25, which is roughly twice as metal
rich as the Sun. The bulge is filled with stars of a wide variety of ages. This may
be due to a series of star formation epochs or intermixing between the bulge and the
disk. The bulge rotates with a peak velocity of 75 km/s and exists as an intermediate
between a purely rotating disk and the much more kinematically hot halo (Minniti &
Zoccali, 2007) In general it is relatively hard to work on things in the bulge because
there is a large amount of interstellar dust along the line of sight toward the galactic

center, which blocks our view, especially in the optical wavelengths, although recent

advances with infrared observatories have opened up the field.

2.1.2 DISK

The disk is the part of the galaxy most familiar to the public. Its spiral arms perhaps
being the component that most catches the eye. Of course there are no face—on images
of the Milky Way, since the Sun is located in the disk, so our mental construct of what
the disk looks like is formed from a combination of observational data and analogy to
other spiral galaxies. The disk itself is composed of a series of substructures of different
ages and components. The young thin disk is very metal-rich with [Fe / H] > +0.1,
with a scale height of 50pc, and is the site of current star formation. The old
thin disk has a scale height of 325pc and is a little more metal poor with —0.5 <
[Fe / H] < +0.3. The final component is the thick disk which is more metal-poor with
an ——1.6 < [Fe/H] — 0.4 and a scale height of 1.4 kpc (Ostlie & Carroll, 1996).

2.1.3 HALO

The halo is one of the oldest and most metal-poor, although arguably the most
complex, component of the galaxy. It is roughly spherical and is kinematically dis-
tinguished by being mostly non-rotating and having high velocity dispersions. For
some time it has been known that the halo consists of two parts, an inner and an
outer halo, although the origins and distinction between them has been in dispute.
In general, the inner halo is taken to be flattened with an axial ratio between 0.6
and 0.7 (Helmi, 2008). The outer halo is much more spherical with an axial ratio
of z 1. There is also a significant dark matter component to the halo. The dark
matter component of the galaxy cannot radiate away its energy since it only interacts
gravitationally, and as a result it is the dominant source of mass in the halo. One of
the major differences between older and more recent models of halo formation is the

consideration of the dark matter component of the halo. In the following sections, we

will discuss the formation and structure of the halo.

2.2 FORMATION MODELS OF THE MILKY WAY HALO

The question of how the Milky Way halo formed is still an open area of research,
and is the question that this work will ultimately address. There are few fields in
science where one can find referenced work back to the 18th century, but this is
one such place. In particular, Immanuel Kant in his 1755 treatise Universal Natural
History and Theory of the Heavens discussed the formation of the galaxy from a more
dispersed state under the influence of gravity. Since then, there have been a number
of theories over the years and in the following section we will discuss the canonical

papers and more recent work.

2.2.1 ELS

One of the first major attempts to understand the formation history of the halo based
on observations of halo stars was by Eggen, Lynden-Bell & Sandage (1962) or ELS.
Eggen et al. (1962) put forward the idea that the interaction time scale between stars
and the galaxy is long compared to the age of the galaxy. Thus it should be possible
to reconstruct the original dynamics of the proto-galaxy by looking at old stars. They
examined the full space motions (U,V,W) of 221 dwarf stars in the solar neighborhood
using information about their orbits and their metallicity to determine where they
initially formed.

The metallicity of a star is a measurement of the fraction all elements heavier than
hydrogen and helium in that star. These heavier elements are referred to collectively
as metals. Knowing the metallicity of a star tells one two things; first it gives some
suggestion as to the interstellar medium that the star was formed from. Different
stellar formation sites tend to have a particular metallicity, for instance all the stars

in a given globular cluster have the same metallicity. The second is that metallicity

6

can be used as an age dating device. Since only hydrogen and helium (with a small
amount of lithium and other light elements) were created in the Big Bang, all other
elements must have been created by processes since then. The primary site for this
is the nuclear fusion that powers stars. These elements are then released into the
surrounding medium by supernova explosions. As time progresses, more metals are
injected into the interstellar medium (ISM) and the stars that form from this ISM
have a higher metallicity. It should be noted that the metallicity of a star does not in
general change over time, so the metallicity is a measurement of the metal abundance
of the local ISM when and where the star was born. Thus on average low metallicity
stars tend to be old, and high metallicity stars tend to be young.

To get metallicities, ELS used a technique called the ultra-violet excess, 6(U-B),
which compares the difference in the (U-B) color of a star against what would be
expected from a (U-B) versus (B-V) color-color diagram of the Hyades open cluster.
A higher 6(U-B) is indicates a lower metallicity. Their first area of study was to
look at the eccentricities of the orbits of their sample stars. They observed was that
6(U-B) was highly correlated with the eccentricity of a star’s orbit which implied that
eccentricity is correlated with age. Furthermore, they found the maximum Z, vertical
height above the disk, for metal—rich stars tends to be near the plane (disk) of the
Milky Way, while metal-poor stars can be found at all heights.

In their model for Galaxy formation, the Galaxy starts as a large self-gravitating
cloud of gas that condenses down forming stars and ultimately the disk of the Milky
Way. Once a star forms, it only interacts by gravity, so its orbit tends to maintain the
kinematics of the gas it formed from. Gas, on the other hand, can interact through
friction, and so tends to lose energy and angular momentum over time.

Using this information, ELS came up with the following scenario. A large gas
cloud roughly 1010 years ago began to collapse. The first stars began to form, as

well as the globular clusters, and as the stars formed they would have eccentric orbits

depending on their location in the cloud. As time progressed, the remaining gas
continued to contract while radiating away energy, and centrifugal forces led to the
formation of the disk. This then is where the second batch of stars formed with higher
metallicity and now relatively circular orbits.

This scenario predicts that there should be a relationship between [Fe/ H] and
galactic radius and that overall there should be a small range in ages for halo objects.
However, more recent work has suggested that there was perhaps a bias in the sample

of stars used by ELS (Chiba & Beers, 2000 and references therein).

2.2.2 SZ

ELS reigned as the primary description of the formation of the Galaxy until a paper
by Searle & Zinn (1978), SZ, came out. In this paper, they analyzed low-resolution
spectra of 177 red giant stars located in 19 globular clusters in the halo of the Milky
Way. They used an interstellar-reddening independent method of determining the
metallicity of the giants and thus the globular clusters in which they are found. One
expectation from an ELS galactic formation scenario is that there should be a metal-
licity gradient as a function of galactocentric radius, RG0, with metal-rich globulars
having small RG0 and metal-poor ones having large RG0. When SZ combined their
work with results from the literature they realized that their were in fact two distinct
populations of globulars. They define the inner halo clusters to be at a RGC < 8 kpc
and outer halo clusters to be RGC > 8 kpc. The inner halo globulars do appear to
have a metallicity gradient, but the outer halo ones do not. This suggests that the
outer halo clusters were not formed from a quickly, 108 years, contracting gas cloud,
rather they were formed in some other fashion.

They also found another difference in the two groups of clusters, the morphology
of the horizontal branches of these clusters was different (for a description of the

horizontal branch see Section 3.1). It has been known for some time that the shape of

the horizontal branch is a function of metallicity and some unknown second parameter
(van den Bergh, 1967; Sandage & Wildey, 1967). For more information on the second
parameter problem see Section 3.5). SZ found that the correlation between metallicity
and horizontal branch morphology had very low dispersion for the inner halo, but
the outer halo showed a large spread in morphologies at a given metallicity. They
postulate that the second parameter may in fact be age, and that the outer halo
globulars may have formed well after the initial contraction of the galaxy.

One final argument in SZ, suggests that the observed chemical abundance, metal-
licity, distribution can not be explained by a homogeneous collapse of a single cloud.
Rather an accretion of smaller clouds muld free-fall into the larger cloud. This, com-
bined with their two previous observations, led them to a theory of the formation of
the halo by accretion of smaller clouds onto the already collapsed inner region. The
stars and clusters formed in these accreted clouds would then dynamically mix with
the Galaxy and form the constituents of the outer halo. Another result of this model
is that there could be a much larger spread in stellar ages than in the homogeneous
collapse scenario. This scenario continued to be explored in a series of papers by Zinn

and collaborators, most notably Zinn (1980, 1986).

2.2.3 FROM THE GROUND UP HIERARCHICAL HALO FORMATION

Both the ELS model and SZ model dominated the theoretical landscape for quite some
time, but each had its own failings, which became more apparent as larger samples of
stars with good kinematics became available (Chiba & Yoshii, 1998; Chiba & Beers,
2000). In particular the relationship between eccentricity of stellar orbits and [Fe/ H],
which had been one of the major motivators of ELS, was found to be non-existent
in larger samples and may have been an effect of their the proper-motion selected
sample (Chiba & Beers, 2000). Also the metallicity gradient as a function of galactic

radius predicted by the ELS model, did not materialize. As for SZ, the existence of a

relationship between rotation velocity and distance above the plane, and observations
that their appeared to be two distinct halos, seemed to be at odds with their idea
of chaotically merging sub-galactic fragments. Also there are issues with forming a
quickly rotating disk in the SZ scenario (Freeman, 1996).

While better observations were challenging these canonical models, there were ma-
jor changes occurring in cosmology. In the 1990’s a new paradigm for the formation of
the universe was taking hold. The new model called lambda-cold-dark—matter,ACDM,
which favored a formation of galaxies through hierarchical assembly, described a uni-
verse in which small dark matter halos formed in filaments called the cosmic web,
originating from smaller perturbations in the original distribution of matter in the
universe. In this scenario, since dark matter can only interact through gravity, larger
halos of dark matter were built up slowly from accreting smaller halos, thus leading
to the idea of galaxies being built from the bottom up (White & Rees, 1978; Peacock,
1999)

Building on the large kinematic studies of Chiba & Yoshii (1998) and Chiba &
Beers (2000), a new theoretical framework was proposed by Bekki & Chiba (2001).
They used a dark matter simulation consistent with ACDM and allowed it to evolve
from a redshift of 25 to the present. Into this model, baryonic gas was added along
with a prescription for changing that gas into stars, and for increasing the metallicity
of the gas as a function of star formation. By using the complete kinematic and
metallicity information from their model, they were able to track the formation of
the three primary parts of the galaxy: the bulge, the disk and the halo. In Bekki
& Chiba (2001), the stars initially form in small clumps and some of these clumps
slowly meld together to form larger clumps. Ultimately they had two relatively large
sub-galactic fragments surrounded by some smaller clumps. The smaller clumps are
disrupted by the larger fragments and spread their stars into what will become the

outer halo. The two sub-galactic fragments then merge which produces an inner

10

halo. During this merger there is also a massive in-fall of gas into the center creating
a starburst that eventually becomes the bulge. After the merger, more small clumps
accrete onto the inner halo, dumping their gas into a disk producing a primordial
thin—disk where more star formation occurs. The production of this disk has the side
effect of taking the initially spherical inner halo and flattening it. At a later point,
one or more relatively large clumps collide with the disk puffing it up into a thick
disk. The rest of the gas falls back into the plane creating a new thin disk, and the

galaxy as we know it has formed.

2.2.4 STELLAR STREAMS

The galactic formation model described by Bekki & Chiba (2001) is very good at
explaining the structures that are seen in the galaxy today. However, there is one
component that is not accounted for. The simulation created by Bekki & Chiba
(2001) is entirely self-contained and, as a result, does not take into account the
possibility of accretion of dwarf galaxies, in the present day. It has been known for
some time that the Milky Way and the Andromeda galaxy are surrounded by, though
not necessarily always gravitationally bound to, a large number of dwarf galaxies, the
whole amalgamation being referred to as the local group. It is clear that over time,
the smaller dwarf galaxies should be consumed by the much larger spirals as they
gravitationally interact (Bullock & Johnston, 2005).

The first indication that this actually did occur was found by Ibata et al. (1994)
when they discovered the Sagittarius stream. This stream of stars with similar kine-
matics was then fully described by Majewski et al. (2003). In the years since, many
more streams have been found (Newberg et al., 2002; Belokurov et al., 2006). Even
more recently, work with the Sloan Digital Sky Survey, SDSS, has discovered several
more dwarf galaxies within the halo of the Milky Way (Belokurov et al., 2007).

Although the ACDM and the galactic formation models discussed are generally

11

in agreement, the specifics are still in dispute. One major issue is the missing halos
problem. Simulations by Klypin et al. (1999) and Moore et al. (1999) suggest that
there should be many more dwarf galaxies than have been found. A number of
attempts have been made to figure out these difficulties (Bullock & Johnston, 2005),

but there are still issues to resolve. '

2.3 THE DUAL HALO AND BEYOND

The concept of the halo being composed from two distinct parts, an inner and an
outer, has been around since SZ. Zinn (1993) did a good job of summarizing the
situation up to that point, focusing primarily on the properties of globular cluster
systems. One of the most recent works on this subject was done by Carollo et al.
(2007). In their work they used over 20,000 stars with spectroscopy out of the SDSS
survey (see Section 4.1). They found strong evidence for the existence of two chem-
ically and kinematically distinct halos, the properties of which are shown in Table
2.1.

Table 2.1: Properties of the Inner and Outer Halos

 

 

 

Property Inner Outer
Location Distance < 10-15 kpc Distance > 15-20 kpc
Rotational Velocity 0 to 50 km / s -40 to -70 km/s
Peak metallicity -1.6 -2.2
Axial Ratio ~ 0.6 ~ 0.9 — 1.0

 

These two halos have very different histories and the formation scenario is similar
to Bekki & Chiba (2001). In this case, the inner halo is comprised from the dissipative

accretion of sub-galactic fragments. Since these fragments still have interstellar gas

12

in them, star formation continues and the mean metallicity increases. Ultimately
the disk forms, which leads to an overall flattening of the halo. Later dwarf galaxies
are accreted onto the galaxy and form the outer halo. Since the dwarf galaxies have
already made all of their stars, the merger is dissipationless. Also, there is less star
formation and thus less metal enrichment in the dwarf galaxies compared to the
Galaxy so their overall metallicity is lower. Finally, the retrograde motion may be
a result of the stronger dynamical friction for stars on prograde as versus retrograde
orbits (Quinn & Goodman, 1986).

The finding of a statistically significant retrograde orbit was one of the primary
results of the Carollo et al. (2007) study. To put this into context, their Table 1
summarizes the previous work done on determining the retrograde motion of the
outer halo. The final result being, that although previous studies provided hints of
an outer halo, Carollo et al. (2007) were able to provide the reliable numbers found

in Table 2.1

13

 

CHAPTER 3:
RR LYRAE STARS

 

RR Lyrae (RRL) stars are intrinsically pulsating variable stars, stars that change in
brightness over short timescales, that have a long history in the study of astronomy
dating back to the end of the nineteenth century. These stars were eventually real-
ized to be of extraordinary usefulness in a variety of astronomical and astrophysical
pursuits. These stars have proven immensely useful in the understanding of distances
in the Milky Way and beyond. Easily identifiable by their unique lightcurve shape,
RRL stars are excellent standard candles because of their relatively uniform absolute
magnitude. Also, their relatively high intrinsic brightnesses makes them visible out
to the edges of the Local Group. The pulsational properties of RRL have also been
used to test aspects of stellar evolution theory. In particular, the masses of RRL stars
that pulsate simultaneously in two distinct modes can be determined from pulsation
theory and compared with the expectations of stellar evolution models.

In the following sections, the underlying nature of RRL stars will be discussed as
well as the basic techniques used to analyze these stars. Furthermore, the source of

their variability and their use as kinematic tracers will be discussed.

3.1 BASIC PROPERTIES

RRL stars are core-helium burning stars that are radially pulsating due to opacity-
driven mechanisms (see section 3.4). They are older (population II) low mass stars

which have a range of metallicity, 0.0 > [Fe/ H] > —2.5. This should not be surprising

14

since they were originally found in globular clusters, although they have been found in
more metal-rich places like the thick disk. Regardless, they are generally considered
representative of Baade’s population 11 stars. RRL stars are periodic variables with
periods between 0.2 and 1.2 days and amplitudes in the V magnitude between 0.2 and
2.0 mags (Kholopov et al., 1998). The properties of RRL are summed up in Table 3.1.
RRL are giant stars that lie on the horizontal branch (HB) in a Hertzsprung-Russell
(HR) diagram. An HR diagram originally plotted the absolute magnitude,MV, versus
spectral type, but it has evolved over the years to include other quantities. For
instance, a more theoretical HR diagram can be used to deduce the evolutionary state
of a star by plotting luminosity versus photospheric surface temperature. In another
variant called the color-magnitude diagram (CMD), the theoretical quantities are
replaced by observable substitutes of a magnitude in some photometric pass-band, V
for instance, for the luminosity and the subtraction of two pass-bands to yield a color,
for example B-V, as a proxy for the temperature. A schematic example of this can
be seen in Figure 3.1. The horizontal branch is so named because it is a relatively
narrow band that lies roughly horizontal in the HR diagram, which is one of the first

indications that RRL stars can be used as standard candles.

Table 3.1: Properties of RR Lyrae Stars (Smith, 1995)

 

 

 

Property Value
[Fe/H] 0.0 to -2.5
log 9 2.5 to 3.0
My +0.6 :1: 0.2 mag
TC” 7400 K to 6100 K

Amplitude 0.2 to 2.0 mag in V

Mass z 0.7 MG
Period 0.2 to 1.2 days
Radius 4 to 6 Re

 

l5

Table 3.1: Properties of RR Lyrae stars. (continued)

Property Value

 

 

Spectral Type A through F

 

    
   

 

 

 

r r I I I
Asymptotic Giant
Branch
—2 l-' —i
RR Lyrae Stars Rod Giant
M, +2 - V _
Horizontal . <""" Summits
Branch
+4 __ ..
Main
+6 __ Sequence -
L , I I I I
+8 0.0 0.4 0.8 1.2 ‘l .6.
(Br-Via -

Figure 3.1 A schematic HR diagram showing the position of RRL stars in the insta-
bility strip. Figure from Smith (1995).

RRL stars appear at the cross-section of the horizontal branch and another dis-
tinct region in the HR diagram called the instability strip. This nearly vertical strip
represents those areas in the HR diagram where stars become unstable to pulsation.

As a result, it is home to many classical variable star types including Cepheids and

16

main-sequence pulsators such as 6 Scuti and SK Phe stars. A more in depth discussion
of the instability strip is in section 3.4.

Although there is some dispute about who originally discovered RRL stars, one
astronomer who is particularly associated with them is Solon I. Bailey. In 1893, he
began a photographic study of globular clusters at the Harvard College Observatory
in Arequipa, Peru (Smith, 1995). Through this project he came to discover many
hundreds of RRL stars, and in his 1902 paper on the variable stars in Omega Cen-
tauri he describes the major breakdown of RRL into three discrete groups based on
lightcurve shape: RRa, RRb and RRc, now referred to as the Bailey types (Bailey
& Pickering, 1902). These are shown in Figure 3.2. It was later realized that the
RRa and RRb stars were in fact the same class of stars and they became known as
RRab. RRab stars pulsate in the fundamental radial mode, and tend to have a saw-
tooth like lightcurve shape. They also have larger amplitudes and periods on average
compared to the RRc stars. As seen in Figure 3.2, the RRc stars have a much more
sinusoidal shape and have a smaller amplitude on average than the RRab. The RRc
stars pulsate in the first overtone radial mode and thus in more recent works RRab
are sometimes called RR(O) and RRc, RR(1). The General Catalogue of Variable
Stars defines RRab stars as having periods from 0.3 to 1.2 days and amplitudes from
0.5 to 2 magniutes in the V filter. For RRc stars, the periods range from 0.2 to 0.5
days with amplitudes no greater than 0.8 magnitudes in V (Kholopov et al., 1998).

RRL stars are intrinsically variable, so they physically grow larger and smaller
as they go through there pulsation cycle. This has a number of effects on the mea-
surements of the star. In the 1980’s several attempts were undertaken to get a more
complete picture of RRL stars. These programs were interested in using the Baade-
Wesselink method of determining stellar radii. As part of these programs, multicolor
photometry, and high resolution spectroscopy were taken to determine magnitudes,

temperatures, and radial velocities. An example of this is work on the RRL star RS

17

 

 

 

 

 

 

 

| l l l l I
‘L 3
1’»
1.3
0
'0
3
.5, v
m Jr b
E
> 0.9
8
N10};
I I I I I I
Phase

Figure 3.2 This figure shows the three Bailey types of RRL stars originally described
in Bailey & Pickering (1902). Figure from Smith (1995).

B00 by Jones et al. (1988). The photometry in the B,V, and K filters can be seen in
Figure 3.3. The lightcurves move from the blue to the red part of the spectrum as one
goes from B to K (for more information about the B and V filters look in section 4.1).
The K filter is a broad band filter that is centered on 22,152 A out in the infrared.
Clearly the amplitude of the lightcurve changes significantly as one proceeds to the
red part of the spectrum, and perhaps more importantly the shape of the lightcurve
is different in each filter. This means that RRL stars go through a significant color
change as they go through their pulsation cycle. As expected, this is mirrored in their

spectral type determined by hydrogen absorption lines where an RRab can go from

18

A7 or A8 at maximum to F5 or F6 at minimum light. RRc stars are a little bluer

and have a smaller amplitude in general going from A7 or A8 at maximum to F1 or

F2 at minimum (Smith, 1995).

 

 

 

 

 

 

9.6 .2. + + 4:
1O " + ++++ + .1
m 10.4 ' ++ ++ 4
10.8 ' +++ ++ + .,
+ ++ +
11.2 - ”t + ++ ++ ++ ++ +. ...++“ -
9.6 ‘Pfi. + 4..“
++ + +
10 ' +++++ + d
> 10.4 - ’* ++.++ + 4
++

10.8 ' IHH- + ++ ++ ++ ++ ++ ++g++++ ..
92 :11: + + + ++ .-
9.4 - + + + +++ ++ ++ In

5‘ 9.6- ” W"““t1++++«-...++.+

9.8 -
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Phase

Figure 3.3 Photometry of the RRL star RS B00 in the B,V, and K filters. This figure
is based on data from Jones et al. (1988).

Since RRL stars are physically pulsating it is also possible to measure their
Doppler shift by taking spectra. The resultant radial velocity curve is largely the
inverse of the photometric lightcurve. For example, the radial velocity lightcurve of
RS B00 is shown in Figure 3.4. The phase in this figure is the same as in Figure 3.3.
The radial velocity curve reaches its minimum at the maximum of the photometric
lightcurve, which means that the star is already shrinking by the time maximum
light is reached. Ultimately the pulsation in the star represents a change in radius of

roughly 15% when compared to the median radius for the star. This intrinsic radial

19

velocity change will become important when using RRL as kinematic tracers, and

must be corrected for .

 

30 I Y I I I T I I 1

20

1O

Vrad (ans)
a

 

 

 

 

-2o
-30
-4o
-50 I l l l l l I l l
o 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Phase

Figure 3.4 Radial velocity curve for RS Boo phased in the same way as Figure 3.3.
This figure is based on data from Jones et al. (1988).

3.2 METHODS OF ANALYSIS

The fact that RRL stars are variable is both what makes them uniquely powerful tools
and also makes them much harder to study than the more static objects, which make
up much of the rest of astronomy. Working with objects in the time-domain presents
several interesting challenges. The first is defining a consistent way of treating time.
At first pass it would seem that using days, months, and years would be an adequate
way of dealing with time, but there are some significant difficulties including the fact

that months do not have the same number of days and that leap years exist. Dealing

20

with these peculiarities can be cumbersome especially when dealing with data over
the course of many years. To deal with this, the idea of a Julian Date (JD) was
devised. The Julian date tells you how many days, including fractions of a day, have
passed since noon Monday, January 1, 4713 BC. There are a number of standards for
measuring the time of day, which can be used to make Julian dates. Two of them that
are used in this project are the International Atomic Time (TAI) and the Coordinate
Universal Time (UTC), which is the solar mean time in Greenwich, England. These
two times differ by roughly 30 seconds, and as such are used interchangeably to make
Julian dates as is convenient in this analysis.

There is a more subtle issue when trying to measure the exact time of an observa-
tion, which plays an important part in making observations of short period variable
stars. This issue arises from the finite speed of light. When we measure the time that
an observation took place we implicity assume that the time we measure on Earth is
directly related to the time at the object we are studying. This is not true because
as the Earth moves around the Sun, there is up to a 16 minute travel time difference
depending on where the Earth is in its orbit. Since 16 minutes is a significant fraction
of certainly the lower period RRL, this must be corrected for. The corrected obser-
vation time is called the Heliocentric Julian Date (HJD ). The HJD is the date and
time of the observation if it had been taken at the Sun, instead of on Earth. This
time correction needed to go from the JD to HJ D is called the heliocentric correction.

When observing a variable star, the intention is to get a complete lightcurve for
it. A problem is that the periods of many RRL are long enough that a single night
is not long enough to measure a whole lightcurve. Beyond that, in a practical sense,
one would want to combine data from multiple nights, and so there is a method called
phasing that puts all of the observations onto one period. Lightcurves for periodic
variables like RRL are plotted from 0 to 1 in phase where this represents one complete

cycle of pulsation. This is shown in Figure 3.5. Phase, like any cyclic function, can be

21

Shifted by a constant and still be correct, so often the HJ D of maximum brightness,
epoch of maximum, is used as the zero-point for the phase. In order to phase the

observations, it is necessary to have an accurate period for the variable.

 

 

 

 

 

i i
. z .- I .
i i
.. 4- .
I i
- i _. I -
r I
- r r .. I; .
i i
r r
r E
. ; -‘- II '
I i I I? i H i; if f
54034 54037 54040 54043 O 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
HJD-2400000 Phase

Figure 3.5 These two plots shows an example variable star. On the left is a plot of
magnitude as a function of HJD. These same points are then phased using a period
of 0.6114 days producing the phase diagram on the right.

3.3 METHODS FOR PERIOD DETERMINATION

In general, determining the period of a periodic variable star is relatively difficult,
and so there have been an number of methods created to handle this, four of the
most prevalent will be discussed here. The Lomb-Scargle periodogram (Lomb, 1976;
Scargle, 1982) is one of the original ways of determining periods, P (P = 27r f where

f is the frequency). It is based on the “classic” periodogram which uses a discrete

22

Fourier transform to determine the most likely frequencies which can then be turned
into periods. Scargle (1982) advanced upon this by developing a new statistic that
was equivalent to a least-squares fit of a sine wave to the data. This was further
advanced by Press & Rybicki (1989) who developed a method of rapidly calculating
the Lomb-Scargle periodogram using Fast Fourier Transforms, FFT. The downsides to
this method are that it does not use photometric errors to calculate the periodogram
and thus requires a relatively uniform set of data, and that its frequency estimation is
highly dependent on how well a cosine describes the shape of the lightcurve (Reimann,
1994)

Another prevalent method is the Phase Dispersion Minimization (PDM) method
described in Stellingwerf (1978). This method carves up a phase interval into a series
of bins. It then calculates a O statistic to describe the amount of scatter in the points.
The bin with the lowest scatter, which is the one with the lowest 6, should be the
one with the correct period. The PDM method is quite commonly used because it
is distributed with the popular astronomy software IRAF. The drawback to PDM,
from the point of view of a survey, is that it is very user intensive.

A more recent method uses cubic splines to determine the period. The method
as discussed in Akerlof et al. (1994), uses a series of knots spaced over the phased
lightcurve and then calculates cubic splines in between the knots. The X2 is then
taken between the spline and the data. The period that gives the best fitting between
the splines and the data is the period of the lightcurve. The choice of knots is very
important because too few results in a poor fit to the Shape of the lightcurve and too
many can over constrain the fit. In the cubic spline method, the knots are equally
space along the phase interval, which can lead to numerical issues with unequally
spaced data (Reimann, 1994).

The final method is the one used in this survey, called Supersmoother (Reimann,

1994). It is part of the smoother class of period finding methods, which includes

23

methods like running means or running linear regressions. Supersmoother uses a
variable-span linear smoother that calculates a short, medium, and long smooth and
then uses the best fit to the data. The period is determined by whichever frequency
gives the best sum of absolute residuals. Supersmoother has the benefit of being
entirely automatic, and does not make assumptions about the shape of the lightcurve.
The use of variable spans, allows it to perform well even with unevenly sampled data.

For a more detailed discussion of the Supersmoother algorithm see Reimann (1994).

3.4 PULSATION THEORY

Understanding the radial pulsation of RRL stars, and other intrinsically variable
stars, got one of its first major pushes by August Ritter in 1873 (Smith, 1995). He

put forward an equation that relates period to the density of the star.

Px/p/po=Q

Where P is the period of the variable in days, p is the density of the star and p53
is the density of the Sun. Q is a constant, which is usually around 0.04 for RRab
stars. Although, there was discussion over the years as to the exact nature of RRL
stars, the next major discussion of possible pulsation mechanisms was in The Internal
Constitution of the Stars by Eddington (1926). Using polytropic models of stellar
structure Eddington showed that without a driving mechanism the decay time of an
oscillation would be on the order of 8000 years, suggesting that it would be nearly
impossible to see a star in this state (King & Cox, 1968). He went on to suggest a
thermodynamic model by which the pulsation could be driven. This is now referred to
as an Eddington “valve”. He envisioned the valve as either being something that put
heat into the star when it was at full compression and diminished at full expansion,

or the opposite, a heat leak which is at a minimum during compression and maximum

24

during expansion. He suggested that the opacity of the star could change and act as
the source of this leak with the highest opacity occurring at maximum compression
(Smith, 1995).

Eddington initially put the valve at the center of the star suggesting that it was
based on the fusion rate in the core. It is now known that the actual location of the
valve is in the outer envelope of the RRL star, in a region where helium is doubly
ionized as was suggested by Zhevakin (1953) and independently by Cox & Whitney

(1958). The valve is related to the Rosseland mean opacity, 11:, which is defined as:

H = IiopnT's
where p is the density and T is the temperature. In regions of stars without a
dominant element being ionized, the values of n a: 1 and s z 3.5. This clearly does
not work as a valve because the point of maximum compression is also a maximum in
temperature, and thus the opacity would be low. In the case of a region with partial-
ionization of a dominant element, however, the 3 drops to 0 or even slightly negative.
This means that opacity is primarily a function of density providing exactly what
we need for an Eddington valve, a maximum in opacity near maximum compression.
This is referred to as the It-mechanism (King & Cox, 1968).

Although the n-mechanism is dominant in RRL, there is another significant driver
of pulsation. This mechanism is based on the way heat behaves in a medium with a
dominant element partially ionized. Heat added to this layer goes into ionizing more of
the element rather than changing the temperature, meaning that the partially-ionized
region remains cooler than neighboring regions. During compression, this results in
more energy being dumped into the partially-ionized region as heat flows from hot to
cool, resulting in another driving mechanism for pulsation. This is generally referred
to as the ’y-rnechanism (King & Cox, 1968).

Using this knowledge about the pulsation mechanisms in RRL we can then ask

25

the question: Why does the instability strip exist in the HR diagram, and what are
the red and blue edges of this strip? The answer lies in the properties of the partial-
ionization zone. In particular, it is the location of this zone which has the most effect
on whether a star begins to pulsate (Smith, 1995). If the temperature of the star
is too high, then the partial-ionization zone gets driven to the surface of the star,
and eventually there is not enough material to drive pulsation. This defines the blue
edge of the instability strip. The red edge is somewhat less well understood, but the
current best scenario is based on energy transport within the star. In hotter giant
stars, like RRL, the cores of the star move energy via convection, and as one progresses
out toward the surface the dominant mechanism becomes radiative transport. Since
opacity only affects radiative transport, the idea is that the red edge of the instability
strip occurs when the star becomes cool enough for convective transport to become
significant in the envelope of the star where the partial-ionization zone is located.
This means that a significant amount of energy can be transported by convection,

which effectively disrupts the Eddington valve.

3.5 OOSTERHOPF DICHOTOMY

As work continued in the early part of the twentieth century, it was soon noticed that
RRL stars in different globular clusters had different mean properties for the periods
and for the ratio of RRab to RRc type variables. This was in particular noticed by
P. Th. Oosterhoff (1939) as he studied a handful of globular clusters. He found that
there appeared to be two distinct groups. The first group, later called Oosterhoff
type I (001) had shorter average periods for the RRab stars, and much lower fraction
of RRc stars. The second group Oosterhoff type II (0011), had a higher average
period for RRab stars and roughly double the fraction of RRc stars. A prototypical
globular clUster for OoI is M3 and for 0011 is M15. It was later realized that the

Oosterhoff dichotomy was also a dichotomy in metallicity with the 001 clusters being

26

more metal—rich than the 0011 clusters. Table 3.2 shows the basic parameters of the

two Oosterhoff types (Smith, 1995).

Table 3.2: Properties of the Oosterhoff Groups

 

 

Group <Pab> <Pc> nRRc/nRRL [Fe/H] Example GC

 

001 0.55 0.32 0.17 > —1.7 M3
OOII 0.64 0.37 0.44 < —1.7 M15

 

When the globular clusters in the Milky way are plotted up on an average RRab
period versus metallicity plot, it becomes clear that there is a zone of avoidance in
both period and metallicity as shown in Figure 3.6a. This clustering of the globular
clusters in this diagram suggests that they were perhaps formed in diflerent settings
and could give clues to the formation of the halo (Lee 82. Carney, 1999).

The existence of the Oosterhoff dichotomy is not well understood, and to add to
this issue it appears that may be particular to the Milky Way itself. A number of
globular clusters in the Local Group have been studied and they do not appear to
Show the same Oosterhoff dichotomy that the Milky Way ones did. Moreover, looking
at Figure 3.6b it becomes abundantly clear that the clusters in the dwarf galaxies of
the local group in fact prefer to fill the Oosterhoff gap. This has implications for the
formation of the Milky Way halo, and is subject that is currently being explored.

There have been a number of theories as to the origin of the Oosterhofl dichotomy.
Some of these include the existence of a hysteresis zone in the instability strip that
keeps blueward evolving RRab stars and redward evolving RRc stars from changing
their pulsation mode (van Albada & Baker, 1973). This would suggest that 001
clusters are evolving red to blue and 0011 clusters the other way, but there has not

been sufficient evidence to support this. Another possibility is that the brightnesses

27

 

0.80IIIIIIIIIIIIIIIIIIlll
a)

Ill'llllllllI'IIIIIIIll

    
 

  

TIIIIIIIIIIIITI

   

 

 

 

 

 

IIJIIIIIIIIIIIIIIIIII

 

I I I I I I I I I I I I l I I I l I I I I
-0.5 -1.0 -1.5 -2.0 -2.5 -0.5 -1.0 -1.5 -2.0 -2.5
[Fe/H] [Fe/H]

Figure 3.6 Globular clusters from the Milky Way (circles and squares) plotted with
globulars from the Local Group (all other symbols). Figure a) includes only the
globular clusters from the Milky Way. Figure b) adds in the clusters from the Local
Group including Fornax, Sagitarius, and the Canis Majoris dwarf spheroidal galaxies.
There are also some clusters from the LMC. The clusters from the Local Group appear
to favor the traditional gap seen in the Milky Way. The period gap is shown as a bar.
Figure from Catelan (2005).
of the horizontal branches of globular clusters in each group are different. Sandage
has a number of papers dealing with the way in which period and intrinsic luminosity
are a function of metallicity. In two of his more recent papers (Sandage, 1993a,b) he
goes into more detail and takes into account some of the criticism of this method.
Ultimately, this leads into the large question of horizontal branch morphology
and the question of the second parameter problem. It is well known that metal-poor
clusters have longer bluer HBS and metal-rich ones have more of a red clump. There
does appear to be some other factor, the second parameter, that affects HB morphol-
ogy. One possibility could be helium enrichment through a variety of mechanisms
(Sweigart & Catelan, 1998). This issue percolates into our understanding of recently

discovered Oosterhoff III clusters, which are very metal-rich bulge globular clusters,

that do not follow the period-metallicity trends of the other two groups Pritzl et al.

28

(2001).

3.6 USEFULNESS As KINEMATIC TRACERS OF THE HALO

RRL stars are very good standard candles, which means that they provide a rare bit of
information, their distance. Furthermore, they are especially good for understanding
the nature of the halo because they are population 11 stars, in general greater than
10 Gyrs in age, so they trace the old populations of the Galaxy. This fact has been
recognized for some time, and the primary limitation in using them is that they are
relatively rare stars, which are hard to identify without multi-epoch observations. As
a result, other HB stars were used instead. These stars should be standard candles
as well, although they occupy a larger color range, so there is a relationship between
color and My. There are also some contamination issues with main-sequence (MS)
stars. An example where both Blue Horizontal Branch (BHB) stars and RRL were
used was in Preston et al. (1991). In this paper, they showed a decrease in stellar
age as a function of increasing galactic radius. They also used BHB stars and RRL
to investigate the shape of the halo.

Layden (1995a,b) undertook a very comprehensive study of the galactic structure
using RRL. He used a sample of 302 RRab with good photometry, radial velocity
measurements, and metallicity determinations (Layden, 1994) to work out rotational
velocities for both disk and halo components of the Milky Way. Combining this
information with proper motions from the literature he was able to get full space
motions, which allowed him to compare his results with a number of previous studies
including Morrison et al. (1990); Beers & Sommer-Larsen (1995). Layden did not see
a retrograde motion for the halo, but rather a slight prograde one, V4, = 18 :I: 13.
His sample, however, was highly limited in spacial extent, with the outer-most RRL
being only 4 kpc from the Sun.

Another approach was used by Lee & Carney (1999) to look at the kinematics

29

of the halo using RRL in globular clusters. They subscribed to the hysteresis ex-
planation of the Oosterhoff groups discussed in section 3.5, with metallicity being
the primary parameter that determines which direction the stars in a cluster will
evolve. Furthermore, they suggested that the Oosterhoff groups actually represent
difl'erent formation epochs. In this case the metal-poor 0011 comes from an ELS
like scenario forming in the proto—galaxy, whereas the metal-rich 00] clusters come
from an epoch of accretion like SZ. They found a possible retrograde motion for the
OoI clusters, < th = —68 :l: 56km/s, and a prograde motion for the 0011 clusters,
< th >= +94 :l: 47 km/s. They saw the Signature of accretion in the 001 globu-
lar clusters with a slight retrograde motion. The OoII globular clusters had a slight
prograde motion. This study had the benefit of sampling at a wide range different
galactocentric radii, but it was limited to w 40 globular clusters.

Both Layden (1995a) and Lee 81. Carney (1999) use the same method to deter-
mine the bulk rotation of the halo. This method, originally developed by Frenk &
White (1980) and summarized by Zinn (1985), determines bulk motions using radial
velocities, positions, and distances only. This is extremely useful because proper mo-
tions, necessary for full-space motions, are often hard to come by and generally are
available only for nearby or fast moving objects. At first look, it appears that there is
a discrepancy between these two papers. This may not be the case, however, because
Layden ( 1995a) samples only the local solar neighborhood, which is likely primarily
disk and old halo, whereas Lee & Carney (1999) sample well out into the newer outer
halo.

A very recent paper by Kinman et al. (2007) presents a more complex view.
Kinman’s group analyzed 26 RRL stars and 52 BHB stars and found that the two
groups may actually trace different components of the halo. They have full proper
motion and radial velocity information for these stars and were able to determine

full U, V, W space-motions. Furthermore, these stars are located in the SA 57 field,

30

which is located toward the North Galactic Pole (N GP), which Should minimize disk
contamination. Ultimately, they found the BHB stars had no net rotation whereas,
the RRL had a net retrograde rotation. Their sample was limited to within Z < 8kpc
and apparent V magnitude < 16.

More recently a number of larger RRL surveys have taken place. These surveys,
usually the result of looking for other transient objects, have provided significantly
more information than has been previously available. The results of these surveys

and their individual benefits and pitfalls will be discussed in section 4.4.

31

 

CHAPTER 4:
THE SLOAN DIGITAL SKY SURVEY
AND OTHER SURVEYS

 

4.1 OVERVIEW

The Sloan Digital Sky Survey, SDSS,(York et al., 2000) is the largest photometric
and spectroscopic survey to date. The survey saw first light in May of 1998 and
began regular operations April 2000 (Adelman-McCarthy et al., 2008). SDSS was de-
signed initially as a quasar survey; it has far exceeded that original mission with work
extending over many different projects relating to both galactic and extra-galactic as-
tronomy. The survey uses a 2.5 meter telescope located at Apache Point, NM. The
original survey had two distinct parts, a photometric survey and a spectroscopic sur-
vey. The photometric survey used a 120 megapixel camera consisting of 30 2048x2048
CCD imagers spaced in six colurrms and five rows (Gunn et al., 1998). Due to the
extra-galactic focus of the original survey a new photometric system, 11’ g’ r' i’ z’ , was
used instead of the more common Johnson-Cousins system. The u’g’r’i’z’ system is
a broad-band photometric system that covers from 3000 to 10,000A (Fukugita et al.,
1996; Stoughton et al., 2002). The u’ g' r’ z" 2’ filter system is compared to the Johnson-
Cousins system in Figure 4.1. When discussing the ugrz’z filter system there are two
slightly different systems that exist. The un-primed ugrz’z system refers to the filters

on the 2.5 meter telescope. The primed system refers to the filters used on any other

telescope and are calibrated to the 1.0 meter USN O telescope at the Flagstaff station

32

(Smith et al., 2002). For the most part these two designations differ by only a small
amount and the two can be transformed back and forth with relative ease.

The telescope was used in a drift scan mode, which drifts at the sidereal rate, where
the sky is imaged in arcs of great circles where each object is integrated through each
of the five filters for 54.1 seconds as shown in Figure 4.2. This results in a 50 detection
limit with I” seeing of 22.3, 23.3, 23.1, 22.3, and 20.8 magnitudes in the u’, g’, r’, t’,
and z' filters respectively (York et al., 2000). Each of these scans is referred to as a
strip and each strip is observed in pairs with the paired strip being offset from the
first by 93% of the CCD width. The combined pair of strips is referred to as a stripe
and is 2.54 degrees wide (York et al., 2000).

 

Normalized Transmission

 

 

 

3000 4000 5000 6000 7000 8000 9000 1000011000
Wavelength (A)

Figure 4.1 Comparison of the SDSS ugriz and the Johnson-Cousins UBVRI filters.
This figure is adapted from Clem (2005).

The results of the photometric survey are then used to define subsets of objects to
be studied in further detail with the spectroscopic survey. The spectroscopic survey

uses two fiber-fed spectrographs each with a blue and a red channel. The blue channel

33

 
 
     
     
 

  

CJCJCJCJCJCJ
DODDDD
DDDDBU
DIODE—JD
IIEIDEIQ

C
C

Figure 4.2 A schematic layout of the ugrz’z filters on the SDSS survey camera. The
arrows show how the sky passes over the camera in drift scan mode. Schematic is
based on description in (Gunn et al., 1998)

runs from 3800A to 6500A and in red from 5800A to 9200A (York et al., 2000). Each
spectrograph has 320 fibers leading to a total of 640 fibers which are attached to
pre—drilled plates. The resolution (SA/A varies as a function of wavelength from 1850
to 2200.

AS the SDSS Survey and it’s followup survey, SDSS—II, have progressed, there
have been a number of data releases culminating with the most recent Data Release
6, DR6, (Adelman-McCarthy et al., 2008). As of DR6, over 9582 sq degrees of the
northern sky have been imaged resulting in photometry of 287 million unique objects.
The photometry of these objects has been calibrated to 1% accuracy for g,r,2',z and 2%
for u. The spectroscopic survey has also produced a massive amount of data, covering
7425 sq. degrees of sky with over 1.27 million spectra including stars, galaxies, quasars

and calibration spectra (Adelman-McCarthy et al., 2008).

34

4.2 SDSS-II SN SURVEY

After the completion of the original SDSS survey in June 2005, now called SDSS-I, an
extension to that survey called SDSS-II was created. SDSS-ll consists of three distinct
surveys: the SDSS Legacy Survey, SEGUE, and the Sloan Supernova Survey (SN
Survey). The SN Survey was designed to study intermediate redshift (0.5 S 2 S 0.35)
type Ia supernovae (Frieman et al., 2008). The SN survey repeatedly images an
equatorial stripe using a two day cadence. The imaged region, known as Stripe
82, runs from 20” to 4” in right ascension and from 1.25 to -1.25 in declination.
This creates a 2.5 degree wide stripe that leads to a total imaged area of 300 sq.
degrees (Frieman et al., 2008). For a definition of right ascension and declination, see
Section 7.3. This particular stripe was chosen because it is equatorial and thus can be
reached by ground-based telescopes in both the northern and southern hemispheres,
but also because Stripe 82 has been used as the photometric calibration stripe since
the beginning of the SDSS and as such provided them with enough data to co-add
the images together to make a very good template image to use in image subtraction
(Frieman et al., 2008).

The two day cadence of the observations is due to the nature of the SDSS imaging
camera. As seen in Figure 4.2, the CCDS are arranged in six columns with a significant
gap in between them. As a result, in order to get a continuous stripe, two interleaving
strips, a north and a south one, must be taken. As in the case with the SDSS-I survey,
the camera tracks at the Sidereal rate, so the individual strips are alternated from
night to night leading to the two day cadence. This observation schedule was followed
from Sept lst to Nov 30th for 2005, 2006, and 2007 (Frieman et al., 2008).

Since the SN Survey used the same drift scan rate as the original survey, they have
comparable photometric accuracies. However, whereas the SDSS-I survey limited
itself only to’photometric nights, the SN Survey had a much higher tolerance for poor

observing conditions, operating in non-photometric and bright moon time. This was

35

not a major issue for them, because they were interested in differential photometry

as versus absolute photometry.

4.3 VARIABLE STAR CATALOGS

The SN Survey produced large numbers of repeated observations of Stripe 82, how-
ever, they used image subtraction techniques to find and quantify their supernovae.
This meant that the images taken by the SN Survey were not put through the normal
photometric routines that the rest of SDSS imaging had gone through. As a result,
it was left to two separate teams, one located at Cambridge and the other at the
University of Washington, to create photometric catalogs for all the observed epochs.

Each of these lightcurve catalogs will be taken in turn.

4.3.1 CAMBRIDGE CATALOG

The Light-Motion Curve Catalogue (LMCC) was produced by Bramich et al. (2008)
at Cambridge. This catalog took data from the 2005 season of the SN Survey as well
as the previous calibration data going back to September of 1998. The catalog has two
main purposes; the first is to perform epoch by epoch photometry on all the objects
in Stripe 82 complete to a limiting magnitude of r ~ 21.5. The second is to make a
proper motion catalog for both point and extended sources down to 18th magnitude
in r, getting ~ 32 mas and ~ 35 mass RMS accuracy (Bramich et al., 2008). The
resulting catalog contains over 4 million objects distributed over a slightly smaller
range than the whole of Stripe 82 going from a right ascension of 20.7” to 3.3" and
a declination of -1.26 to 1.26 degrees as can be seen in Figure 4.3.

The LMCC has a counterpart catalog that contains mean properties such as the
proper motions, reddening, and variability indices. This catalog is call the Higher-
Level Catalogue (HLC). The HLC consists of a series of binary fits files providing

229 photometric and astrometric parameters (Bramich et al., 2008). Both of these

36

catalogs are publicly available through SDSS—II. The most current version and the

one used in this project is version 2.

4.3.2 WASHINGTON CATALOG

The LMCC and HLC catalogs are very useful for searching for variable objects, but
they have one major downside, they only include data from 2005 and before. Increas-
ing the number of epochs in a lightcurve increases the chances of determining the
type, period, etc. To this end, another catalog from the University of Washington
was used. This catalog, developed by Ivezié et a1. (2007) and Sesar et al. (2007),
contains data through 2006. It also used a different algorithm for determining the
photometric zero points, which helped increase the overall photometric accuracies of
the data. This new photometric accuracy had particular effect on the u filter allowing
many more It epochs to be included in the lightcurves. Since the LMCC catalog was
used for the initial detection of the variable stars, the limits in right ascension and

declination are the same.

4.4 OTHER RRL SURVEYS

In recent years, there have been a number of large scale variable star surveys, that
have either piggy backed on surveys designed for another purpose, QUEST-I and
LONEOS, or were companion projects for another program, N SVS. They each have
their benefits, either wide sky coverage or deep exposures, and they each have provided
a catalog of stars that can be used to explore questions about the structure of the
galaxy. As such, they will be discussed during the analysis of this survey’s data in

Chapter 8.

37

 

 

 

Figure 4.3 The locations in galactic coordinates of the SN Survey, in blue running
through negative galactic latitudes, and the QUEST survey, in red primarily running
through positive galactic latitudes with an extra section below the galactic equator
near galactic longitude of 210.

4.4.1 NSVS

The Northern Sky Variability Survey (NSVS) (Wozniak et al., 2004) was conducted
on the Robotic Optical Transient Search Experiment (ROTSE—I) telescope (Akerlof
et al., 2000). The survey found 1197 RRab stars down to magnitude of V = 14
Kinemuchi et al. (2006). The survey was conducted on 206 16.4 x 16.4 degree fields,
in a nonstandard filter that roughly corresponds to Cousins R. For a map of the
fields, consult Akerlof et al. (2000) Figure 1. The project was a companion project
to a program designed to do optical follow-ups of Gamma-Ray Bursts. The program
benefited from extensive sky coverage, but was limited by their relatively bright cut-

off magnitude.

38

4.4.2 QUEST-I

The Quasar Equatorial Survey Team (QUEST-I) was originally designed for a large
scale quasar survey. A number of scientific projects were done on this telescope each
with its own filter set, the filter that was common to all of these was V. As a result,
the variable star detection and lightcurves are done in the V filter only. The QUEST-
I survey is perhaps the most similar to the SN Survey in the sense that it is also a
drift-scan equatorial stripe. The position of QUEST-I can be seen in relation to the
SDSS Survey in Figure 4.3. The survey produced 498 RRL stars 395 RRab and 103
RRc, down to a limiting magnitude of V = 19.5 (Vivas et al., 2004). The QUEST-I
survey provides a strong compliment to the SN Survey in that it clearly covers a

region of the galaxy not sampled by the SN Survey to a similar depth.

4.4.3 LONEOS

The Lowell Observatory Near Earth Object Survey (LON EOS) was designed to dis-
cover near earth asteroids (Bowell et al., 1995). The LON EOS survey is unfiltered
and covers 1430 deg2 in 5 deg2 fields. A map of the fields can be found in Figure 1
of Miceli et al. (2008). In their survey they found 838 RRab stars down to R ~ 18.5
(Miceli et al., 2008). The data reduction pipeline for this survey is discussed in Rest
(2002)

39

 

CHAPTER 5:
PHOTOMETRIC ANALYSIS

 

5.1 VARIABLE STAR SELECTION

As was mentioned in a previous chapter, the LMCC catalogs, both versions one
and two, contain on order four million objects, most of which are stars. Since we
are looking for on order a few hundred stars we need several powerful methods to
distinguish the variable from the non-variable stars. One of the primary ways in
which this is done is to develop a variability index. Variability indices all have the
same underlying prinicipal of looking for large deviations from the mean magnitude
of the star compared to the photometric error for each point. The Welch-Stetson
index is an example of one of these (Stetson, 1996). These sort of indices tend to
fail, however, when one approaches the limiting magnitude of a survey. This occurs
because often times the stated error in the magnitudes tends to be an underestimate.
This leads to false positives which render the index less effective. To compensate for
that one can use color cuts. Since RRL stars must exist in the instability strip and
are giant stars, one can devise a series of color cuts to separate them out from the
rest of the stars. One such attempt was made in Ivezié et al. (2005).

Once the candidate variable stars are found, the next step is to determine which
stars are actually RRL and which are not. The first issue is to determine the period of
the variable stars. There are a number of programs which can do this, but the one we
used is called supersmoother (Reimann, 1994). Supersmoother attempts to determine

the period by finding the period that creates the smoothest looking fit to the data.

40

For RRL stars, this works pretty well, although there are cases where aliases Show
up, or the smoothest line is not the true period due to errors in the photometry. At
this point, the two best periods are looked at and are classified by eye into variable
star type, and into ab or c type if they are determined to be RRL stars. To further
help with classification, we used Layden’s (1998) RRL templates. These templates
were fit to each of the 15 best periods found by supersmoother, and the best two fits
were then examined. Finally, the results were plotted on a period-amplitude diagram

and final classifications were chosen at that point.

5.1.1 WELCH-STETSON INDEX

When searching for variability in a sample of stellar photometry, one is by definition
looking for a change in magnitude, which is outside of the specified errors. One can
quantify this variability by creating an index that for example becomes larger as
more deviations from the mean magnitude are found. One can further increase the
robustness of the index by looking at pairs of observations. The idea being that if their
is a variation in two separate filters, or paired observations in the same filter, for a
given epoch that it is more likely that this is true variation as versus bad photometry,
cosmic rays etc. An early example of an index that used this method can be found in
Welch & Stetson (1993). In this early index using paired b and v observations (5.1),
b,- and v,- are the photometric observations, 6 and 0 are the weighted means in each
filter, 017,1 and am- are the photometric errors, and n is the number of paired b, u

observations (Welch & Stetson, 1993; Stetson, 1996).

1: 07771—1) n (1:3) (vial—iv) (5])

i=1

 

 

This original Welch-Stetson index was improved upon in Stetson (1996) by general-

izing this process to any number of observations in any number of filters per epoch.

41

This new variability index is given in equation (5.2). In this index, the observations
taken at a given epoch are grouped into k pairs where each pair is product of the nor-
malized residual of each observation Pk = 6,0,)63'06). These pairs are then weighted

by a value wk depending on how many distinct frames go into the pair.

J ___ 22:1 wk SSn(Pkl\/ lPkl (52)

22:1 wk

In the ideal case, one has pair constructed from two observations, and this will have
a weight, wk, of 1. In some instances, there will be no pair for a given frame, so
one will have to pair a frame with itself. In that case, one uses a lower weight of
0.5 to account for the loss of information compared to a pair constructed from two
frames. In the case of this study, three frames at the same epoch a, b, and c were
paired together. This generates three pairs ab, be, and ac each with a weight of 2 / 3.
When these three pairs are added to the sum, their combined weights 2/3 at 3 gives
a combined weight of 2 thus accounting for the more information provided by using
three and making them twice as good as a pair of two frames which is twice as good

as a pair of a frame and itself. Pairs are built from normalized residuals:

5: ln v—v
n—l 00

This formulation is similar to equation (5.1), but has an added statistical factor of

 

 

E to account for the fact that this observation was used in creating the mean.
This normalized residual is then used to create a pair. The variability index depends
on the fact that for random error the expectation value of (6,117) is zero, and for true
variation tends towards a positive number. In the case where one pairs a frame with
itself, z'(lc) = j(k), the expectation value for random error tends to unity for (51-2).

To put both types of pairs on even ground, i.e. going to 0 for non-variable stars the

42

pairs function Pk is defined as:

51(k)5i(j)a if WC) 7420');
6,?er ifz'(k)=z’(j).

This variability index is useful for detecting a wide variety of variable stars, but
it can be made more sensitive to the periodic stars that are the focus of this study.
Stetson (1996) developed a kurtosis index shown in equation (5.3). In this equation,
N is the total number of observations and 6,- is the normalized residual from before.
This index is created such that for Gaussian noise the index tends to 0.798. For a
pure sinusoid K —+ 0.900 and for a sawtooth K —» 0.866. In the case of a single bad

measurement, K —> 0 as N —> oo.

K:\/1_/N=N 2:: 12—12

To take advantage of both the J and K indices a final combined L index was

 

created. Stetson’s version of it is found in equation (5.4). There is a division by
0.798 in this equation that causes the L index to equal the J index for a Gaussian
distribution of magnitudes, but there is a slight enhancement for periodic variables
like RRL since they have a more sawtooth or sinusoidal shape. In equation (5.4)
Zw/wau is a weighting factor where wall is the number of possible frames a star
could appear in and 2w is the number of frames the star was actually detected
in. Given that the data in this project was taken in stripes in the sky, which were of
variable length depending on conditions and time of year, it was not straightforwardly
possibly to calculate this weighting factor. As a result Zia/wan = 1 for the Welch-

Stetson L index used in this project.

—(——)(77)

43

 

The LMCC catalog provides u, g, r, 1', and z lightcurves for each object in the
catalog. The quality of the photometry, however, varies significantly depending on
the filter. As a result, only 9, r, and 2' observations were used to create the L index.
The pairs varied from three down to one observation per epoch as the photometry

allowed, and the weights were assigned as previously discussed.

 

 

 

20 T I I I I

18 - ' -

16 - j

14 - ‘ ' ‘ -

12 - - ' . . . ~ ‘ ..

_I 10 - _ c1

.4. . .

8 '- . a

6 .- 35"; -I
2‘.- l _ 1.3;;

4 - '.C_'::i.-"‘?”."¥. ’ L -
" .

 

 

 

 

Figure 5.1 This is a 1/ 10 sample of stars from the LMCC catalog. The solid line is
a magnitude cut 9 of 20.0 and the dotted line is cut at g of 21.5. There is a clear
over-density of stars between these cuts.

In figure 5.1 the L variability index has been plotted as a function of g magnitude
for stars from the LMCC catalog. It is apparent from this plot that as one goes to
dimmer magnitudes, the number of variable candidates increases. This effect is due
primarily togthe underestimate of photometric errors at high magnitude. As a result,
the contamination by non-variable stars of a variable star sample selected using L is

magnitude dependant. The number of candidates drops down after g of 21.5 because

44

the database is not complete beyond this magnitude (Bramich et al., 2008).

5.1.2 FINDING THE MEAN

One of the critical steps in creating a robust variability index is developing a mean
magnitude to compare individual observations against. The first inclination in a
situation like this is to use a weighted mean weighting the observations based on the
photometric errors, 1/02. This suffers from the generic problem of means that they
are overly influenced by outliers. This is especially true in the case of variable stars,
since the photometry by definition is outside of the photometric errors. There are
a number of methods that have been developed to deal with this issue including 30
clipping. Any sort of clipping method is undesirable, because insignificant fluctuations
in photometry near the clipping limit can make the difference between a point be
included or not. Stetson came up with a solution to this issue by creating a smoothly
varying iterative process for finding the mean (Stetson, 1996, 1987). In the first
iteration, a weighted mean using photometric errors is used. For all other iterations,
a smooth function based on the normalized residuals from section 5.1.1 is used and is
shown in equation (5.5). In this case, a = b = 2 was used, although the result tends
to be insensitive to the choice of values for a and b. The iterations were continued
until the mean converged to a tolerance of 0.0001 mags or until 50 iterations were

[1.0311 (5.5)

completed.

5.1.3 COLOR CUTS

The variability index, L, is very effective at picking out the variable stars especially
at the brighter end of the magnitude range. In the case of this project, however, we
are interested in the RR Lyrae in particular, and not as interested in other variables

like eclipsing binaries. Since the instability strip puts relatively narrow constraints on

45

the colors that an RRL star can have, one can exclude many non—rrl stars by making
appropriate color cuts. Such cuts were explored by Ivezic’: et al. (2005). This paper
used 153 RRL stars from the QUEST survey (Vivas et al., 2004), of both ab and c

type, to define a color box in SDSS colors. This lead to the set of equations (5.6).

0.99 < uo — go < 1.28,

—0.11 < 0—r0< 0.31,
g (5.6)

—0.13 < 7‘0 — to < 0.20,

-—0.19 < 2'0 — 20 < 0.23.

To determine how well the color cuts of (5.6) worked, a search for RRL stars
in version 1 of the LMCC catalog using only variability index L, g magnitude, and
classification as a star. To limit the number of false positives a very limited range of
L and g were searched (11 2 L 2; 5 with a g < 18). The 109 RRab and 63 RRc that
resulted from this search are shown in figure 5.2.

It quickly became apparent from figure 5.2 that a large number of RRL stars
were being excluded from the sample, primarily due to a lack of u information in
the LMCC catalogs. To help correct for this issue, a much less stringent set of color
cuts was developed. These cuts, shown in equations (5.7), do not rely on the no — go
color, which is a powerful discriminator, but one that relies on the often poor u filter.
Due to incomplete sampling of the lightcurves, the average colors used to select RRL
candidates missed a few on the edge, so the new cuts were expanded by 10% to bring
in close cases. The old and new color cuts are compared in figure 5.2 where the solid
lines represent the original color cuts from (5.6) and the dotted lines represent the

expanded cuts from (5.7).

46

—0.13 < 90 - r0 < 0.33,
—0.14 < r0 — 2'0 < 0.23, (5-7)

—0.21 < i0 — :50 < 0.25.

5.1.4 FINAL CANDIDATE LIST

Optimizing RRL candidate selection requires balancing the two methods for finding
RRL in a large sample of stars: the variability index and color cuts. As described
in subsection 5.1.1, the variability index L is very good at picking out variable stars.
There are two ways, though, that non-RRL stars can appear in the sample. The first
is that their are other types of variable stars, the most important in a contamination
context are binary stars. Eclipsing binaries, can have period aliases that look very
much RRc stars, and as such are picked up by the a variability index search. They
are also much more common than RRL stars are, and therefore represent a significant
contamination source.

The other issue is that photometric errors tend to become under estimated at
dimmer magnitudes. This leads to a greater increase of false positives just to random
error in the photometry. This issue is exacerbated by the fact that some of the 2005
SN data was taken under non-photometric conditions. This can lead to deviations
from the mean magnitude, which are significant enough to appear in a variability
index. These two issues mean that the variability index alone is not good enough
to pick out a significant number of RRL, without also finding many contaminating
stars.

The other method of picking out RRL based on their unique colors due to their
location on the HR Diagram also has some issues related to limitations in the LMCC
data set. The most problematic issue is that, as discussed in subsection 5.1.3, the
u filter is missing or inadequately sampled in the LMCC. Unfortunately, no — go is

the most stringent of the color cuts. This is issue is made more difficult because not

47

Table 5.1. RRL Candidate Search Parameters.

 

Search L Typea 3 go # Epochs g, r, 2' Color Cuts # Stars

 

First 2 8 > 4 20.0 2 5, 2 5, 2 5 None 812
Second 2 5 > 4 20.0 2 5, 2 5, 2 5 Eq. (5.7) 1228
Third 2 2 > 4 21.5 2 5, 2 5, Z 5 Eq. (5.6) 438
Combined . . . .. . . . . . .. 1675

 

8‘This is the average of the star or galaxy designation that SDSS gives each
measurement where a galaxy is 3 and a star is 6.

all of the RRL lightcurves are sampled fully. This can lead to average colors that lie
somewhat outside of the expected ranges.

To compensate for these two competing issues, a three-tiered system for selecting
RRL candidates was used. The first variability search is the most inclusive in color
space. As shown in table 5.1, it depends only on the Welch-Stetson variability index
L and some basic quality assurance parameters. One important but often overlooked
parameter is the object type. In SDSS all objects are typed as stars or galaxies. Stars
have a value of 6 and galaxies have a value of 3, and each epoch for a given object
is typed independently. The parameter I call Type in table 5.1 is the average of the
SDSS type over all the epochs. This is less than 6, because stars can be misidentified
if they are crowded in bad seeing conditions.

The second tier search uses the relaxed color cuts described in equation (5.7). This
allows for candidates with a lower L value while cutting back on contamination from
non-variables and non-RRL stars. The minimum L for each cut was chosen based
primarily on star counts. As can be seen in figure 5.3 the number of stars rapidly
increases as one goes below L = 8 in the no color cut situation and L = 5 in the
relaxed color cut search.

The third and final tier search using the color cuts specified in (5.6) provided a

means to get down to a much lower L and also push to a much dimmer go magnitude.

48

As can be seen in figure 5.3, the third search is very tight, and is resilient against
significant contamination. This is at the cost of losing stars that may have poor u
magnitudes or unusual colors due to incompleteness. These three searches were then

combined to produce a total of 1675 candidates as shown in table 5.1.

5.2 VARIABLE IDENTIFICATION

Once the RRL candidates have been identified it is necessary to use visual identifi-
cation to separate out the true RRL stars from the other variable and non-variable
stars. The first step in doing this is determing the period of each of the stars. As
was discussed in section 3.3, there are a number of methods for finding the period
of a periodic variable star. The period finding method that was ultimately decided
upon was a program out of the MACHO project called supersmoother. The two most
likely periods for each candidate from supersmoother are then used to phase the can-
didate’s lightcurve (as discussed in section 3.3). The resulting phased lightcurves are
then typed by eye. This is done by looking for the characteristic shapes and periods
of RRL stars. The stars identified as RRL from this process have their lightcurves
fit to known RRL templates. This process is done for two reasons, one it can help
fill in gaps in the lightcurve, and two it provides another way of typing the RRL. Fi-
nally, the results are plotted on a period-amplitude diagram. Stars that deviate from
their expected position on the diagram are then reanalyzed more closely to check for

incorrect identifications or periods.

5.2.1 FINDING THE PERIOD: SUPERSMOOTHER

The supersmoother algorithm for finding periods was developed by Reimann (1994)
for the MACHO project. This project was interested in finding gravitational mi-
crolensing events caused by massive, compact, halo objects (MACHOs) to get an

estimate on the amount of baryonic dark matter in the Galaxy. A gravitational mi-

49

crolensing event occurs when a MACHO passes in front of a distant star. Light from
that star is lensed by the gravity of the MACHO causing two images of the distant star
to appear. These two images have very small angular separation, so it appears that
the star itself has simply become brighter (Cook et al., 1995). As a result of looking
for these transient events, the MACHO project took data that was also appropriate
for finding variables. The supersmoother program was developed to find periods for
many different kinds of variables, and thus does not make assumptions about the
shape of lightcurves. It belongs to a class of period finders known as smoothers.
These period finders, attempt to fit linear smooths through the data, and then find
the period that minimizes the dispersion

For this project, four different period searches were used to attempt to find the
appropriate period for the candidate variables. Each of searches used the same su-
persmoother configuration file with only the period range changing for each one. The
configuration file, phase.par, is shown in appendix A.1. The different searches were
designed to account for the many different types of variable stars that might be in

the sample. These searches are summarized in table 5.2.

Table 5.2: RRL Candidate Period Searches

4— - —
! ‘—

Search Period Range (in days) Candidate Type x-Axis

 

First 0.1 - 100 All Phase
Second None Aliased HJD

Third 0.01 - 3 Aliased & Short Phase
Fourth 0.1 - 3 RRL Phase

 

The first search was designed to pick up most types of known periodic variables

including RRL and Cepheids. All of the candidate variables were included in this

50

search. The two most probable periods for each candidate were plotted and the
variables were typed using the metric shown in table 5.3. Based on the type given
to each candidate, the next three searches were run in turn on the candidate type

specified in table 5.2. After each search, the types were updated.

Table 5.3: Variable Typing Metric

 

 

 

Abbrev. Type Characteristics # Found
a Aliased Period around 1 day 823
ab RRab Period approx. 0.4-1.0 days 305
c RRc Period approx. 0.2-0.4 days 116
ev Eclipsing variable Two eclipses 93
i Interesting Periodic w/ good lightcurve 13
lpv Long period variable Periods of hundreds of days 5
ms Main sequence Period 0.01-0.2 days 4
n Not variable One or two bad points 86
o Other variables Lightcurve is indeterminate 73
5 Short Period < 0.2 days not clearly an ev or ms 133
u Unclassifiable Too few points to type the lightcurve l4
type+B Blazhko effect Star shows scatter near max or min. 35
type+? Uncertain type Low confidence of assigned type 24

 

In the second search, supersmoother was not used, rather the one day aliased can-

didates were plotted as a function of Heliocentric Julian Date (HJD). In many cases, it
became obvious that they were in fact long period variables, which supersmoother was
unable to fit due to the 100 day maximum period limit in the first search. The third
search concentrated on the short period variables and remaining aliased variables. In

this case, supersmoother was run with a period range including the low period end

51

between 0.01 and 0.1, in general searches this is not done because it greatly increases
the time it takes to find a correct period. This low period searching picked up many
main-sequence pulsators such as 6 Scuti and SX Phoenicis stars. It also did a better
job of picking out periods for eclipsing variables. The fourth and final search was
run on the stars that were marked. as RRL of some type. By constraining the period
search to the much smaller range of 0.1 to 3 days compared to the first search, the
initial search grid used by supersmoother was much finer increasing the chance of

finding good periods.

5.2.2 FINDING THE PERIOD: TEMPLATE FITTING

Using the most common settings supersmoother produces up to 15 different possible
periods. In most cases, the best period is listed first or second in the .per file. In
those situations when this is not the case due to, for instance, a strong alias, another
technique must be used to find the best period. One such technique is template
fitting. In template fitting, a series of generalized lightcurve templates are fitted to
the phased data. The best fitting template / period combination provides not only the
best RRL period for a given variable, but also is useful for typing RRL that have
incomplete lightcurves.

The RRL templates used in this project come from Layden (1998). For the RRab
stars he used 3 to 4 well sampled RRL lightcurves from Nikolov et al. (1984) for
each template. Each lightcurve was normalized from 0 to 1 in phase and 0 to 1
in magnitude. He also included an RRc template, a sine wave and two binary star
templates. These are shown in Figure 5.4. These templates are fit to the lightcurves
using a Levenberg-Marquardt method for minimizing X2 (examples found in Press et
al. (1992)). The fitter was run over all 15 available periods and all 10 templates to find
the lowest X2 value. In general the fit with the lowest X2 value was taken, although,

in some cases there were bad points are underestimated errors that made the best fit

52

not have the lowest X2. Ultimately all of the RRL candidates were examined by eye
to ensure that the automatic template fitting routine worked correctly. By fitting the
lightcurves to templates, not only was there an independent source of variable typing
information, but also the epoch of maximum could be determined as well as the mean

magnitude (both intensity and magnitude weighted) and amplitude.

5.2.3 PERIOD-AMPLITUDE DIAGRAM

Once all of the period finding and template fitting is complete, there is one more
tool that can be used to determine how well the fits worked. This tool is the period-
amplitude diagram. In this case the 9 filter was used for measuring the amplitude and
is shown in Figure 5.5. The figure is divided into four distinct regions allowing for the
selection of aberrant points. The RRab stars and RRc stars clearly cluster in their
own respective parts of the diagram. Those points that were significantly deviant
from their respective locus (i.e. appeared in an incorrect region) were reexamined in
detail. The primary source of issue with most of the deviant points was that an alias
had been selected for the period, or an incorrect type had been assigned due to poor
template fitting. Once these concerns were corrected, the resulting period-amplitude
diagram is found in Figure 5.6.

The final period-amplitude diagram has the lines appropriate to two different
possible sequences for the Oosterhoff groups. The standard linear sequences from
Clement & Shelton (1999); Clement & Rowe (2000) are based on least squared fits
to the period-amplitude diagrams for “normal” RRab lightcurves in 001 and 0011
clusters respectively. The derived equations for these lines can be found in equations
(5.8). More recently, Cacciari et al. (2005) revisited M3 in particular, but also other
globular clusters in the literature and found that a quadratic line actually fit the
RRab in the period amplitude diagrams better. These lines are also described in

equation (5.8). These equations are defined in the V filter system, so they needed to

53

be transformed to 9. To do this, we used the transformations based on the HK survey

in Zhao & Newberg (2006). This lead to a ratio of amplitudes of Ag/AV = 1.17.

AV = —7.1314log(P) — 0.85687 OoI Clement
AV = —4.93961 log(P) + 0127973 0011 Clement (5 8)
AV = —2.627 — 22.046log(P) — 30.876(log(P))2 OoI Cacciari

AV = —1.415 — 18.341 log(P) — 30.876(log(P))2 OoII Cacciari

Looking at Figure 5.6, it is not clear which set of lines is necessarily the better fit
to the data, but what is clear is that there is a very strong OoI component to the halo
sample as has been seen in previous work (Lee & Carney, 1999). There also appears
to be a far number of Oosterhoff intermediate RRab stars. The photometric results
for the individual stars are in Table 0.2 located in Appendix C. Example lightcurves
are located in Appendix B. We will return to the period-amplitude diagram for a

more thorough analysis in Chapter 8.

54

 

 

 

 

 

 

 

 

 

 

 

r I TTI I I I I I I I I I I I I I I I I
0.4 —
’C 0-2 T P:
| :x
O“ '- : x
v - A
0 " 3. ‘
-0.2 I I I I I I L I I I I L I I I I I I I I I
-20 -15 -10 -5 0
(u-g)
0.3 .— I I I I I I I I I I I I I _.
Z '- - - "I —————— I :1
0.2 :- I v I "j
- I I _
3 ' l I
f: 0.1 _— : x x x x X 5| ° ”1
A C I 3“ "h " x ' Z
0 _ I A A éflq I: K .4
I I If“ I." e a ' j
I- I A | .1
-0.1 .— I ‘ A : x 1
_0'2 - I I I I I I I LI I I I I ‘
-0.2 0 0.2 0.4

(9-0

 

IIIIIIIIIIIIIII'IIII

 

0.4 — " A "‘
' -I
— A x .l
x
’— x * xx fi X "
:9; x
02 '— x A x —-I
I x x x x‘ I J
.. x “'4‘“

 

 

 

 

 

 

_02 IIIIIIIIIJIIIIIIIIIII
0.4 0.6 0.8 1 1.2 1.4

(u-g)

 

 

 

 

 

 

 

 

0'3 _IIIIIIIIIlrII']I'IllleI_
I ----------- n I

:_ ' I _4

0.2 _ . I _
: l I j

01 _— I I _—
' I l I 3

- l I _

I 'b 4 I 1

.. I I _I

'0.1 "_- l I -:
_ l I -

I ' I 1

'02 _“ I— —————————— J E
'IIIIIIIIIIIIII_IIIIIIIIII

 

P
o:

-0.2 -0.1 0 0.1 0.2

0-0

Figure 5.2 Plots in color space of RRL stars selected without color cuts with x for
RRab and triangles for RRc. The solid lined boxes are color cuts from Ivezié et al.
(2005). The dashed lined boxes are the expanded boxes designed to pick up more

RRL.

55

 

600 ,
No Color Cuts (FIrst) ——

Relaxed Cuts (Second) ----------
lvezic’ (2005) Cuts (Third) — ---------------

Number of Stars

 

 

 

 

 

 

 

Figure 5.3 The three histograms are of the stars selected by the three variability
searches described in table 5.1. The two vertical lines show the L cutoffs for the first

and second searches at L = 8 and L = 5 respectively.

56

 

'U'I'II'IIII'III'III' IIII'IIIIIIIIIIIIIIII 'UII'I'I'VUI'II‘IIII' IIUIIIIIIT'IIUIIIIII' 'I'l'III'III'IIIl'II'

' RRa1 " RRaZ " RRa3 " RRb1 "' RRbZ 1

-1 ° :3. *4: ,1"; '5'. .44
u: up: ”III-z qua ::b: .4
I... .q-.. 4.: 'db. In: J
b. an . .Il- ‘ q- '8 '. I '-
05 1 1 '2 1 1
O . I: a" I; a" I: "- I: ‘i t a.
5 -o.5 - a -- g -- z -- 5‘s -- 2. - -o.5
: I, .1: .3: ~‘I I ‘:

lllllljlllllllll|lll|
I I I I U I I I

- AlmI .-

 

Illl'lllllll'lll'llll Illl'llllllllllllllll IlllIllllllllllllllll
I 1 III I Y I I I I I I I

I l l U I I I I
‘ RRb3 ‘ ' RRC ‘ ' Sine ‘

 

 

 

 

 

 

 

.1 Q. :‘a 3' ' r- u”. a -—— —- ,1
-: u..l. 2 :.,. I. : q:- : z : 3' .I- .
h ' .CD ‘ I db 2 : d. ' I . ' fi- I
b 1 .- z ' u . . n u I u I 1.3 83 a.
31 - a I n. as; ' u t; 5 u a x a a n ‘
2 .05 I— “a I—I— I: x -—-— ‘ ' --u a u u——u . n- -05
- x uII- “I : ll ‘ ' «InI ' ' '1.“ '-
: ':: u: as ::- -- it :
o _ 1., -_. "U _- 1.; _.: 2': z. . 0
I I l l I l l I l l I I I l l l [A] l l I I I 141*] I I l l I l l Lglil I l I I l I l I l I l l I l I I l
0 0.5 10 0.5 1 0 0.5 1 0 0.5 1 0 0.5 1
Phase Phase Phase Phase Phase

Figure 5.4 The ten templates used to fit the RRL lightcurves. These include six for
RRab type and one for RRc type. Also a sine wave and templates for binary stars
were also included. All of the templates were normalized to between 0 and -l.

57

 

 

 

 

 

 

 

I I I ‘|‘.I' I RRab +
1| I: RF“: x
‘ '-, oou ---------
‘.
15 . § ’3‘ . -
U a; E.
.4? g g x=o.1
x=-o.31v=1.1 13%;.
<9 *3 '. ‘
1 - + . . ‘ -
' ‘2 vii Z‘
- : 41.5 X=0.08
X = -0.69 frfixfi x‘ .*§‘ I "
x . x ”10;“ x 11 "
xx ,k xx #1 i '
X :xlxx “(xxx I‘ :
o 1 I I I I I
-1 -0 8 -0.6 -0.4 -O.2 O 0.2
L090”)

Figure 5.5 This period-amplitude diagram shows the RRL stars divided by type. The
RRab and RRc clearly cluster together, although there are a few clearly not in the
right part of the diagram. The diagram is divided into four blocks used to check for
aberrant points. The mean RRab lines for 001 and 0011 are plotted.

 

 

 

 

 

2 I I I I I I I I I I I 2
. . RRab + RRab +
. -_ RRc x an? x
I I‘ I _ _ _ _ - _ _ -
oon --------- oon ---------
15 » if - €15 - 15
43 t“ * +4 as +
g» 3 i; »
ti? L. 4F + 1*
it-“ :‘fi‘t' 4: I
«P 1 - + 22;. ‘aa -- , 3? - 1
a4 éfix 41’ 3+
ifl ?; It +*+ + I
)e‘xwxié, ’* iv )a’x‘fixas ’§
0.5 - m $131+“ -- W + 13.; - 0.5
x x x fl t I. 3&1 x x 1:, J
x “xx X ’ ‘ 'I. x ”(xx " ‘ I'.
:x §>g x x t ‘ I. :x §X x x at .'
x XXX x wxxx ‘ 1". X xXx x ”xxx 1 n
1 1 1 1 1 1 ‘1 1‘9. 1 1 1 1 1 1 4 I L 1 1
-0.8 -0.7 -O.6 -0.5 -0.4 -0.3 -0.2 -O.1 0 0.1 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1
L°9(P) L090”)

Figure 5.6 The final period-amplitude diagrams for the SDSS variables. The left panel
shows the 001 and 0011 lines from Clement & Rowe (2000). The right panel shows

the 001 and 0011 quadratic lines from Cacciari et a1. (2005). Each of these lines were
transformed into the 9 band by using Ag/AV = 1.17

58

 

CHAPTER 6:
SPECTROSCOPIC OBSERVATION AND
DATA REDUCTION

 

In order to use the RRL stars as tracers of the Milky Way halo, it is necessary to
have kinematic information about these stars. There is a certain amount of this
information available from the HLC proper motion catalog, but this is by its very
nature limited to relatively close stars. Ultimately, one needs radial velocities and
metallicities to determine both the distances and kinematics of RRL stars. For this
project, that information comes from two sources. The first is an observing run
specifically designed to get radial velocities and metallicities from these stars at the
Cerro Tololo Inter-American Observatory (CTIO) four meter Blanco telescope. These
observations were synchronized with the lightcurves from the LMCC catalog to get
observations at the best possible phase.

The second source of radial velocity and metallicity information is from the SDSS
spectroscopic surveys. In this case, the RRL were not specifically targeted, but instead
were happened upon serendipitously as part of other campaigns. These spectra have
the benefit of being able to reach much deeper than the 18th magnitude limit in
V of the Blanco data, but are limited in usefulness in some cases. This is because
they were taken at random phase, and perhaps more problematically are actually
composite spectra made from the summation of a few spectra, which again were not
necessarily taken at the same phase. This, however, was a fixable situation because

the individual spectra that go into making the composite spectra became available.

59

6.1 CTIO BLANCO 4M DATA

6.1.1 OBSERVATIONS

The spectroscopy done at the Blanco 4m telescope at CTIO was obtained on the
medium-resolution R-C Spectrograph, RCSPEC. The RCSPEC is a single-slit spec-
trograph, which uses a grating to disperse the light from a target object onto a Loral
3K CCD, which is a 3000 x 1000 pixel camera. For this project, the optical setup
was a slit width set to 1.5 arcsec, the decker set to 2, and the collimator set to blue.
Since the primary spectral lines of interest for this survey were the Call H and K lines
as well as the Balmer lines, the KPGLF grating was used at a tilt of 49.928 which
resulted in a wavelength range of 3505A to 5118A with a 0.51 A/pixel plate scale. A
CuSO4 filter was used to limit light leakage on the red end of the spectrum. The final
resolution of the spectra was R w 2300. The targets were observed over the course of

five nights during Aug/Sep of 2007. This resulted in 120 spectra of RRL stars.

6.1.2 IMAGE REDUCTION AND SPECTRAL EXTRACTION

The spectroscopic images were calibrated and reduced using one of the standard astro-
nomical software suites called image Reduction and Analysis Facility, [RAFl (Tody,
1986, 1993). The methods used were based on A User’s Guide to CCD Reductions
with IRAF by Massey (1997). The images were processed using the normal meth-
ods of bias subtraction, flat fielding and bad pixel removal employing primarily the
CCDPROC task in the IMRED and CCDRED packages. Bias subtraction is done
to remove electrical zero point from the digital image. Flat fielding is a process by
which an image of a uniform object, usually the twilight sky or an illuminated white

spot in the dome, is used to calibrate out sensitivity variations across the CCD, as

 

1IRAF is distributed by the National Optical Astronomy Observatories, which are operated by
the Association of Universities for Research in Astronomy, Inc., under cooperative agreement with
the National Science Foundation.

60

well as deal with some kinds of optics issues. The spectral shape of the flat lamp was
removed using the RESPONSE task.

Once the images were calibrated they were then processed to transform the two
dimensional images into l-d spectra. The process we used to extract the spectra
was based on the document A User’s Guide to Reducing Slit Spectra with [RAF by
Massey et al. (1992). The background was removed from the object spectra by using
the task APALL, part of the KPNOSLIT package. During this process the light that
is in the object aperture is summed to make a 1 dimensional spectrum. In order
to use the l-d spectra it is necessary to calibrate the wavelengths versus a known
source. For this purpose a number of Helium-Neon-Argon comparison lamps were
observed. The calibration lamps must be taken after any significant movement of the
telescope because flexure in the spectrograph itself varies as one moves across the sky.
The IDENTIFY task was used to assign wavelength values to the known lines of the
comparison lamp spectra. Finally a wavelength solution was applied to each object
spectrum using the associated comparison lamp spectrum. This was done by the task
DISPCOR, which also linearizes the final object spectrum. The resulting spectra had
cosmic rays removed by hand. An example spectrum sampled at various stages from

beginning to end of this process is shown in Figure 6.1.

6.1.3 RADIAL VELOCITIEs

Now that the spectra were fully processed, it became possible to determine the quan-
tities of interest: the radial velocities and the metallicities. The radial velocities are
found by looking for Doppler shifts in the spectral lines. The Doppler shift is defined
by the change in the central wavelength of an absorption line from the laboratory
value, AA, due to the radial velocity, the velocity of the star along the line of sight,

of a star through the equation:

AA

/\0

Old

61

300

 

 

 

 

 

 

 

 

 

 

 

 

200 _ .r ‘IN”wwt’fi’éfihfiafiit??? [*‘1.ylj"‘;.%t.wm m “We ..
E 100 W I l W HVPWN\W1MM,A _
0 _ l l f l MAW
_100 l l l I 1
0 500 1000 1500 2000 2500
Pixels
400 l l l I T T I
X 300 l' w, Wamwwl‘fiam(“W"I'Al‘wl‘iffi‘i’*‘ll'l“‘”if‘w“i‘“wim““q ”WWW“;
E 200 _ . #2“, WM A?” l
100 R.A,/~.,:“*-,r l ,1 -
3800 4000 4200 4400 4600 4800 5000
Wavelength (Angtroms)
1.2 [H 1‘ I . I l I V I II
x 0.8 Vii W, ,1” A r‘~*‘*"“A M‘twwfl M"WWivMA-a~'~‘t*W=wMM,WWW”!
2 l ‘ 1 ‘f l! l V
“- 0.4 -‘ 1 l u l -
o l I L I L l l
3800 4000 4200 4400 4600 4800 5000

Wavelength (Angtroms)

Figure 6.1 Examples of CTIO spectra at different stages of reduction. Top: Raw l-d
spectrum. Middle: Spectrum with wavelength solution and bad columns removed.
Bottom: Final continuum flattened spectrum.

where u is the velocity of the star, c is the speed of light, and A0 is the laboratory rest
wavelength of the line. To determine the radial velocities two different methods were
employed. The first was to use a line by line measurement method and the second
was to match up the whole spectrum to a template spectrum using a cross-correlation
algorithm. For the metallicities a line index method based on measurements of the
CaII K line were used.

The first method to determine the radial velocity of each star was based on the
individual lines shown in Table 6.1. For each line a 10A search radius around the
lab frame value is used to try and find the line peak in the object spectrum. Then
a Gaussian is fit to the line taking into account asymmetries. After this has been

done for each line, the lines are taken together and a weighted mean and standard

62

deviation is found. If a single line deviates significantly from the rest, then it is
removed and the mean is recalculated. This is the same basic techniques as in Beers
et al. (1990). This method benefits from being straightforward, but is subject to
issues like line misidentifications. Also having one or two bad line measurements,
can strongly influence the final value of the mean, although this is relatively easy to

check and correct for. The heliocentric correction for these spectra was done using

the RVCORRECT task in the ASTUTILS package.

Table 6.1: Spectral Lines used to Determine Radial Velocity

 

r

Name Wavelength (A) Series Element

 

H3 3798.329 Balmer H
H 5 3888.896 Balmer H
K 3933.669 Fraunhofer CaII
H 5 4101.637 Balmer H
H, 4340.319 Balmer H
H 3 4861.255 Balmer H

 

The second method for determining the radial velocity involves doing a cross-
correlation fit against template spectra. The template spectra represent a grid of
metallicity and temperature for horizontal-branch stars. The template spectra are
synthetic and were created using the ATLASQ model atmospheres from Kurucz and
using the spectral synthesis program SPECTRUM (Gray & Corbally, 1994) and were
provided to us by Wilhelm (private communication). The first step is to find the
appropriate template for each star. This is done in a non-interactive mode of the
FXCOR task. A lorentzian function was applied to the correlation peaks for each
template. Once the best template was found, the FXCOR task was run interactively

using a parabola to find the radial velocity with the best correlation peak. This

63

method has the benefit that it uses the entire spectra to determine the radial velocity.
The caveat to this is that low signal to nose and defects in the spectra can cause poor
radial velocity measurements.

Once both velocities were determined for each star, a comparison was made be-
tween the two velocities. If the absolute difference between the two was less than
15 km/s, then they were weighted averaged together. If the absolute difference was
larger than 15 km/s, then the fits for both methods were examined and if the line
by line method returned consistent velocities for each line it was used, otherwise the

template method’s velocity was used.

6.1.4 DETERMINING METALLICITY

Determining the metallicity of RRL stars requires extra care compared to non-pulsating
stars. This is because as the RRL star goes through it’s pulsation cycle the temper-
ature of the star changes. This is reflected in the strength of the hydrogen Balmer
lines for these stars. As was mentioned in Section 3.1 the spectral type measured by
these lines varies from A7 or A8 at maximum to F5 or F6 at minimum light for RRab
and for RRc stars A7 or A8 at maximum to F1 or F2 at minimum. An example of
how the spectrum of a star can change through its pulsation can be seen in Figure
6.2. The situation is even more complicated in RRab stars where significant shocks
can form in the atmosphere causing line doubling near maximum brightness.

One common method for determining metallicity is to measure the strength of
the Call K line. There is relation between [Ca/H] and [Fe/H] and the Call line is
much easier to measure in low to medium resolution spectra than the more numerous,
but weaker, iron lines. Using this basic principle Preston (1959) developed the AS
system for measuring the metallicity of RRL stars. The AS is the difference between
the spectral type as determined by the Balmer H7 and H5 lines and from the Call

line. In metal poor stars, AS is large and in metal rich stars it is small. The AS

64

F '1
IAIDE-ZH - m l“ (I:

i.eoE-eu - ) fl

 

 

 

 

 

 

l. DOE-2'! _ A

 

 

 

 

 

 

8.00E-25 P

 

 

 

 

 

F114)

6.00E-25 p _

1.005-25 _ .‘

  
   

    

2.00E-25 -

| l
1 I 1&1 § 7 B .
'I

I J I 41 J 1 l
3600: 3800.

 

 

 

ooo. uzoo. uuoo. neon. usoo. sooo.
LRMBDF!

Figure 6.2 An example of how the spectrum of the RRab star changes as a function of
phase. The above spectra are of the RRab star RW Dra. The top most spectra is the
star near maximum and the bottom one is the star at minimum. The line strength
of Call K versus the Balmer lines clearly changes. Figure from Smith (1995).

system was defined for RRab stars at minimum light although it was extended to
other phases, although still avoiding maximum, by Butler (1975) and Smith (1986).
Kemper (1982) then extended the AS system to include RRc stars. There have been
a number of papers dealing with alterations to the basic AS system, but there have
also been parallel methods that are very similar. For instance, Freeman & Rodgers
(1975) developed a method that is based on pseudoequivalent widths of the Balmer
lines and Call K. The method we followed, builds on that study and was developed
by Layden (1994).

We measured the equivalent widths of H7 and Call K, using a Voigt profile fitting

65

routine as in Wilhelm et al. (1999). The metallicities were then determined following
Layden (1994) and were used as the first pass to determine a distance to each of our
stars using the methods describe in 7.1.3. Then the equivalent widths were corrected
for interstellar CaII contributions using Beers (1990). These corrected equivalent

widths were then used to get a final metallicity measurement.

6.2 SDSS DATA

SDSS has a wide variety of spectroscopic programs and has taken a large number of
stellar spectra. As it happens, RRL stars have been accidentally measure from time
to time as part of other programs. It soon became apparent that over 300 spectra had
been taken of our stars! On the downside, these spectra were observed without any
of the precautions usually taken when dealing with RRL stars. This issue is further
complicated by the fact that SDSS spectra are not in themselves single observations,
but rather are composites of several single observations, which in some cases are
taken over multiple nights. This makes metallicity determinations difficult since in
many cases it is impossible to determine what different phases the combined spectrum
was created from, and this, as shown in Figure 6.2, would lead to highly inaccurate
metallicity determination.

Relatively recently the single-epoch spectra data has become available, although
only as two dimensional spectra. Processed one dimensional single-epoch spectra
became available from West (2008). These spectra only contained flux and error
as a function of wavelength, although they were processed using the same software
as Adelman-McCarthy et al. (2008). To analyze these spectra further, the headers
from the combined spectra were copied to each of single-epoch spectra to create a
starting point for further analysis. It soon became apparent that the signal to noise
ratio on many of the single-epoch spectra was too low to able to determine radial

velocities and metallicities. As a result, we recombined the single-epoch data taking

66

into account the phase of each spectra, and an example of this is shown in Figure 6.3.
This recombination was done by taking the weighted mean of each of the spectra,

removing the highest point at any given wavelength to help correct for cosmic-rays.

 

400

300

200

 

Flux

 

 

100

 

AAAAAAAAAA 4

8000 9000

 

 

 

Flux

AgAAAAAAAAka

Wavelength (Angstroms)

Figure 6.3 An example of the recombination of three single-epoch spectra (top graph)
for a given star at roughly the same phase into a spectrum with much better signal
to noise (bottom graph), and with most of the cosmic rays removed.

Once the recombination was done, the spectra were processed through the SEGUE
Stellar parameter pipeline (SSPP) described in Lee et al. (2007). This pipeline cre~

67

ated as part of SDSS 11 determines stellar parameters including temperature, effective
gravity, metallicity and radial velocity through a number of different methods. The
two quantities that we are interested in of course are metallicity and radial veloc-
ity. The radial velocities that would normally come out of the SDSS spectroscopic
reduction pipeline (Stoughton et al., 2002) are not usually reliable because they are
found using the normal combined spectra. The SSPP run on the recombined spec-
tra redetermines the radial velocities using strong absorption lines (Lee et al., 2007).

Since radial velocities need to be corrected for phase, we will discuss them further in

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Section 7.2.
0 u
-0.5 - -
F! '1 ' '
I
B
E:
9
'5 -1.5 - -
I l
-2 - -
K,
-2.5 - L l A I J
-2.5 -2 -1.5 -1 -0.5 0
SDSS[Fe/H]

Figure 6.4 A comparison between the metallicities from the SDSS spectra processed
using the SSPP and the CTIO spectra processed using the method of Layden (1994).
The solid line is the unity line. The dashed line is the least squares fit.

The metallicity determinations in the SSPP use up to 77 separate measurements

of line indices. These methods are fully described in Lee et al. (2007). For our

68

purposes the primary issue is to ensure that the RRL stars are near minimum for
these measurements. As described in the previous section, the metallicity of an RRL
star is defined at minimum, since the SSPP has no corrections built into it to deal
with different phases. For comparing the CTIO spectra and the SDSS spectra we
restricted ourselves to the phase ranges of 0.4 to 0.8 for RRab and 0.3 and 0.7 for
RRc. Also, we removed stars that showed signs of Blazhko effect since this could lead
to poor phase determination. The result of this comparison is shown in Figure 6.4.
The dotted line shows the least-squares fit to the data points. As one can see, the
relationship is linear, and only removed from the line of unity by a small offset. The

equation for this fit is shown in (6.1).
[Fe/HlCTIO = 1.013(2l:0.278) >< [Fe/HlSDSS + 0.275 i 0.077 (6.1)

The fit was done in this fashion because the error bars on the CTIO data are larger
than on the SDSS data. However, the CTIO metallicity scale is based on a poorly
defined zero-point, and so the metallicity system that we will use is the SDSS one. All
of the CTIO metallicities will be transformed to the SDSS scale by using the inverse

of Equation (6.1) shown in Equation (6.2) with errors calculated using Equation (6.3)

2 _.:. UCTIO 2
UlFe/Hl _ ( 1.013 ) (6'3)

All metallicities in the rest of this document will be on the SDSS system.

69

 

CHAPTER 7:
PRODUCTION OF DERIVED
PARAMETERS

 

In chapter 6 the observation and reduction of spectra from both the both the SDSS
survey and the CTIO Blanco 4m was discussed, however, that was only the first step.
The spectra taken represent only 241 of the total 421 RRL stars found in this survey.
Using the information gleaned from the reduction of those stars, in this chapter we’ll
create relationships that will relate photometric observations to metallicity. Also, the
radial velocities will be corrected and transformed into frames of reference, where

they can be used to explore the kinematics of the halo.

7.1 DETERMINING METALLICITIES & DISTANCES

In general, it is much more difficult to get spectroscopic information about a star
than it is to get photometric data. As such, a number of attempts have been made to
relate photometric parameters to metallicity. There are two techniques that are of in-
terest to us, and they are Fourier Coefficient fitting and Period-Amplitude-Metallicity
relations. Both of these methods attempt to use information about the shape of the
lightcurve to determine the metallicity of an RRL star. As a consequence of this,
they are primarily used with type RRab stars, which have more variation in their
lightcurve shape. Although progress has been made on the Fourier method with RRc

stars in Morgan et al. (2006) there are theoretical reasons to believe that the Period-

70

Amplitude-Metallicity relation may not be useful for RRc stars (Bono et al., 2007).

So for this work both of these methods will be restricted to RRab stars.

7 . 1 . 1 FOURIER METALLICITIES

Throughout much of the history of the study of variable stars, lightcurve shape was
often discussed in very qualitative terms. In an attempt to bring a more quantifiable
method to describing the shape of a light curve, Simon & Lee (1981) decided to use
Fourier cosine series to describe the shape of Cepheid variable lightcurves. Simon
expanded this work to include RRL stars in Simon & Teays (1982). More recently,
Jurcsik & Kovacs (1996) used Fourier sine series to develop a method for determining
metallicities. They used a Fourier sine series as shown in (7.1). In this equation
A,- is the amplitude of each order, (p,- is the frequency offset, t is the HJD, and w,-
is the frequency. They needed a set of parameters that could be used to relate the
different orders of the sine series. The adopted convention was to create a parameter
Rij = Ai/Aj to relate the amplitudes and ¢ij = jab,- - idj to relate the frequency
offsets.

n
V = A0 + 2 Ai sin[27rwit + 902'] (7.1)

i=1
Since the various papers on this subject use both sine and cosine series to describe the
lightcurves, it is worth noting that the Rij remains the same regardless of whether
a sine or cosine series is used. As for the 45,5, it can be changed between sine and
cosine by adding the factors as appropriate from Table 7.1. The table is cyclic, so to

get conversions for higher 4517' just start again from the top.

Table 7 .1: (1331 Conversion Chart

Parameter Factor Parameter Factor

 

Cosine to Sine Sine to Cosine

 

71

Table 7 .1: $31 Conversion Chart (continued)

 

 

Parameter Factor Parameter Factor

 

$21 —% $21 +§
$31 +7T $31 —7T
$41 -37“ $41 +3?"
$51 0 $51 0

 

Jurcsik & Kovacs (1996) produced a relationship relating the parameters (1)31 and

period to metallicity that is shown in equation (7.2).
J[Fe/H] = -5.038 - 5.394P + 1.3450531 (7.2)

This relationship is defined for the Johnson V filter, and as such is not of much use
for this project, but it does provide a template for how to approach this relation. In
this case, we are interested in creating a relation based on the g lightcurves. We used
the metallicities from the SDSS spectra and an 8th order Fourier Sine series fit to the
lightcurve data to create a relationship between metallicity, (1531, and period shown
in Figure 7.1. In Jurcsik & Kovacs (1996) they used a 15th order sine series, but we
reduced the order to prevent ringing in the fit. The relatively small number of stars
(32), is because only the best fitting Fourier series were used.

Clearly the (1)31 parameter is the less certain of the two as to be expected, because
although our lightcurves are fairly complete, (1)31 is very sensitive to gaps and scatter
in the magnitudes. This fit resulted in the metallicity relationship showed in equation

(7.3) with an error in the metallicity given by (7.4).

[Fe/H]SDSS = —4.181(:l:0.090) — 4.653(i1.043)P + 1.043(i0.270)¢31 (7.3)

72

 

6 r I E I ~ 1.8

- f i ‘ j; i a 1.6
-65 - i g f, f . T ,, ‘» ‘3
5 f .. 1‘13; 17. 1 _ f i i i f -l. 1.4 '5
. 7 .; ff} 1 T w ,. ”3% i 1 1,2 1::
g I 1?le 1- . if ’ g
uh" f g —.\" T .1 ; * f "I, .. d 1 IL
75 - ‘ if} I U H

QI- * i - 0.4

 

 

 

-8.5 l I I l l I I l I I I I I
0.5 0.55 0.6 0.65 4.7 4.8 4.9 5 5.1 5.2 5.3 5.4 5.5

POI'IOd $31

 

Figure 7 .1 The dashed line shows the least squares fit. The plots show the dependence
of the fit on the two independent variables, Period (in days) on the left and ¢31 on
the right.

0[2Fe/H] g (1.0430,,31)2 (7.4)

7. 1.2 PERIOD-AMPLITUDE-METALLICITY

The Fourier metallicity method is powerful tool for getting metallicites for stars
without spectra; Unfortunately it is limited to RRL stars that have fairly complete
lightcurves, so another method is necessary. Sandage (2004) found a relationship
between the amplitude in the Johnson V filter, the period, and the metallicity of an

RRab star. His relationship is shown in equation (7 .5).
[Fe/ H] = —1.453(:l:0.027)AV — 7.990(i0.091) logP — 2.145 :l: 0.025 (7.5)

In general the period-amplitude relation for determining metallicity is has much more
scatter in it relative to the Fourier method. This only makes sense given that it uses

amplitude, which is much less sensitive to lightcurve shape than (1)31 (Bono et al.,

73

2007). On the other hand, amplitude is a much easier quantity to measure and can
be used on relatively incomplete lightcurves. Again we used our own metallicities and

the g lightcurves to create a period-amplitude relation. The fit can be seen in Figure

 

 

 

 

 

7.2.
0.2 I r u I l x 2
O " [ “ E ‘ -2.2
.02 j} .. ._ , . -2.4
. T I .
-0.4 A; x I; _ j J: T g « -2.6
<9 -o.a + liTlQ iii E T ti 4 - -2.8 E
. l 1., l I i “xxxfi g
E -o.a t i ii ~. f i ._ ., %} if}? i £7 -3 g
_ .1 t } E’# ‘r f E E}? f “:32 u.
-1.2 L .Ifiz‘l \ .- i } ix“- 4.4.4
-1.4 - i i - 4,:- 1 I V‘x‘} -3.6
-1.6 - { “xx:- i i - -3.8
-1.8 ' 1 ' 1 ‘ L .4
03 ~025 -O.2 -O.15 06 08 1 12 14 -
L090”) Ag

Figure 7.2 The dashed line shows the least squares fit. The plots show the dependence
of the fit on the two independent variables, base ten logarithm of the period (in days)
on the left and the amplitude in g on the right.

Clearly there is a lot of scatter in the relation, but this method does provide a
rough estimate of the metallicity for those stars with poor lightcurves and no spec-
troscopy. The new period-amplitude relation can be found in equation (7.6) with the

errors in the metallicity found in equation (7.7).
[Fe/H]SDSS = —5.921(:}:1.204) loglo P — O.816(:l:0.191)Ag — 2.289 :1: 0.056 (7.6)

ogre/H] = (1.204log10 P)2 + (0.191149)2 + 0.0562 (7.7)

74

7.1.3 DISTANCE

One of the most important properties of RRL stars for the purpose of studying the
structure of the halo is the fact that they are very good standard candles. As discussed
in Section 3.6, we can use the fact that we know the intrinsic luminosity of the RRL
star and the measured brightness to determine the distance to the star using the

distance modulus formula:

mg — Mg = 5log10(d) — 5

Where mg is the apparent magnitude in the 9 filter, M9 is the absolute magnitude in
the 9 filter, and d is the distance to the star in parsecs. There are two issues that need
to be address before the distance modulus can be used. The first is that the absolute
magnitude for RRL is normally described in the Johnson V band. The second is that
there is some metallicity dependence to the intrinsic brightness of RRL.

There have been a number of papers on trying to determine the absolute mag—
nitude of RRL stars. We followed the work of Wilhelm et al. (2007) to create our
M9 vs. [Fe/ H] relationship. To start with, we used the zero point of My = 0.55 for
[Fe/ H] = -1.6 from Demarque et a1. (2000). This was then transformed to the SDSS
9 band by using a B-V color for RRL stars to be 0.293 and Fukugita et al. (1996)
gives M9 = 0.594 at [Fe/H] = -1.6. Combining this with the slope of 0.214 from the
My vs. [Fe/ H] relation by Clementini et al. (2003) gives the final absolute magnitude

equation shown in equation (7.8).

M9 = 0.214[Fe/H] + 0.94 (7.8)

In order to be able use equation (7.8), we had to have the metallicity for each of our
RRL stars. A metallicity was assigned for each RRab and most of the RRc stars by
combining the results from the CTIO and SDSS spectroscopy and the photometric

75

methods. The results of this are shown in Table 7.2 with N being the number of stars

that used that method, and type being the type of RRL star the method applies to.

For the RRc stars that had no metallicity determination from spectroscopy, the mean

and standard deviation metallicity from the RRc with spectroscopy was adopted for

the metallicity and the error respectively. The methods are self-explanatory except

for group (C). In that case, since there is no correction for phase, we used a weighted

mean of the metallicity from spectroscopy and from the Fourier method when the

lightcurve was good enough. The order of the groups from A-E for RRab and F-I

for RRc is the order of quality of the metallicity determination with group (I) being

used only for distance determination and not for any of the analysis in Chapter 8.

Table 7 .2: Methods used for Determining Metallicity for RRL Stars

 

Group Type Method N
(A) RRab SDSS Spectra from 0.5 to 0.8 in phase 52
(B) RRab CTIO Spectra 68
(C) RRab SDSS Spectra all other phases except maximum w/ Fourier method 46
(D) RRab Fourier method 86
(E) RRab Period-Amplitude method 53
(F) BBC SDSS Spectra from 0.3 to 0.6 in phase 29
(G) RRc CTIO Spectra 15
(H) RRc SDSS Spectra all other phases except maximum 15
(I) RRc Set to mean of other RRc, [Fe/H] = —1.52 :l: 0.55 57

 

7 .2 CENTER OF MASS CALCULATIONS

One issue unique to using variable stars as kinematic tracers is that they have a

component of their radial velocity that is due to the mechanism that makes them

76

variable. In the case of RRL stars, their pulsation is the source of this added radial
velocity as discussed in Section 3.1. Before they can be used as tracers, their radial
velocity must be corrected to the center of mass frame. For RRab stars, there is a
method using a synthetic radial velocity curve developed by Liu (1991). The synthetic
curve can be seen in Figure 7.3. The normalized function S(¢) is scaled by the
amplitude in the V filter to get the radial velocity correction for any given phase. To

transform our 9 amplitudes to V, we just used the relation from Section 5.2.3.

 

 

 

 

 

1 . . . . a . 1
0.9 - - - 0.9
0.8 - -- - 0.8
0.7 - - - 0.7
A 0.6 - -- - 0.6 ...
(7)6" 0.5 - ~ - 0.5 g
0.4 - l" - 0.4
0.3 ~ - - 0.3
0.2 - -- 0.2
0.1 - - 0.1
O I I I I I J L I
0 0.2 0.4 0.6 0.3 o 0.2 0.4 0.6 0.8
phase phase

Figure 7.3 The two normalized radial velocity templates were used to correct our
radial velocities to the center of mass. The one on the left is for RRab type, the one
on the right for RRc type.

As for the RRc stars, the issue is a little less clear cut. There is no standard
method for correcting RRc stars, so we followed the lead of Jeffery et al. (2007). We
used the radial velocity data from Jones et al. (1988) for DH Peg to create a radial
velocity template by fitting the curve with a smoothed spline. Then a simple scaling
relation was used to relate V amplitude to radial velocity amplitude. The resulting
radial velocity curve is shown in Figure 7.3.

Once the radial velocities have been corrected to the center of mass velocity, it is

now possible to compare the radial velocities from the CTIO and the SDSS spectra.

77

Figure 7.4 shows the fit between the two systems. The transformation equation

between the two systems is shown in equation (7.4) with errors in equation (7.10). It

is clear from equation 7.4 that there is an insignificant offset and only a slight slope

from unity. All the radial velocities were transformed onto the SDSS system.

SDSS Radial Velocity (km/s)

200

100

-600

 

l I I l I I I
'T' i
I”
’I
I’ A
I
I
’I
q
i d
I
’ 'l
I”
I
-
’1
’I
’I
I’
I
1” .4
I
I’
I
’I
I”
I -
I
I A I I I I I

 

 

 

-500 -400 -300 -200 -100 0 100 200
CTIO Radial Velocity (km/s)

Figure 7.4 The dashed line shows the least squares fit between the SDSS and CTIO
radial velocity systems, the solid black line is the line of unity.

VSDSS = 0.931(i0.0258)VCT10 — 3.765 :l: 3.792 (7.9)

a?) = (093100770)2 + (0.0258VCT10)2 + 3.7922 (7.10)

78

7 .3 FRAME OF REFERENCE

One of the fundamental issues in trying to understand the kinematics and structure of
galaxy is that we observe from a moving platform. For the same reasons that we had
correct the Julian date to the heliocentric frame in Section 3.2, so too do the radial
velocities have to be corrected to the heliocentric system to remove the motion of
Earth’s orbit from the radial velocities. This is good enough for comparing different
observations, but to look at the kinematics of the galaxy we had to go a few steps
further.

First of all, one needs to understand the coordinate systems that are used in
astronomy. One of the most common systems used for describing observations is the
equatorial system. The equatorial system is effectively a grid system of the longitude
and latitude for the sky where the longitudinal coordinate is called Right Ascension,
RA, and the latitudinal coordinate is Declination, Dec. In general, unless the object
has high proper motion, each star has its own unique RA and Dec. The equatorial
system is aligned with the Earth with plus and minus 90° being directly overhead,
the zenith, at the north and south poles. RA is often measured in hours going from
0 to 24, with 0 hours being defined at the moment of the vernal equinox, although
degrees are also used running from 0 to 360. As one moves to the east on the sky,
the RA increases, and at any given time at any given location only 12 hours of RA
can be seen from the east to west horizon. For every hour that passes, the RA that
passes through the zenith advances by one hour going through the full 24 in a day.
Every night at local midnight, the RA is different from the previous night by roughly
4 minutes, completing a full rotation once a year due to the revolution of the Earth
about the Sun.

Although the equatorial system is useful for observation, for our purposes, the
galactic coordinate system is the most useful. This system where b is the galactic

latitude and l is the galactic longitude is shown in Figure 7.5a. These coordinates are

79

analogous to latitude and longitude on Earth. In the l coordinate, 0° points towards
the center of the galaxy whereas 180° points towards the anti-center. In b, the North
Galactic Pole (NGP) is at +90° and likewise the South Galactic Pole (SGP) is at
—90°. One note of caution, l and b are defined at the position of the Earth not the
center of the galaxy like some of the other coordinate systems. Another system that
is often used is the Cartesian coordinate system (X, Y, Z). In this system X points
from the galactic center to the Earth, Y points in the direction of galactic rotation,
and Z points toward the NGP. This is a galactic center based coordinate system,
although there are other definitions where this is not true.

When talking about velocity, often times a cylindrical system based at the center
of the galaxy is used. Figure 7.5b shows the positional components (r, (25, z). The
velocity components for this system would be (VT, V4,, and V2). Another frame
of reference that is often used is the space velocity taken with respect to the Local
Standard of Rest (LSR). The LSR is defined as a circular orbit in the plane of the
galaxy at the radius of the Sun. These velocities are (U, V, W ) and their direction is
shown in Figure 7.5b. For more information on coordinate systems consult Mihalas
& Binney (1981).

With these coordinates in mind, we then proceeded to correct for the Sun’s pecu-
liar velocities with respect to the LSR. Since the Sun’s orbit is slightly non-circular,
we need to correct to this frame. Equation (7.11) transforms from heliocentric radial
velocities to LSR radial velocities (Allende Prieto et al., 2006). The Sun’s peculiar
velocities (Ug, V9, W9) = (-9.0, 12.0, 7.0 ) km/s (Mihalas & Binney, 1981).

VLSR = Vhelio — U9 cosbcosl + VG cosbsinl + W9 sinb (7.11)

Once we had the radial velocities in the LSR frame we went one more step, and that
was to put the radial velocities into the galactic standard of rest (GSR). This way

the LSR’s orbit around the Galaxy is removed. This is a simple transformation; one

80

merely subtracts off the LSR’s rotational velocity 8L5 R = 220 km/s along the line
of sight as shown in equation (7.12). Once the radial velocities are in the GSR frame,

we were able to begin analyzing the kinematics of the RRL stars.

V6153: VLSR+9LSRCOSbSinl (7.12)

81

a)

 

  
     
 

 

 

 

 

fr
/1 Rotation
I A
b s
a c ’ I : Galactic Center
1800 Su 00
D)
W
u “(A
4‘ Rotation
A

 

r
?\ Galactic Center

 

 

 

 

Figure 7.5 Frame a) shows the galactic coordinate system describing l and 0. Frame
b) shows the cylindrical coordinate system with the U, V, and W space vectors.

82

 

CHAPTER 8:
TRACING THE HALO

 

Having created a catalog of RRL stars with photometric and kinematic quantities
we were then able to begin looking for signs of different populations in the Halo.
There were three different methods used to separate out the populations: photometric
properties, kinematic properties and metallicity groups. Each of these methods have
their benefits and weaknesses, but together they should provide a clear picture of the

halo. Each of these will be taken in turn in the following sections.

8.1 PHOTOMETRIC PROPERTIES

8.1.1 SPACIAL DISTRIBUTION

One of the benefits of looking at the spacial distribution of the RRL stars was that we
were able to use all 421 stars. This is especially useful when one has just an equatorial
strip of sky because that allows us to only worry about the RA component of the
location. One issue that can affect the interpretation of a spacial distribution map is
the change in extinction as a function of RA. The extinction is plotted in Figure 8.1
and comes from the reddening maps of Schlegel et a]. (1998). It is worth noting that
there is as much as 0.5 magnitudes worth of extinction in the 9 band, which could
cut down on the detection of low amplitude RRL in the most distant parts of the
diagram at those RAS.

The map of all 421 RRL stars is shown in Figure 8.2. It is clear that the dis-

83

0.7

 

 

 

+
0.6 -
i.
0.5 " + 4‘."
+ ++ + +
it W:
0.4 ’fif 3“. +
O + *1' ++ +
< + f + i +
its + +» is
0.3 "I +++ + 1+ 4' : ++
14+: $+ + 431:.- + 4'
ft++# + 4* *+t+: + "I.
0.2 ' g... g; is: ffi" + 41-
+ +
+ + it £38» *{L 1 + ++fi= +4: +
* *4: + ++ “+15 3”“
0'1 I. + #1. ##4-fiw j, +* ++
O I I I I I I I I I
31 O 320 330 340 350 360 370 380 390 400
RA (Degrees)

Figure 8.1 An extinction map of the SDSS-II SN stripe as a function of RA. The RAS
between 0° and 60° have an additional 360° added to make a more continuous graph.
It is clear that there is up to 0.5 magnitudes of extinction at the ends of the stripe

82.

tribution of stars is not uniform, but it is not immediately obvious Whether that is
due to stellar streams or just to the observational bias of the sample. What is clear
is that there is a deficiency in the number of stars at large distances at the edges of
the stripe, and this is to be expected given Figure 8.1. In order to be able to make a
more precise statement about the existence or non-existence of stellar streams in this

sample, we intend to use the methods outlined in Vivas & Zinn (2006). However, this

is beyond the scope Of this work and we will return to it in a later paper.

84

 

 

 

 

 

 

 

 

 

 

A 2 ._ [‘3' 0° .. A
8. ':. 1,1 "i . ' .1' §
m '5..." . m
0 ,2" O
c :. . -.._- c
s - s
a) - ......... " ,-' _. pg
0 4o D
............. 352:0 - ..
60 .. ........... -.%o,.-"
80 .............

Figure 8.2 Shows the spacial location of the all 421 RRL stars in this survey as a
function of RA. The both maps are identical except that the map on the right has
the distance error bars included.

8.1.2 OOSTERHOFF PROPERTIES

One of the many benefits of having used RRL stars as tracers of halo populations was
that we could take advantage of the Oosterhoff classification as a way of separating
stellar components. As discussed in Section 3.6, Lee & Carney (1999) believed that
OoII type clusters were associated with an inner halo and that 001 type clusters were
associated with an outer accreted halo. In our case, we did not have globular clusters
to work with, so we needed another way to distinguish between 001 and Doll RRL
stars. One method uses a parameter called A log P which applies only to RRab stars

and is defined in equation (8.1).
A log P = log Pobs — log P(OOI P-A relation) (8.1)

This equation uses the 001 period-amplitude line from Clement & Rowe (2000) shown
in Figure 5.6 and described in equation 5.8. The amplitude of the RRab star is put into
this relation and the resulting log period was subtracted from the observed log period.

The result is shown in Figure 8.3. In Figure 8.3a, the 001 and 0011 populations are

85

clearly visible with the 001 peaking at 0 by definition and the 0011 peaking at A log P
of 0.061. This is at a somewhat lower than the 0.075 that the LONEOS survey found
(Miceli et al., 2008), but it is very possible that this difference can be explained
through binning and the fact that we are using a different filter. Also, our 0011
peak is not as well defined, which puts our plot as an intermediate one between the

QUEST-I plot where only a tail is visible and LONEOS with it’s well defined peak.

 

 

 

 

 

 

 

 

 

 

 

 

 

45 l I I I l l- 1 I I 11 I 1

40 - a) AIIIZI - b) - 8>I2I * ‘4

35 - -r ‘ 12
g 30 r -- - 10
(D 25 - -_ - 8
n I-

g 20 1,, .4 6
a; 15 t " 4
10 - JLLL -' ‘

5 _ _- +4 . 2
30 r" : l .+ . i n" . : ' +7! J. H 40
25 F C) 15>|Z|>8 1' d) 15<|Z| a 35
9 ' n 30
g 20 r ‘- 1 25
g 15 ' "' 'l 20
a: 10 - -' _r ‘ ‘5
n: - - 10
5 - LL, ‘- - 5

O {—1 l_ L l l— l L I I 0

-0.1-0.05 0 0.05 0.1 0.15 -0.1-0.05 0 0.05 0.1 0.15
ALog(P) ALog(P)

Figure 8.3 These four plots show Alog P as a function of [2] below the plane. The
dashed line shows the out between 001 and 0011 types and is located at 0.045.

One way to look at the different components of the halo is to look at stellar
properties as a function of distance away from the plane of the disk. In general, one
would like to use IZmax| the maximum distance away from the plane that a star
reaches in its orbit. This is a strong selection parameter because it separates out not

just different components of the halo, but also removes mixing in the sample between

86

the thick disk stars and halo stars. Thick disk stars stay confined to the thick disk
and thus have a smaller |Zmax| than halo stars. If there are two components to the
stellar halo, then they would also separate out to some degree in Zmax. We do not
have good proper motions for the RRL stars in this survey, so we cannot get full
orbital solutions for them. This means that we cannot determine |Zmaxl, but we
can get a relatively good proxy for it |Z|. Stars spend most of their time near the
maximum of their orbit, so the current distance |Z| away from the plane is correlated
with Zmax; although this will introduce some scatter into our samples because there
will be halo stars away from their maximum with low |Z|.

Figures 8.3 b, c and (I show how the distributions of A log P change as a function
of |Z|. Clearly the 0011 population is strongest near the plane of the galaxy, and
drops off significantly at higher |Z|. That being said, it is unclear whether the OoII
component disappears completely, because it is not known whether the distribution
of A log P should be Gaussian. It may very well not be Gaussian.

When Suntzeff et al. (1991) plotted A log P versus [Fe / H] they saw a gap between
the 001 stars and the 0011 stars. This would be the expectation at first pass because
the Milky Way globulars show a separation in mean RRab period as discussed in Sec-
tion 3.5. They determined their metallicities using the AS method. Later Kinemuchi
et al. (2006) found that their version of the plot had no clear separation, although
in their case they did not have spectroscopy and were instead using the Jurcsik &
Kovacs (1996) method to obtain metal abundances. As can be seen in Figure 8.4a, we
also see very little separation, but there are definitely two main groups in our sample
of RRab. To check and make sure that this is not due to the Jurcsik & Kovacs (1996)
method, Figure 8.4b uses only RRab with SDSS spectra, and still there appears to
be only a thinning out of stars near where the gap should be, although it is possible

that the gap is being lost due to our error bars.

87

 

 

 

 

 

061 + Ool +
a) 00" X b) 00" x
0.15 - i» - -0.15
'01 - "’ + + " + 4. " '01
9:4 + + +++""' + + + 9.;
8’ -0.05 ' + + -- + ++ + ‘ -0.05 8’
a. ++4+$1+ 4+1"- + a]
+* .. + . tat:
+ 4. ‘0' + 4-
Oh +1 n++ + .- ++ffif1 + ' 0
+ + + 1I: + + 4* It
+ ‘1' +++
+ ++++ 3+ ++ + :1.
005 - xgfi x * -- Xx *+ 005
. "xwx Xxxxx)$‘ 1 '
7% X x x x
x Xx Xx
x )5 2‘ x
0.1 I I fix I I I l I I I 0.1
-3.5 '3 -2.5 '2 -1.5 -1 -0.5 '0 '2.5 '2 -1.5 -1 -0.5 0
[Fe/H] [Fe/H]

Figure 8.4 These two plots show the relationship between A log P and [Fe/ H] and are
similar to plots in Suntzeff et a1. (1991) and Kinemuchi et al. (2006). Plot a) shows
all the RRab, and plot b) is limited to just RRab with SDSS spectra (Groups (A)
and (C)).

8.2 KINEMATIC PROPERTIES

8.2. 1 RADIAL VELOCITIES

Clearly Figure 8.3 is suggestive that there are different populations of stars at different
heights below the plane. One way of looking for those different populations is by
looking at the radial velocities. As discussed in Section 2.3, in the dual halo model
of the galaxy described by Carollo et al. (2007), the inner halo has a small prograde
rotational velocity in the cylindrical coordinate, V¢ and the outer halo has a net
retrograde motion. Unfortunately, we would need good proper motions for our RRL to

derive V¢, so we have to instead use a proxy parameter. Although it is not immediately

88

obvious, the radial velocity in the galactic center of rest, Vgsr actually tracks V¢
relatively well. For more on this, see Section 8.2.2.

One of the other issues we must be aware of is that their is a mixing in our sample
between the thick disk stars and halo stars. This means that for our lowest |Z| bin
there is a component to V93,» that is coming from stars moving at the LSR. velocity
of 220 km/s. For the rest of the |Z| cuts, it is not a significant issue.

In Figure 8.5, we see the distribution of radial velocities with progressive cuts in
|Z|. Figure 8.5a shows a very broad distribution of velocities ranging from -300 to 300
km/s. As we progress through plots b and c in Figure 8.5, it is clear that the entire
distribution shifts from more positive to more negative as one goes higher in |Z I.
Ultimately in 8.5d, there appears to be two separate velocity populations. To help
quantify this shift, the parameters of the Gaussian fits are shown in Table 8.1. This
is what would be expected in a dual halo scenario, because as one progress outward
in |Z| the dominance of the inner halo over the velocity distribution should slowly be

replaced by the outer halo.

Table 8.1: Parameters for Gaussian Fits to Radial Velocity Populations

 

|Z| N Gauss Type Amp Amp Er, Mean Meangr, a 03,,-

All 1 Data 15.5 1.0 -4.62 10.3 138 11

|Z| < 8 1 Data 10.5 1.2 61.5 16.8 131 17

15 > |ZI > 8 1 Data 8.82 0.57 16.0 9.4 126 10
15 < |Z|(B1ue) 2 Data 14.1 0.9 -115 3 42.6 3.2
15 < |Z|(Black) 2 Data 8.55 0.98 41.9 4.6 33.6 4.7

 

Due to the dependence of the relationship between Vgsr and Vphi, we applied

a distance, D < 30 kpc cut to Figure 8.6 and the corresponding Table 8.2. This

89

 

RRL Stars
8

 
  
  

l:
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l

l

 

I
o
l
I
l
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v
-

~~
.
~
\
..
f
a
l
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l

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A

 

RRL Stars

 

010-th
T

 

 

l
l
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l
l
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." l

  
   

.
.
.
.
- I

 

 

-150 0

150 300

Vgsr (km/s)

-300 -150 0 150 300
V93, (km/s)

RRL Stars

RRL Stars

Figure 8.5 These four plots show the radial velocity in the gsr frame for the RRL
stars as a function of |Z| below the plane. Each plot has a Gaussian fit to it, with
plot (1) using a combination of two Gaussians.

significantly reduces the star counts, but the overall shape of the graphs remain

the same, suggesting that the distance cut is not affecting our overall results. In

particular, Figure 8.6d still maintains the dual populations although the star counts

suffer the greatest loss in this |Z| bin.

Table 8.2: Parameters for Gaussian Fits to Radial Velocity Populations with D < 30 kpc

 

 

 

|Z | N Gauss Type Amp Amp Er, Mean Mean Er,- a OE".

All 1 Data 13.8 1.0 10.2 11.8 139 12
_ All 1 Model 43.3 3.1 83.1 10.1 121 10.8
|Z| < 8 1 Data 10.5 1.2 61.5 16.7 131 17
|Z| < 8 1 Model 80.1 7.7 134 7.8 70.2 7.8

 

90

Table 8.2: Gaussian Parameters to Vgsr Fits with D < 30 kpc (continued)

 

|Z| N Gauss Type Amp AmpErr Mean Mean Err a GE"-
15 > |Z| > 8 1 Model 18.9 1.2 -0.6 7.7 103 8
15 > |Z| > 8 1 Data 8.72 0.57 18.6 9.5 126 10

15 < |Z|(Blue) 2 Data 9.47 1.58 -120 6.7 31.6 6.8
15 < |Z|(Black) 2 Data 4.13 1.0 33.1 24.9 70.7 28.9
15 < |Z| 2 Model NA NA NA NA NA NA
15 < |Z| 1 Model 10.9 1.1 11.7 11.9 106 13

 

8.2.2 A QUESTION OF BIAS

One of the first questions that must be dealt with in the context of radial velocities
is whether we are seeing a bias due to the sampling methods. In the case of this
survey, there is even more doubt than usual in that regard because we are looking at
a very narrow stripe that cuts through a wide range of galactic latitude. One way to
examine this issue is to look at a model version of our galaxy. The Besganon model
galaxy from Robin et al. (2003) provides an excellent way to test our observations.
The Besganon model uses a Single kinematically consistent halo, so any effects in
velocity space seen in both our observed and the model histograms are likely to be
due to observational bias. The corollary to that is that anything that is different may
tell us something about the nature of the halo.

The Bescanon model is available at http://bz'son.obs-besancon.fr/modcle/ and the
parameters used to generate the model that we used can be found in Appendix A2.
A few of the most pertinent constraints are that we only used giant stars between
A0 and F9 with My between 0 and 1 to select out only horizontal branch stars. The

model was submitted in two parts, the first going from RA of 300° to 360° and the

91

 

 

 

 

 

RRL Stars
2;
RRL Stars

 

 

(7 15L |Z| >' a P

4
' u
‘\
|
\
‘
0
\
\
u
l
I ‘.
I l I!
l
l ‘5
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n 7‘
l 'I
h I I
\
" “
' u
o '.
, ‘_
-.--’I L I I I "-.

~3OO -150 0 150 300
V95, (km/s)

     

l
1

RRL Stars
RRL Stars

 

 

 

ems-mm
1

 

 

Figure 8.6 These four plots show the radial velocity in the gsr frame for the RRL stars
as a function of |Z| below the plane. Each plot has a Gaussian fit to it, with plot (1)
using a combination of two Gaussians. All of these velocities are for stars with a D
< 30 kpc.

second from a BA of 0° to 60° in steps of 5°. The Dec for both parts ran from —1.25°
to 125° in steps of 125°. Once the model was created it was cut down so that only
stars with a distance between 3.9 kpc and 52.4 kpc were used, keeping the models
consistent with our observations.

The first test of our methods is to check the use of V93,» as a velocity parameter.
The Bescanon model gives us a perfect test bed for determining this because it pro—
vides us with the velocities U, V, and W, which allow us to generate V¢. In Figure
8.7, the relationship between Vgsr and Va, is explored. It is clear from Figure 8.7
that the relationship between V4, and V93,» is basically one to one, although there is

significant scatter. The scatter increases with increasing distance, which makes sense

92

because one has less of a lever arm on the projection of the transverse motion as one
gets farther out. For the purposes of this analysis, 30 kpc appears to be the best
compromise between scatter in the relation and star counts. The relation is centered
near 0 in both V,» and Vgsr, so it is unlikely to have much bearing on the mean of our
velocity distributions, although the scatter in this relation will add to the standard

deviation of our distributions.

 

 

 

 

 

 

300 -+a) ++A|J Dist + r b) Dist<10kpc . 300
+
++ m +
U) 150 _ + z ++ ++ + _‘_ + +++I ++ .. 150 u)
E + # t: + f "’1 +++ E
|— O '- :++$$1 III: .I. + + +I+++ ++ + +141 .- 0 \—
w + + + :l: + m
>0 ‘1' {9V ++ '1' + >0
'150 ' +1» $ #+ 4* 4+ “ 4:" ‘ '150
+4 :4- #1
-300 - .- - -300
300 t c) Dist<20kpc .. d) Dist<30kpc+ . 300
U) 150 * +++Mr#+ ++ + .. + + + ”ti“: q 150 m
E O _ +++ ++ +191” .. 4++++ 41’” _ 0 E
g I 1 + 4,, + + I + at; + + g
> ~150 " 1; #+fi++ "' '1’ #4.. 44* + '4 -150 >
-300 - -- - -300
-300 -150 0 150 300 -300 -150 0 150 300
V, km/s V¢ km/s

Figure 8.7 These four plots show the relationship between Vgsr and V¢ in the model
galaxy. It is clear that the linear relationship becomes less clear as distance increases.
The line is the unity line.

It is now possible to test our observed velocities versus the model of the single
halo. The model data was processed in the exact same way as the observed data
and the resulting histograms are shown in Figure 8.8. The Gaussian fits to the
model data use the exact same starting values and the results are shown in Table

8.2. Qualitatively the Figures 8.6b,c and 8.8b,c appear to be very similar, which is to

93

be expected, since they should be primarily representative of the inner halo. Figures
8.6d and 8.8d, on the other hand appear to be quite different, with 8.6d appearing
to be two separate populations, where 8.8d is clearly one distribution, so much so
that the second Gaussian would not converge. The Table 8.2 also contains the single
Gaussian fit for 8.8d.

One way to test the likelihood that these are similar graphs is to use the Kolmogorov-
Smirnov test (KS-test). This statistical test is nice because it is relatively insensitive
to the underlying distribution that one is trying to look at, but it is not as sensitive to
differences as tests that assume a Gaussian. The null hypothesis, that both the data
and the model come from the same distribution, was the object of this test. To test
this, the model and observational data were compared to each other using KS-test.
The results of this are shown in Table 8.3. The D K S is the maximum difference be-
tween the distributions in the cumulative distribution plot and P K S is the probability
that the hypothesis is true with 0 being 0% and 1 being 100%. The table strongly
suggests that in the farthest |Z| bin the observations are not compatible with the null
hypothesis. It is also interesting that the nearest |Z| bin is also incompatible, owing
to the large positive Vgsr peak in the model. This will be explored more thoroughly
in the next section. The central bin is reasonably compatible, as would be expected,
since it is far from the plane of the galaxy and it should still sample the inner halo

strongly in the dual halo model.

Table 8.3: The Results of the KS-Test of the Null Hypothesis

 

|Z| DKS D < 30 kpc PKS D < 30 kpc DKS PKS

All 02168 0.000 0.1808 0.000

|Z| < 8 0.2534 0.000 0.2534 0.000
15 > |Z| > 8 0.1215 0.435 0.1102 0.518
15 < |Z| 0.2175 0.026 0.3371 0.000

 

94

 

 

  

  

65 I I l I r ‘ 100
a) j 90
55 2 80
a 45 - f = 3 7o 12
55 35 - h 3 6° 3
.1 F55 : 5° _.
CC 25 - ., - 40 II
1: If ; 30 a:
5 . .' '1 10
25 41‘ ’1 1 l [r I I I 4 16
20 C) 15 > |Z| > 8 - 14
.. " 21 - 12
E 15 .. Ml ‘ 10 g
(D l' _ 8 (D
—I F'“ a _1
E 10 - - - 6 E
5 - J -4- . _ . - ‘ """""""" - 4
.' - _______ 2. - ............ a 2
0 .1.-"’ 1 1 1 ‘ 1 """ 1 4 1 O

 

 

 

 

-300 -150 0 150 300 -300 -150 0 150 300
Vgsr (km/s) Vgsr (km/s)

Figure 8.8 These four plots show the radial velocity in the gsr frame for the model
horizontal branch stars as a function of |Z | below the plane. Each plot has a Gaussian
fit to it, with plod d) using a combination of two Gaussians, although the fit did not
converge. The D < 30 kpc cut was applied.

8.2.3 THE THICK DISK

Although the discrepancies between the model galaxy in Figure 8.8 and the data in
Figure 8.6 are understandable in the context of the dual halo except for the |Z| i 8
bin shown in the b) panels. This bin should be dominated by the inner halo, and thus
look like a single halo model, but there is a large asymmetry on the positive Vgsr side
in the model which is not reflected in the data. To understand this more difference, it
is necessary to look at which populations the model stars are being derived from. In

Figure 8.9a, panel b) from Figure 8.8 is redone, but this time the stellar populations

95

that the stars belong to are shown. It is clear that the thick disk is the source of
the asymmetry. To test this further, the populations as a function of Z are shown in
Figure 8.9b. There is a exchange in the dominant population around Z of -4. Using
this as another constraint on this |Z| bin in order to remove the effect of thick disk
stars as much as possible, Figures 8.9c,d show the model and the data. Running a
KS-test on the two shows an increase of the probability of the null hypothesis to .20.
This is still not as high of a P value as is seen in the 15 < |Z| < 8 bin, but this may

be attributed to low numbers of stars and residual thick disk contamination.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

l l I l l l I l 100
100 r _ Halo - b) . Halo
9 80 _ Tthk DISK"; """"" 4 r Tthk DISK """ ‘ 80 2
6% 60 __ 8>IZ| .4 " :..:m: " 60 g
E' 40 - ' ‘ - ' ' ' 4° “-5
m 20 a - - 20 a:
‘ *‘ ‘ fight». 2
0 1 _.JIJ— .11: 4 l 1 1"-..1 1 '-—r'1_' 0
-300 -150 O 150 300 -8 -6 -4 -2
Vgsr Z
I I 7 17 r T 1 T I l 10
c) Model8>|Z|>4 d) Data8> |Z|>4
20 ' ‘ - 1 8
E EFL, F E
S 15 - Ira 4 - - 5 S
(I) L- ' l CD
g; 1° . . . - #1ij - 4 g
5 - L ~ — D J 2
O L [_IJJ l l I 1 0
-300 -150 O 150 300 -300 -150 0 150 300
V957 VQST

Figure 8.9 These four plots show the analysis of the |Z| < 8 bin from Figure 8.8b.
Plot a) shows which model population the stars come from. Plot b) shows how the
two populations change as a function of Z. Plot 0) and d) use a much more narrow
range in |Z| than before to try and exclude thick disk stars.

The question that then presents itself is: Why does sampling the thick disk cause

disagreement with the model? The model galaxy is supposed to be a model the

96

thick disk as well as the halo, so there should be no disagreement. The most likely
cause of the issue is that the model produces non-variable horizontal branch stars,
and the data is composed of variable RRL stars. An underlying assumption of this
analysis is that the ratio of variable to non-variable stars is fixed. Within a given
population this should be roughly true, but there is no guarantee that the ratio
will be the same in between populations. In fact, work by Layden (1995a,b) found
that there is a difference in the ratios of RRL stars to horizontal branch stars in
these two populations. This should not be an unexpected result, since the ratio
in each population should be dependent on the horizontal branch morphology of
that population, which itself is dependent on properties such as metallicity. Since
metallicity clearly differs between the thick disk and the halo, it makes. sense that

their ratios should be different as well.

8.3 COMPOSITION PROPERTIES

The final way to analyze the data was by looking at the metallicity distributions.
In the dual halo scenario described by Carollo et al. (2007), they saw a relationship
between Z Max, and the metallicity distribution of their sample. They saw their
sample slowly become more metal poor as they went to higher Zmax. In their Figure
3, they used V¢ cuts to help exclude the prograde stars, and the stronger their cut in
V¢ the more pronounced the shift. In our case, we do not have enough stars to make
such cuts, but we still see the slow decline of metal rich stars with increasing |Z| as
shown in Figure 8.10. We applied a very modest cut of Vgsr < 50 km/s in Figure 8.11,
and the result is a more stark version of Figure 8.10. This especially for the b and 0
panels, where the missing higher metallicity component iS quite noticeable. In the case
of panel d in both figures, we do not see the Significant Shifts to lower metallicities seen
in Carollo et a1. (2007). This may be due to the metallicity dependence of horizontal

branch morphology. In Section 3.5, we discussed how metallicity can affect the shape

97

of the horizontal branch, and it is possible that very metal-rich and metal-poor RRL
are missing because the horizontal branches that would generate them do not pass

through the instability strip.

 

 

 

 

 

 

 

 

 

 

3° ' All |Z| ‘_
25 - -
g 20 + ~_ §
(1) E (7)
_1 15 ' 5 "_ _l
a: E D:
(I: 10 - 1. CE
25 4 5. 1 1 P“ n
c) 15>|Z|>8 -
20 - 5 ._
m 3 _ 9
5 15 ~ 5 - g
a) g ‘ U)
a! 10 - é -' E'
c: g ' a:
5 - rl—if --
0 1 1 1 1 1 1
-2.5 -2 -1.5 -1 -o.5 -2.5 -2 -1.5 -1 -o.5
[Fe/H] [Fe/H]

Figure 8.10 These four plots Show the metallicity RRL stars as a function of |Z| below
the plane. A vertical line is drawn at [Fe/ H] = -2.0.

One final way to look at the metallicity distribution of our sample is to cut on
velocity. In Figure 2 of Carollo et al. (2007), they plotted the metallicity distribution
as a function of the velocity V which is the orbital motion with respect to the lsr.
Since we do not have access to the orbital velocities, we again use a proxy parameter,

in this case it is a parameter we call V220 and is defined as:

V220 = Vgsr — 220

This effectively takes our gsr velocity and shifts it so that it is at rest with respect to

98

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 I E I I I l E I j I 8
a) All |Z| b) 8 > |Z|
15 . -- - 6
U) : : U)
E a s E
(I) : d : _ (D
a: . a:
c: z n:
5 - .- - 2
12 1 : P 1 : . 1' 16
10 _ :7 15>|Z|>8 .' d) 15<|Z| . :42;
m 5 q' f (n
In} 8 ' s - a - 10 a
a) _ s ._ 5 q a)
_1 6 ' E 8 _1
I 4 - . 1- ' -l 6 m
I I 1_ E -1 4 m
2 " [— ‘P f L ' 2
O I J I I I l :l l I l 0
-2.5 -2 -1.5 -1 -0.5 -2.5 -2 -1.5 -1 -0.5
[Fe/H] [Fe/H]

Figure 8.11 These four plots Show the metallicity RRL stars as a function of |Z| below
the plane. This plot is the same as Figure 8.10 except there is a Vbsr < 50 kIn/s cut
applied to weed out the prograde stars. A vertical line is drawn at [Fe/ H] = -2.0.

the lsr. The resulting graph in shown in Figure 8.12. We see very broad distributions
in [Fe/H] in panels a, b, and 0, although by the time we get to panel (1 there is a
definite loss of stars on the metal-rich end. Also, the distribution has a longer tail
on the metal-poor end. We don’t see the complete shifting to lower metallicities a

function of V220, but this again may be an issue of horizontal branch morphology.

99

1 1 12

 

15 I I

 

 

  

 

 

a) V220>-145km/s _ b) ~145§>V220>-242km/s . 1o
2 1o - -- , - 8 12
a 2 a
a ' ‘ 6 s
cc 5 u -- 1 4 1:
‘11—, F l—l—J—U l 2
I I I—III—I I I I I I 18

 

 

 

 

 

 

8 1. i I I I I I I I ' I
7 _ c) 520} V220>-242 km/s .. a) vzzooszo km/ . 15
U) 6 1. I .."' I ' 14 (I)
5 5- -' ‘ ‘2 5
a) 4 _ _.' ‘ 10 a)
_1 - - 8 _1
I 3 " '1. d 6 I
a: 2 a:
- -_ . 4
1 " ‘1- '- 2
0 1 Fr!— 1 1 1"]. 0
-2.5 -2 -1.5 -1 -0.5 -2.5 -2 -‘I .5 -1 -O.5
[Fe/H] [Fe/H]

Figure 8.12 These four plots Show the metallicity RRL stars as a function of V220 =
V93,» — 220. V220 is a proxy for the kinematic quantity V. The normal cut of dist < 30
kpc has been applied. This graph is similar to Carollo et al. (2007) Figure 2.

100

 

CHAPTER 9:
FINAL ANALYSIS AND CONCLUSIONS

 

The final catalog produced in this work contains 421 RRL stars of which 305 are RRab
and 116 are RRc. Of these, 241 stars have measured radial velocities, and 364 stars
have metallicities either from spectroscopy or from the photometric relations defined
in Chapter 7. The full catalog ranges in go magnitude from 13 to 20 which covers a
distance of 3 to 95 kpc from the sun. Beyond that, each of the 421 RRL stars has full
lightcurves in the ugrz’z filters, although coverage varies from star to star and filter to
filter, which is a significant step since very few lightcurves exist in this filter set. This
is even more important because of the use of the ugriz filter system in future large
surveys. Using these lightcurves and metallicities, photometric parameters including
distance are Shown in Table C.2. Using the radial velocities from the 241 RRL stars
with spectra, kinematic parameters were derived and can be found in Table 0.3. A
description of the quantities in both these tables can be found in Table (3.1. Finally
some example lightcurves can be found in Appendix B.

The Oosterhoff dichotomy provided a powerful tool for looking at different pop-
ulations. The main strength of using the Oosterhoff groups is that they do not rely
on kinematic information to interpret and thus provide an independent method of
searching for halo structure. Since the sample of RRL stars comes from the field, the
A log P method was used to determine Oosterhoff type. The RRL clearly separated
into the two distinct Oosterhoff groups, and there was a clear dependence on |Z| with

0011 being Similar in peak height to the Col RRL in the |Z| bin closest to the plane,

101

with the 00] becoming dominant farther away. This clearly suggests different origins
for the RRL at different |Z| heights.

AS part of the Oosterhoff study, we also checked to see if the gap in the Alog P
versus metallicity plot existed for field RRL as it does for the Milky Way globulars.
The plot revealed a definite thinning out near where the gap would be, but it is
possible that our errors in metallicity are blurring over the gap itself.

The kinematic information also provided a very interesting look at the halo. That
being said, there was also a danger of biases in the sample of RRL due to the way
in which the survey cuts across the sky. Primarily the study was limited by the lack
of full space velocities, which are difficult to get given that the very accurate proper
motions given by the HLC were not sufficient for the RRL stars in this sample. Instead
radial velocities had to be used, and although they were corrected to the gsr, they
can ultimately only give one dimension of a three dimensional problem. That being
said, we were really only interested in the Vphi component of the star’s velocity and
that actually tracks relatively well with V937. over the survey area. It does add a large
amount of scatter to the observations, especially at large distances, but it did not
appear to introduce offsets or other peculiarities into the analysis.

This was further compensated for by using the Bescanon model galaxy, which uses
a single halo model. By comparing the observations with model data out in the same
way with the same parameters, the differences between the velocity distributions were
probably due to the data’s deviation from the Single halo model. A large asymmetry
appeared in the nearest |Z| bin that was not attributable to difference in the halo,
but was in fact related to contamination of the sample by thick disk RRL stars. After
further analysis it appeared likely that the ratio of variable to non-variable horizontal
branch stars varies between the thick disk and the halo. The strongest appearance of
the dual halo Signature was in the farthest |Z| bin, where the data separated into two

distinct velocity groups, while the model remained a single Gaussian. This strongly

102

suggests the need for at least a two-component model for the halo.

The analysis of the metallicity, although not entirely separate from the Oosterhoff
analysis, did provide some interesting results. Although the other two analyses found
distinct groups, the metallicity analysis did not. The means of metallicity distribu-
tions did not change Significantlyas a function‘of |Z| nor V220, although there was
definite change in the wings of the distribution, with more metal-rich stars vanishing
at more distant |Z|. Since it was expected from previous work that there would be
a relationship between metallicity and the different kinematic groups, this suggests
the possibility that RRL stars do not make good tracers in that respect. This may
be due to the way in which the horizontal branch is affected by metallicity. If the
metallicity of the horizontal branch becomes too metal-rich or too metal-poor, then
it will not pass through the instability strip and thus not be seen in this sample.

This work shows how RRL stars can be effectively used to understand the nature
of the halo. Furthermore, there is now a catalog of RRL stars that can be used to
investigate other areas of study from clumping due to stellar streams to understanding
lightcurve morphologI in the ugrz’z system. This provides a testbed for techniques
that will become important as new surveys come on-line that use the ugriz filter

system.

103

APPENDICES

104

 

APPENDIX A:
CONFIGURATION FILES

 

A. 1 EXAMPLE PHASE.PAR

TYPE HARM BASS KNOT INPU NAME MAXN SUBN SUBP FULL MIN? MAX? RATE NMIN FHID FNUM SHOW
2 1 0.0 20 0 1 800 -1 0.0 1 0.1 100 12.0 20 1.0 100 15

 

phase.par: Parameter settings for phase.run. Change values above

 

TYPE: Type of curve: 1 - trigonometric polynomial
2 - supersmoother
3 8 cubic spline
HARM: If TYPEPI, The number of harmonics in the trigonometric model
Takes values 1,2,3,...
BASS: It TYPE-2, Bass parameter in supersmoother - range is [0,10]
0 recommended.
KNOT: If TYPEs3, Number of knots in cubic spline - need at least 8.
INPU: Format of data file:
0 8 Data in three columns
1 . Kem short data
2 - MACHO data file
3 . MACHO .fit file
NAME: Naming convention for output files:
1 - Append suffixes like ".per" and ".fit" after the first period
in the name of the data file. e.g. lcb1.dat -> lcb1.per

2 = Append suffixes after the end of the data file name.

105

MAXN:
SUBN:

SUBP:

MINP:

MAXP:

RATE:

NMIN:

FNID:

FNUM:

SHOW:

e.g. var.101 -> var.101.per
Maximum number of observations in data input file
Number of observations in subset: 80-100 recommended
Set this to be negative if you don’t want subsetting
Periods greater than this are fit with the full data,
periods less than this with the subset data.
1 8 Refine diSplayed estimates with full data.
0 8 Do not refine with full data before d18playing
Minimum period considered:
Halving this quantity almost doubles run-time
Maximum period considered:
If <80, maximum period is length of the data
Sampling rate in the initial estimation grid: (6 or more)
Doubling this quantity almost doubles run-time
Number of intermediate estimates: 10-20 recommended
Width of final grids in standard units(1/span): 1-1.5 recommended
Change at your own risk!
Number of values in each final grid: 100-200 recommended
Change at your own risk!

Number of period estimates displayed in output

A.2 PARAMETER FILES POR BESACON MODEL

A.2.1

PARAMETERS FOR PART A

Model of stellar population synthesis of the Galaxy

Catalogue simulation with kinematics, Johnson-Cousins photometric system

The resulting parameter file 1203092479.951603.par has been built;

a message will be sent to address when the simulation is complete

new access to the model form:

Summary of the parameters you supplied

A mail will be sent to address: when the simulation is completed

106

The name of the file containing the result of the simulation
will be: 1203092479.951603.resu

Field of view:

distance step mode: progressive

large field

coordinate system: equatorial coordinates; equinox: 2000.0

minimum maximum step number of bins
right ascension: 0 60.00 5.00 12
declination: -1.25 1.25 1.25 2

solid angle: 149.988096623
Extinction:
diffuse absorption: 0.700 mag/kpc

no discrete clouds

parameter min max step
Absolute V magnitude 0 1 0.50
Spectral type 3 4.9 0.1
Luminosity class 3 3 1.00
Population-age 1 10 1

[Fe/H] -4.20 1.50 0.30

Distance 0.000000 100.000000 2.00
V app. magnitude -99.00 99.00 1

B-V -99.00 99.00 10.00

U-B -99.00 99.00 10.00

V-I -99.00 99.00 10.00

V-K -99.00 99.00 10.00

alpha proper motion -10.00 10.00 100

delta prOper motion ~10.00 10.00 100

Radial velocity -900.00 900.00 500
u velocity -500.00 500.00 10.00

v velocity -500.00 500.00 10.00

w velocity -500.00 500.00 10.00

Total proper motion 0.00 1000.00 10.00

107

LogTeff 3.00 6.00 0.10
Errors:
error modezparabolic error 8 A + B*m + C*m2

photometric errors:

photometric band A B, C

V 0.000000 0.000000 0.000000
B 0.000000 0.000000 0.000000
U 0.000000 0.000000 0.000000
R 0.000000 0.000000 0.000000
I 0.000000 0.000000 0.000000
J 0.000000 0.000000 0.000000
H 0.000000 0.000000 0.000000
K 0.000000 0.000000 0.000000
L 0.000000 0.000000 0.000000
proper motion errors:

A B C

0.000000 0.000000 0.000000

radial velocity errors:

A B C

0.000000 0.000000 0.000000

A.2.2 PARAMETERS FOR PART B

Model of stellar population synthesis of the Galaxy

Catalogue simulation with kinematics, Johnson-Cousins photometric system
The resulting parameter file 1203094459.295218.par has been built;

a message will be sent to address when the simulation is complete

new access to the model form:

Summary of the parameters you supplied
A mail will be sent to address: when the simulation is completed
The name of the file containing the result of the simulation

will be: 1203094459.295218.resu

108

Field of view:
distance step mode: progressive
large field

coordinate system: equatorial coordinates; equinox: 2000.0

minimum maximum 7 step number of bins
right ascension: 300 359.99 5.00 11
declination: -1.25 1.25 1.25 2

solid angle: 149.963098606
Extinction:
diffuse absorption: 0.700 mag/kpc

no discrete clouds

parameter min max step

Absolute V magnitude 0 1 0.50
Spectral type 3 4.9 0.1

Luminosity class 3 3 1.00
Population-age 1 10 1

[Fe/H] -4.20 1.50 0.30

Distance 0.000000 100.000000 2.00
V app. magnitude -99.00 99.00 1

B-V -99.00 99 00 10.00

U-B -99.00 99.00 10.00

V-I -99.00 99.00 10.00

V-K -99.00 99.00 10.00

alpha proper motion -10.00 10 00 100

delta proper motion -10.00 10 00 100

Radial velocity -900.00 900.00 500
u velocity -500 00 500.00 10.00

v velocity -500.00 500.00 10.00

w velocity -500.00 500.00 10 00

Total prOper motion 0.00 1000.00 10.00
LogTeff 3.00 6.00 0.10

Errors:

109

error modezparabolic error

photometric errors:

photometric band

V 0.000000

B 0.000000

U 0.000000

R 0.000000

I 0.000000

J 0.000000

H 0.000000

K 0.000000

L 0.000000

proper motion errors:

A B C
0.000000 0.000000
radial velocity errors:
A B C
0.000000 0.000000

OOOOOOOOO>

.000000

.000000
.000000
.000000
.000000
.000000
.000000
.000000

B

.oooooo ‘

0.000000

0.000000

0000000000

A + B*m + C*m2

.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000
.000000

110

 

APPENDIX B:
EXAMPLE LIGHTCURVES

 

RRab Period: 0.6470 Days

RRc Pen'od: 0.33863 Days

 

 

 

 

 

 

 

 

 

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O

0.5 1

Phase

1.5 2

1
Phase

0.5 1.5

[0

Figure B.1 These are two example lightcurves one RRab and one RRc showing how
the lightcurves change with filter.

111

Example Lightcurves from SDSS-ll

 

 

 

 

 

 

 

 

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- l I I I I I I I I I I I I I I I I—r 1 --._I I I I I I I I I I I I I I I I I I I: 16
- :1- :- :: - - 1
15.5 :- ' t ' s -.- Ir 5; ’1' -‘-: -_ 16.2
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0‘ : z I; z t: ::- _; L, _3 1; 16.4
16 - E 'I. E =- —: ‘0 '1
5-4 '\_-¢ 'fifi j, ‘1. ,1; -: 16.6
16.5 E::1:}::1:l::::1:+::€.':H:'::::}H:.L,L::+F.16'8
18 .- - - 1:- -: 16.8
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If: '5‘ 9 fig I" 1' .55 ’1' ,1. E .
I_l I l I I I I I I I I I I I I I I I ;:-I L l l I I I I I I I I l I I I I I I: 17.6
195 _ I I I I I I‘r r1 I r I I I I I I I I _II I I I I I I I I I I I I I I I I T 1:
: ix: :1 :: _: 19.8
20 : g 1 1_— I H: I H I -I 20
05 5: II: I: III fII 11:1 11 :1 _I
20.5 :‘E { -:_ 131 I, :20.2
: 12155111; I [Eli-13111;} :11 Eff lf 20.4
-_ k“ “17 A: 3:511:51 _:_ 3
21 ' I 1 1 1 I i I 1 1 I 1 1 1 1 1 Ji 1 1 1 ' 1 1 1 1 I 1 1 1 1 I 1 1 I l I 1 I 1 1 - 20.6
o 0.5 1 1.5 2 0.5 1 1.5 2
Phase Phase

Figure B.2 These are example lightcurves displaying how the lightcurve quality is
affected by apparent magnitude, with the brightest RRab and RRc in blue and the
top and the dimmest RRab and RRC at the bottom in red.

112

 

APPENDIX G:
DATA TABLES

 

Table 01: Description of Parameters in Data Tables

 

 

 

 

 

Parameter Description Table
Variable Name of the star in the catalog G2, 0.3
Type Bailey type 02
P Period in days 0.2
Amp 9 Amplitude in magnitudes 02
Max Epoch The HJD - 2450000 of maximum light 0.2
< M > The magnitude weighted mean magnitude in g 0.2
< I > The intensity weighted mean magnitude in g 0.2
A The extinction due to dust in g in magnitudes C.2
< 90 > The dereddened intensity weighted mean 0.2
M The absolute magnitude in g C .2
A log P Described in equation (8.1) G2
00 Oosterhoff Group (defined only for RRab stars) 0.2
[Fe/ H] Metallicity of the star 02
Grp Metallicity group defined in Table 7 .2 G2
RA Right Ascension of the star in degrees C.3
Dec Declination of the star in degrees 03
l Galactic latitude in degrees 03
b Galactic longitude in degrees G3

 

113

Table C .1: Parameters in Data Tables (continued)

 

 

 

Parameter Description Table
X, Y, Z Cartesian coordinates from the galactic center 0.3
R Radius from the galactic center using X and Y only 0.3
helio Heliocentric radial velocity in km/s C.3
lsr Radial velocity in the local standard of rest in km/s C.3
gsr Radial velocity in the galactic standard of rest km/s C.3

 

114

 

 

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116

 

 

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144

 

 

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145

 

 

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146

 

 

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@852232208 4mm mem .20 82828222028— 02882222262 2nd @388

147

 

 

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148

 

 

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149

 

 

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N

Aguascoov 4mm mem -.o 828230-282 028822282258 5.0 @3289

150

 

 

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151

 

 

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152

 

 

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153

 

 

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154

 

 

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155

 

 

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156

 

 

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157

 

 

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158

 

 

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159

 

 

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888 282 82288 82 8 8 2 hm 88222 8 2 82 <2 82882882,

Agasscoov 4mm mem .20 mos-2220.28— 0288828222 2.82.0 @3288

 

 

160

 

 

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888.28- 888.882- 888.88- 888.82 882.82- 888.8 882.8 88.8 88.82 28.88- 888.88 8882.2- 8888.888 828888
888.88 288.88- 888.88 888.82 888.88- 882.82 888.8 88.8 88.88 888.28- 888.88 88828 888.88 8828888
888.8- 888.882- 888.8- 888.8 882.8- 288.8 882.8 88.8 88.82 888.88- 882.88 8288.8 8888.888 8888288
<2 <2 <2 888.88 888.88- 888.88 888.8- 88.2 88.88 82.28- 888.88 8888.8 8888.888 8882288
888.8- 888.882- 888.82- 828.82 882.82- 888.8 888.8 88.8 88.82 88.88- 288.88 8828.8- 8888.888 8888888
888882 888.88 888.882 888.82 888.82- 888.8 828.22 82.8 88.82 888.88- 888.822 8828.8- 828882 888888
<2 <2 <2 888.8 888.8- 888.8 228.8 88.8 28.8 888.28- 888.88 28888- 8888.888 8888888
<2 <2 <2 282.82 888.22- 888.8 828.8 88.8 88.82 888.88- 888.88 88828- 8888.888 8888828
<2 <2 <2 888.88 288.88- 888.88 888.8 88.8 28.88 888.88- 288.88 8882.2 8288.888 8888828
<2 <2 <2 888.88 888.28- 888.88 888.8- 88.8 88.88 888.88- 888.88 288.8. 8888888 8888828
882.88- 888882. 888.88- 828.82 882.88- 288.8 888.82 82.8 88.88 828.28- 888.822 8882.2 8888.82 888828
288 282 822822 82 8 > x am 28282 8 2 82 <8 82882885

 

 

 

 

 

288228282282 28882 88228 28 8828888882 8228888282 H8.0 82888

 

 

161

 

 

<2 <2 <2 888.82 888.88- 888.82 828.8 88.8 88.88 888.88- 888.88 8882.2 8888.888 8828888
888.88 828.88- 882.88 888.8 888.22- 882.8 888.8 88.8 28.82 28.88- 888.88 888.2. 8888.888 8888888
<2 <2 <2 888.82 888.88- 888.82 888.8 28.8 88.88 888.88- 888.28 8888.8 8888.888 8888888
888.882- 888.888. 888.882- 888.82 888.82- 882.82 888.8 82.8 28.28 288.88- 888.88 82888 2822888 2888888
<2 <2 <2 888.28 888.28- 888.28 888.8- 88.8 88.88 882.88- 888.8 8888.8 8888.888 8822888
888.88- 888.882- 888.88- 888.82 888.88- 828.8 888.82 82.8 88.88 88.88- 888.822 2888.8- 8882.22 288888
828.88 888.88- 828.28 888.82 888.88- 888.82 888.8 88.8 88.88 882.88- 82.88 8888.2- 8888.888 8888888
<2 <2 <2 888.8 888.82- 888.8 888.8 82.8 88.82 888.88- 888.88 8888.8 8888.888 2288888
888.88- 888.882- 888.88- 888.82 888.82- 888.8 882.8 88.8 88.82 888.88- 888.88 888.8. 8888.888 8888888
828882. 888.888- 822.282- 888.88 888.88- 888.82 888.8 88.8 82.88 828.88- 88.88 8888.8 8888.888 8888888
<2 <2 <2 882.88 888.88- 882.88 288.2- 88.8 88.88 888.88- 88.88 8882.2- 2888.888 888888
888.282- 888.888- 888882. 882.82 888.22- 888.8 888.8 22.8 88.82 228.88- 88.88 8882.2- 8888.888 8888888
<2 <2 <2 828.88 888.88- 82888 288.8 88.2 88.88 8.88- 882.88 8888.8- 8888.888 8888888
<2 <2 <2 882.88 888.88- 888.88 288.8- 88.8 88.88 888.88- 882.88 8888.8- 8288.888 8888888
<2 <2 <2 888.88 888.88- 888.88 888.8 88.8 88.88 888.88- 882.88 8888.2 8888.888 8822888
888.888- 888.888- 282.228- 888.82 888.28- 888.82 888.8 88.8 28.88 888.88- 888.88 888.8. 8288888 8888288
288 .282 82288 28 8 > x :82 88282 8 2 82 <8 828.82-88>

228222222888 28828 88828 28 8828888882 828822282222 H8.0 82888

 

 

162

 

 

888.28 888.88- 288.28 888.82 888.88- 828.82 888.8 28.8 88.88 888.88- 288.88 8888.2- 2282.888 8888888
<2 <2 <2 882.88 288.88- 88888 288.8 88.2 88.88 888.88- 888.8 8888.8- 8888.888 8888888
888882. 888.888- 888.882- 888.82 828.82- 888.82 888.8 88.2 28.82 888.88- 882.8 8882.2 8888.888 2888888
288.882- 828.888- 888.882- 888.8 888.8- 888.8 888.8 88.8 88.8 888.88- 888.88 8888.8 8888.888 2888888
282.88 888.88- 888.88 888.8 888.8- 2888 288.8 88.8 28.8 888.88- 888.88 888.8. 8888.888 2888888
888.222- 888.288- 888.822- 288.88 888.88- 282.88 888.8 88.2 88.28 888.88- 828.88 8888.8 8888.888 8888888
828.28- 888882- 888.88- 888.28 888.88- 888.88 888.8 88.8 88.88 888.88- 888.88 88828 8228.888 8888888
888.88 888.88- 882.88 888.88 288.88- 888.88 888.8 88.8 88.88 888.88- 888.88 28888- 8888.888 8888888
<2 <2 <2 888.88 888.88- 28888 288.8 88.8 88.88 888.88- 888.88 8882.2 8888.888 8888888
28888 888.88- 888.88 888.82 888.88- 888.82 888.8 28.8 88.88 888.88- 288.8 8888.8- 8828.888 8288888
<2 <2 <2 888.82 882.82- 888.82 882.8 88.8 88.82 828.88- 888.88 88882 8888.888 8288888
888.88 888.88- 828.88 882.8 888.8- 888.8 888.8 88.8 88.82 888.88- 88.88 8888.2- 8288.888 8888888
<2 <2 <2 888.82 282.88- 888.82 822.8 82.2 28.28 888.88- 88.88 8888.8 8828.888 8888888
288.88- 888882. 888.88- 888.8 288.8- 888.8 888.8 88.8 88.8 228.88- 288.88 8888.2 8882.888 8822888
88888- 888.888- 88.88. 88822 888.82- 888.8 888.8 82.8 82.82 88.88- 888.88 8888.8 8888.288 8888888
<2 <2 <2 822.88 888.88- 882.88 882.2 88.2 88.88 888.88- 888.8 8888.8- 8888.288 8888888
888 882 82288 m 8 > 2 :82 88282 8 2 882 <82 8288285

2288822882 8282 88828 88 882288882 8288582222 H8.0 8288,2-

 

 

163

 

 

<2 <2 <2 888.88 288.88- 888.88 888.88 88.8 88.88 888.88- 888.882 2888.8- 828882 888888
888.88- 888.882- 828.88- 828.22 888.8- 888.8 888.22 88.8 88.22 288.88- 888.882 8888.8- 2288.82 888888
<2 <2 <2 288.82 888.82- 828.8 888.22 88.8 82.82 828.88- 828.882 8888.2- 888282 888888
888.88- 888.882- 888.88- 888.88 888.88- 888.22 888.82 88.8 88.88 888.88- 882.882 8888.8- 8288.82 888888
<2 <2 <2 882.82 288.82- 288.8 888.22 82.8 82.82 828.88- 82.882 8888.8- 8888.82 888888
288.88- 888882. 882.88- 888.82 288.88- 288.8 888.82 88.8 88.88 88.88- 888.882 8888.8- 8888.82 888888
<2 <2 <2 828.88 888.88- 82882 888.88 88.2 82.88 288.88- 888.882 8288.8- 8888.82 888888
828.88- 828.882- 888.88- 288.82 888.82- 888.8 882.82 88.8 88.28 882.88- 8.282 8888.8- 8888.82 888288
888.88 882.88- 888.88 828.8 888.8- 888.8 888.8 88.8 28.8 282.88- 288.88 888.8 8888.888 888888
888.82 288.88- 888.82 888.82 888.22- 888.8 888.8 88.8 88.82 888.88- 88.88 2888.2- 8888.888 8888888
888.88- 888.888- 88.88. 888.82 888.88- 888.82 888.8 82.8 88.88 8.88- 888.88 2888.8 8888.888 2888888
<2 <2. <2 888.8 288.8- 828.8 828.8 22.8 88.8 88.88- 888.88 8888.2 8888.888 8888888
88.28. 882882- 882.28- 888.82 882.82- 888.8 888.8 88.8 88.82 888.88- 888.88 8888.8 8888.888 8888888
882.88- 888.882- 888.88- 888.8 888.8- 828.8 888.8 88.8 88.8 888.88- 822.28 2888.8- 8888.888 8888288
888.8- 888.822- 288.8- 882.82 888.88- 888.82 288.8 82.8 88.88 888.88- 88.88 8888.8- 2288.888 8288888
<2 <2 <2 888.88 888.28- 228.88 882.8 88.8 88.88 888.88- 888.88 8888.8- 8888.888 8288888
.288 882 82288 8 8 > x :82 88222 8 2 882 <82 828828888

 

 

22882222222288 8.8282 8828 .28 8828288282 8228828822 8.0 828.88

 

 

164

 

 

<2 <2 <2 888.82 888.82- 888.8 288.82 88.2 82.82 882.88- 8.882 8888.2 8288.88 888888
<2 <2 <2 888.88 888.88- 888.88 888.82 88.8 82.88 888.88- 288.882 888.8 8882.8 88288
<2 <2 <2 888.88 888.88- 888.8 888.82 88.8 88.88 888.88- 88.282 8888.2 8888.88 888888
828.882- 288.882- 888.882- 888.82 888.28- 888.8 888.82 28.8 28.88 888.88- 888.282 8888.8 8888.28 888888
<2 <2 <2 888.88 888.88- 282.82 888.88 82.8 88.88 828.88- 888.882 8882.2 8288.28 888888
<2 <2 <2 888.88 888.88- 888.88 288.82 88.2 88.88 888.28- 888.8 8888.2- 8828.8 88888
888.88 888.8 288.88 888.22 888.8- 888.8 888.22 88.8 88.8 888.88- 888.882 8888.8- 8888.88 282228
<2 <2 <2 828.82 888.28- 288.8 882.82 88.8 88.88 888.88- 888.882 8888.2 8888.88 8828
888.88 288.8- 888.88 888.82 888.82- 888.8 888.82 82.8 28.82 888.88- 888.882 8882.2- 8888.88 888888
<2 <2 <2 888.88 888.88- 888.88 888.28 88.8 88.88 288.88- 828882 8888.2- 8828.88 888888
288.88 888.8- 888.88 888.22 888.8- 888.8 888.22 88.8 88.82 888.28- 88.882 8888.8 8888.82 288888
<2 <2 <2 888.22 888.8- 888.8 882.22 88.8 88.8 888.28- 828.882 8888.2 888882 888888
888.88- 888.882- 288.88- 888.82 888.88- 888.8 888.82 88.8 88.88 28.88- 288.282 8888.8- 8888.82 288888
888.828 888.282 288.288 828.8 888.8- 888.2 288.8 88.8 88.8 888.88- 888.882 8888.8- 8288.82 828288
888.88- 882.822- 888.88- 888.82 888.82- 888.8 888.82 88.8 88.82 888.88- 888.882 8888.8- 8888.82 882288
888.88- 888.882- 888.88- 888.82 888.88- 88882 888.82 88.8 88.88 888.88- 888.882 8888.8- 888.82 288888
888 882 822822 82 8 > 2 am 88222 8 2 882 <82 82882888,

 

 

 

 

 

32222282803 4mm mem mo 88.28-288820882 oSmEofix 282.0 @589

 

165

 

 

888.882- 888.888- 888882. 888.82 888.88- 828.8 882.82 88.8 28.88 888.88- 888.882 8888.8 8888.88 888888
828.88- 888.282- 288.88- 828.22 888.8- 888.2 288.22 88.8 88.8 888.88- 828.882 8888.8 288.88 882888
<2 <2 <2 828.28 888.88- 888.8 288.88 88.8 88.88 888.88- 888.882 8288.8- 8888.88 888888
288.822- 888.882- 888.222- 888.82 888.28- 888.8 888.82 82.2 88.88 888.88- 888.882 8888.8- 888.88 888828
882.2- 888.88- 288.8 288.82 888.8- 888.8 888.82 88.8 88.22 888.88- 888.882 8888.8- 288288 888828
<2 <2 <2 888.82 888.82- 888.8 882.82 88.8 88.28 882.88- 88.882 8888.2 8888.88 888888
888.882- 888.882- 888.88- 888.88 888.88- 888.8 888.28 82.8 88.88 888.88- 88.882 8882.8- 8228.88 888888
<2 <2 <2 888.88 288.88- 888.88 888.82 88.2 88.88 882.88- 828.882 8888.2 8888.8 82888
888.28- 888.882- 888.88- 888.88 888.88- 888.8 888.28 28.8 88.88 888.88- 888.882 8882.8 8888.88 888888
888.882- 888.882- 888.882- 888.82 888.82- 888.8 888.82 82.8 88.88 28.88- 288.882 8888.8- 8888.88 888288
<2 <2 <2 888.8 888.8- 828.8 888.8 88.8 88.8 828.88- 888.882 8882.2 8888.8 88888
888.28 888.82- 888.88 888.82 888.82- 888.8 888.82 88.8 88.82 888.88- 288882 8888.8 8888.88 888888
8888- 888.822. 888.8- 888.8 888.8- 288.8 888.8 88.8 28.8 888.28- 888.882 8888.8- 8888.8 88288
888.88- 882.882- 888.28- 888.22 888.8- 888.8 828.22 88.8 88.8 882.28- 888.882 888.8. 8888.88 888888
288.88- 888.282- 888.88- 888.82 888.28- 888.8 882.82 88.8 88.88 888.88- 288.882 888.8 8888.88 888888
228.882 888.882 888882 888.82 888.82- 888.8 888.82 82.8 88.28 888.28- 888.882 8888.8- 8888.88 888888
888 28.2 82288 m 8 8 x :82 88222 8 2 82 <2 8288228>

2228228282288 .252 88828 28 88228888 82882228822 8.0 8288-8

 

166

 

 

<2 <2 <2 888.82 882.82- 888.8 888.8 88.8 88.82 888.28- 882882 88888- 88828 88888
<2 <2 <2 882.82 882.82- 222.8 888.82 82.8 88.82 888.88- 888.882 8288.8 8828.88 888888
288.882- 888.882- 888882- 888.82 888.82- 828.8 888.82 82.8 88.88 888.88- 888.882 8888.8- 8828.88 888888
<2 <2 <2 888.88 828.88- 888.8 228.88 88.8 88.88 888.88- 888.882 8888.8- 8882.88 288888
<2 <2 <2 888.88 888.88- 822.8 888.88 88.2 88.88 88.88- 88.882 8282.2 8888.28 888888
888.88- 888.882- 888.88- 888.88 888.88- 828.8 888.88 88.8 28.88 888.88- 888.882 8888.8 8888.28 888888
888.88- 282.882- 888.88- 888.88 828.88- 888.8 888.88 82.8 88.28 288.88- 88882 888.8 8288.28 888828
288.88- 288.88- 288.82- 888.88 288.88- 888.8 882.88 82.8 88.88 888.88- 828.882 8888.8 2882.28 8288
888.882- 888.882- 28.822. 882.88 882.82- 888.8 888.82 88.8 88.88 888.88- 288.882 8888.8 882888 88888
888.8 888.88- 888.82 888.82 888.82- 888.8 882.82 88.8 88.88 828.88- 882.882 8888.8- 828888 888888
888.882- 888.882- 888882. 288.82 888.82- 888.8 888.82 88.8 88.88 888.88- 888.882 888.8 2282.88 888888
<2 <2 <2 888.82 888.82- 8828 882.82 28.2 88.82 288.88- 288.882 8888.2 8888.88 888888
888.88- 888.882- 888.88- 888.82 888.88- 882.82 888.8 88.8 88.88 828.88- 888.88 8882.2 8888.8 8888
882882 888.88 888882 828.82 888.82- 888.8 888.8 88.8 88.82 288.28- 288.282 882.8 8888.8 8828
888.88- 888.88- 882.82- 828.88 288.88- 888.8 888.88 88.8 88.88 888.88- 888.882 8888.2 8888.88 888888
828.88- 882.822- 282.88- 888.88 888.88- 888.8 882.88 82.8 88.88 822.88- 888.882 2888.8 8888.88 828888
888 882 82288 m 8 > 2 am 28282 8 2 882 <82 828822828

 

 

 

Agascoov 4mm mem 80 8828888205 02882282202 2.82.0 @3289

 

167

"I:

 

 

828.88 288.28. 888.88 888.88 288.88- 882.8 888.88 88.8 88.88 888.88- 288.882 8888.8- 8888.88 888888
<2 <2 <2 882.82 888.88- 888.82 822.22 88.8 88.28 888.28- 882.882 2888.8- 8888.8 88888
888.88 888.28 888882 888.82 882.82- 888.8 888.82 88.8 88.82 828.88- 888.282 8888.8 8888.88 882888
888.88- 822.882- 88.88. 888.82 888.8- 888.8 888.8 28.8 28.8 882.28- 288.882 8828.8 8888.8 88888
<2 <2 <2 828.88 888.82- 888.8 828.82 88.8 88.88 888.88- 888.282 8882.8 8888.88 888888
888 882 82288 2 8 > 22 :2 28222 8 2 82 <2 8288288>

2882222228882 222 88228 28 88228822282 8228582222 U8.0 82882.

 

 

168

BIBLIOGRAPHY

169

 

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