Citation
Analysis of gamma² Velorum photometry from the South Pole

Material Information

Title:
Analysis of gamma² Velorum photometry from the South Pole
Added title page title:
Analysis of Gamma 2 Velorum photometry from the South Pole
Creator:
Taylor, MaryJane, 1961-
Publication Date:
Language:
English
Physical Description:
vii, 181 leaves : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Carbon ( jstor )
Datasets ( jstor )
Emission spectra ( jstor )
Helium ( jstor )
Hell ( jstor )
Photometry ( jstor )
Spectroscopy ( jstor )
Stellar spectra ( jstor )
Telescopes ( jstor )
Wolf Rayet stars ( jstor )
Gamma 2 Velorum (Binary star) -- Observations ( lcsh )
Wolf-Rayet stars ( lcsh )
Wolf-Rayet stars -- Observations -- Technique ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis (Ph. D.)--University of Florida, 1988.
Bibliography:
Includes bibliographical references.
General Note:
Typescript.
General Note:
Vita.
Statement of Responsibility:
by MaryJane Taylor.

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
024602698 ( ALEPH )
AFL4116 ( NOTIS )
19924492 ( OCLC )
AA00004828_00001 ( sobekcm )

Downloads

This item has the following downloads:


Full Text











ANALYSIS OF -y2 VELORUM PHOTOMETRY FROM
THE SOUTH POLE














BY

MARYJANE TAYLOR


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1988




LIBRARIES




ANALYSIS OF y2 VELORUM PHOTOMETRY FROM
THE SOUTH POLE
BY
MARYJANE TAYLOR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
jO OF E LIBRARIES.


ACKNOWLEDGEMENTS
At this time, I would like to express my gratitude to
those people without whose contributions this work would
have been far more frustrating.
First, I would like to thank the chairman of my Ph.D.
committee, Dr. John P. Oliver, for his dedication and
expertise in the construction of the South Pole Optical
Telescope (SPOT), and for his knowledge in the development
of the software necessary to automate the SPOT system. I
thank him for the many hours that he has dedicated to this
project and for the endless headaches incurred, especially
near the beginning of each observing season.
I would also like to express my sincere appreciation
to Dr. Kwan-Yu Chen, principal investigator of the SPOT
project, for his contributions to this research. He has
provided valuable advice in various aspects of this
research, and has read this dissertation with a very keen
eye. Dr. Chen has also supplied me with references to
current papers in the literature on Wolf-Rayet stars, and on
72 Velorum.
Thanks also go to Dr. Frank Bradshaw Wood, principal
investigator of the project in its infancy, for suggesting


the topic of this dissertation.
Dr. Wood has also made me
aware of current topics in the literature, and has carefully
read this dissertation.
In addition, I extend my thanks to the two other
members of my Ph.D. committee, Dr. Jerry L. Weinberg and
Dr. Bruce T. Edwards, for reading this work and making
constructive criticisms which have improved its quality.
Thanks also go to Dr. Stephen T. Gottesman, chairman
of the Astronomy Department at the University of Florida,
for taking the time to share his knowledge of the
interpretation of power spectra with me, despite not being a
formal member of my Ph.D. committee.
I would like to express my sincere appreciation to
Dr. John E. Merrill for his devotion to the South Pole
project, and for always keeping the members of the team on
track. His experience and insight into many of the problems
has been extremely useful.
I would like to extend ray sincere thanks and
appreciation to Dr. Lance Erickson for the long discussions
on the application of the methods of Fourier Transform and
least squares analysis as they pertain to this work. I
would also like to thank him for the weekends that never
were and for helping me to maintain my sanity these last few
months .
Finally, warmest thanks go to my father whose never-
ending patience, encouragement, and support for the last
i i i


twenty-one years have made me always strive for the best. I
owe my love of astronomy to him. In addition, he is to be
thanked for the long hours devoted to the tedious process of
formatting this dissertation in accordance with Graduate
School regulations, and for the use of his Hewlett-Packard
printer .
The Steller Photometry Program was supported by grants
from the Division of Polar Programs of the National Science
Foundation.
IV


TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT vi
CHAPTERS
1 BACKGROUND 1
2 INSTRUMENTATION AND DATA ACQUISITION SYSTEM ... 17
Instrumentation 17
Data Acquisition 28
3 DATA REDUCTION TECHNIQUES 31
4 DATA ANALYSIS 52
Spectral Photometry of the Hell and CIII
Emission Lines 52
Data Setl 52
DataSetll 65
Data Set III 76
DataSetIV 86
DataSetV 110
Data Set VI 122
Data Set VII 146
B and V Photometry 155
5 DISCUSSION AND CONCLUSIONS 165
REFERENCES 176
BIOGRAPHICAL SKETCH 179
v


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANALYSIS OF y2 VELORUM PHOTOMETRY FROM
THE SOUTH POLE
BY
MARYJANE TAYLOR
DECEMBER, 1988
Chairman: John Parker Oliver
Major Department: Astronomy
Several hundred photometric observations of the Wolf-
Rayet star, y2 Vel, have been obtained with the South Pole
Optical Telescope using filters centered on the Hell and
CIII emission lines at 4686 and 5696, respectively. The
observations are reduced to intensities using a region of
the continuum centered at 4768 as the comparison source.
Two independent techniques are used to determine
periodic behavior of variations in the strengths of these
emission lines. The first of these methods uses power
spectrum analysis based on Deeming's method of Fast Fourier
Transforms. The second process involves fitting a first-
order sine wave to the data using 1east-squares analysis.
vi


The results of the analysis of the Hell feature
indicate a "fundamental period" of 1.20 hours with an
amplitude of fluctuation of a few percent. Several
harmonics and subharmonics of this period are also detected.
The presence of a 1.20 hour period of variability has
certain theoretical implications which are discussed in the
context of recent theoretical developments. It is suggested
that these spectral changes are due to radial pulsations of
72 Vel .
An attempt is made to arrive at satisfactory models to
describe intensity variations of the CIII emission line.
Despite the weakness of this feature, periods are determined
which indicate slightly more rapid fluctuations than those
found to describe variations in the corresponding helium
data. It is suggested that this result may provide clues
indicating the relative locations of line formation in the
extended wind.
In addition to the analysis of narrowband photometry,
observations in the visual and blue filters are reduced to
photometric magnitudes using comparison star HR3452. The
analyses of these data do not reveal any evidence of an
optical eclipse in the y2 Vel system, at least within the
limitations of these data.
vi i


CHAPTER 1
BACKGROUND
In 1867, C. Wolf and G. Rayet discovered several stars
which exhibited very unusual spectral characteristics;
namely, numerous broad emission features superimposed on a
faint continuum. At that time, the only star known to have
an emission line in its spectrum was 7 Cassiopeia. Normal
stellar spectra consisted of a continuum marked by
absorption rather than emission lines. In fact, the only
astronomical objects which were known to have spectra
dominated by emission features were gaseous nebulae. Wolf
and Rayet both realized that the spectral characteristics of
the stars they had discovered were considerably different
from either those of 7 Cas or gaseous nebulae. Although
originally classified in the Henry-Draper Catalogue as
spectral type 0, these peculiar objects have now been
assigned the separate spectral classification of W and are
called Wolf-Rayet stars.
In general, the spectrum of a star is used as an
indication of the evolutionary state of that star. However,
even today, over a century after their discovery, neither
the evolutionary status, nor many of the physical,
geometrical and chemical properties of Wolf-Rayet objects
1


2
are well understood. It follows then, that in the case of
Wolf-Rayet stars, an understanding of the nature of the
emission line mechanisms producing such a peculiar spectrum
is needed before establishing an appropriate evolutionary
scenario. It is this unusual emission line spectrum which
is commonly referred to as the "Wolf-Rayet phenomenon." The
most currently accepted interpretations of observations of
such an intriguing class of stellar objects are presented in
the following few paragraphs.
Analysis of spectroscopic observations of Wolf-Rayet
stars allows us to state with considerable confidence, the
characteristics of the spectra. Very succinctly, the
spectra are dominated by emission lines superimposed on a
continuum which is characteristic of an 0 or an early B type
star. Only in a few cases are intrinsic absorption features
observed. Generally, the only absorption features in a
Wolf-Rayet stellar spectrum are the P Cygni absorption
components of certain emission lines in some stars. Such
P Cyg absorptions indicate expansion of material surrounding
the Wolf-Rayet star. The width of the emission lines can be
used to infer the velocities of expansion. Although the
breadth of the emission lines varies for each ion, probably
because different ions are formed at different depths of the
atmosphere, these widths correspond to velocities ranging
between hundreds and thousands of kilometers per second.


3
Spectral observations have led astronomers to realize
that there are two main sequences of Wolf-Rayet stars. The
WN type Wolf-Rayet star has an optical spectrum dominated by
nitrogen and helium ions. Several of the WN subtypes
exhibit traces of carbon: in particular, the CIV lines at
5801 and 5812 in the optical region of the spectrum and at
1550 in the ultraviolet region. The enhanced abundances of
helium and nitrogen, as well as the lower than "normal"
abundances of carbon and oxygen, may be explained by the
fact that these stars expose material which has been
processed in the CNO cycle. The second sequence of Wolf-
Rayet stars are the WC types. As their nomenclature
indicates, these stars have spectra which are almost
entirely dominated by carbon ions, but which may also
display helium and oxygen lines as well. Unlike the WN
types, the presence of nitrogen in WC stars is very weak.
The possible blends of NIII ions with strong carbon features
have been identified (Underhill 1959, Bappu 1973) at optical
wavelengths and blends of NIV and NV ions with other carbon
emissions may be present also at ultraviolet wavelengths
(Willis 1980). The (WC) class of Wolf-Rayet stars exposes
material which has probably been processed by a helium
burning convective core. In this case, however, the
presence of carbon and oxygen is enhanced at the expense of
nitrogen and helium. Recently, Barlow and Hummer (1982)
have identified a third class of Wolf-Rayet stars, WO. The


4
WO types represent stars which exhibit strong amounts of
oxygen rather than carbon and are probably a result of
extreme helium burning.
The ratio of hydrogen to helium in Wolf-Rayet stars
remains a controversial topic. Based on analyses of optical
spectra, the H/He ratios are quite low; generally
significantly less than unity. It is not yet known if this
is a reflection of the true chemical composition of these
stars. The ratio of abundances of these elements is often
determined by using measurements of the Hell Pickering
series. Since even quantum number Hell lines occur at
nearly the same wavelength as the hydrogen Balmer lines,
significant amounts of hydrogen could be inferred from an
increase in the strength of the Hell line intensities which
produces a non-smooth Pickering decrement. According to
Smith (1973), an analysis carried out in this manner
resulted in N(H)/N(He) =1-2 for late WN Wolf-Rayet stars
(WNL): namely, WN7 and WN8. She found considerably lower
ratios, N(H)/N(He) =0 for earlier WN types (WNE). These
results indicate that stars which exhibit the Wolf-Rayet
phenomenon may indeed be chemically evolved stars with WNE
types more evolved than WNL types. A similar study was
carried out for WC stars by Rublev (1972) who concluded,
N(H)/N(He) < 0.08. This result must be regarded with a
little more caution than those results for WN types, because
of the severe blending effects of the Hell Pickering lines


with several carbon and oxygen transitions. However, it is
the apparent position of this class of stars on the
5
Hertzsprung-Russe11 diagram which contradicts the
observational evidence for low hydrogen abundances. In
fact, this is the primary reason that Underhill (1982) is
still of the opinion that the low H/He ratios do not
necessarily indicate a true absence of hydrogen: it is still
her assertion that Wolf-Rayet stars are chemically similar
to our Sun with 3 < H/He < 10. The temperatures and
luminosities which have been derived for these stars place
them in the hydrogen burning band of the Hertzsprung-Russe11
diagram. More specifically, the late type WN stars are
expected to lie near the BO supergiants while the early WN
types and the WC stars seem to reside near the BO giants.
In addition, current stellar models, which have been
calculated for He-rich stars with masses and luminosities
similar to those for Wolf-Rayet stars, indicate considerably
higher effective temperatures than are observed for Wolf-
Rayet stars. One could agree with Underhill, who states
that perhaps certain physical processes occur in the
atmospheres of these complicated stars which may explain the
low amounts of detected hydrogen relative to helium. It is
true that if the temperatures or densities of the atmosphere
are not appropriate, hydrogen emission may not be observed
in the assumed amounts. Hydrogen emission occurs in the
Balmer lines as a result of a recombination event. Such a


6
deionization can occur only if the electron temperatures are
low enough and the density of material is adequate to allow
a sufficient number of recombinations to occur in a small
enough period so that the transitions involved appear in the
Balmer series. Underhill contends that the densities and
temperatures may not be appropriate to allow hydrogen to
emit readily. Additional observational evidence indicates
that this is not the case. Since the x-ray flux is
relatively low, one would infer that the temperatures of
Wolf-Rayet stellar atmospheres are not high enough to allow
hydrogen emissions to occur. Given current theories of the
evolution of hot, massive stars, it is not difficult to
imagine that these objects eject much of the material from
their atmospheres during their long-term evolutionary
development. The spectral observations can be easily
misinterpreted since many of the emission features in the
spectra of Wolf-Rayet objects are either very broad or the
result of blending of several different species.
One representative of the Wolf-Rayet class of stars,
72 Velorum, is located in the southern sky at right ascension
8^09m12s, and declination -4718' (1988.5). The y2 Vel
system is the brightest Wolf-Rayet star in the sky with a
visual magnitude of 1?76. The star is an intriguing system
for both observational and theoretical astronomers. It has
been the object of many investigations in the last twenty
years or so and is the subject of this dissertation.


7
The y2 Vel system was discovered to be a spectroscopic
binary by Sahade in 1955. The system consists of a Wolf-
Rayet star of spectral classification WC8 and a hot
supergiant component of spectral type 09 (Conti and Smith
1972). Ganesh and Bappu (1967) carried out the first
spectroscopic study in order to determine the orbital
parameters of the y2 Vel system. Using the CIII-IV blend at
4650 which is assumed to be formed in the envelope of the
Wolf-Rayet component, and the hydrogen absorption line of
the 0 star at 4340, they found an orbital period of 78^5.
A more accurate estimate of the period was accomplished by
Niemela and Sahade (1980). They determined a radial
velocity curve using the strong violet-shifted absorption
line, Hel at 3888. The position of this feature was
measured relative to the H8 absorption line of the 09
supergiant. They determined a period of revolution of
78^5002 with an eccentricity of 0.40. Moffat et al. (1986)
carried out another radial velocity analysis of the y2 Vel
system in an attempt to define the orbital elements of the
system more precisely. They used radial velocity curves
obtained for several emission features but were unable to
improve upon the 78^5002 period or the 0.40 eccentricity.
However, their refined ephemeris coupled with the period of
Niemela and Sahade, is given by the following equation:
E0 = JD2445768.96 + 78^5002.
(1-D


8
With the aid of equation (1-1), the 09 star is found to be
in front of the WC8 star at phase 0.0 and behind the Wolf-
Rayet component near phase 0.5. According to the study by
Moffat et al. (1986) the epoch of periastron passage is
determined to be JD2445802.6. Although an eclipse has been
detected in the ultraviolet region of the spectrum (Willis
and Wilson 1976) at phase 0.65, no definite eclipse effects
have been recorded at optical wavelengths. Gaposchkin (1959)
reported a 16d2334 period with an amplitude of variation
ranging between 0?19 and 0?13 using visual and photographic
measurements, but to date no other work has been able to
confirm these results.
The masses of the individual components can be
determined from the simple relation
^abs
^em
mWr
M09
(1-2)
where Kabs and Kem are the amplitudes of the absorption and
emission radial velocity curves and Myg/MQg represents the
ratio of the mass of the WC8 star to the 09 supergiant.
According to the study by Moffat et al., the best values for
Kabs and Kem are given by
Kabs = 70 2 km/sec, and
Kem = 130 6 km/sec.


9
These values are in good agreement with independently
determined results by Niemela and Sahade but have smaller
uncertainties. They indicate that the mass of the Wolf-
Rayet companion is approximately 0.54 times that of the 09
star, with minimum masses of 17M for the WC8 component and
32Mq for the supergiant.
In an attempt to measure the radius of the individual
components of -y2 Vel as well as the separation distance of
this binary system, Brown et al. (1970) obtained
interferometry measurements with the stellar intensity
interferometer at the Narrabri Observatory. Their
measurements in the continuum at 4430 included the effects
of both y2 Vel and 71 Vel, which is located 41" away from -y2 .
However, the analysis is simplified by the fact that the
assumed contribution from 71 Vel is negligible. These data
indicate an angular diameter of the Wolf-Rayet component of
0'.'44 0705 x 10'3, and an angular semi-major axis of the
orbit of 473 075 x 10 3. Using spectroscopic observations
of Ganesh and Bappu (1967), together with these angular
measurements, the distance modulus for the 7 Vel system is
7.7 0.3 or about 350pc. A simple trigonometric
computation yields a radius for the WC8 component of 17RQ. A
more conventional method of measuring HB indices of several
stars in the vicinity of 7 Vel results in a distance of
457pc, and hence, a radius for the Wolf-Rayet star of 22RQ.


10
Since an eclipse in the optical region of the spectrum
has not been detected, the inclination of the system can be
assumed to be approximately 70. Using the angular semi
major axis of the orbit obtained by Brown et al., the
projected semi-major axis is 2.09 0.05 x 1013cm, or
assuming i=70, 2.22 + 0.13 x 1013cm. These measurements
imply a separation distance of approximately 319RQ.
Brown et al. also obtained measurements of the CIII-IV
emission line of y2 Vel at 4650. These data include effects
of both components of y2 Vel in addition to the effects from
the material surrounding the Wolf-Rayet component. The
interferometry measurements give an angular diameter of the
region emitting at the CIII-IV frequency of 2 0 5 019 x
10'3. If one assumes the 350pc distance established by
Brown et al., these data indicate that the CIII-IV blend is
formed in a region around the WC8 component, and at a
distance of 76 10Rq from that star. Wood (1941) derived a
relationship between the dimensions of Roche equipotential
surfaces, relative to the semi-major axis, as a function of
the mass ratio. Given the described dimensions for y2 Vel,
the radius of the Roche lobe is 83RQ. Hence, evidence
indicates that this emission feature at 4650 is formed in
the outer regions of the circumstellar material, and in
fact, in material which nearly fills the Roche lobe of the
72 Vel binary system. These and other properties of this
intriguing system are summarized in Table 1-1. In light of


11
the discussion presented earlier, it is interesting to note
that H/He < 0.02 for 72 Vel; certainly much different from
the chemical composition of the Sun.
The y2 Vel system is a very peculiar binary which has
attracted much attention in the last two decades. One of
the most intriguing features of this system is the
indication of possible variability in some of the emission
lines which are present in the spectrum of the Wolf-Rayet
component. It is this primary characteristic which has made
72 Vel the subject of so many photometric and spectroscopic
studies. As early as 1918, variation in the shapes of
certain emission lines were reported (Perrine 1918). West
(1972) was the first author to assert that variation in line
intensities might also exist. The results of 24 hours of
observation with the Orbiting Astronomical Observatory,
0A0-2, indicated no short term variations in the CIII
emission line at 1909. Since then however, several authors
have presented evidence of variations in both line profile
and line strength.
Jeffers et al. (1973a), first reported a 6 minute
periodicity in both line intensity and line shape using
photometric measurements of the Hell (4686) and CIII-IV
(4650) features. A later re-analysis of their data (1973b)
revealed even more rapid fluctuations on the order of 154
3 seconds with an amplitude of only about 2%.


12
Table 1-1
Properties of the 7 Velorum System
Spectroscopic binary WC8 + 091
a)
P =
78^5002
b)
V =
176
c )
mwr
* sin3i > 17M
O
d)
M09
* sin3i > 32M
O
e )
q =
0.54
f )
rwr
= 17 3R
0
g)
R0 9
= 76 1 OR
O
h)
a *
sin(i) > 319R
0
i)
H/He
= 0.02


13
Austin, Schneider, and Wood (1973) carried out a very
extensive photometric study of y2 Vel at the Mt. John
Observatory in New Zealand. They used six narrowband
filters with AA = 10. Three of these filters isolated the
emission lines identified as Hell at 4686, CIII at 5696,
and CIV at 5812; the other three filters were centered on
the continuum at 4804, 5302, and 6106. These authors
found that the continuum remained essentially constant as
did the triply-ionized carbon feature, at least in the time
interval spanned by their observations. Austin et al.
detected definite night-to night variations in the strengths
of the CIII and Hell features. The amplitude of variation
for the doubly-ionized carbon line amounted to about 0I?12.
Fluctuations of also occurred in a time period of less
than 2 hours. The singly ionized helium feature exhibited an
increase in brightness of 006 in less than 90 minutes, but,
Austin et al. also note that this line appears to remain
quite stable (i.e., within 0^02) for relatively long time
intervals. Austin et al. carried out a more intensive y2 Vel
observing program, concentrating only on variations in the
Hell line. Their results were inconclusive; y2 Vel exhibited
no detectable variations in a 20 minute time period on the
first night of observation. On the second night, however,
the ionized Hell line increased in brightness by as much as
OPlO in 20 minutes, and again later that night, brightened
by the same amount in only 2 minutes. In summary, Austin


14
et al. confirmed previous results that rapid variations can
occur, although not with the rapidity of the degree reported
by Jeffers et al. They found these variations to be
temporal and not periodic.
Another photometric study carried out by Lindgren
et al. (1975) did not provide evidence for stable variations
between 1 and 10 minutes in length, in either the CIII-IV
line at 4650 or the Hell line at 4686. However, Lindgren
et al. did confirm nightly variations in emission line
strength, in particular, a 0^03 to 0?05 change in the 4650
feature. The observations that were utilized for this
investigation covered an interesting phase of 7 Vel: that is,
when the 0 star was in front of the Wolf-Rayet component.
These results may indicate an eclipse of some part of the
supergiant by a portion of the circumstellar material
associated with the WC8 star.
Bahng presented results of several spectrophotometric
studies of the y2 Vel system, studying the short term
variations in emission line strengths using a photoelectric
spectrum scanner in the spectral range 4600 to 4720. In
1973, Bahng reported on results of spectrum scans which were
acquired on four different nights. The emission lines of
particular interest were CIII-IV (4650) and Hell (4686).
Although a power spectrum analysis did not yield evidence of
a periodic phenomenon, Bahng did find variations in the
equivalent widths of these lines. These changes amounted to


15
a 2% variation for CIII-CIV, and a 4% variation for Hell and
occurred on time scales of 4 to 20 minutes. Later, Bating
(1974) recorded a 6 hour spectrophotometry observation in
which variations were detected between 10 and 20 minutes.
Again, however, no predominant periodicity was found. In
his 1975 paper, Bahng analyzed spectroscopic scans which
were obtained in 1973 and 1974. He compares the theoretical
rms deviations computed from photon counting statistics with
the rms deviations of the measured equivalent widths of the
emission lines of interest. Since the rms deviations of the
data exceed the theoretical rms deviations by more than a
factor of three, Bahng considers these variations to be
statistically significant, indicating that real short term
variations in the emission lines of the y2 Vel system do
exist. Although there is no predominant periodicity, a
power spectrum analysis indicates considerable power near a
frequency corresponding to a period of about 1 minute.
Using integrated magnitudes, Bahng finds evidence of
variation with a s emi ampl i tude of O'iOl with a periodicity
of 339.5 seconds. In addition, a significant peak at
16.2 minutes was also present in the data.
We have presented only a brief summary of some of the
more notable observational studies of y2 Vel which have
appeared in the literature over the last few years. As
pointed out by Haefner et al. (1977), ultra-short period
variations (i.e., those on the order of minutes) seem to be


16
detected only in observations which do not exceed much more
than 30 minutes in length. Fluctuations of this nature are
not generally detected in longer runs of continuous data,
but variations in both emission line strength and profile
are almost always detected on a nightly basis.


CHAPTER 2
INSTRUMENTATION AND DATA ACQUISITION SYSTEM
Instrumentation
The data used for this research were obtained with the
automated optical telescope located at the Amundsen-Scott
South Pole Station on Antarctica. This instrument is a two-
mirror siderostat zenith telescope with an f/6, 7.8cm
achromatic lens (Figure 2-1). As one would expect, certain
special design considerations are necessary to ensure normal
operation in the extremely 1ow-temperature, harsh
environment which prevails at the South Pole. A specially
insulated building measuring 12' x 8' x 8' was constructed
on site by carpenters of ITT Antarctic Services during the
1985 austral summer. The telescope occupies an 8' x 8'
section of the building, while the computer system and
control electronics are housed in an adjacent room measuring
4' x 8. This smaller room can be completely isolated from
the telescope room and heated when necessary. Under normal
circumstances, the telescope room temperature is maintained
at approximately -20C.
The optical head of the telescope contains two
optically-f1 at front-surfaced mirrors which act to redirect
17


18
Figure 2.1 South Pole Optical Telescope.


19
incoming light onto the objective lens and eventually to the
photometer. It is this section of the telescope which
protrudes from the roof of the building and is therefore
exposed to ambient conditions. In order to prevent blowing
snow and other sources of moisture from entering the optical
head, a transparent window is used to completely seal the
interior of the telescope. During the 1986 observing
season, the formation of frost on this window and on the
mirrors in the optical head interfered with some of the
observations. Since this time, however, upgrades to the
system have been installed and all of the optics are
maintained frost-free. This was accomplished by pumping
existing dry polar air (the absolute humidity is equivalent
to about 0.15mm of precipitable water vapor) through a
cylindrical tube containing a desiccant, and into the
optical head of the telescope. Here, the air is heated so
that a temperature gradient of about 5C is maintained
between the air in the optical head and the air immediately
outside of the entrance window, with the inside air being
the warmer.
Of the entire design of the telescope, the only moving
part which is directly exposed to ambient conditions is the
elevation axle. The gear system which moves the telescope in
the east to west direction, also moves the entire telescope
tube in both azimuthal directions. The azimuth-motion
resides well within the telescope room, and a worm gear


20
drive is controlled by a stepper motor assembly. This same
technique provides motion to the field stop and filter
wheels. With the siderostat design, the elevation motor
must move only the optical head in the north to south
direction, and it is therefore necessarily mounted outside
on the optical head assembly. The elevation axis joint is
located between the two portions of the optical head which
house the two optically-flat mirrors. This "up and down"
motion of the telescope is provided with an assembly of
teflon ball bearings.
The optical design of the telescope (Figure 2-1) is
straight forward. Incoming light rays strike the objective
after being deflected by the two front surfaced mirrors in
the optical head; each is positioned at about 45 with
respect to the normal. As the photons emerge from the lens,
they pass through a diaphragm, a filter, and a fabry lens
which images the objective of the telescope onto the
photomultiplier. It is the photometer which measures the
signal and converts it into an equivalent number of photons.
The South Pole Optical Telescope (SPOT) is equipped
with several diaphragms and filters allowing the selection
of any one of a number of combinations depending upon the
brightness of the source, the tracking rate of the
telescope, and the type of observation desired. The current
system has diaphragms which restrict the field of view to
1, 5', 2', or 1' regions of the sky.
The 1 field of view


21
is used only in the initial star pointing procedure while
the 5', 2', and 1' diaphragms are used in the star centering
procedure and for the actual measurement of the object's
brightness; the smaller the diaphragm, the smaller the
contribution from sky background, and the fainter the
limiting magnitude of the telescope. The filters which are
currently mounted in the filter wheel include the Johnson
standard B and V filters, a neutral density filter, and four
narrowband filters. The neutral density filter is a ND5 and
is used in the star find procedure. This filter's sole
purpose is to prevent the telescope from observing a source
which is too bright: that is, one which could saturate, and
subsequently damage the photomultiplier. Of the narrowband
filters, one is used to monitor auroral and/or sky
variations. This filter is centered on the very strong 01
emission line at 5577. The remaining three narrowband
filters were chosen specifically for our research on the
Wolf-Rayet star, 72 Vel. Two of these filters isolate
emission features in the spectrum of Vel; the Hell
emission line at 4686 and the CIII emission line at 5696.
The third filter isolates the continuum region at 4768.
More specific details of each of these filters are presented
in Table 2-1. Also, the response curves for six of the
filters are presented in Figures 2-2 through 2-6. In
addition to the filters, one position on the filter wheel
contains a mirror which directs the light beam perpendicular


Table 2-1
Filter Specifications
Peak
Wavelength
(A)
Half-Power
band-width
(A)
Integration
time
(secs)
Purpo se
4400
900
2
B
5500
900
2
V
4686
32
4
Hell
5696
32
8
cm
4768
92
2
Continuum
5577
100
8
01


Transmittance
o
Wavelength ( A )
Figure 2-2 Response curves for the U (O), B (A), and V (0)
filters.
CO


Transmittance
50
40
30
T
o
o
o
o
o
o
20
10 -
0
4640
o
o
o
4660 4680
o
Wavelength ( A )
o
4700
4720
Figure 2-3 Response curve for the helium emission filter.
F-


Transmittance
60
50
40
o
o
o
30
o
20
o
10
o
0 L-
5650
5670
1 i
5690 5710
o
Wavelength ( A )
o
o
o
i
o
5730
5750
Figure 2-4 Response curve for the carbon emission filter.
ro
Ln


Transmittance
60
50
40
30
20
10
o
5500 5550 5600 5650 5700
1
1
1
1
o
o o
o
o
o
o
o
-
o
o
-
o

o
o
-
-
o
o
-
o
o
o
1_
1
1
1
o
Wavelength ( A )
Figure 2-5 Response curve for the oxygen emission filter. ^


Transmittance
o
Wavelength ( A )
Figure 2-6 Response curve for the continuum filter.
ro


28
to the optical axis where it can be viewed through an
eyepiece. This section of the telescope is maintained at a
temperature below -20C in an effort to reduce the dark
current in the system. During the three years in which the
telescope has been operating, this level has remained nearly
constant at 3.7 counts/sec. Other regions of the telescope
are also maintained at optimum temperatures according to a
thermal design analysis carried out by Esper (1986). These
temperatures are achieved with thermal insulation, heaters,
and residual heat generated by stepping motors and other
electronic components. The entire thermal design of the
telescope works in concert with ambient temperatures ranging
from -40C to -80C.
Data Acquisition
The SPOT system is controlled with a modified Dynatem
RM-65 computer system with an 8-bit 6502 microprocessor.
The computer system controls the motions of the telescope in
azimuth and elevation, as well as the motions of the
component parts including the field stop and filter wheels.
The necessary special functions and interfacing of the
telescope are provided by custom built circuits supplied by
the electronics shop at the University of Florida. The
software is written in 6502 assembler language and FORTH.
FORTH is a computer language designed specifically for
instrument control processes. As mentioned previously, the


29
computer system is located in the SPOT building. However,
in some cases, the telescope can be commanded from a remote
terminal. The data storage media which are currently
employed are 8-inch floppy disks.
During an observing session, photometric and
engineering data are accumulated until a one kilobyte region
of memory is filled. At that time, the buffered data are
written onto a disk file. At the beginning of each hour,
the telescope suspends the acquisition of photometric data
and records the status of various aspects of the telescope.
Such things as the azimuth and elevation positions of the
telescope, the Julian Date and Sidereal Time, and the
temperatures of various sections of the telescope are stored
as engineering records. After these data are obtained, the
telescope resumes its observing program. Observing programs
instruct the telescope with respect to pointing, diaphragm,
appropriate filtering, and number of seconds to integrate
for each filter. In addition, in the case of stellar
photometry, the telescope is given the number of steps and
directional information for sky readings. Observing
programs can be modified, deleted, and added from the remote
terminal. Depending upon the observing program and sky
conditions, a given floppy disk may fill in a week to ten
days. At this time, the winter-over scientist changes the
disks, copies the contents of the filled disk to tape for
archival purposes, and transmits the data to a mainframe


30
computer in Malabar, Florida, via the ATS-3 satellite
communication link. These data are retrieved from the VAX
system in Malabar, and stored on the ATS-VAX at the
University of Florida where the reduction of the data
proceeds.


CHAPTER 3
DATA REDUCTION TECHNIQUES
During the 1986, 1987, and 1988 observing seasons,
SPOT obtained data for several different research projects.
In order to optimize telescope time and to make the best use
of clear skies, the data for more than one research project
were usually acquired within the same observing program.
The research project that is the subject of this
dissertation has two major objectives: 1) the search for
variations in emission line strengths using the Hell, CIII,
and continuum filters; and, 2) the search for eclipse
effects requiring observations in the B and V filters, as
well as integrations on the comparison star, HR3452. In
some observing programs, both y2 Vel and HR3452 were observed
using all available filters. In other cases, an increase in
the time-resolution of the data was accomplished by removing
HR3452 from the observing program and omitting the B and V
filter observations. Therefore, several different observing
sequences were required to obtain the data necessary for
this investigation. Specific details of these programs will
be presented in Chapter 4 where individual data sets are
discussed. In general however, the data acquired for this
project were obtained with the use of the 5' diaphragm in
31


32
conjunction with the B, V, Hell, CIII, 01, and continuum
filters .
As one would expect, the photometric integration times
necessary to obtain adequate signals for a given star and
filter varies from star to star, and from filter to filter.
The required integration length depends on a particular
star's color and its distance from the zenith. For example,
to obtain comparable deflections in B and V filters for a
red star, it is necessary to integrate for a longer time
period in the blue region than in the yellow. In addition,
the increased scattering of blue light at large zenith
distances requires the use of longer integration times for
the blue filter than for the visual filter. The optimum
integration times for the stars and filters used in this
study are based on this knowledge, as well as experience
that was acquired during the early portions of the 1986
observing season. Integration times are presented in
Table 3-1.
Since a star does not wholly fill the observing
diaphragm, each stellar deflection is actually the sum of
the star and surrounding sky emissions. In order to obtain
a net star reading, it is necessary to remove an appropriate
value for the sky intensity from each of the stellar
measurements. In the case of a single channel photometer,
this is easily accomplished by integrating on a region of
sky which is devoid of stars within the limitations of the


33
Table 3-1
Optimum Integration Times for y2 Vel and HR3452
Star Filter Integration time
(secs)
72 Vel B 2
V 2
Hell 4
Cl 11 8
Continuum 2
01 8
HR3452 B 10
V
10


34
telescope but which is reasonably close to the star under
observation. In an automated telescope system, it is
necessary to make certain that both of these criteria are
satisfied. A search of the Palomar Sky Survey (1954) prints
indicates that a displacement of 15' in the positive azimuth
direction from the star is an appropriate displacement for a
typical sky reading. In very general terms, each stellar
deflection is accompanied by a corresponding sky deflection
which is used in the data reduction procedures.
During the observing season, data are periodically
transmitted from the South Pole to the VAX 11/750 computer
at the University of Florida. Each of the programs used in
the reduction and analysis of the y2 Vel photometry was
written in Fortran-77. In order to optimize computer run
time, the data were initially passed through a program which
unpacked the data from hexadecimal to decimal format and
divided the data into two files: an engineering file
containing information relating to the design and operation
of the telescope, and a separate photometric file containing
the data relating to the stellar deflections. The raw
stellar deflections were then reduced to photometric
magnitudes using standard reduction techniques (Henden and
Kaitchuck 1982). Since each deflection is recorded as a
counts-per-1ime quantity, and since an integration time
varies from filter to filter, it is necessary to convert all
measurements to the same unit; a counts per second quantity.


35
Once this is done, the next step is to remove the sky
background from each stellar measurement. This is
accomplished through a simple linear interpolation of the
sky readings to the time of the star reading. The number of
counts (per second) due to the background are then
subtracted from the total counts (per second) due to the
star. In the case of the B and V data, HR3452 was chosen as
the comparison star. HR3452 is very close in color to y2 Vel
with a spectral classification of B1V and is located only
24' from the variable. For these reasons, and the fact that
HR3452 has not been reported to be variable, it meets the
criteria for a reasonably good comparison star. A second
linear interpolation scheme is then used to determine the
intensity of the comparison star, HR3452, at the time of the
variable star reading. Finally, equation (3-1) is used to
give the differential photometric magnitude
Am = -2.5 log
(3-1)
where Am is the differential magnitude between y2 Vel and
HR3452 and Dv and Dc are the deflections (in counts per
second) of y1 Vel and of HR3452, respectively.
A major portion of this research is the search for
periodic intensity variations in the Hell and CIII emission
lines of y1 Vel. One technique which is commonly employed in


36
period determination analyses uses the method of Fourier
transformation of time and intensity measurements. In
particular, Deeming's method (1975) of Fast Fourier
transforms is well suited for data which are unequally
spaced in time. In general, Fourier transform analysis is
used to transform data from the time domain into the
frequency domain, or vice-versa. For instance, a continuous
function of time, f(t), can be transferred into frequency
space according to the formula
F(j/) = FT [ f ] = f(t) exp (-2nii/t) dt. (3-2)
The inverse Fourier transform is given by
f(t) = FT [ F ] = F(v) exp (2wii/t) du. (3-3)
In equation (3-2), F(t/) represents a spectrum of
frequencies. Each peak in F(v) corresponds to the relative
strength of each frequency in the data, f(t). In most
cases, and in this data, f(t) is not a continuous function,
but rather a sampled-set determined at discrete times, t^.
Hence, equation (3-2) can be re-written as a discrete
transf orm
N
F(v) = 2 E [f(t) exp (-2rii/t)],
1
(3-4)


37
with N equal to the number of data points in f(t). The
periodic exponential function in equation (3-4) indicates
that the Fourier transform of f(t) has both real and
imaginary components. Since it is the power as a function
of frequency, rather than the Fourier transform that is of
interest, the derivation must be carried one step further.
If f(t) is assumed to be real, then F(v) must satisfy the
relation
F(-/) = F* (i/),
(3-5)
where F* (u) is the complex conjugate of F(y). Then, according
to Rayleigh's theorem which states:
| f(t) |2 dt
(3-6)
we can define the power, P(i/), as the product of F(i/) and its
complex conjugate, which leaves only the real quantity
P (i/) | F(i^) |2 = C2 (v) + S2 (i/).
(3-7)
C(i/) and S(y) in equation (3-7) are the cosine and sine
components, respectively, and are defined as
C() = ^ S [f(t) cos (2jn/t)], and
(3-8)


38
S(i/) = | Z [f (t) sin (2nvt) ] (3-9)
Since the data used in this investigation are acquired
over a finite span of time and are not sampled at equal time
intervals, the Fourier transform is contaminated with the
sampling function. This spectral window or "beam", as it is
frequently referred to, is given by the relation,
W(i/) G2 (v) + H2 (j/) (3-10)
In (3-10), G(i/) and H(i/) are defined in the following manner:
G(y) S [cos (2iri/t)], and (3-11)
H(^) | S [sin (2iri/t) ] (3-12)
According to equations (3-10) to (3-12), the spectral
window function is normalized so that W(0) = 1. The Fourier
convolution theorem states that the Fourier transform of the
signal, FT[f], is the convolution of the data with the
sampling function (i.e., beam). The fact that the data are
unequally spaced over a finite time interval has several
consequences. The sampling period of the data puts a
constraint on the lowest frequency which is resolvable in
the data. The frequency resolution in a discrete data set
corresponds to the width of the beam (W) at u = 0 and is


39
related to the sampling period by equation (3-13)
Su 1/T, (3-13)
where T is the length of the data set. Therefore, the
minimum frequency which is retrievable from a given data set
is given by the Nyquist frequency
'min 1 / ( 2 AT). (3-14)
It is the data spacing which puts an upper limit on the
frequency that can be recovered from a given data set. In
the case of a function consisting of points spaced at equal
time intervals At, the maximum frequency is given by the
Nyquist frequency. Since the data used in this study are
unequally spaced in time, At is not a constant. For purposes
of this investigation, it was deemed appropriate to
calculate the average time interval for a given data set and
use that value for At, according to equation (3-15):
'max l/(2At). (3-15)
In addition to upper and lower limits being placed on
the frequencies which can be recovered from the data, the
data are also not represented by a continuous function of
time. The discrete nature of the data contributes several


40
features to the spectral window (beam) which adds
considerable complexity to the nature of the spectrum.
Since the Fourier transform of the data is a convolution of
the spectrum with the window, the resulting "dirty" spectrum
may be subsequently contaminated with spurious sampling
features. This aliasing can be masked as either a damping
of real features or an enhancement of "false" features,
making interpretation difficult. For this reason, a one
dimensional deconvolution (CLEAN) algorithm (Hogbom 1974)
has been adopted for use in this study. The CLEAN algorithm
is that used by Roberts, Lehr, and Drehar (1987) which
deconvolves the sampling function from the dirty spectrum to
give a better representation of the true spectrum. This
deconvolution is accomplished in the following manner:
First, the largest peak in the dirty spectrum is located and
stored. The spectral window (beam), is then superimposed
onto the dirty spectrum so that the main peak in the beam
coincides with the largest peak in the dirty spectrum. A
given percentage of the spectral window is subtracted from
the corresponding dirty spectrum. The peak of the spectrum
which is removed from the data spectrum is stored as a CLEAN
component. The spectrum which results after subtracting the
spectral window is also stored in a residual file. This
procedure is repeated for a given number of deconvolutions
(peaks). A gaussian representation of the spectral window
is then convolved with each CLEAN spectra. Finally, in


41
order to preserve the noise which existed in the original
spectrum, the residual values are added back to the
convolved CLEAN components giving the CLEAN spectrum. The
input parameters for the CLEAN algorithm include the number
of CLEANS to be performed (peaks) and the gain. It is the
gain which governs the percentage of the spectral window
which is removed from the dirty spectrum during each CLEAN
ite ration.
The CLEAN procedure is used to improve the strongest
features of a spectrum and is of limited use in data which
may have a small signal-to-noise ratio. In fact, experience
has shown that over-CLEANing data can actually introduce
spurious features in the clean spectrum. Figures 3-1 to 3-6
are used to demonstrate the hazards that can result from the
improper use of this algorithm. Figures 3-1 and 3-2
represent the spectral window, and the dirty spectrum,
respectively, for a sample data set. Figures 3-3 to 3-6
represent the clean spectra after 10, 100, 500, and 1000
iterations of CLEAN. It is quite clear from these graphs
that the peak at t/ = 40 (cycles per day) in Figure 3-3
decreases in strength as the number of CLEANS increases,
while the peak at i/ = 235d 1 gets progressively stronger.
In fact, in Figure 3-6 this is the only peak which remains
in the spectrum with considerable amplitude. It is likely
that a misinterpretation of the data would result in this
type of over-application. In this study, care has been


Relative Power
Frequency ( d -1)
Figure 3-1 Spectral window function for the test data set.


Relative Power
Frequency (d 1)
Figure 3-2 Power spectrum for the test data set before
execution of the clean algorithm.


Relative Power
Figure 3-3 Power spectrum for the test data set after
10 iterations of the clean algorithm.


Relative Power
Figure 3-4 Power spectrum for the test data set after
100 iterations of the clean algorithm.


Relative Power
Figure 3-5 Power spectrum for the test data set after
500 iterations of the clean algorithm.


Relative Power
Figure 3-6 Power spectrum for the test data set after
1000 iterations of the clean algorithm.


48
taken to minimize the introduction of such spurious features
into the power spectrum. This has been accomplished by
restricting the CLEAN procedure to data above the 3crlevel.
In addition, more stability is provided in the CLEAN
algorithm through use of a smaller gain size and more
iterations. In our case, a gain parameter of 0.25 was
adopted.
Because of the possible uncertainties of the power
spectrum analysis, an independent method of period
determination was also used. This method is based on
fitting models to the data with the aid of the method of
1east-squares. The intensity variations which may exist in
the y2 Vel system are probably much more complex than that
which can be described as a simple cosine or sine function.
However, the main purpose of this research is to use the
long, continuous observational runs of y2 Vel to establish
whether variations in the strengths of the Hell and CIII
emission lines do, in fact, occur. It is not necessarily our
purpose to mathematically describe such variability
precisely. A shift in phase could certainly cause a poor
fit to the data. In any event, the variations which would
exist in such a complicated system are probably aperiodic at
best. So, for purposes of this investigation, a first-
order sine function of the following form is assumed:
D(tj_) = A0 + At sin
P
(3-16)


49
In equation (3-16), D(t^) represents the calculated
deflection at time t^, A0 is the baseline in intensity
units, Aj is the amplitude of the sine wave expressed in
intensity units, t0 is the "phase" or time of maximum, and P
is the period. It should be noted at this point that P is
expressed in days divided by In radians, a representation
that was chosen so that when one complete cycle had elapsed,
that is, t t0 = P, then cos (t tQ)/P = 1. In the
1east-squares procedure, four parameters are adjusted. They
are: A0, At t0, and P. A first-order Taylor expansion is
computed using the partial derivatives of equation (3-16)
with respect to each of these variables. The routine used
in this context is that of Banachiewicz (1942) and uses
Cracovian calculus. If the errors which were calculated in
the 1 east squares routine exceed the convergence criteria,
corrections are computed and added to A0 At t and P to
give the new quantities:
Ao + AA0,
A, + AA>,
t0 + At0, and
P + AP.
The convergence criteria are the following:


50
E(A0) < 1 x 10'\
E(At) < 1 x 10'3,
E(t0) < 5 x 1 O'3. and
E (P) < 5 x 10'4.
In these relations, E represents the difference between the
new and old values for each quantity considered.
If all of the parameters do not converge within 30
iterations, it is assumed that a satisfactory model could
not be found within the given input parameters. In the case
of the Hell photometry, an initial intensity-amplitude of
0.01 and a baseline of 0.35 were used. For the CIII data, an
amplitude of 0.01 and a baseline of 0.10 were used as
initial inputs. These two parameters were not as sensitive
to small variations as were the period and phase. To account
for the increased sensitivity in P, an initial and final
period were entered along with a desired incremental step
value. Upper and lower limits on the period assumed two
samples per cycle in accordance with the Fourier transform
analysis. To account for increased sensitivity (to small
perturbations) in the phase parameter, a similar approach is
adopted. That is, a desired increment in the phase is
entered as an input parameter. In every case, the time of
the first observation was used as an initial "guess" of the
value for t0. The parameter was incremented from this
initial value to the initial value plus 2n.


51
Weather reports are recorded at the South Pole every
six hours. Photometric data recorded between two
consecutive periods of reported clear skies were used in
this investigation of ~/2 Vel In addition, since the
telescope does not have the capability to monitor the star
and nearby sky simultaneously, data recorded during times of
auroral activity were omitted from the study. These
omission criteria resulted in the selection of seven
individual data sets obtained during the 1986 austral
winter. Individual observations in each data set were
omitted if the observation in the continuum filter and the
observation in one or more of the emission line filters
deviated by more than 10%. The analysis of the data sets
that resulted is presented in the following chapter.


CHAPTER 4
DATA ANALYSIS
Spectral Photometry of the Hell and CIII Emission Lines
Data Set I.
This data set includes observations of y1 Vel that
extend from JD2446577.6847 through JD2446577 7792. These
(81) data-points were obtained for each of the helium,
carbon, and continuum filters with an average spacing
between successive readings with a given filter of 0^0012.
Both y2 Vel and HR3452, the comparison star used to reduce
the blue and visual photometry, were observed. The exact
sequence of observations is presented in Table 4-1. In the
table, an "X" is used to indicate that an integration was
accomplished with the corresponding filter listed at the top
of the table.
These data were reduced to a ratio of intensities
using methods described in Chapter 3. The data were first
transformed into frequency space using Deeming's method of
Fast Fourier Transforms and assuming a minimum of two
samples per cycle. Given the data spacing of 0^0012 and the
2.3 hour time interval spanned by this data, the minimum and
maximum frequencies which can be retrieved from the data set
with a reasonable amount of confidence, are i/m^n = lid'1 and
"max = ^l^d'1. Figure 4-1 represents the sampling
5 2


53
Table 4-1
Observing Program 1
Star B V He C 0
C*1
CS2
C*
V*3
V*4
V*
V*
vs
v*
v*
v*
v*
c*
CS
c*
V*
v*
v*
V*
v*
v*
V*
v*
V*
v*
v*
v*
X X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X X
X X
X X
XXX
XXX
XXX
XXX
XXX
XXX
XXX
XXX
XXX
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
1 C* = comparison star
2CS = comparison sky
3V* = variable star
4 VS = variable sky
Continuum
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X


Relative Power
Figure 4-1 Spectral window function for data set I.


55
function for both the helium and the carbon observations.
The peaks at u = 52, 158, 364, and 470d'1 are the odd
harmonics i/p 1/3, V5, and v-j, and the peak at 1/ = 318d'1 is the
even harmonic, i/g. The convolution of the helium data with
the spectral window function gives the spectrum in Figure
4-2. The noise level is denoted in the lower left-hand
portion of the plot at 5.16 x 10'6. All features to the
right of the arrow satisfy the Nyquist criterion of a
minimum of two samples per cycle. The peaks at v = 170, 323,
and 377d_1 represent statistically significant features at
the 4ct^j level. A closer look at the spectrum reveals that
the peak at 1/ = 377d 1 is part alias with that at v = 323d'1
and the peak at v = 170dl is part alias with a peak at
v = 8d_1. This last feature is shown in Figure 4-2 but since
it does not satisfy the Nyquist criterion, it is discarded
in this study. Spurious features at or above the 3a^ level
are removed from this spectrum with a deconvolution process;
the resulting power spectrum is depicted in Figure 4-3.
Here, we see that the peak at v = 323d'1 is the only feature
which remains above the 4ct^ level, at 4.25a^. This
frequency corresponds to a period of 4.5 minutes. According
to the least squares routine, the best fit to the data,
within the limitations of the Nyquist criterion, occurs for
P = 4.4 minutes, and is described by:


Relative Power
Figure 4-2
Power spectrum for the helium observations of
data set I before execution of the clean
algorithm.


Relative Power
0 50 100 150 200 250 300 350 400
Frequency ( d 1)
Figure 4-3 Power spectrum for the helium observations of
data set I after execution of the clean
algorithm.


58
D(tt) 0.35099 + 0.00813 sin
r t 0.18518 *\
0.00049 J
(4-1)
The model computed from this equation is presented in
Figure 4-4 and seems to fit the observations quite well.
However, because there are only an average of 2.5 points per
cycle, one should regard this result with some skepticism.
A visual inspection of a plot of intensity as a function of
time indicates that perhaps the data are changing at a more
gradual rate. The least squares routine was executed for
periods on the order of the time interval spanned by the
data set; convergence occurred for a period of 2.40 hours.
The model given by
D(tt) = 0.34986 + 0.00724 sin
r 0.10507 -x
0.01592
(4-2)
is graphically presented in Figure 4-5. The amplitude of
the variation of this model is only slightly lower than that
for the more rapid fluctuations depicted in Figure 4-4.
The power spectrum for the carbon data obtained in
this time interval is shown in Figure 4-6. Again, the arrow
in the lower left of the plot represents the noise level at
1.287 x 10'*. It should be noted that the peak at v = 70d_1
is part alias with that at v = 20d'1 and should, in
principle, be removed with the CLEAN algorithm (Figure 4-7).


Hell : Continuum
0.400
0.380
0.360
0.340
0.320
0.300
0.180 0.200 0.220 0.240 0.260 0.280
Julian Date (2446577.5+)
Figure 4-4 Model of the variations of the helium data for
data set I with P 4.43 minutes. Each plotted
point represents the average intensity within a
0^0040 interval.
Ln


Hell : Continuum
Figure 4-5 Model of the variations of the helium data for
data set I with P = 2.40 hours. Each plotted
point represents the average intensity within a
0*?0040 interval.
CTs
O


Relative Power
Figure 4-6 Power spectrum for the carbon observations of
data set I before execution of the clean
algorithra.


Relative Power
Figure 4-7 Power spectrum for the carbon observations of
data set I after execution of the clean
algorithm. as


63
This is indeed the case, and the only peak which survives
the CLEANing process is that at v = 20d1 corresponding to
0^050 ( 1.20 hours). The signal-1o noise ratio for this peak
is quite large at 8.04ctjj. The 1 e a s t s quar e s routine
converges to a set of quantites with a similar period,
P = 0^0459 (1.10 hours), according to the following
relation:
D(tt) = 0.11117 + 0.00231 sin
r tL 0.45362
L 0.00731
(4-3)
The fit to the data using equation (4-3) is presented in
Figure 4-8. Although the amplitude is quite low, this
period may be considered as a satisfactory representation to
the fluctuations since both the power spectrum analysis and
the 1east-squares routine give similar results.
In summary, the helium data seem to follow two
different periods. The shortest period of about 4.4 minutes
represents very rapid fluctuations and was arrived at by
both the power spectrum analysis and the 1east-squares
solution. In addition, models were computed for periods
corresponding to the length of the data set with the best
fit solution having a period of P = 2.40 hours. Both fits
seem to be quite satisfactory. The intensity variations of
the CIII emission line, although low in amplitude, can be
modeled with a period of 1.10 hours.


Clll : Continuum
0.165
Figure 4-8 Model for the variations of the carbon data for
data set I with P 1.10 hours. Each plotted
point represents the average intensity within a
0.0040 interval.


65
Data Set II.
One-hundred and eighty-nine observations of
72 Vel were obtained in the helium and carbon filters using
the observing sequence presented in Table 4-2. The time
interval spanned by these data extends from JD2446606.6048
through JD2446606.9553. The average spacing between
successive observations for a given filter is 0t0019. The
decrease in time-resolution of these data relative to the
time-resolution of the data discussed in the previous
section, is an artifact of the new observing program. As
can be seen from Table 4-2, several new objects have been
added to the original observing program (Table 4-1). This
ultimately limits the high frequency components which are
recoverable in the data analysis.
The spectral window for these data is given in Figure
4-9, and includes first, second, third, and fourth harmonics
at v = 23, 43, 64, and 84d'1, respectively. In general, as
in this case, spectral windows for unequally sampled data
tend to be more complex at high frequencies. The power
spectrum of the helium data is computed for frequencies
between i/m^n = 6d* and i/max = 265d'1 (Figure 4-10). The
noise level in this figure is denoted in the lower left-hand
portion of this plot at 9.976 x 10'7. Given this level of
noise, the peaks at v = 16, 22, 42, and 218d_1 might be
classified as statistically significant. However, when one
superimposes the spectral window on the power spectrum, it


66
Table 4-2
Observing Program 2
Star B V He C 0 Continuum
V*1 X X
V* X X
V* X X
V* X X
V*
V*
V*
V*
VS2 X X
V*
v*
v*
v*
V* X X
V* X X
V* X X
V* X X
C*3 X X
CS4 X X
C* X X
V* X X
C* X X
CS X X
C* X X
V* X X
V* X X
V* X X
V* X X
V*
V*
v*
v*
VS X X
V*
v*
v*
v*
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X'
X
X
X
X
X


67
Table 4 2 --continued
Star B V He C 0 Continuum
V* X X
V* X X
V* X X
V* X X
C* X X
CS X X
C* X X
Sky B5 X X
a Gru X X
a Gru S6 X X
a Gru X X
fi Gru X X
B Gru S' X X
B Gru X X
Sky A8 X X
X
X
X
X
X
X
X
XX X
XX X
XX X
XX X
X
X
X
X
X
X
1 V* = variable star
2 VS = variable sky
3 C* = comparison star
4CS = comparison sky
5 Sky B = 1 region of sky
6 q Gru S = a Gru sky
7 B Gru S = B Gru sky
8 Sky A = 1 region of sky
180 from variable star
180 from Sky B


Relative Power
Figure 4-9 Spectral window function for data set II.
a\
co


Relative Power
1.0E-5 -
7.5E-6
5.0E6
2.5E-6
0.0
50
100
150 200
Frequency ( d 1)
250
Figure 4-10 Power spectrum for the helium observations of
data set II before execution of the clean
algorithm.


70
is apparent that the peaks at u = 22d_1 and v = 62d-1 are part
aliases of the considerably stronger peak at v = 42d l .
The CLEAN algorithm has been used in an attempt to eliminate
these spurious features, and the results are shown in Figure
4-11. Now, only two peaks have survived the CLEANing
process while remaining statistically significant. They
include the feature at u = 42d_1 (P = 0^0238 ) with a signal-
to-noise ratio of 7.7crjq and the feature at v = 238d-1
(P = 0^0042) which has a weaker signal of 4.5a^.
Although the least-squares routine converges for
periods corresponding to each of these frequencies, the
largest amplitude of variation and the lowest observed minus
calculated (0-C) residuals occurs for P 0^0476 (¡/ = 21d').
It is interesting to note that this period is exactly twice
the period concluded from the power spectrum analysis and
perhaps this is an indication of the presence of harmonics
of a fundamental period. In order to determine whether or
not one of these periods fits the data more satisfactorily
than another, models are computed for each. Results
indicate that these data are most adequately represented by
P = 0^0476 (Figure 4-12) according to the equation
D(tL) = 0.34430 + 0.00379 sin
r 0.48459
^ 0.00757
(4-4)
Perhaps this period reflects the fundamental mode of


Relative Power
0 50 100 150 200 250
Frequency ( d 1)
Figure 4-11 Power spectrum for the helium observations of
data set II after execution of the clean
algorithm. -J


Hell : Continuum
Julian Date (2446606.5+)
Figure 4-12 Model for the variations of the helium data for
data set II with P 1.14 hours. Each plotted
point represents the average intensity within a
0^0040 interval.


73
variation while the prominent feature in the power spectrum
may represent the first harmonic. The carbon data have also
been analyzed in a manner comparable to that used in the
analysis of the helium data. The dirty and clean spectrums
are presented in Figures 4-13 and 4-14, respectively.
Although the feature at v = 42d'L is considerably weaker than
the corresponding peak in the power spectrum for the helium
data, it still remains the strongest feature in the plot at
5.8cjfj where ajq = 1.680 x 10'7. Three other peaks rise just
above the 4a^j level; they occur at
¡/ = 12, 57, and 204d'1.
The 1east-squares routine is executed for periods
ranging between 0^0038 (v = 263d1) and 0^1759 (v = 6d').
Solutions with periods corresponding to each of the
frequencies, found from the power spectrum analysis, result.
Once again, the strongest convergence occurs for a period of
0^0477 (v = 21d1). One must realize that in all of these
models, the amplitude of variation of the CIII emission line
is so small that results cannot be considered to represent
significant intensity changes. The largest variation in
this case is only 1% of the average deflection. It is for
this reason that no specific model is suggested for this
data .
In summary, power spectrum analysis gives a period of
0^0238 for both helium and carbon data. The 1east-squares
routine, although still converging for periods on this


Relative Power
1.2E-6
8.0E-7
4.0E-7
0.0
50
100
150
200
250
Frequency ( d ~1)
Figure 4-13 Power spectrum for the carbon observations of
data set II before execution of the clean
algo rithm.


Relative Power
Figure 4-14 Power spectrum for the carbon observations of
data set II after execution of the clean
a1 go rithm.
Ln


76
order, give a best-fit with a period equal to twice this
period, or, P = 0^0476.
Data Set III.
This data set spans the time interval that extends
from JD2446607.9853 through JD2446608.1921 The set includes
90 observations of 72 Vel for the helium, carbon, and
continuum filters.
The observing sequence used to obtain these data is
that shown in Table 4-2, with At = 0^0023. Assuming a
minimum of two samples per cycle and a minimum of two cycles
per data set, frequencies from i^m^n = lOd'1 to vmax = 217d'1
can be tested. Figure 4-15 shows the sampling function
which results from the acquisition of these data. The peak
at v = 20d 1 has considerable power amounting to about one-
third that of the main peak at i/ = 0d l. The second harmonic
at v = 41d'1, and the third harmonic at v = 62d~l are also
identified. In addition, a broad feature centered at
i/ = 33d 1 has a second harmonic at u = 55d'1 The convolution
of the helium data with the beam is determined and presented
in Figure 4-16 in terms of relative power. The noise level
of these data is indicated by an arrow at the bottom left of
the plot at 2.719 x 10'6. The features at u = 11, 73, and
215d-1 rise above the 4ctj^ level, but the peak at v = 73d 1 is
part alias with the stronger feature at v = lid'1. The
CLEANing process leaves two of the three peaks mentioned
above as statistically significant (Figure 4-17). The


Relative Power
0 50 100 150 200
Frequency ( d "1)
Figure 4-15 Spectral window function for data set III.


Relative Power
Figure 4-16 Power spectrum for the helium observations of
data set III before execution of the clean
algorithm.


Relative Power
O 50 100 150 200
Frequency (d1)
Figure 4-17 Power spectrum for the helium observations of
data set III after execution of the clean
algorithm.


80
feature at v = lid'1 has a relative power of 4.9a^ while the
peak at 1/ = 215d has the slightly lower s i gnal t o no i s e
ratio of 4.6cr^ and represents more-rapid fluctuations in
intensity. The 1east-squares routine converges to two sets
of parameters with almost the same amplitude in each case.
The first of these models has a period of 2.00 hours and
corresponds to the peak at u = lid'1 in Figure 4-17. The
model for this period is shown in Figure 4.18, and has been
computed with the relation
D(ti) = 0.34410 + 0.00354 sin
r t 0.50673
0.01328
(4-5)
The second set of parameters which results from the least-
squares routine has a period of 2.75 hours and is described
by equation (4-6):
DCti) = 0.34442 + 0.00329 sin
r tt 0.59261
L 0.01820 J
(4-6)
Even though this period does not correspond to any prominent
feature in the power spectrum, the model, pictured in Figure
4-19, seems to fit the data as well as that described by
equation (4-5). Even though the 0-C residuals computed from
equation (4-5) are slightly smaller than the residuals
determined with equation (4-6), gaps in the data make it


Hell : Continuum
Julian Date (2446607.5+)
Figure 4-18 Model for the variations of the helium data for
data set III with P 2.00 hours, each plotted
point represents the average intensity within a
0^0040 interval.
oo


Hell : Continuum
Julian Date (2446607.5+)
Figure 4-19 Model for the variations of the helium data for
data set III with P 2.74 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
00
K>


83
difficult to determine which fit is better. Since the first
of the two periods is supported by both methods of analysis,
P = 2.00 hours may be the more appropriate period.
The Fourier transform of the carbon data is converted
to power, and Figures 4-20 and 4-21 show the results before
and after the removal of spurious features. The la noise
level of these data is at 5.685 x 10'7. Since the carbon
emission feature is considerably weaker than the helium
line, it is more difficult to discern intensity variations
in the carbon feature from random noise. In this particular
case, the CLEANing process reduces the strongest peaks in
Figure 4-20 below the 4a^¡ level. The 1 east squares routine
has the strongest convergence for a period of 34.4 minutes.
However, since the amplitude of variation is only a very
small fraction of the total deflection in that filter, one
cannot regard this result with much confidence.
In summary, the intensity variations of the helium
emission line are probably best described with a period of
2.0 hours. This period was selected by both power spectrum
and 1east-squares analytic techniques. In addition, the
data have been modeled with a period of 2.75 hours, which
was a solution of the 1east-squares method only. Relatively
large gaps in the data make it difficult to determine which
period, if either, is the more appropriate. The power
spectrum of the carbon data indicates no strong periodicity.
A 1 east squares solution with a period of 34.4 minutes is


Relative Power
Figure 4-20 Power spectrum for the carbon observations of
data set III before execution of the clean
algorithm.


Relative Power
Figure 4-21 Power spectrum for the carbon observations of
data set III after execution of the clean
algorithm.


86
found, but because of the low amplitude of the variation,
this period is not considered to be a true description of
the variations within this weak feature.
Data Set IV.
This data set consists of 18.8 hours of photometry of
72 Vel extending from JD2446608.7789 through JD2446609.5624.
The observing program displayed in Table 4-2 was used to
obtain a total of 347 observations through each of the
helium, carbon, and continuum filters. The average time
interval between successive observations in any one of these
filters is 0^0023.
The spectral window is depicted in Figure 4-22 where
peaks at u = 17d'1 and u = 35d_1 are separated by the same
amount as the peaks at u = 38d_1 and v = 56d_1. The
convolution of the intensity measurements of the helium
emission line with this somewhat more complicated sampling
function gives the power spectrum in Figure 4-23. The laN
noise level at 8.850 x 10'' indicates that peaks at i/ = 11.5,
35, and 166.5d-1 are statistically significant features. A
strong feature also occurs at i/ = 1.5d'x. but this frequency
does not satisfy the Nyquist criterion of two samples per
cycle. The peak at v = 35dl is part alias with that at
v = l.Sd1, and should, in principle, be reduced in strength
by the CLEAN algorithm. Figure 4-24 shows the power
spectrum that results after the spurious features are
removed from the dirty spectrum. The only feature which


Relative Power
Figure 4-22 Spectral window function for data set IV.


Relative Power
0 50 100 150 200
Frequency ( d1)
Figure 4-23 Power spectrum for the helium observations of
data set IV before execution of the clean
algorithm.


Relative Power
0 50 100 150 200
Frequency ( d 1)
Figure 4-24 Power spectrum for the helium observations of
data set IV after execution of the clean
algorithm.


90
remains statistically significant is the peak at v = 1.5dl,
but as mentioned previously, the period which corresponds to
this frequency does not satisfy the Nyquist criterion, and
thus cannot be considered in the context of this research.
Although no peaks survive the CLEANing process, the least-
squares routine is still executed. Convergence to a set of
parameters with P = 0^208 (5.0 hours) is given by
DCti) = 0.34736 + 0.00250 sin
r t 0.22157
L 0.03316
(4-7)
This fit, depicted in Figure 4-25, has a very low
amplitude of variation and is certainly not a very good
representation of the changes in intensity. This
unsatisfactory fit is confirmed by the low power in the
peaks of the CLEANed power spectrum (Figure 4-24). An
attempt was made to improve the results obtained with these
data by dividing the long data set into two subsets. The
first subset contains observations made between
JD2446608.77 89 and JD2446609.1646, and has a time resolution
which is exactly the same as that for the whole data set.
The shorter span of this subset allows for frequencies
between = 5d-1 and ^max = 217d' to be tested. The
spectral window function for these data, depicted in Figure
4-26, shows similar, but slightly stronger, peaks as those
from Figure 4-22. Dirty and clean spectra are shown in


Hell : Continuum
Julian date (2446607.5+)
Figure 4-25 Model for the variations of the helium data of
data set IV with P 5.00 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
VO


Relative Power
1.000
Figure 4-26 Spectral window function for subset I of data
set IV.
ro


Full Text
xml version 1.0 encoding UTF-8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchema-instance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID ELETESAXQ_76UU2Y INGEST_TIME 2011-10-27T12:39:27Z PACKAGE AA00004828_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES




ANALYSIS OF 72 VELORUM PHOTOMETRY FROM
THE SOUTH POLE
BY
MARYJANE TAYLOR
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
fD OF E LIBRARIES.

ACKNOWLEDGEMENTS
At this time, I would like to express my gratitude to
those people without whose contributions this work would
have been far more frustrating.
First, I would like to thank the chairman of my Ph.D.
committee, Dr. John P. Oliver, for his dedication and
expertise in the construction of the South Pole Optical
Telescope (SPOT), and for his knowledge in the development
of the software necessary to automate the SPOT system. I
thank him for the many hours that he has dedicated to this
project and for the endless headaches incurred, especially
near the beginning of each observing season.
I would also like to express my sincere appreciation
to Dr. Kwan-Yu Chen, principal investigator of the SPOT
project, for his contributions to this research. He has
provided valuable advice in various aspects of this
research, and has read this dissertation with a very keen
eye. Dr. Chen has also supplied me with references to
current papers in the literature on Wolf-Rayet stars, and on
72 Velorum.
Thanks also go to Dr. Frank Bradshaw Wood, principal
investigator of the project in its infancy, for suggesting

the topic of this dissertation.
Dr. Wood has also made me
aware of current topics in the literature, and has carefully-
read this dissertation.
In addition, I extend my thanks to the two other
members of my Ph.D. committee, Dr. Jerry L. Weinberg and
Dr. Bruce T. Edwards, for reading this work and making
constructive criticisms which have improved its quality.
Thanks also go to Dr. Stephen T. Gottesman, chairman
of the Astronomy Department at the University of Florida,
for taking the time to share his knowledge of the
interpretation of power spectra with me, despite not being a
formal member of my Ph.D. committee.
I would like to express my sincere appreciation to
Dr. John E. Merrill for his devotion to the South Pole
project, and for always keeping the members of the team on
track. His experience and insight into many of the problems
has been extremely useful.
I would like to extend ray sincere thanks and
appreciation to Dr. Lance Erickson for the long discussions
on the application of the methods of Fourier Transform and
least - squares analysis as they pertain to this work. I
would also like to thank him for the weekends that never
were and for helping me to maintain my sanity these last few
months .
Finally, warmest thanks go to my father whose never-
ending patience, encouragement, and support for the last
i i i

twenty-one years have made me always strive for the best. I
owe my love of astronomy to him. In addition, he is to be
thanked for the long hours devoted to the tedious process of
formatting this dissertation in accordance with Graduate
School regulations, and for the use of his Hewlett-Packard
printer .
The Steller Photometry Program was supported by grants
from the Division of Polar Programs of the National Science
Foundation.
IV

TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT vi
CHAPTERS
1 BACKGROUND 1
2 INSTRUMENTATION AND DATA ACQUISITION SYSTEM ... 17
Instrumentation 17
Data Acquisition 28
3 DATA REDUCTION TECHNIQUES 31
4 DATA ANALYSIS 52
Spectral Photometry of the Hell and CIII
Emission Lines 52
Data Setl 52
DataSetll 65
Data Set III 76
DataSetIV 86
DataSetV 110
Data Set VI 122
Data Set VII 146
B and V Photometry 155
5 DISCUSSION AND CONCLUSIONS 165
REFERENCES 176
BIOGRAPHICAL SKETCH 179
v

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
ANALYSIS OF y2 VELORUM PHOTOMETRY FROM
THE SOUTH POLE
BY
MARYJANE TAYLOR
DECEMBER, 1988
Chairman: John Parker Oliver
Major Department: Astronomy
Several hundred photometric observations of the Wolf-
Rayet star, y2 Vel, have been obtained with the South Pole
Optical Telescope using filters centered on the Hell and
CIII emission lines at 4686Á and 5696Á, respectively. The
observations are reduced to intensities using a region of
the continuum centered at 4768Á as the comparison source.
Two independent techniques are used to determine
periodic behavior of variations in the strengths of these
emission lines. The first of these methods uses power
spectrum analysis based on Deeming's method of Fast Fourier
Transforms. The second process involves fitting a first-
order sine wave to the data using 1east-squares analysis.
vi

The results of the analysis of the Hell feature
indicate a "fundamental period" of 1.20 hours with an
amplitude of fluctuation of a few percent. Several
harmonics and subharmonics of this period are also detected.
The presence of a 1.20 hour period of variability has
certain theoretical implications which are discussed in the
context of recent theoretical developments. It is suggested
that these spectral changes are due to radial pulsations of
72 Vel .
An attempt is made to arrive at satisfactory models to
describe intensity variations of the CIII emission line.
Despite the weakness of this feature, periods are determined
which indicate slightly more rapid fluctuations than those
found to describe variations in the corresponding helium
data. It is suggested that this result may provide clues
indicating the relative locations of line formation in the
extended wind.
In addition to the analysis of narrowband photometry,
observations in the visual and blue filters are reduced to
photometric magnitudes using comparison star HR3452. The
analyses of these data do not reveal any evidence of an
optical eclipse in the y2 Vel system, at least within the
limitations of these data.
vi i

CHAPTER 1
BACKGROUND
In 1867, C. Wolf and G. Rayet discovered several stars
which exhibited very unusual spectral characteristics;
namely, numerous broad emission features superimposed on a
faint continuum. At that time, the only star known to have
an emission line in its spectrum was 7 Cassiopeia. Normal
stellar spectra consisted of a continuum marked by
absorption rather than emission lines. In fact, the only
astronomical objects which were known to have spectra
dominated by emission features were gaseous nebulae. Wolf
and Rayet both realized that the spectral characteristics of
the stars they had discovered were considerably different
from either those of 7 Cas or gaseous nebulae. Although
originally classified in the Henry-Draper Catalogue as
spectral type 0, these peculiar objects have now been
assigned the separate spectral classification of W and are
called Wolf-Rayet stars.
In general, the spectrum of a star is used as an
indication of the evolutionary state of that star. However,
even today, over a century after their discovery, neither
the evolutionary status, nor many of the physical,
geometrical and chemical properties of Wolf-Rayet objects
1

2
are well understood. It follows then, that in the case of
Wolf-Rayet stars, an understanding of the nature of the
emission line mechanisms producing such a peculiar spectrum
is needed before establishing an appropriate evolutionary
scenario. It is this unusual emission line spectrum which
is commonly referred to as the "Wolf-Rayet phenomenon." The
most currently accepted interpretations of observations of
such an intriguing class of stellar objects are presented in
the following few paragraphs.
Analysis of spectroscopic observations of Wolf-Rayet
stars allows us to state with considerable confidence, the
characteristics of the spectra. Very succinctly, the
spectra are dominated by emission lines superimposed on a
continuum which is characteristic of an 0 or an early B type
star. Only in a few cases are intrinsic absorption features
observed. Generally, the only absorption features in a
Wolf-Rayet stellar spectrum are the P Cygni absorption
components of certain emission lines in some stars. Such
P Cyg absorptions indicate expansion of material surrounding
the Wolf-Rayet star. The width of the emission lines can be
used to infer the velocities of expansion. Although the
breadth of the emission lines varies for each ion, probably
because different ions are formed at different depths of the
atmosphere, these widths correspond to velocities ranging
between hundreds and thousands of kilometers per second.

3
Spectral observations have led astronomers to realize
that there are two main sequences of Wolf-Rayet stars. The
WN type Wolf-Rayet star has an optical spectrum dominated by
nitrogen and helium ions. Several of the WN subtypes
exhibit traces of carbon: in particular, the CIV lines at
5801Á and 5812Á in the optical region of the spectrum and at
1550Á in the ultraviolet region. The enhanced abundances of
helium and nitrogen, as well as the lower than "normal"
abundances of carbon and oxygen, may be explained by the
fact that these stars expose material which has been
processed in the CNO cycle. The second sequence of Wolf-
Rayet stars are the WC types. As their nomenclature
indicates, these stars have spectra which are almost
entirely dominated by carbon ions, but which may also
display helium and oxygen lines as well. Unlike the WN
types, the presence of nitrogen in WC stars is very weak.
The possible blends of NIII ions with strong carbon features
have been identified (Underhill 1959, Bappu 1973) at optical
wavelengths and blends of NIV and NV ions with other carbon
emissions may be present also at ultraviolet wavelengths
(Willis 1980). The (WC) class of Wolf-Rayet stars exposes
material which has probably been processed by a helium¬
burning convective core. In this case, however, the
presence of carbon and oxygen is enhanced at the expense of
nitrogen and helium. Recently, Barlow and Hummer (1982)
have identified a third class of Wolf-Rayet stars, WO. The

4
WO types represent stars which exhibit strong amounts of
oxygen rather than carbon and are probably a result of
extreme helium burning.
The ratio of hydrogen to helium in Wolf-Rayet stars
remains a controversial topic. Based on analyses of optical
spectra, the H/He ratios are quite low; generally
significantly less than unity. It is not yet known if this
is a reflection of the true chemical composition of these
stars. The ratio of abundances of these elements is often
determined by using measurements of the Hell Pickering
series. Since even quantum number Hell lines occur at
nearly the same wavelength as the hydrogen Balmer lines,
significant amounts of hydrogen could be inferred from an
increase in the strength of the Hell line intensities which
produces a non-smooth Pickering decrement. According to
Smith (1973), an analysis carried out in this manner
resulted in N(H)/N(He) =1-2 for late WN Wolf-Rayet stars
(WNL): namely, WN7 and WN8. She found considerably lower
ratios, N(H)/N(He) =0 for earlier WN types (WNE). These
results indicate that stars which exhibit the Wolf-Rayet
phenomenon may indeed be chemically evolved stars with WNE
types more evolved than WNL types. A similar study was
carried out for WC stars by Rublev (1972) who concluded,
N(H)/N(He) < 0.08. This result must be regarded with a
little more caution than those results for WN types, because
of the severe blending effects of the Hell Pickering lines

with several carbon and oxygen transitions. However, it is
the apparent position of this class of stars on the
5
Hertzsprung-Russe11 diagram which contradicts the
observational evidence for low hydrogen abundances. In
fact, this is the primary reason that Underhill (1982) is
still of the opinion that the low H/He ratios do not
necessarily indicate a true absence of hydrogen: it is still
her assertion that Wolf-Rayet stars are chemically similar
to our Sun with 3 < H/He < 10. The temperatures and
luminosities which have been derived for these stars place
them in the hydrogen - burning band of the Hertzsprung-Russe11
diagram. More specifically, the late type WN stars are
expected to lie near the BO supergiants while the early WN
types and the WC stars seem to reside near the BO giants.
In addition, current stellar models, which have been
calculated for He-rich stars with masses and luminosities
similar to those for Wolf-Rayet stars, indicate considerably
higher effective temperatures than are observed for Wolf-
Rayet stars. One could agree with Underhill, who states
that perhaps certain physical processes occur in the
atmospheres of these complicated stars which may explain the
low amounts of detected hydrogen relative to helium. It is
true that if the temperatures or densities of the atmosphere
are not appropriate, hydrogen emission may not be observed
in the assumed amounts. Hydrogen emission occurs in the
Balmer lines as a result of a recombination event. Such a

6
deionization can occur only if the electron temperatures are
low enough and the density of material is adequate to allow
a sufficient number of recombinations to occur in a small
enough period so that the transitions involved appear in the
Balmer series. Underhill contends that the densities and
temperatures may not be appropriate to allow hydrogen to
emit readily. Additional observational evidence indicates
that this is not the case. Since the x-ray flux is
relatively low, one would infer that the temperatures of
Wolf-Rayet stellar atmospheres are not high enough to allow
hydrogen emissions to occur. Given current theories of the
evolution of hot, massive stars, it is not difficult to
imagine that these objects eject much of the material from
their atmospheres during their long-term evolutionary
development. The spectral observations can be easily
misinterpreted since many of the emission features in the
spectra of Wolf-Rayet objects are either very broad or the
result of blending of several different species.
One representative of the Wolf-Rayet class of stars,
72 Velorum, is located in the southern sky at right ascension
8^09m12s, and declination -47°18' (1988.5). The y2 Vel
system is the brightest Wolf-Rayet star in the sky with a
visual magnitude of 1?76. The star is an intriguing system
for both observational and theoretical astronomers. It has
been the object of many investigations in the last twenty
years or so and is the subject of this dissertation.

7
The y2 Vel system was discovered to be a spectroscopic
binary by Sahade in 1955. The system consists of a Wolf-
Rayet star of spectral classification WC8 and a hot
supergiant component of spectral type 09 (Conti and Smith
1972). Ganesh and Bappu (1967) carried out the first
spectroscopic study in order to determine the orbital
parameters of the y2 Vel system. Using the CIII-IV blend at
4650Á which is assumed to be formed in the envelope of the
Wolf-Rayet component, and the hydrogen absorption line of
the 0 star at 4340Á, they found an orbital period of 78^5.
A more accurate estimate of the period was accomplished by
Niemela and Sahade (1980). They determined a radial
velocity curve using the strong violet-shifted absorption
line, Hel at 3888Á. The position of this feature was
measured relative to the H8 absorption line of the 09
supergiant. They determined a period of revolution of
78^5002 with an eccentricity of 0.40. Moffat et al. (1986)
carried out another radial velocity analysis of the y2 Vel
system in an attempt to define the orbital elements of the
system more precisely. They used radial velocity curves
obtained for several emission features but were unable to
improve upon the 78^5002 period or the 0.40 eccentricity.
However, their refined ephemeris coupled with the period of
Niemela and Sahade, is given by the following equation:
E0 = JD2445768.96 + 78^5002.
(1-D

8
With the aid of equation (1-1), the 09 star is found to be
in front of the WC8 star at phase 0.0 and behind the Wolf-
Rayet component near phase 0.5. According to the study by
Moffat et al. (1986) the epoch of periastron passage is
determined to be JD2445802.6. Although an eclipse has been
detected in the ultraviolet region of the spectrum (Willis
and Wilson 1976) at phase 0.65, no definite eclipse effects
have been recorded at optical wavelengths. Gaposchkin (1959)
reported a 16d2334 period with an amplitude of variation
ranging between 0?19 and 0?13 using visual and photographic
measurements, but to date no other work has been able to
confirm these results.
The masses of the individual components can be
determined from the simple relation
^abs
^em
mWr
M09
(1-2)
where Ka]_)S and Kem are the amplitudes of the absorption and
emission radial velocity curves and Myg/MQg represents the
ratio of the mass of the WC8 star to the 09 supergiant.
According to the study by Moffat et al., the best values for
Kabs and Kem are given by
Kabs = 70 ± 2 km/sec, and
Kem = 130 ± 6 km/sec.

9
These values are in good agreement with independently
determined results by Niemela and Sahade but have smaller
uncertainties. They indicate that the mass of the Wolf-
Rayet companion is approximately 0.54 times that of the 09
star, with minimum masses of 17M for the WC8 component and
32Mq for the supergiant.
In an attempt to measure the radius of the individual
components of -y2 Vel as well as the separation distance of
this binary system, Brown et al. (1970) obtained
interferometry measurements with the stellar intensity
interferometer at the Narrabri Observatory. Their
measurements in the continuum at 4430Á included the effects
of both y2 Vel and 71 Vel, which is located 41" away from -y2 .
However, the analysis is simplified by the fact that the
assumed contribution from 71 Vel is negligible. These data
indicate an angular diameter of the Wolf-Rayet component of
0'.'44 ± 0705 x 10'3, and an angular semi-major axis of the
orbit of 473 ± 075 x 10 3. Using spectroscopic observations
of Ganesh and Bappu (1967), together with these angular
measurements, the distance modulus for the 7 Vel system is
7.7 ± 0.3 or about 350pc. A simple trigonometric
computation yields a radius for the WC8 component of 17RQ. A
more conventional method of measuring HB indices of several
stars in the vicinity of 7 Vel results in a distance of
457pc, and hence, a radius for the Wolf-Rayet star of 22RQ.

10
Since an eclipse in the optical region of the spectrum
has not been detected, the inclination of the system can be
assumed to be approximately 70°. Using the angular semi¬
major axis of the orbit obtained by Brown et al., the
projected semi-major axis is 2.09 ± 0.05 x 1013cm, or
assuming i=70°, 2.22 + 0.13 x 1013cm. These measurements
imply a separation distance of approximately 319RQ.
Brown et al. also obtained measurements of the CIII-IV
emission line of y2 Vel at 4650Á. These data include effects
of both components of y2 Vel in addition to the effects from
the material surrounding the Wolf-Rayet component. The
interferometry measurements give an angular diameter of the
region emitting at the CIII-IV frequency of 2” 0 5 ± 019 x
10'3. If one assumes the 350pc distance established by
Brown et al., these data indicate that the CIII-IV blend is
formed in a region around the WC8 component, and at a
distance of 76 ± lOR^ from that star. Wood (1941) derived a
relationship between the dimensions of Roche equipotential
surfaces, relative to the semi-major axis, as a function of
the mass ratio. Given the described dimensions for y2 Vel,
the radius of the Roche lobe is 83RQ. Hence, evidence
indicates that this emission feature at 4650Á is formed in
the outer regions of the circumstellar material, and in
fact, in material which nearly fills the Roche lobe of the
72 Vel binary system. These and other properties of this
intriguing system are summarized in Table 1-1. In light of

11
the discussion presented earlier, it is interesting to note
that H/He < 0.02 for 72 Vel; certainly much different from
the chemical composition of the Sun.
The y2 Vel system is a very peculiar binary which has
attracted much attention in the last two decades. One of
the most intriguing features of this system is the
indication of possible variability in some of the emission
lines which are present in the spectrum of the Wolf-Rayet
component. It is this primary characteristic which has made
72 Vel the subject of so many photometric and spectroscopic
studies. As early as 1918, variation in the shapes of
certain emission lines were reported (Perrine 1918). West
(1972) was the first author to assert that variation in line
intensities might also exist. The results of 24 hours of
observation with the Orbiting Astronomical Observatory,
0A0-2, indicated no short term variations in the CIII
emission line at 1909Á. Since then however, several authors
have presented evidence of variations in both line profile
and line strength.
Jeffers et al. (1973a), first reported a 6 minute
periodicity in both line intensity and line shape using
photometric measurements of the Hell (4686Á) and CIII-IV
(4650Á) features. A later re-analysis of their data (1973b)
revealed even more rapid fluctuations on the order of 154 ±
3 seconds with an amplitude of only about 2%.

12
Table 1-1
Properties of the 7 Velorum System
Spectroscopic binary WC8 + 091
a)
P =
78^5002
b)
V =
1“76
c )
mwr
* sin3i > 17M
O
d)
M09
* sin3i > 32M
O
e )
q =
0.54
f )
rwr
= 17 ± 3R
0
g)
R0 9
= 76 ± 1 OR
O
h)
a *
sin(i) > 319R
0
i)
H/He
= 0.02

13
Austin, Schneider, and Wood (1973) carried out a very
extensive photometric study of y2 Vel at the Mt. John
Observatory in New Zealand. They used six narrowband
filters with AA = 10Á. Three of these filters isolated the
emission lines identified as Hell at 4686Á, CIII at 5696Á,
and CIV at 5812Á; the other three filters were centered on
the continuum at 4804Á, 5302Á, and 6106Á. These authors
found that the continuum remained essentially constant as
did the triply-ionized carbon feature, at least in the time
interval spanned by their observations. Austin et al.
detected definite night-to - night variations in the strengths
of the CIII and Hell features. The amplitude of variation
for the doubly-ionized carbon line amounted to about 0I?12.
Fluctuations of also occurred in a time period of less
than 2 hours. The singly ionized helium feature exhibited an
increase in brightness of 0â„¢06 in less than 90 minutes, but,
Austin et al. also note that this line appears to remain
quite stable (i.e., within 0^02) for relatively long time
intervals. Austin et al. carried out a more intensive y2 Vel
observing program, concentrating only on variations in the
Hell line. Their results were inconclusive; y2 Vel exhibited
no detectable variations in a 20 minute time period on the
first night of observation. On the second night, however,
the ionized Hell line increased in brightness by as much as
O’PlO in 20 minutes, and again later that night, brightened
by the same amount in only 2 minutes. In summary, Austin

14
et al. confirmed previous results that rapid variations can
occur, although not with the rapidity of the degree reported
by Jeffers et al. They found these variations to be
temporal and not periodic.
Another photometric study carried out by Lindgren
et al. (1975) did not provide evidence for stable variations
between 1 and 10 minutes in length, in either the CIII-IV
line at 4650Á or the Hell line at 4686Á. However, Lindgren
et al. did confirm nightly variations in emission line
strength, in particular, a 0^03 to 0?05 change in the 4650Á
feature. The observations that were utilized for this
investigation covered an interesting phase of 7 Vel: that is,
when the 0 star was in front of the Wolf-Rayet component.
These results may indicate an eclipse of some part of the
supergiant by a portion of the circumstellar material
associated with the WC8 star.
Bahng presented results of several spectrophotometric
studies of the y2 Vel system, studying the short term
variations in emission line strengths using a photoelectric
spectrum scanner in the spectral range 4600Á to 4720Á. In
1973, Bahng reported on results of spectrum scans which were
acquired on four different nights. The emission lines of
particular interest were CIII-IV (4650Á) and Hell (4686Á).
Although a power spectrum analysis did not yield evidence of
a periodic phenomenon, Bahng did find variations in the
equivalent widths of these lines. These changes amounted to

15
a 21 variation for CIII-CIV, and a 4% variation for Hell and
occurred on time scales of 4 to 20 minutes. Later, Bahng
(1974) recorded a 6 hour spectrophotometry observation in
which variations were detected between 10 and 20 minutes.
Again, however, no predominant periodicity was found. In
his 1975 paper, Bahng analyzed spectroscopic scans which
were obtained in 1973 and 1974. He compares the theoretical
rms deviations computed from photon counting statistics with
the rms deviations of the measured equivalent widths of the
emission lines of interest. Since the rms deviations of the
data exceed the theoretical rms deviations by more than a
factor of three, Bahng considers these variations to be
statistically significant, indicating that real short term
variations in the emission lines of the y2 Vel system do
exist. Although there is no predominant periodicity, a
power spectrum analysis indicates considerable power near a
frequency corresponding to a period of about 1 minute.
Using integrated magnitudes, Bahng finds evidence of
variation with a s emi - ampl i tude of O'iOl with a periodicity
of 339.5 seconds. In addition, a significant peak at
16.2 minutes was also present in the data.
We have presented only a brief summary of some of the
more notable observational studies of y2 Vel which have
appeared in the literature over the last few years. As
pointed out by Haefner et al. (1977), ultra-short period
variations (i.e., those on the order of minutes) seem to be

16
detected only in observations which do not exceed much more
than 30 minutes in length. Fluctuations of this nature are
not generally detected in longer runs of continuous data,
but variations in both emission line strength and profile
are almost always detected on a nightly basis.

CHAPTER 2
INSTRUMENTATION AND DATA ACQUISITION SYSTEM
Instrumentation
The data used for this research were obtained with the
automated optical telescope located at the Amundsen-Scott
South Pole Station on Antarctica. This instrument is a two-
mirror siderostat zenith telescope with an f/6, 7.8cm
achromatic lens (Figure 2-1). As one would expect, certain
special design considerations are necessary to ensure normal
operation in the extremely 1ow-temperature, harsh
environment which prevails at the South Pole. A specially
insulated building measuring 12' x 8' x 8' was constructed
on site by carpenters of ITT Antarctic Services during the
1985 austral summer. The telescope occupies an 8' x 8'
section of the building, while the computer system and
control electronics are housed in an adjacent room measuring
4' x 8’. This smaller room can be completely isolated from
the telescope room and heated when necessary. Under normal
circumstances, the telescope room temperature is maintained
at approximately -20°C.
The optical head of the telescope contains two
optically-f1 at front-surfaced mirrors which act to redirect
17

18
Figure 2.1 South Pole Optical Telescope.

19
incoming light onto the objective lens and eventually to the
photometer. It is this section of the telescope which
protrudes from the roof of the building and is therefore
exposed to ambient conditions. In order to prevent blowing
snow and other sources of moisture from entering the optical
head, a transparent window is used to completely seal the
interior of the telescope. During the 1986 observing
season, the formation of frost on this window and on the
mirrors in the optical head interfered with some of the
observations. Since this time, however, upgrades to the
system have been installed and all of the optics are
maintained frost-free. This was accomplished by pumping
existing dry polar air (the absolute humidity is equivalent
to about 0.15mm of precipitable water vapor) through a
cylindrical tube containing a desiccant, and into the
optical head of the telescope. Here, the air is heated so
that a temperature gradient of about 5°C is maintained
between the air in the optical head and the air immediately
outside of the entrance window, with the inside air being
the warmer.
Of the entire design of the telescope, the only moving
part which is directly exposed to ambient conditions is the
elevation axle. The gear system which moves the telescope in
the east to west direction, also moves the entire telescope
tube in both azimuthal directions. The azimuth-motion
resides well within the telescope room, and a worm gear

20
drive is controlled by a stepper motor assembly. This same
technique provides motion to the field stop and filter
wheels. With the siderostat design, the elevation motor
must move only the optical head in the north to south
direction, and it is therefore necessarily mounted outside
on the optical head assembly. The elevation axis joint is
located between the two portions of the optical head which
house the two optically-flat mirrors. This "up and down"
motion of the telescope is provided with an assembly of
teflon ball bearings.
The optical design of the telescope (Figure 2-1) is
straight forward. Incoming light rays strike the objective
after being deflected by the two front - surfaced mirrors in
the optical head; each is positioned at about 45° with
respect to the normal. As the photons emerge from the lens,
they pass through a diaphragm, a filter, and a fabry lens
which images the objective of the telescope onto the
photomultiplier. It is the photometer which measures the
signal and converts it into an equivalent number of photons.
The South Pole Optical Telescope (SPOT) is equipped
with several diaphragms and filters allowing the selection
of any one of a number of combinations depending upon the
brightness of the source, the tracking rate of the
telescope, and the type of observation desired. The current
system has diaphragms which restrict the field of view to
1°, 5', 2', or 1' regions of the sky.
The 1 field of view

21
is used only in the initial star pointing procedure while
the 5', 2', and 1' diaphragms are used in the star centering
procedure and for the actual measurement of the object's
brightness; the smaller the diaphragm, the smaller the
contribution from sky background, and the fainter the
limiting magnitude of the telescope. The filters which are
currently mounted in the filter wheel include the Johnson
standard B and V filters, a neutral density filter, and four
narrowband filters. The neutral density filter is a ND5 and
is used in the star find procedure. This filter's sole
purpose is to prevent the telescope from observing a source
which is too bright: that is, one which could saturate, and
subsequently damage the photomultiplier. Of the narrowband
filters, one is used to monitor auroral and/or sky
variations. This filter is centered on the very strong 01
emission line at 5577Á. The remaining three narrowband
filters were chosen specifically for our research on the
Wolf-Rayet star, 72 Vel. Two of these filters isolate
emission features in the spectrum of Vel; the Hell
emission line at 4686Á and the CIII emission line at 5696Á.
The third filter isolates the continuum region at 4768Á.
More specific details of each of these filters are presented
in Table 2-1. Also, the response curves for six of the
filters are presented in Figures 2-2 through 2-6. In
addition to the filters, one position on the filter wheel
contains a mirror which directs the light beam perpendicular

Table 2-1
Filter Specifications
Peak
Wavelength
(A)
Half-Power
band-width
(A)
Integration
time
(secs)
Purpo s e
4400
900
2
B
5500
900
2
V
4686
32
4
Hell
5696
32
8
cm
4768
92
2
Continuum
5577
100
8
01

Transmittance
o
Wavelength ( A )
Figure 2-2 Response curves for the U (O), B (A), and V (0)
filters.
CO

Transmittance
50
40
30
T
o
o
o
o
o
o
20
10 -
0 —
4640
o
o
o
4660 4680
o
Wavelength ( A )
o
4700
4720
Figure 2-3 Response curve for the helium emission filter.
â– F-

Transmittance
60
50
40
o
o
o
30
o
20
o
10
o
0 l—
5650
5670
1 i
5690 5710
o
Wavelength ( A )
o
o
o
i
o
5730
5750
Figure 2-4 Response curve for the carbon emission filter.
ro
Ln

Transmittance
60
50
40
30
20
10
o
5500 5550 5600 5650 5700
1—
1—
1—
1
o
o o
o
o
o
o
o
-
o
o
-
o
—
o
o
-
-
o
o
-
o
o
o
1_
1
1
1
o
Wavelength ( A )
Figure 2-5 Response curve for the oxygen emission filter. ^
ON

Transmittance
o
Wavelength ( A )
Figure 2-6 Response curve for the continuum filter.
ro

28
to the optical axis where it can be viewed through an
eyepiece. This section of the telescope is maintained at a
temperature below -20°C in an effort to reduce the dark
current in the system. During the three years in which the
telescope has been operating, this level has remained nearly
constant at 3.7 counts/sec. Other regions of the telescope
are also maintained at optimum temperatures according to a
thermal design analysis carried out by Esper (1986). These
temperatures are achieved with thermal insulation, heaters,
and residual heat generated by stepping motors and other
electronic components. The entire thermal design of the
telescope works in concert with ambient temperatures ranging
from -40°C to -80°C.
Data Acquisition
The SPOT system is controlled with a modified Dynatem
RM-65 computer system with an 8-bit 6502 microprocessor.
The computer system controls the motions of the telescope in
azimuth and elevation, as well as the motions of the
component parts including the field stop and filter wheels.
The necessary special functions and interfacing of the
telescope are provided by custom built circuits supplied by
the electronics shop at the University of Florida. The
software is written in 6502 assembler language and FORTH.
FORTH is a computer language designed specifically for
instrument control processes. As mentioned previously, the

29
computer system is located in the SPOT building. However,
in some cases, the telescope can be commanded from a remote
terminal. The data storage media which are currently
employed are 8-inch floppy disks.
During an observing session, photometric and
engineering data are accumulated until a one kilobyte region
of memory is filled. At that time, the buffered data are
written onto a disk file. At the beginning of each hour,
the telescope suspends the acquisition of photometric data
and records the status of various aspects of the telescope.
Such things as the azimuth and elevation positions of the
telescope, the Julian Date and Sidereal Time, and the
temperatures of various sections of the telescope are stored
as engineering records. After these data are obtained, the
telescope resumes its observing program. Observing programs
instruct the telescope with respect to pointing, diaphragm,
appropriate filtering, and number of seconds to integrate
for each filter. In addition, in the case of stellar
photometry, the telescope is given the number of steps and
directional information for sky readings. Observing
programs can be modified, deleted, and added from the remote
terminal. Depending upon the observing program and sky
conditions, a given floppy disk may fill in a week to ten
days. At this time, the winter-over scientist changes the
disks, copies the contents of the filled disk to tape for
archival purposes, and transmits the data to a mainframe

30
computer in Malabar, Florida, via the ATS-3 satellite
communication link. These data are retrieved from the VAX
system in Malabar, and stored on the ATS-VAX at the
University of Florida where the reduction of the data
proceeds.

CHAPTER 3
DATA REDUCTION TECHNIQUES
During the 1986, 1987, and 1988 observing seasons,
SPOT obtained data for several different research projects.
In order to optimize telescope time and to make the best use
of clear skies, the data for more than one research project
were usually acquired within the same observing program.
The research project that is the subject of this
dissertation has two major objectives: 1) the search for
variations in emission line strengths using the Hell, CIII,
and continuum filters; and, 2) the search for eclipse
effects requiring observations in the B and V filters, as
well as integrations on the comparison star, HR3452. In
some observing programs, both y2 Vel and HR3452 were observed
using all available filters. In other cases, an increase in
the time-resolution of the data was accomplished by removing
HR3452 from the observing program and omitting the B and V
filter observations. Therefore, several different observing
sequences were required to obtain the data necessary for
this investigation. Specific details of these programs will
be presented in Chapter 4 where individual data sets are
discussed. In general however, the data acquired for this
project were obtained with the use of the 5' diaphragm in
31

32
conjunction with the B, V, Hell, CIII, 01, and continuum
filters .
As one would expect, the photometric integration times
necessary to obtain adequate signals for a given star and
filter varies from star to star, and from filter to filter.
The required integration length depends on a particular
star's color and its distance from the zenith. For example,
to obtain comparable deflections in B and V filters for a
red star, it is necessary to integrate for a longer time
period in the blue region than in the yellow. In addition,
the increased scattering of blue light at large zenith
distances requires the use of longer integration times for
the blue filter than for the visual filter. The optimum
integration times for the stars and filters used in this
study are based on this knowledge, as well as experience
that was acquired during the early portions of the 1986
observing season. Integration times are presented in
Table 3-1.
Since a star does not wholly fill the observing
diaphragm, each stellar deflection is actually the sum of
the star and surrounding sky emissions. In order to obtain
a net star - reading, it is necessary to remove an appropriate
value for the sky intensity from each of the stellar
measurements. In the case of a single channel photometer,
this is easily accomplished by integrating on a region of
sky which is devoid of stars within the limitations of the

33
Table 3-1
Optimum Integration Times for y2 Vel and HR3452
Star Filter Integration time
(secs)
72 Vel B 2
V 2
Hell 4
Cl 11 8
Continuum 2
01 8
HR3452 B 10
V
10

34
telescope but which is reasonably close to the star under
observation. In an automated telescope system, it is
necessary to make certain that both of these criteria are
satisfied. A search of the Palomar Sky Survey (1954) prints
indicates that a displacement of 15' in the positive azimuth
direction from the star is an appropriate displacement for a
typical sky reading. In very general terms, each stellar
deflection is accompanied by a corresponding sky deflection
which is used in the data reduction procedures.
During the observing season, data are periodically
transmitted from the South Pole to the VAX 11/750 computer
at the University of Florida. Each of the programs used in
the reduction and analysis of the y2 Vel photometry was
written in Fortran-77. In order to optimize computer run¬
time, the data were initially passed through a program which
unpacked the data from hexadecimal to decimal format and
divided the data into two files: an engineering file
containing information relating to the design and operation
of the telescope, and a separate photometric file containing
the data relating to the stellar deflections. The raw
stellar deflections were then reduced to photometric
magnitudes using standard reduction techniques (Henden and
Kaitchuck 1982). Since each deflection is recorded as a
counts-per-1ime quantity, and since an integration time
varies from filter to filter, it is necessary to convert all
measurements to the same unit; a counts per second quantity.

35
Once this is done, the next step is to remove the sky
background from each stellar measurement. This is
accomplished through a simple linear interpolation of the
sky readings to the time of the star reading. The number of
counts (per second) due to the background are then
subtracted from the total counts (per second) due to the
star. In the case of the B and V data, HR3452 was chosen as
the comparison star. HR3452 is very close in color to y2 Vel
with a spectral classification of B1V and is located only
24' from the variable. For these reasons, and the fact that
HR3452 has not been reported to be variable, it meets the
criteria for a reasonably good comparison star. A second
linear interpolation scheme is then used to determine the
intensity of the comparison star, HR3452, at the time of the
variable star reading. Finally, equation (3-1) is used to
give the differential photometric magnitude
Am = -2.5 * log
(3-1)
where Am is the differential magnitude between y2 Vel and
HR3452 , and Dv and Dc are the deflections (in counts per
second) of y1 Vel and of HR3452, respectively.
A major portion of this research is the search for
periodic intensity variations in the Hell and CIII emission
lines of y2 Vel. One technique which is commonly employed in

36
period determination analyses uses the method of Fourier
transformation of time and intensity measurements. In
particular, Deeming's method (1975) of Fast Fourier
transforms is well suited for data which are unequally
spaced in time. In general, Fourier transform analysis is
used to transform data from the time domain into the
frequency domain, or vice-versa. For instance, a continuous
function of time, f(t), can be transferred into frequency
space according to the formula
F(j/) = FT [ f ] = f(t) * exp (-2n\ut) dt. (3-2)
The inverse Fourier transform is given by
f(t) = FT[F] = F(v) * exp (2wii/t) du. (3-3)
In equation (3-2), F(t/) represents a spectrum of
frequencies. Each peak in F(v) corresponds to the relative
strength of each frequency in the data, f(t). In most
cases, and in this data, f(t) is not a continuous function,
but rather a sampled-set determined at discrete times, t^.
Hence, equation (3-2) can be re-written as a discrete
transf orm
N
F(v) = 2 E [ f (t) * exp (-2?rivt)],
1
(3-4)

37
with N equal to the number of data points in f(t). The
periodic exponential function in equation (3-4) indicates
that the Fourier transform of f(t) has both real and
imaginary components. Since it is the power as a function
of frequency, rather than the Fourier transform that is of
interest, the derivation must be carried one step further.
If f(t) is assumed to be real, then F(v) must satisfy the
relation
F( -«/) = F* («O,
(3-5)
where F* (u) is the complex conjugate of F(y). Then, according
to Rayleigh's theorem which states:
| f(t) |2 dt
(3-6)
we can define the power, P(i/), as the product of F(i/) and its
complex conjugate, which leaves only the real quantity
P (i/) - | F(i^) |2 = C2 (v) + S2 (i/).
(3-7)
C(i/) and S(y) in equation (3-7) are the cosine and sine
components, respectively, and are defined as
C(„) = ^ £ [f(t) * cos (2jn/t)], and
(3-8)

38
S(i/) = | Z [f(t) * sin (2ni/t) ] . (3-9)
Since the data used in this investigation are acquired
over a finite span of time and are not sampled at equal time
intervals, the Fourier transform is contaminated with the
sampling function. This spectral window or "beam", as it is
frequently referred to, is given by the relation,
W(i/) - G2 (u) + H2 (j/) . (3-10)
In (3-10), G(u) and H(i/) are defined in the following manner:
G(y) S [cos (2ni/t) ] , and (3-11)
H(y) - | S [sin (2ni/t) ] . (3-12)
According to equations (3-10) to (3-12), the spectral
window function is normalized so that W(0) = 1. The Fourier
convolution theorem states that the Fourier transform of the
signal, FT[f], is the convolution of the data with the
sampling function (i.e., beam). The fact that the data are
unequally spaced over a finite time interval has several
consequences. The sampling period of the data puts a
constraint on the lowest frequency which is resolvable in
the data. The frequency resolution in a discrete data set
corresponds to the width of the beam (W) at u = 0 and is

39
related to the sampling period by equation (3-13)
Su « 1/T, (3-13)
where T is the length of the data set. Therefore, the
minimum frequency which is retrievable from a given data set
is given by the Nyquist frequency
«'min “ 1 / ( 2 AT). (3-14)
It is the data spacing which puts an upper limit on the
frequency that can be recovered from a given data set. In
the case of a function consisting of points spaced at equal
time intervals At, the maximum frequency is given by the
Nyquist frequency. Since the data used in this study are
unequally spaced in time, At is not a constant. For purposes
of this investigation, it was deemed appropriate to
calculate the average time interval for a given data set and
use that value for At, according to equation (3-15):
«'max “ l/(2At). (3-15)
In addition to upper and lower limits being placed on
the frequencies which can be recovered from the data, the
data are also not represented by a continuous function of
time. The discrete nature of the data contributes several

40
features to the spectral window (beam) which adds
considerable complexity to the nature of the spectrum.
Since the Fourier transform of the data is a convolution of
the spectrum with the window, the resulting "dirty" spectrum
may be subsequently contaminated with spurious sampling
features. This aliasing can be masked as either a damping
of real features or an enhancement of "false" features,
making interpretation difficult. For this reason, a one¬
dimensional deconvolution (CLEAN) algorithm (Hogbom 1974)
has been adopted for use in this study. The CLEAN algorithm
is that used by Roberts, Lehár, and Drehar (1987) which
deconvolves the sampling function from the dirty spectrum to
give a better representation of the true spectrum. This
deconvolution is accomplished in the following manner:
First, the largest peak in the dirty spectrum is located and
stored. The spectral window (beam), is then superimposed
onto the dirty spectrum so that the main peak in the beam
coincides with the largest peak in the dirty spectrum. A
given percentage of the spectral window is subtracted from
the corresponding dirty spectrum. The peak of the spectrum
which is removed from the data spectrum is stored as a CLEAN
component. The spectrum which results after subtracting the
spectral window is also stored in a residual file. This
procedure is repeated for a given number of deconvolutions
(peaks). A gaussian representation of the spectral window
is then convolved with each CLEAN spectra. Finally, in

41
order to preserve the noise which existed in the original
spectrum, the residual values are added back to the
convolved CLEAN components giving the CLEAN spectrum. The
input parameters for the CLEAN algorithm include the number
of CLEANS to be performed (peaks) and the gain. It is the
gain which governs the percentage of the spectral window
which is removed from the dirty spectrum during each CLEAN
ite ration.
The CLEAN procedure is used to improve the strongest
features of a spectrum and is of limited use in data which
may have a small signal-to-noise ratio. In fact, experience
has shown that over-CLEANing data can actually introduce
spurious features in the clean spectrum. Figures 3-1 to 3-6
are used to demonstrate the hazards that can result from the
improper use of this algorithm. Figures 3-1 and 3-2
represent the spectral window, and the dirty spectrum,
respectively, for a sample data set. Figures 3-3 to 3-6
represent the clean spectra after 10, 100, 500, and 1000
iterations of CLEAN. It is quite clear from these graphs
that the peak at t/ = 40 (cycles per day) in Figure 3-3
decreases in strength as the number of CLEANS increases,
while the peak at i/ = 235d 1 gets progressively stronger.
In fact, in Figure 3-6 this is the only peak which remains
in the spectrum with considerable amplitude. It is likely
that a misinterpretation of the data would result in this
type of over-application. In this study, care has been

Relative Power
Frequency ( d -1)
Figure 3-1 Spectral window function for the test data set.

Relative Power
Frequency (d 1)
Figure 3-2 Power spectrum for the test data set before
execution of the clean algorithm.

Relative Power
Figure 3-3 Power spectrum for the test data set after
10 iterations of the clean algorithm.

Relative Power
Figure 3-4 Power spectrum for the test data set after
100 iterations of the clean algorithm.

Relative Power
Figure 3-5 Power spectrum for the test data set after
500 iterations of the clean algorithm.

Relative Power
Figure 3-6 Power spectrum for the test data set after
1000 iterations of the clean algorithm.

48
taken to minimize the introduction of such spurious features
into the power spectrum. This has been accomplished by
restricting the CLEAN procedure to data above the 3crlevel.
In addition, more stability is provided in the CLEAN
algorithm through use of a smaller gain size and more
iterations. In our case, a gain parameter of 0.25 was
adopted.
Because of the possible uncertainties of the power
spectrum analysis, an independent method of period
determination was also used. This method is based on
fitting models to the data with the aid of the method of
1east-squares. The intensity variations which may exist in
the y2 Vel system are probably much more complex than that
which can be described as a simple cosine or sine function.
However, the main purpose of this research is to use the
long, continuous observational runs of y2 Vel to establish
whether variations in the strengths of the Hell and CIII
emission lines do, in fact, occur. It is not necessarily our
purpose to mathematically describe such variability
precisely. A shift in phase could certainly cause a poor
fit to the data. In any event, the variations which would
exist in such a complicated system are probably aperiodic at
best. So, for purposes of this investigation, a first-
order sine function of the following form is assumed:
D(tj_) = A0 + At * sin
P
(3-16)

49
In equation (3-16), D(t^) represents the calculated
deflection at time t^, A0 is the baseline in intensity
units, Aj is the amplitude of the sine wave expressed in
intensity units, t0 is the "phase" or time of maximum, and P
is the period. It should be noted at this point that P is
expressed in days divided by In radians, a representation
that was chosen so that when one complete cycle had elapsed,
that is, t¿ - t0 - P, then cos (t¿ - tQ)/P = 1. In the
1east-squares procedure, four parameters are adjusted. They
are: A0, A1 , t0, and P. A first-order Taylor expansion is
computed using the partial derivatives of equation (3-16)
with respect to each of these variables. The routine used
in this context is that of Banachiewicz (1942) and uses
Cracovian calculus. If the errors which were calculated in
the 1 east - squares routine exceed the convergence criteria,
corrections are computed and added to A0 , A , t , and P to
give the new quantities:
Ao + AA0,
A, + AA],
t0 + At0, and
P + AP.
The convergence criteria are the following:

50
E(A0) < 1 x 10'\
E(At) < 1 x 10\
E(t0) < 5 x 1 O'3. and
E (P) < 5 x 10'4.
In these relations, E represents the difference between the
new and old values for each quantity considered.
If all of the parameters do not converge within 30
iterations, it is assumed that a satisfactory model could
not be found within the given input parameters. In the case
of the Hell photometry, an initial intensity-amplitude of
0.01 and a baseline of 0.35 were used. For the CIII data, an
amplitude of 0.01 and a baseline of 0.10 were used as
initial inputs. These two parameters were not as sensitive
to small variations as were the period and phase. To account
for the increased sensitivity in P, an initial and final
period were entered along with a desired incremental step
value. Upper and lower limits on the period assumed two
samples per cycle in accordance with the Fourier transform
analysis. To account for increased sensitivity (to small
perturbations) in the phase parameter, a similar approach is
adopted. That is, a desired increment in the phase is
entered as an input parameter. In every case, the time of
the first observation was used as an initial "guess" of the
value for t0. The parameter was incremented from this
initial value to the initial value plus 2n.

51
Weather reports are recorded at the South Pole every
six hours. Photometric data recorded between two
consecutive periods of reported clear skies were used in
this investigation of ~/2 Vel . In addition, since the
telescope does not have the capability to monitor the star
and nearby sky simultaneously, data recorded during times of
auroral activity were omitted from the study. These
omission criteria resulted in the selection of seven
individual data sets obtained during the 1986 austral
winter. Individual observations in each data set were
omitted if the observation in the continuum filter and the
observation in one or more of the emission line filters
deviated by more than 10%. The analysis of the data sets
that resulted is presented in the following chapter.

CHAPTER 4
DATA ANALYSIS
Spectral Photometry of the Hell and CIII Emission Lines
Data Set I.
This data set includes observations of y1 Vel that
extend from JD2446577.6847 through JD2446577.7792. These
(81) data-points were obtained for each of the helium,
carbon, and continuum filters with an average spacing
between successive readings with a given filter of 0^0012.
Both y2 Vel and HR3452, the comparison star used to reduce
the blue and visual photometry, were observed. The exact
sequence of observations is presented in Table 4-1. In the
table, an "X" is used to indicate that an integration was
accomplished with the corresponding filter listed at the top
of the table.
These data were reduced to a ratio of intensities
using methods described in Chapter 3. The data were first
transformed into frequency space using Deeming's method of
Fast Fourier Transforms and assuming a minimum of two
samples per cycle. Given the data spacing of 0^0012 and the
2.3 hour time interval spanned by this data, the minimum and
maximum frequencies which can be retrieved from the data set
with a reasonable amount of confidence, are i/m^n = lid'1 and
"max = ^l^d'1. Figure 4-1 represents the sampling
5 2

53
Table 4-1
Observing Program 1
Star B V He C 0
C*1
CS2
C*
V*3
V*4
V*
V*
vs
v*
v*
v*
v*
c*
CS
c*
V*
v*
v*
V*
v*
v*
V*
v*
V*
v*
v*
v*
X X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X X
X X
X X
XXX
XXX
XXX
XXX
XXX
XXX
XXX
XXX
XXX
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
1 C* = comparison star
2CS = comparison sky
3V* = variable star
4 VS = variable sky
Continuum
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X

Relative Power
Figure 4-1 Spectral window function for data set I.
Ln

55
function for both the helium and the carbon observations.
The peaks at u = 52, 158, 364, and 470d'1 are the odd
harmonics ¡/^, 1/3, V5, and v-¡, and the peak at 1/ = 318d'1 is the
even harmonic, i/g. The convolution of the helium data with
the spectral window function gives the spectrum in Figure
4-2. The noise level is denoted in the lower left-hand
portion of the plot at 5.16 x 10'6. All features to the
right of the arrow satisfy the Nyquist criterion of a
minimum of two samples per cycle. The peaks at v = 170, 323,
and 377d_1 represent statistically significant features at
the 4ct^j level. A closer look at the spectrum reveals that
the peak at 1/ = 377d 1 is part alias with that at v = 323d'1
and the peak at v = 170dl is part alias with a peak at
v = 8d_1. This last feature is shown in Figure 4-2 but since
it does not satisfy the Nyquist criterion, it is discarded
in this study. Spurious features at or above the 3a^j level
are removed from this spectrum with a deconvolution process;
the resulting power spectrum is depicted in Figure 4-3.
Here, we see that the peak at v = 323d'1 is the only feature
which remains above the 4ct^ level, at 4.25a^. This
frequency corresponds to a period of 4.5 minutes. According
to the least - squares routine, the best fit to the data,
within the limitations of the Nyquist criterion, occurs for
P = 4.4 minutes, and is described by:

Relative Power
Figure 4-2
Power spectrum for the helium observations of
data set I before execution of the clean
algorithm.

Relative Power
Figure 4-3 Power spectrum for the helium observations of
data set I after execution of the clean
algorithm.

58
D(tt) - 0.35099 + 0.00813 * sin
r t± - 0.18518 *\
0.00049 J
(4-1)
The model computed from this equation is presented in
Figure 4-4 and seems to fit the observations quite well.
However, because there are only an average of 2.5 points per
cycle, one should regard this result with some skepticism.
A visual inspection of a plot of intensity as a function of
time indicates that perhaps the data are changing at a more
gradual rate. The least - squares routine was executed for
periods on the order of the time interval spanned by the
data set; convergence occurred for a period of 2.40 hours.
The model , given by
D(tt) = 0.34986 + 0.00724 * sin
r t± - 0.10507 -x
^ 0.01592
(4-2)
is graphically presented in Figure 4-5. The amplitude of
the variation of this model is only slightly lower than that
for the more rapid fluctuations depicted in Figure 4-4.
The power spectrum for the carbon data obtained in
this time interval is shown in Figure 4-6. Again, the arrow
in the lower left of the plot represents the noise level at
1.287 x 10'*. It should be noted that the peak at v = 70d_1
is part alias with that at u = 20d'1 and should, in
principle, be removed with the CLEAN algorithm (Figure 4-7).

Hell : Continuum
0.400
0.380
0.360
0.340
0.320
0.300
0.180 0.200 0.220 0.240 0.260 0.280
Julian Date (2446577.5+)
Figure 4-4 Model of the variations of the helium data for
data set I with P - 4.43 minutes. Each plotted
point represents the average intensity within a
0^0040 interval.
Ln

Hell : Continuum
Figure 4-5 Model of the variations of the helium data for
data set I with P = 2.40 hours. Each plotted
point represents the average intensity within a
0*?0040 interval.
CTs
O

Relative Power
Figure 4-6 Power spectrum for the carbon observations of
data set I before execution of the clean
algorithra.

Relative Power
Figure 4-7 Power spectrum for the carbon observations of
data set I after execution of the clean
al gor i thm .

63
This is indeed the case, and the only peak which survives
the CLEANing process is that at v = 20d‘1 corresponding to
0^050 ( 1.20 hours). The signal-1o - noise ratio for this peak
is quite large at 8.04ctjj. The least - squares routine
converges to a set of quantites with a similar period,
P = 0^0459 (1.10 hours), according to the following
relation:
D(tt) = 0.11117 + 0.00231 * sin
r tL - 0.45362
L 0.00731
(4-3)
The fit to the data using equation (4-3) is presented in
Figure 4-8. Although the amplitude is quite low, this
period may be considered as a satisfactory representation to
the fluctuations since both the power spectrum analysis and
the 1east-squares routine give similar results.
In summary, the helium data seem to follow two
different periods. The shortest period of about 4.4 minutes
represents very rapid fluctuations and was arrived at by
both the power spectrum analysis and the 1east-squares
solution. In addition, models were computed for periods
corresponding to the length of the data set with the best
fit solution having a period of P = 2.40 hours. Both fits
seem to be quite satisfactory. The intensity variations of
the CIII emission line, although low in amplitude, can be
modeled with a period of 1.10 hours.

Clll : Continuum
0.165
Figure 4-8 Model for the variations of the carbon data for
data set I with P - 1.10 hours. Each plotted
point represents the average intensity within a
0.0040 interval.

65
Data Set II.
One-hundred and eighty-nine observations of
72 Vel were obtained in the helium and carbon filters using
the observing sequence presented in Table 4-2. The time
interval spanned by these data extends from JD2446606.6048
through JD2446606.9553. The average spacing between
successive observations for a given filter is 0“0019. The
decrease in time-resolution of these data relative to the
time-resolution of the data discussed in the previous
section, is an artifact of the new observing program. As
can be seen from Table 4-2, several new objects have been
added to the original observing program (Table 4-1). This
ultimately limits the high frequency components which are
recoverable in the data analysis.
The spectral window for these data is given in Figure
4-9, and includes first, second, third, and fourth harmonics
at v = 23, 43, 64, and 84d'1, respectively. In general, as
in this case, spectral windows for unequally sampled data
tend to be more complex at high frequencies. The power
spectrum of the helium data is computed for frequencies
between i/m^n = 6d* and i/max = 265d'1 (Figure 4-10). The
noise level in this figure is denoted in the lower left-hand
portion of this plot at 9.976 x 10'7. Given this level of
noise, the peaks at v = 16, 22, 42, and 218d_1 might be
classified as statistically significant. However, when one
superimposes the spectral window on the power spectrum, it

66
Table 4-2
Observing Program 2
Star B V He C 0 Continuum
V*1 X X
V* X X
V* X X
V* X X
V*
V*
V*
V*
VS2 X X
V*
v*
v*
v*
V* X X
V* X X
V* X X
V* X X
C*3 X X
CS4 X X
C* X X
V* X X
C* X X
CS X X
C* X X
V* X X
V* X X
V* X X
V* X X
V*
V*
v*
v*
VS X X
V*
v*
v*
V*
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X'
X
X
X
X
X

67
Table 4 - 2 --continued
Star B V He C 0 Continuum
V* X X
V* X X
V* X X
V* X X
C* X X
CS X X
C* X X
Sky B5 X X
a Gru X X
a Gru S6 X X
a Gru X X
fi Gru X X
B Gru S' X X
B Gru X X
Sky A8 X X
X
X
X
X
X
X
X
XX X
XX X
XX X
XX X
X
X
X
X
X
X
1 V* = variable star
2VS = variable sky
3C* = comparison star
4CS = comparison sky
5 Sky B = 1° region of sky
6 q Gru S = a Gru sky
7 B Gru S = B Gru sky
8 Sky A = 1° region of sky
180° from variable star
180° from Sky B

Relative Power
Figure 4-9 Spectral window function for data set II.
a\
co

Relative Power
1.0E-5 -
7.5E-6
5.0E—6 ■
2.5E-6 •
0.0
50
100
150 200
Frequency ( d “1)
250
Figure 4-10 Power spectrum for the helium observations of
data set II before execution of the clean
algorithm.

70
is apparent that the peaks at v = 22d_1 and v = 62d-1 are part
aliases of the considerably stronger peak at v = 42d l .
The CLEAN algorithm has been used in an attempt to eliminate
these spurious features, and the results are shown in Figure
4-11. Now, only two peaks have survived the CLEANing
process while remaining statistically significant. They
include the feature at u = 42d_1 (P = 0^0238 ) with a signal-
to-noise ratio of 7.7crjq and the feature at v = 238d-1
(P = 0^0042) which has a weaker signal of 4.5a^.
Although the least-squares routine converges for
periods corresponding to each of these frequencies, the
largest amplitude of variation and the lowest observed minus
calculated (0-C) residuals occurs for P = 0^0476 (¡/ = 21d').
It is interesting to note that this period is exactly twice
the period concluded from the power spectrum analysis and
perhaps this is an indication of the presence of harmonics
of a fundamental period. In order to determine whether or
not one of these periods fits the data more satisfactorily
than another, models are computed for each. Results
indicate that these data are most adequately represented by
P = 0^0476 (Figure 4-12) according to the equation
DUi) = 0.34430 + 0.00379 * sin
r - 0.48459
^ 0.00757
(4-4)
Perhaps this period reflects the fundamental mode of

Relative Power
0 50 100 150 200 250
Frequency ( d “1)
Figure 4-11 Power spectrum for the helium observations of
data set II after execution of the clean
algorithm. -J

Hell : Continuum
Julian Date (2446606.5+)
Figure 4-12 Model for the variations of the helium data for
data set II with P - 1.14 hours. Each plotted
point represents the average intensity within a
0^0040 interval.

73
variation while the prominent feature in the power spectrum
may represent the first harmonic. The carbon data have also
been analyzed in a manner comparable to that used in the
analysis of the helium data. The dirty and clean spectrums
are presented in Figures 4-13 and 4-14, respectively.
Although the feature at v = 42d'L is considerably weaker than
the corresponding peak in the power spectrum for the helium
data, it still remains the strongest feature in the plot at
5.8cjfj where = 1.680 x 10'7. Three other peaks rise just
above the 4a^j level; they occur at
¡/ = 12, 57, and 204d'1.
The 1east-squares routine is executed for periods
ranging between 0^0038 (v = 263d1) and o4l759 (v = 6d').
Solutions with periods corresponding to each of the
frequencies, found from the power spectrum analysis, result.
Once again, the strongest convergence occurs for a period of
0^0477 (v = 21d1). One must realize that in all of these
models, the amplitude of variation of the CIII emission line
is so small that results cannot be considered to represent
significant intensity changes. The largest variation in
this case is only 1% of the average deflection. It is for
this reason that no specific model is suggested for this
data .
In summary, power spectrum analysis gives a period of
0^0238 for both helium and carbon data. The 1east-squares
routine, although still converging for periods on this

Relative Power
1.2E-6
8.0E-7
4.0E-7
0.0
50
100
150
200
250
Frequency ( d ~1)
Figure 4-13 Power spectrum for the carbon observations of
data set II before execution of the clean
algo rithm.

Relative Power
Figure 4-14 Power spectrum for the carbon observations of
data set II after execution of the clean
a1 go rithm.
Ln

76
order, give a best-fit with a period equal to twice this
period, or, P = 0^0476.
Data Set III.
This data set spans the time interval that extends
from JD2446607.9853 through JD2446608.1921 . The set includes
90 observations of 72 Vel for the helium, carbon, and
continuum filters.
The observing sequence used to obtain these data is
that shown in Table 4-2, with At = 0^0023. Assuming a
minimum of two samples per cycle and a minimum of two cycles
per data set, frequencies from = lOd'1 to ^max = 217d'1
can be tested. Figure 4-15 shows the sampling function
which results from the acquisition of these data. The peak
at v = 20d 1 has considerable power amounting to about one-
third that of the main peak at i/ = 0d l. The second harmonic
at v = 41d'1, and the third harmonic at v = 62d_1 are also
identified. In addition, a broad feature centered at
i/ = 33d 1 has a second harmonic at u = 55d'1 . The convolution
of the helium data with the beam is determined and presented
in Figure 4-16 in terms of relative power. The noise level
of these data is indicated by an arrow at the bottom left of
the plot at 2.719 x 10'6. The features at u = 11, 73, and
215d-1 rise above the 4ct^ level, but the peak at v = 73d 1 is
part alias with the stronger feature at u = lid'1. The
CLEANing process leaves two of the three peaks mentioned
above as statistically significant (Figure 4-17). The

Relative Power
0 50 100 150 200
Frequency ( d "1)
Figure 4-15 Spectral window function for data set III.
'•'j

Relative Power
Figure 4-16 Power spectrum for the helium observations of
data set III before execution of the clean
algorithm.

Relative Power
O 50 100 150 200
Frequency (d“1)
Figure 4-17 Power spectrum for the helium observations of
data set III after execution of the clean
algorithm.

80
feature at v = lid'1 has a relative power of 4.9a^ while the
peak at 1/ = 215d’ has the slightly lower s i gnal - t o - no i s e
ratio of 4.6cr^ and represents more-rapid fluctuations in
intensity. The 1east-squares routine converges to two sets
of parameters with almost the same amplitude in each case.
The first of these models has a period of 2.00 hours and
corresponds to the peak at u = lid'1 in Figure 4-17. The
model for this period is shown in Figure 4.18, and has been
computed with the relation
D(ti) = 0.34410 + 0.00354 * sin
r t± - 0.50673
0.01328
(4-5)
The second set of parameters which results from the least-
squares routine has a period of 2.75 hours and is described
by equation (4-6):
DCti) = 0.34442 + 0.00329 * sin
r tL - 0.59261
L 0.01820 J
(4-6)
Even though this period does not correspond to any prominent
feature in the power spectrum, the model, pictured in Figure
4-19, seems to fit the data as well as that described by
equation (4-5). Even though the 0-C residuals computed from
equation (4-5) are slightly smaller than the residuals
determined with equation (4-6), gaps in the data make it

Hell : Continuum
Julian Date (2446607.5+)
Figure 4-18 Model for the variations of the helium data for
data set III with P - 2.00 hours, each plotted
point represents the average intensity within a
0^0040 interval.
oo

Hell : Continuum
Julian Date (2446607.5+)
Figure 4-19 Model for the variations of the helium data for
data set III with P - 2.74 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
00
K>

83
difficult to determine which fit is better. Since the first
of the two periods is supported by both methods of analysis,
P = 2.00 hours may be the more appropriate period.
The Fourier transform of the carbon data is converted
to power, and Figures 4-20 and 4-21 show the results before
and after the removal of spurious features. The la noise
level of these data is at 5.685 x 10'7. Since the carbon
emission feature is considerably weaker than the helium
line, it is more difficult to discern intensity variations
in the carbon feature from random noise. In this particular
case, the CLEANing process reduces the strongest peaks in
Figure 4-20 below the 4a^¡ level. The 1 east - squares routine
has the strongest convergence for a period of 34.4 minutes.
However, since the amplitude of variation is only a very
small fraction of the total deflection in that filter, one
cannot regard this result with much confidence.
In summary, the intensity variations of the helium
emission line are probably best described with a period of
2.0 hours. This period was selected by both power spectrum
and 1east-squares analytic techniques. In addition, the
data have been modeled with a period of 2.75 hours, which
was a solution of the 1east-squares method only. Relatively
large gaps in the data make it difficult to determine which
period, if either, is the more appropriate. The power
spectrum of the carbon data indicates no strong periodicity.
A 1 east - squares solution with a period of 34.4 minutes is

Relative Power
Figure 4-20 Power spectrum for the carbon observations of
data set III before execution of the clean
algorithm.

Relative Power
Figure 4-21 Power spectrum for the carbon observations of
data set III after execution of the clean
algorithm.

86
found, but because of the low amplitude of the variation,
this period is not considered to be a true description of
the variations within this weak feature.
Data Set IV.
This data set consists of 18.8 hours of photometry of
72 Vel extending from JD2446608.7789 through JD2446609.5624.
The observing program displayed in Table 4-2 was used to
obtain a total of 347 observations through each of the
helium, carbon, and continuum filters. The average time
interval between successive observations in any one of these
filters is 0^0023.
The spectral window is depicted in Figure 4-22 where
peaks at u = 17d'1 and u = 35d_1 are separated by the same
amount as the peaks at u = 38d_1 and v = 56d_1. The
convolution of the intensity measurements of the helium
emission line with this somewhat more complicated sampling
function gives the power spectrum in Figure 4-23. The laN
noise level at 8.850 x 10'' indicates that peaks at i/ = 11.5,
35, and 166.5d-1 are statistically significant features. A
strong feature also occurs at u = l.Sd'1. but this frequency
does not satisfy the Nyquist criterion of two samples per
cycle. The peak at v = 35dl is part alias with that at
v = l.Sd1, and should, in principle, be reduced in strength
by the CLEAN algorithm. Figure 4-24 shows the power
spectrum that results after the spurious features are
removed from the dirty spectrum. The only feature which

Relative Power
Figure 4-22 Spectral window function for data set IV.

Relative Power
0 50 100 150 200
Frequency ( d“1)
Figure 4-23 Power spectrum for the helium observations of
data set IV before execution of the clean
algorithm.

Relative Power
0 50 100 150 200
Frequency ( d “1)
Figure 4-24 Power spectrum for the helium observations of
data set IV after execution of the clean
algorithm.

90
remains statistically significant is the peak at v = 1.5d‘l,
but as mentioned previously, the period which corresponds to
this frequency does not satisfy the Nyquist criterion, and
thus cannot be considered in the context of this research.
Although no peaks survive the CLEANing process, the least-
squares routine is still executed. Convergence to a set of
parameters with P = 0^208 (5.0 hours) is given by
DCti) = 0.34736 + 0.00250 * sin
r t± - 0.22157
L 0.03316
(4-7)
This fit, depicted in Figure 4-25, has a very low
amplitude of variation and is certainly not a very good
representation of the changes in intensity. This
unsatisfactory fit is confirmed by the low power in the
peaks of the CLEANed power spectrum (Figure 4-24). An
attempt was made to improve the results obtained with these
data by dividing the long data set into two subsets. The
first subset contains observations made between
JD2446608.77 89 and JD2446609.1646, and has a time - resolution
which is exactly the same as that for the whole data set.
The shorter span of this subset allows for frequencies
between = 5d-1 and ^max = 217d-1 to be tested. The
spectral window function for these data, depicted in Figure
4-26, shows similar, but slightly stronger, peaks as those
from Figure 4-22. Dirty and clean spectra are shown in

Hell : Continuum
Julian date (2446607.5+)
Figure 4-25 Model for the variations of the helium data of
data set IV with P - 5.00 hours. Each plotted
point represents the average intensity within a
0^0040 interval.

Relative Power
1.000
Figure 4-26 Spectral window function for subset I of data
set IV.
ro

93
Figures 4-27 and 4-28. The noise level in each of these
plots is denoted in the lower left-hand portion of the
figures at 1.737 x 10'6. The CLEAN algorithm removes all
spurious features at or below the 3ct^ level. As can be seen
from Figure 4-28, this process again leaves no statistically
significant features. The 1east-squares solution for this
data set gives a period of 5.79 hours corresponding in
frequency to the strongest peak in Figures 4-27 and 4-28,
even though these peaks do not meet the statistical
significance criterion. This model is described by
DCti) = 0.34527 + 0.00329 * sin
r t± - 1.38569
0.03842
(4-8)
and is shown in Figure 4-29. The 0-C residuals computed
with this equation are marginally better than those
determined with equation (4-7).
The second subset of data set IV covers the time range
that extends from JD2446609.1870 through JD2446610.5624.
The minimum and maximum frequencies retrievable from this
subset are the same as for subset 1. Again, the spectral
window (Figure 4-30) is qualitatively similar to the beam
for the entire data set, but is quantitatively different.
Figures 4-31 and 4-32 represent the power spectra before and
after CLEANing the data. The only peaks which remain above
the 4a^ level after the CLEANing process are those at

Relative Power
Figure 4-27 Power spectrum for the helium observations of
subset I of data set IV before execution of the
clean algorithm.

Relative Power
0 50 100 150 200
Frequency (d~1)
Figure 4-28 Power spectrum for the helium observations of
subset I of data set IV after execution of the
clean algorithm.

Hell : Continuum
0.40
0.38 -
0.36 -
0.34 -
0.32 -
0.30
1.260 1.335 1.410 1.485 1.560 1.635
Julian Date (2446607.5+)
Figure 4-29 Model for the variations of the helium data of
subset I of data set IV with P = 5.79 hours.
Each plotted point represents the average
intensity within a 0^0040 interval.

Relative Power
1.000
Figure 4-30 Spectral window function for subset II of data
set IV.
vO

Relative Power
8.0E—6 -
6.0E-6 -
4.0E-6 -
2.0E-6
0.0
50
100
150
200
Frequency ( d~1)
Figure 4-31 Power spectrum for the helium observations of
subset II of data set IV before execution of
the clean algorithm.
^3
OO

Relative Power
8.0E—6 -
6.0E-6 -
4.0E-6 -
2.0E-6
0.0
50
100
150
200
Frequency ( d “1)
Figure 4-32 Power spectrum for the helium observations of
subset II of data set IV after execution of the
clean algorithm.
VO

100
v = lOd1 and u = 182d1. The first of these features
corresponds to P = 0.100 (2.40 hours), and has a relative
power of about 4.38ct^ where = 1.553 x 10'6. The second
peak which rises above the 4ct^ level in Figure 4-32
represents rapid fluctuations with P = 7.91 minutes, and has
a signal-to-noise ratio of 4.10. According to the least-
squares analysis of subset 2, a period of 0^2631 (6.31
hours) best satisfies the variations present in these data.
This period does not agree with either of the periods found
through power spectrum analysis. However, because it
corresponds to a frequency which is less than this is
not a serious discrepancy. The model which is computed with
the parameters chosen by the least - squares routine, namely
D(tt) = 0.34796 + 0.00307 * sin
r tj_ - 1.76215
L 0.04187
(4-9)
is shown with averaged points in Figure 4-33. In addition
to this solution, a shorter period was also found through
the 1east-squares method of analysis. This solution has a
smaller amplitude of variation, but the period of 2.23 hours
agrees well with the peak in Figure 4-32 at v = lOd1. The
mathematical representation of this model is described by
r t± - 1.72011 ^
D(ti) = 0.34887 + 0.00238 * sin
0.01480
(4-10)

Hell : Continuum
Julian Date (2446607.5+)
Figure 4-33 Model for the variations of the helium data of
subset II of data set IV with P - 6.31 hours.
Each plotted point represents the average
intensity within a 0^0040 interval.
o

102
and is depicted in Figure 4-34. The the O-C residuals
determined from this model are larger than those determined
from equation (4-9). In addition, the amplitude of
fluctuation in equation (4-10) represents a smaller
percentage of the total intensity of the helium line than
does equation (4-9). These considerations lead one to
believe that equation (4-9) is a better representation of
the changes in intensity of this subset.
Analysis of the carbon data are really quite
inconclusive. The power spectra for the entire data set,
before and after execution of the CLEAN algorithm, are shown
in Figures 4-35 and 4-36, respectively. The noise level of
these data, in terms of relative power, is denoted in the
lower left-hand portion of the figures at 1.266 x 10'7.
None of the features in Figure 4-36 are statistically
significant at the 4aj^ level. In an attempt to improve the
interpretation of these data, the data set was again divided
into two subsets which correspond in time to subsets I and
II of the helium data. The dirty and clean power spectra
which result after taking the Fourier transform of the first
subset of these data, are shown in Figures 4-37 and 4-38.
The noise level in each of these plots is at 2.316 x 10'7.
As seen in these figures, three high-frequency
components exceed the 4ct^ level by a small amount. Figures
4-39 and 4-40 are similar plots for the second subset. In
this case, the peak at v = 57d-1 (P = 0^0179) has a larger

Hell : Continuum
Julian Date (2446607.5+)
Figure 4-34 Model for the variations of the helium data of
subset II of data set IV with P - 2.23 hours.
Each plotted point represents the average
intensity within a 0^0040 interval. (-*
o

Relative Power
0 50 100 150 200
Frequency (d~1)
Figure 4-35 Power spectrum for the carbon observations of
data set IV before execution of the clean
algorithm.
104

Relative Power
3.0E-6
T
2.4E-6
1.8E-6
1.2E-6
6.0E-7
0.0
50
100
150
200
Frequency ( d_1 )
Figure 4-36 Power spectrum for the carbon observations of
data set IV after execution of the clean
algorithm.
105

Relative Power
Frequency (cM)
Figure 4-37 Power spectrum for the carbon observations of
subset I of data set IV before execution of the
clean algorithm.
106

Relative Power
1.5E-6
1.0E-6
5.0E-7
0.0
0 50 100 150 200
Frequency ( d“1)
Figure 4-38 Power spectrum for the carbon observations of
subset I of data set IV after execution of the
clean algorithm.
o

Relative Power
Frequency ( d“1)
Figure 4-39 Power spectrum for the carbon observations of
subset II of data set IV before execution of
the clean algorithm.
108

Relative Power
1.6E-6
1.2E-6 -
8.0E-7 -
4.0E—7
0.0
50
100
150
200
Frequency ( d 1 )
Figure 4-40 Power spectrum for the carbon observations of
subset II of data set IV after execution of the
clean algorithm.
109

110
signal-to - noise ratio of 6.9. The 1east-squares routine
converges for a period very close to this frequency, but the
amplitude of the resultant variation is again very small and
therefore is not considered to be significant.
In summary, the variations in the helium intensity
measurements are not well modeled when the entire data set
is analyzed. Separating this data set into two subsets
provides a better understanding of the fluctuations. The
first of these subsets is best described with a period on
the order of 5.79 hours. Although a period of 2.23 hours is
determined from the least - squares method of analysis and
corresponds in frequency to a peak in the power spectrum of
these data, the second subset is best described with a
longer period of 6.31 hours. If variations in the intensity
of the carbon emission line exist, the amplitude of the
fluctuations is too low to be modeled within the limitations
of the s e data.
Data Set V.
The data set investigated in this section includes
over 13 hours of continuous photometry of y2 Vel. The (421)
data points cover the time interval extending from
JD2446615.3343 to JD2446615.8918. The observing program
used to secure these data is that described in Table 4-2.
Successive observations in any single filter are separated
by 0<*0013.

Ill
According to the Nyquist theorem, peaks in the power
spectrum at frequencies between = 3.5d_1 and
‘'max = 385d 1 can be regarded with confidence. The sampling
function for this data set is computed for this range of
frequencies and is shown in Figure 4-41. In the diagram we
note that the complexity of the window increases toward high
frequencies, but the first three odd harmonics at
approximately, u3 = 50d'x, i/3 = lOOd'1 , and ^5 = 150d'*, are
still identifiable. The convolution of the intensities of
the helium emission line with the beam results in Figure
4-42. The noise level at 8.732 x 10‘7 indicates that peaks
at v = 4.5, 12, 19, 131, and 154d_1 are statistically
significant. The peak at u = 154d1 corresponds to a peak in
the spectral window, however the height of the peak in the
beam is not sufficient to exclude the possibility that an
additional component contributes to this feature. The
corresponding peak in the beam is only about 5% as strong as
the main component in that spectrum, while the peak at
v = 154d_1 shown in Figure 4-42 is nearly 75% of the main
feature at 1/ = 4.5d1. Deconvolving the power spectrum
(Figure 4-42) from the spectral window function results in
the CLEANed spectrum depicted in Figure 4-43 where the
features at v = 4.5, 12, and 154dl remain statistically
significant. The strongest of these features at v = 4.5d1,
has a relative power of approximately 8.7a^ and corresponds
to a period of 5.3 hours. The feature at v = 12d'x also

Relative Power
0 50 100 150 200 250 300 350
Frequency (d~1)
Figure 4-41 Spectral window function for data set V.

Relative Power
Figure 4-42 Power spectrum for the helium observations of
data set V before execution of the clean
algorithm. i-4

Relative Power
Figure 4-43 Power spectrum for the helium observations of
data set V after execution of the clean
algorithm.
114

115
rises substantially above the 4ct^j level at 7 . 7u^¡ and
corresponds to P = 2.0 hours. Finally, the peak at
v = 154d' has a relative power equal to 5.5 times that of
the lo-fl level and represents more rapid fluctuations on the
order of 9.35 minutes. The 1east-squares routine converges
to a period which corresponds most closely to the feature at
v = 4.5d'1 in Figure 4-43. The resulting model, described
mathematically as
D(ti_) - 0.34509 + 0.00325 * sin
r t¿ - 0.86470
L 0.03217
(4-11)
is depicted in Figure 4-44. This fit seems to represent the
changes in the observations for the earlier points much more
satisfactorily than for data points near the end of the data
set. The change in the quality of the fit may indicate that
the variations exhibited in these data might better be
understood if the data are divided into two subsets. The
separation point was chosen to coincide with a 32 minute gap
in the data. Using the method of 1east-squares, the best-
fit sine wave for the data obtained between JD2446615.3315
and JD2446615.4851 was accomplished with the equation
f ti - 0.31000 ^
DCti) = 0.34441 + 0.00318 * sin
0.00786
(4-12)

Hell : Continuum
0.400
0.380
0.360
0.340
0.320
0.300 *— ' 1 ' 1 1—
1.830 1.930 2.030 2.130 2.230 2.330
Julian Date (2446613.5+)
Figure 4-44 Model for the variations of the helium data for
data set V with P - 4.85 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
116

117
where P = 1.19 hours. The data spanning the remainder of
the data set, JD2446615.5075 through JD2446615.8918 , results
in a variation which is slightly larger than that for the
first subset, with P = 1.83 hours. The appropriate
parameters selected by the 1east-squares routine results in
equation (4-13) :
DCti) = 0.34472 + 0.00340 * sin
r tt - 0.59330 ^
0.01211
(4-13)
Models for each of these portions of the data are shown in
Figures 4-45 and 4-46, where each plotted point is a 0^0040
average. The accompanying error-bars represent the standard
deviation associated with the individual points within each
average.
Figures 4-47 and 4-48 show dirty and clean power
spectra for the carbon data. Several peaks in each of these
plots have a relative power in excess of 4er^ = 5.940 x 10 '.
Although it is interesting to note that the three strongest
peaks occur at frequencies around u = 12d‘l , it is not
possible to single out the dominance of any one period. The
least - squares routine also has difficulty in converging to a
best overall period which satisfies the variations. In
fact, the routine converges for several periods with the
largest amplitude of variation amounting to only 1% of the
intensity of the carbon emission feature. It is certainly

Hell : Continuum
0.400
0.380
0.360
0.340
0.320
0.300 ' « ■—
1.830 1.880 1.930 1.980
Julian Date (244661 3.5+)
Figure 4-45 Model for the variations of the helium data for
subset I of data set V with P — 1.19 hours.
Each plotted point represents the average
intensity within a 0^0040 interval.
co

Hell : Continuum
0.400
0.380 -
0.360 -
0.340 -
0.320 -
0.300
2.000 2.050 2.100 2.150 2.200 2.250 2.300 2.350
Julian Date (2446613.5+)
2.400
Figure 4-46 Model for the variations of the helium data for
subset II of data set V with P - 1.83 hours.
Each plotted point represents the average
intensity within a 0^0040 interval.

Relative Power
Figure 4-47 Power spectrum for the carbon observations of
data set V before execution of the clean
algorithm.
120

Relative Power
1.2E-6
Figure 4-48 Power spectrum for the carbon observations of
data set V after execution of the clean
a1gorit hm.

122
not possible to discern these small variations from the
intrinsic noise which is present in the data.
This data set is certainly one of the most difficult
to interpret. Neither the power spectrum analysis nor the
least - squares method provide conclusive results. Probably no
single period can be used to describe the fluctuations
present during the time interval covered here. Figures 4-49
and 4-50 are plots of the last six hours of the data set.
Figure 4-49 depicts the intensity of the helium emission
line as a function of time, while Figure 4-50 is a similar
plot of the changes in the carbon emission feature. These
diagrams certainly seem to indicate a progressive change in
the intensities of the emission lines, especially at
JD2446615.82, but the phenomenon does not seem to repeat on
a regular basis .
Data Set VI.
Five-hundred and ninety-eight observations of
-y2 Vel were obtained in the helium and carbon filters from
JD2446621.7110 through JD2446622.3862 . In order to
eliminate gaps in the data set due to observations of
objects other than -y2 Vel and comparison star HR3452, a new
observing program was written. This program (Table 4-3)
consists only of deflections of -y2 Vel and HR3452. The
average spacing between reduced observations in each of the
helium and carbon filters is 0^0011. The time - resolution of
these data taken together with the 16.2 hours spanned by the

Hell : Continuum
0.400
0.380
0.360
0.340
0.320
0.300
2.140 2.190 2.240 2.290 2.340 2.390
Julian Date (2446613.5+)
J 1 1 1 L
Figure 4-49 Plot of the intensity of the helium emission
line for the last six hours of data set V.
ro

Clll : Continuum
Figure 4-50 Plot of the intensity of the carbon emission
line for the last six hours of data set V.
124

125
data set, allows for a wide range of frequencies to be
analyzed: i/m^n = 3d'1 and vmax = 455d '. Figure 4-51 shows
the resulting spectral window function for this group of
data. Although the relative power of the peaks is quite
low, one can still identify first and second harmonics.
After that, the beam becomes contaminated with extremely low
amplitude noise. The strongest harmonic, other than that at
v = Od 1 , is located at about v = 60d-1 and is only about 3.5%
as strong as the main component. Hence, one would expect
that the sampling function may introduce only small
amplitude spurious features into the power spectrum of these
data. Figures 4-52 and 4-53 show the Fourier transform of
the helium intensity measurements in terms of relative power
and the spectrum which results after cleaning at the 3ctj^
level. The noise level for each of these plots is at
1.192 x 10'6, and a number of peaks are statistically
significant. In Figure 4-53, the strongest feature at
v = 3d'1 (P = 8.0 hours) occurs at the beginning of the plot
and corresponds almost exactly to vm^n. This feature has a
signal-to-noise ratio of approximately 12.4. The next
strongest feature is at v = 20d1 (P = 1.20 hours) and has a
relative power equivalent to 8.39ct^. Another reasonably
strong peak is located at u = 323d1. suggesting rapid
variations on the order of 263 seconds. This peak has a
strength which exceeds the noise level by a factor of 5.5,
and must be considered within the context of this analysis.

126
Table 4-3
Observing Program 3
Star B V He C 0 Continuum
C*1 X X
CS2 X X
C* X X
V*3 X X
V* X X
V* X X
V* X X
v*
v*
v*
V*
VS4 X X
V*
V*
V*
v*
V* X X
V* X X
V* X X
V* X X
C* X X
CS X X
C* X X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
1 C* = comparison star
2CS = comparison sky
3V* = variable star
4VS = variable sky

Relative Power
Figure 4-51 Spectral window function for data set VI.
^sj

Relative Power
Frequency (cM)
Figure 4-52 Power spectrum for the helium observations of
data set VI before execution of the clean
algo rithm.

Relative Power
1.6E-5
1.2E-5
8.0E-6
4.0E-6
0.0
0 75 150 225 300 375 450
Frequency ( d~1 )
Figure 4-53 Power spectrum for the helium observations of
data set VI after execution of the clean
algorithm.

130
One last feature which should be mentioned is that at
v = 409d 1 (P = 211 seconds). Although this peak is
certainly not as strong as the three previous features which
have been mentioned, it still remains statistically
significant. The least - squares routine does not converge
for a period this short, but the possible significance of
the presence of this feature must not be overlooked. Very
rapid fluctuations, between 150 and 200 seconds, have been
previously reported (Jeffers et al. 1973a, 1973b) and the
variation found in these data with P = 211 seconds may
support the existence of intermittent rapid fluctuations of
the intensity of the Hell emission line.
The 1east-squares routine converges for two primary
sets of parameters: one has a period of 1.24 hours which
corresponds quite well with the peak at u = 20d_1, from
Figure 4-53. The other period which is identified has a
length of 2.44 hours, almost exactly twice the 1.24 hour
period. This is a good indication of a first harmonic. The
model for the shorter of these periods is shown in Figure
4-54 and is described by equation (4-14):
D(tt) = 0.34712 + 0.00403 * sin
r t± - 0.22991 i
L 0.00825 J
(4-14)
Neither this model nor the model for the 2.44 hour period
seems to be an adequate representation of these data.

Hell : Continuum
Julian Date (2446621.5+)
Figure 4-54 Model for the variations of the helium data of
data set VI with P - 1.24 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
►—4
LO

132
However, a look at the deflections in the helium filter
plotted as a function of time, indicates that perhaps a
change in period or phase may have occurred at approximately
JD2446621.9118. The fluctuations seem to be quite regular
to this point, when a new period seems to become more
dominant.
To investigate this possibility, the original data set
was again divided into two subsets. The first subset
covered the interval between JD2446621.7110 and
JD2446621.9118 , while the second subset covered the
remainder of the data extending from JD2446621.9126 through
JD2446622.3862 .
The sampling function for the first data set is
exactly the same as that pictured in Figure 4-51. According
to the power spectrum analysis procedure, the most dominant
period in this data, 1.26 hours, is the most prominent
feature in both the dirty (Figure 4-55) and the clean
(Figure 4-56) spectra. Although three peaks rise above the
statistically significant level at 4ajj, only one feature
remains significant after eliminating spurious features due
to the sampling function. This peak, located at u = 20d ‘,
has a relative power of 6.6 times the noise level. The
least - squares routine converges to a period of 1.30 hours,
and has one of the largest amplitudes of variation found in
the data so far. The model which corresponds to these

Relative Power
0 75 150 225 300 375
Frequency ( d_1)
Figure 4-55 Power spectrum for the helium observations of
subset I of data set VI before execution of the
clean algorithm.

Relative Power
0 75 150 225 300 375
Frequency ( d_1)
Figure 4-56 Power spectrum for the helium observations of
subset I of data set VI after execution of the
clean algorithm.
134

135
values is shown in Figure 4-57, and is computed with
equation (4-15):
D(t¿) = 0.35033 + 0.00642 * sin
r - 0.22766
0.00861
(4-15)
Although the error-bars from the plot are rather large, the
model seems to represent a periodic variation on the order
of 1.30 hours.
Again, as one might expect, the beam for the second
subset is very similar to that computed for the whole data
set (Figure 4-51). The noise level for these data is in
fact quite low at 1.585 x 10'6, and several peaks in the
power spectrum (Figure 4-58) might be considered as
statistically significant. The CLEAN procedure reduces the
spurious peaks, leaving one dominant feature at u = 3d1, and
two lower amplitude features at u = 9d-1 and v = 166d_1
(Figure 4-59). The feature at v = 3d1 is actually not
within the Nyquist frequency limit for this subset, but the
good agreement with the least - squares routine implies that
this may represent a possible periodicity in these data.
The equation used to describe these variations is
D(tj_) - 0.34457 + 0.00472 * sin
r - 0.51460
0.05324
(4-16)

Hell : Continuum
0.40
0.38
0.36
0.34
0.32 -
0.30
0.200
0.250 0.300 0.350
Julian Date (2446621.5+)
0.400
Figure 4-57 Model for the variations of the helium data for
subset I of data set VI with P — 1.30 hours.
Each plotted point represents the average
intensity within a 0^0040 interval.
136

Relative Power
Figure 4-58 Power spectrum for the helium observations of
subset II of data set VI before execution of
the clean algorithm.
137

Relative Power
0 75 150 225 300 375
Frequency ( d~1)
Figure 4-59 Power spectrum for the helium observations of
subset II of data set VI after execution of the
clean algorithm.
138

139
and the resulting model is shown in Figure 4-60 along with
0^0040 average intensity measurements. A second solution
for the quantities A0, At , tQ , and P is also produced
through the 1east-squares method of analysis. The period,
2.64 hours, corresponds to a frequency of u = 9d-1 and is
also a prominent feature in the power spectrum. The model
for this solution is depicted in Figure 4-61, and is
computed with the relation
DCti) = 0.34565 + 0.00314 * sin
r ti - 0.50998 >1
L 0.01751 J
(4-17)
The two models for this subset of the data indicate
periodic variations in the intensity measurements of the
helium emission feature. In one case a model with a period
of about 8 hours is fit to the data, and in the other case,
a model with a period of 2.64 hours is used.
The power spectra for the carbon data are shown in
Figures 4-62 to 4-65. Figures 4-62 and 4-63 are the dirty
and clean spectra for the first subset, and Figures 4-64 and
4-65 are the corresponding spectral plots for the second
subset. As has been frequently the case in the analysis of
the carbon data, the fluctuations have a very low amplitude
and one cannot be certain of the reality of the results. As
an example, a fit has been computed with the 1east-squares
routine. The strongest variation is described by

Hell : Continuum
0.40
0.38 -
0.36 -
0.34 -
0.32
0.30
0.405 0.480 0.555 0.630 0.705 0.780 0.855
Julian Date (2446621.5+)
Figure 4-60 Model for the variations of the helium data for
subset II of data set VI with P - 8.03 hours.
Each plotted point represents the average
intensity within a 0^0040 interval.
140

Hell : Continuum
Julian Date (2446621.5+)
Figure 4-61 Model for the variations of the helium data for
subset II of data set VI with P - 2.64 hours.
Each plotted point represents the average
intensity within a 0.0040 interval.
141

Relative Power
2.5E-6
Figure 4-62 Power spectrum for the carbon observations of
subset I of data set VI before execution of the
clean algorithm.
142

Relative Power
0 75 150 225 300 375
Frequency ( d_1)
Figure 4-63 Power spectrum for the carbon observations of
subset I of data set VI after execution of the
clean algorithm.
143

Relative Power
2.0E-6
T 1 1 1 T
1.5E-6
1.0E-6
0 75 150 225 300 375
Frequency ( d “1)
Figure 4-64 Power spectrum for the carbon observations of
subset II of data set VI before execution of
the clean algorithm.
144

Relative Power
2.0E-6
1.5E-6
1.0E-6
5.0E-7
0.0
0 75 150 225 300 375
Frequency ( d “1)
Figure 4-65 Power spectrum for the carbon observations of
subset II of data set VI after execution of
the clean algorithm.
145

146
D(t¿) = 0.10728 + 0.00159 * sin
r t± - 0.81853 'i
L 0.01580
(4-18)
Although the model (Figure 4-66) appears to follow the small
changes in the observations, such low amplitude models must
be regarded with skepticism.
In summary, the fluctuations present in the helium
emission line in particular, are better understood when the
data are divided into smaller subsets. In each case, good
agreement is achieved with the independent period search
techniques used in this study. The beginning of the data
set is best described with a period on the order of 1.26
hours. The latter portion of the data are described with
either of two periods, P = 8 hours or P = 2.64 hours.
Although the variations of the carbon emission data are not
conclusive within the limits of this analysis, a model with
a period of 1.71 hours is presented.
Data Set VII.
This data set is the last of the data obtained with
the SPOT during the 1986 observing season which is included
in this dissertation. These data include 243 reduced
intensity measurements of the helium emission line and 251
observations of the carbon emission feature. In each case,
0^0025 elapses between consecutive readings for a specific
filter. The observing program used in the acquisition of
these data is presented in Table 4-2. The instrument

Clll : Continuum
Julian Date (2446621.5+)
Figure 4-66 Model for the variations of the carbon data for
data set VI with P - 2.38 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
147

148
observed for 14.3 hours, from JD2446662.8415 through
JD2446663.4373, before being interrupted by bad weather.
The sampling function for this data set is represented by
the power spectrum presented in Figure 4-67. Here, the
three obvious peaks are the first, second, and third
harmonics at vx = 20d'1, u2 = 40d'1 , and
i/3 = 60d l. Even though a few additional observations of the
carbon emission line were used, the spectral window does not
differ from Figure 4-67 for either the helium or the carbon
data .
Relative power computed from the real and imaginary
components of the Fourier transform of the helium intensity
measurements is plotted in Figure 4-68. Features which rise
above the statistically significant level of 4<7^ , include
peaks at u = 6, 14, 18, 26, 48, 96, and 140d1. Indeed, many
of these features rise only minimally above the 4a^ level
and some are at least part aliases with stronger peaks. The
CLEANing function (Figure 4-69) has reduced the strength of
each feature quite substantially. However, peaks at u = 6,
14, 48, and 140d_1 still need to be considered as possible
periodicities within these data. The peak at v = 6d"'
(P = 4.0 hours) is clearly the strongest feature in both
Figures 4-68 and 4-69 at 8.7ct^; lu^ = 3.224 x 10'6. The
next largest peak occurs at v — 14d_1 (1.71 hours) with a
signal-1o - noise ratio on the order of 7.0. The other two
features in Figure 4-69 correspond to periods of 30 minutes

Relative Power
Figure 4-67 Spectral window function for data set VII.
149

Relative Power
Frequency ( d~1 )
Figure 4-68 Power spectrum for the helium observations of
data set VII before execution of the clean
algo rithu. ^
o

Relative Power
Figure 4-69 Power spectrum for the helium observations of
data set VII after execution of the clean t-*
al gor i t hm .

152
and 10.3 minutes, each with relative powers equal to 4.7ct^.
Execution of the 1east-squares routine to determine
the best-fit sine wave to the data, gives a period of
P = 3.72 hours, or, in terms of cycles per day, u = 6.4d'1.
Because of the agreement between these two methods, a model
is computed according to the 1east-squares parameters which
are expressed by the relation
D(tt) = 0.35098 + 0.00548 * sin
r t¿ - 0.44141
L 0.02428
(4-19)
The model itself is shown in Figure 4-70 together with the
standard deviations associated with 0^0040 averages. A
variation with a slightly smaller amplitude is also found by
the 1east-squares method of analysis. The second
convergence corresponds well with the peak at u = 14d'! in
the power spectrum of Figure 4-69. The model, shown in
Figure 4-71, is computed from equation (4-20):
D(tt) = 0.35094 + 0.00499 * sin
r tL - 0.81915
L 0.01140
(4-20)
Based on the 0-C residuals computed from equations
(4-19) and (4-20), the longer of these two periods seems to
be a better representation for the variations. The least-
squares routine does not converge for periods corresponding

Hell : Continuum
Julian Date (2446662.5+)
Figure 4*70 Model for the variations of the helium data for
data set VII with P - 3.66 hours. Each plotted
point represents the average intensity within a
0^0040 interval.

Hell : Continuum
Julian Date (2446662.5+)
Figure 4-71 Model for the variations of the helium data for
data set VII with P - 1.72 hours. Each plotted
point represents the average intensity within a
0^0040 interval.
154

155
to v = 48dl or v = 140d'1, and so, no model for these periods
is considered successful.
The power spectrum resulting from a convolution of the
carbon emission data with the spectral window (Figure 4-67)
is given in Figure 4-72. The la^ level in this plot is at
3.141 x 10'6, and none of the features rise above the 1 . 5u¡^
level. Hence, the power spectrum analysis finds no
significant periodic variations in these data. Likewise,
any convergence in the 1 east - squares algorithm has a very
low amplitude and cannot be accepted with confidence.
Briefly, according to either the power spectrum method
of analysis or the least - squares analysis procedure, the
carbon data exhibit no periodic variations with substantial
amplitude above the noise. The power spectrum and the
least - squares solutions agree quite well in describing
intensity variations of the Hell emission line. The fits
proposed here are for P = 3.66 hours and P = 1.72 hours.
The intensity changes seem to be better modeled with the
longer period of 3.66 hours.
B and V Photometry
The Wolf-Rayet star, y2 Vel, is a spectroscopic binary
with an orbital period of 78^5002 (Moffat et al. 1986).
Although Gaposchkin (1959) reported a period of 16^2334
based on a set of visual and photographic measurements, no
other investigator has detected an optical eclipse of this
system.

Relative Power
Frequency (d~1)
Figure 4-72 Power spectrum for the carbon observations of
data set VII before execution of the clean
algorithm.

157
Willis and Wilson (1976) have compared ultraviolet
spectroscopic observations of y2 Vel at phases 0.30 and 0.65.
They detect considerable differences in some of the spectral
features at these two phases which they explain as an
eclipse effect. These authors conclude that at phase 0.65,
the geometry of the system is such that the outermost layers
of the extended envelope of material surrounding the WC8
star, eclipse the 091 companion. According to this model,
the radius of the circumste11ar wind must be on the order of
250R^; substantially different from the result obtained by
Brown et al. (1970).
One aspect of this dissertation involves the
acquisition of B and V photometry of y2 Vel with the SPOT in
an attempt to confirm or deny the existence of an optical
eclipse. During the 1986 and 1988 observing seasons at the
South Pole, several different observing programs were used
to obtain B and V photometry of this system. A total of
2,538 observations (JD2446553.6 through JD2447312.6) with
the visual filter, and 2,315 observations (JD2446553.6
through 2447309.0) with the blue filter, were recorded.
These data were reduced to photometric magnitudes using the
techniques described in Chapter 3, with HR3452 as the
comparison source. The time of each observation is
converted to phase using the ephemeris given by equation
(1-1), and average magnitudes are computed at 0.005 phase
intervals. Table 4-4 lists the average AV magnitudes, the

158
Table 4-4
Average AV Magnitudes for 72 Vel
Pha s e
Magnitude
(AV)
RMS
Deviation
N
0.004
-3.100
0.073
38
0.005
-3.100
0.049
71
0.013
-3.086
0.036
91
0.017
- 3.114
0.027
161
0.038
-3.093
0.072
66
0.040
-3.090
0.021
22
0.046
-3.139
0.083
80
0.050
-3.228
0.047
6
0.079
-3.092
0.027
22
0.303
-3.060
0.021
24
0.377
-3.114
0.036
48
0.407
-3.113
0.039
94
0.412
-3.116
0.039
126
0.417
-3.116
0.038
96
0.422
-3 . 102
0.045
99
0.433
-3 . 107
0.061
85
0.436
-3.086
0.060
39
0.444
-3.203
0.024
9
0.603
-3.111
0.018
3
0.606
-3.122
0.062
5
0.618
-3.168
0.060
3
0.643
-3.099
0.057
6
0.648
-3.121
0.061
3
0.652
- 3.089
0.044
6
0.657
-3.112
0.054
9
0.662
-3.075
0.048
8
0.673
-3.091
0.028
101
0.675
-3.092
0.012
9
0.689
-3.117
0.019
23
0.690
-3.032
0.036
23
0.699
-3.120
0.028
34
0.702
-3.123
0.037
118
0.706
-3.117
0.033
89
0.714
-3.092
0.062
39
0.715
-3.101
0.077
50
0.731
-3.107
0.051
72
0.783
-3.082
0.067
60
0.787
-3.126
0.066
126
0.864
-3.122
0.069
45
0.867
-3.094
0.036
180
0.870
-3.110
0.037
60
0.881
-3.122
0.062
66
0.887
-3.087
0.075
108
0.998
-3.126
0.057
98

159
average phase within each interval, the standard deviation
associated with each average, and the number of points used
to determine each average. Table 4-5 contains similar data
for observations obtained with the blue filter.
Figures 4-73 and 4-74 are phase-magnitude diagrams for
the visual and the blue photometry, respectively. The
error-bars in each of these figures are quite large, and
therefore, small changes in the brightness of the system
cannot be detected. The large standard deviations of these
data can probably be attributed to several different
circumstances.
Although y2 Vel is a very bright object, variations in
the sky background will still affect the quality of the data
obtained with the SPOT. Auroral events which occur at such
a southerly latitude are probably more rapid and more
intense than at most other locations on Earth. The spectrum
of this sky brightening phenomenon is contaminated with
several strong emission features at wavelengths which
coincide with our filter bandpasses. Since the half-power
band-width of the B and V filters used in this study is
about 900Á, rapid variations in the intensity of the sky
background cannot be properly accounted for.
Two ways to minimize the effects of variations in the
sky brightness would be to observe the sky more frequently
and to use a smaller field stop. However, most of the data
used in this investigation were obtained during the 1986

160
observing season. At this point in the project, the effects
of variations in the sky brightness were not well understood
and the telescope was not equipped with diaphragms other
than the 5' field stop.
In the acquisition of most of these data, a sky
observation was obtained only once every 30 to 45 minutes.
This is certainly not adequate to account properly for
spatial and/or temporal changes. In addition, since these
data were obtained with a 5' diaphragm, intensity variations
in a portion of sky with large angular diameter contaminated
each stellar measurement.
One must also realize that the telescope used in the
acquisition of these data is operated at a remote site with
virtually no human intervention. Therefore, if the
telescope can find a given star, data are recorded whether
the sky is of photometric quality or not. It is the
responsibility of the analyst to attempt to distinguish
between variations which are intrinsic to the star which is
being observed, and variations which are due to bad weather.
This is not always a simple task. Weather data are recorded
at the South Pole once every six hours, but for astronomical
purposes, this is far from adequate. Further, thin clouds
which can severely contaminate photometric data are probably
not often detected by a visual observer.
These are a few of the problems which arise in the
analysis of the B and V observations obtained with the SPOT.

161
Table 4-5
Average AB Magnitudes for 72 Vel
Phase
Magnitude
(AB)
RMS
Deviation
N
0.004
-3.507
0.072
34
0.005
-3.484
0.069
67
0.013
- 3.461
0.036
79
0.017
-3.485
0.046
153
0.038
-3.469
0.066
54
0.040
-3.434
0 . 105
20
0.047
-3.511
0.079
57
0.050
-3.516
0.080
7
0.079
-3.428
0.047
22
0.303
-3.414
0.027
24
0 . 377
-3.474
0.051
46
0.407
-3.484
0.076
100
0.412
- 3.476
0.080
125
0.417
-3.469
0.078
85
0.422
- 3.482
0.068
99
0.433
-3.480
0.072
64
0.436
-3.435
0.085
36
0.444
-3.558
0.039
9
0.603
-3.498
0.010
3
0.606
-3.521
0.079
5
0.618
-3.518
0.028
3
0.673
- 3.439
0.060
99
0.675
- 3.344
0.024
9
0.689
- 3.489
0.034
23
0.690
-3.389
0.054
22
0.699
-3.472
0.059
33
0.702
-3.460
0.047
116
0.706
-3.489
0.033
89
0.714
-3.458
0.055
39
0.715
-3.463
0.074
46
0.731
-3.491
0.058
72
0.783
-3.481
0.043
36
0.787
- 3.472
0.060
95
0.864
-3.497
0.074
45
0.867
-3.456
0.071
171
0.870
-3.468
0.071
54
0.881
-3.463
0.064
56
0.887
- 3.482
0.062
112
0.998
-3.510
0.087
93

-3.00
Phase
Figure 4-74 Phase-magnitude diagram for the blue photometry
of 72 Vel. Each point represents the average
magnitude within a 0.005 phase interval. The
error-bars show the standard deviations
associated with each average.
162

-2.60
-2.80 -
-3.00 -
-3.20 -
-3.40 -
-3.60
0.7
0.9
0.1
0.3
0.5
0.7
Phase
Figure 4-73 Phase-magnitude diagram for the visual photometry
of y2 Vel. Each point represents the average
magnitude within a 0.005 phase interval. The error-
bars show the standard deviations associated with
each average.
163

164
It is probably some combination of these situations that
leads to the large rms deviations in the reduced data.
Complete phase coverage in the blue and visual filters
has not yet been obtained (Figures 4-73 and 4-74). In fact,
observations at certain critical phase positions are
missing, particularly those just before phase 0.0. In any
case (within the limitations of these data) no eclipse is
seen in either the blue or yellow regions of the spectrum.
If an eclipse of the extended atmosphere does occur, the
change in light would probably be very small.
Additional observations are required before we can
fully ascertain whether or not an optical eclipse does in
fact occur in the Y Vel system. In addition, and perhaps
more importantly, more - accurate observations are necessary.
It seems almost certain that any automated optical telescope
operating at such a remote site must observe the star and
sky simultaneously in order to detect, and subsequently
account for, variations in sky conditions.

CHAPTER 5
DISCUSSION AND CONCLUSIONS
One-hundred and fifty-seven of the stars in our Galaxy
have been classified as Wolf-Rayet stars. Variations in the
spectra of these objects have been detected, but mechanisms
which cause such changes in the continuum and emission line
features are not well understood. At one time, all Wolf-
Rayet stars were assumed to be members of binary star
systems. Until recently, it was this binarity which was the
most common explanation put forward to describe these
spectral variations.
In 1974, Moffat and Haupt stated that the percentage
of Wolf-Rayet stars which are members of binary star systems
"May be much higher than 73% . . .". As recently as 1982,
Bisiacchi et al. reported that up to 100% of the WNE type
Wolf-Rayet stars may be members of binary systems. On the
other hand, a more recent study by Moffat (1986) shows that
the frequency of binarity among this class of stars has
probably been largely over-estimated.
Today approximately 40% of the Wolf-Rayet stars- in the
Milky Way Galaxy are known to be members of binary systems.
This new statistic, taken together with the fact that
spectral variations have also been observed in Wolf-Rayet
165

166
stars which do not have a companion, indicates that another
mechanism must exist to explain the changes in the spectra
of these stars. Theoretical research now indicates that
these changes are probably linked to the high rates of mass
loss (i.e. , ~2 x 10'5Mq /yr) associated with this type of
object.
Although periodic or quasi - periodic variations
resulting from a steady laminar flow of material would
simplify the theoretical interpretation of these stars,
according to Vreux (1987) it is more likely that random
fluctuations in the wind occur. Vreux has suggested that
pulsational instabilities may cause changes in the spectral
characteristics of Wolf-Rayet stars. In his 1987 paper,
Vreux presented a discussion of variations in the spectra of
these objects which have been reported in the literature
thus far. No ultra-short time variations with periods less
than 1 minute have been cited for any Wolf-Rayet star. The
only object which has reported periodic variations between 2
minutes and 2 hours is y2 Vel. Periodic variations on the
order of 150 - 200 seconds have been reported in the
literature by only one group. Vreux and others have noted
that short period changes have been detected only when high¬
speed photometry is recorded for a period of about ten
minutes. Other investigators, who have recorded longer
intervals of observations, report variations of longer
duration. Some report periodic behavior, others report

167
aperiodic behavior. More observations are certainly
required before one is able to confirm or deny such rapid
variations in intensity. Vreux goes on to mention that no
other Wolf-Rayet star, either of the WN or WC type, has been
reported to exhibit periodic variations of this order.
However, variations on the order of minutes have been
detected for some WC type stars, with variability being more
common in the later subclasses.
The last period range that Vreux reports concerns
periods between 0^3 and l4o. As one would expect,
variations with periods of a day, or a multiple of a day,
are difficult to determine from most geographical locations.
Still, possible periods on this order have been reported for
some stars .
Vreux suggests that spectral changes in Wolf-Rayet
stars may be attributed to non-radial pulsations. Until
results of a theoretical study carried out by Noels and
Scuflaire (1987) were published, theory did not support the
existence of non-radial pulsations in he1ium-burning stars.
However, Noels and Scuflaire have considered the
evolutionary sequence of a lOOM^ star and have shown that
non-radial pulsations can produce vibrational instabilities
if regions of hydrogen she11 - burning persist. This theory
pertains most closely to the WN type Wolf-Rayet stars which
have higher H/He ratios than do the WC types. These results
may lend support to Vreux's suggestion that non-radial

168
pulsations may contribute to the changes in the spectra of
WN type Wolf-Rayet stars. According to the models, periods
for these pulsations range from 30 minutes to a few hours,
and are in good agreement with observational studies.
In the case of helium-burning stars which no longer
have regions of hydrogen shell burning, radial pulsations
can actually produce these vibrational instabilities.
However, the star must be a helium star, nearly homogeneous
in structure, and have a mass larger than the critical mass
of 16MQ (Noels and Gabriel 1981, Noels and Gabriel 1984,
Maeder 1985). The periods which result from radial
pulsations are generally less than an hour in length,
considerably smaller than those derived for the WN types.
These theoretical studies suggest that non-radial
pulsations contribute to the large rates of mass loss in the
case of WN type Wolf-Rayet objects, and that radial
pulsations are more appropriate for the WC stars. It is
interesting to note that the temperatures which were
originally derived for Wolf-Rayet stars were not adequate to
provide such intensive winds through radiation pressure.
However, it is now suspected that the temperatures are
actually much higher than originally believed (Cherepashchuk
et al . 1984), and radiation pressure alone can support such
a strong and efficient phenomenon.
Cox and Cahn (1988) have carried out an independent
study involving the theoretical modeling of these stars as

169
well. They have computed both radial and non-radial models
for the non-adiabatic case, for five different masses of
Wolf-Rayet stars. Their results for the linear non-
adiabatic radial modes indicate periods from 22 minutes to
2.10 hours. Results for the linear non-adiabatic, non-
radial modes, indicate longer periods ranging from 2.9 hours
to 20.8 hours.
One of the models considered by these authors was for
an originally 12 0Mq star which had evolved past the hydrogen
shell burning stage. This model was found to be unstable
only against the radial fundamental mode of pulsation, which
agrees well with results of Maeder (1985). Cox and Cahn
conclude that if mass loss is to be the result of
pulsations, it must be only from the radial fundamental mode
which appears in hydrogen-free stars. Their 85MQ model
exhibits radial pulsations which are again in agreement with
Maeder ' s study.
One of the few models which has been put forward in an
attempt to explain spectral variations in y2 Vel includes the
presence of a third body. In 1977, Moffat suggested that
this star may be one member of a triple-star system.
Analysis of narrowband photoelectric - photometry revealed a
5.09 period. In addition, rapid spectrophotometric scans
indicated the presence of a longer period of variation
between 13 and 19 days. It was not until 1985 that Jeffers
et al. pursued Moffat's suggestion, and derived a model for

170
this system containing a neutron-star companion to the WC8
component. According to their investigation, the Wolf-
Rayet component would have a mass of 20MQ and a radius of
13.1 ± 0.2R0, while the neutron-star would have a mass of
1.4 ± 0.2M0.
Assuming a circular orbit for the neutron-star, and,
because of the lack of optical eclipses, an inclination of
70°, the maximum separation distance between the neutron-
star and the WC8 star would be 36R0. Interferometry
measurements made by Brown et al. (1970) indicate that the
extended wind of the Wolf-Rayet component has a radius of
76R0. Hence, the neutron-star would be contained well within
this envelope of material. The orbital period which results
from this situation is about
Since these two stars are reasonably close to one
another, one would expect that the accretion of material
onto the neutron-star would produce a large flux of hard
x-rays. However, observations reveal that very low amounts
of hard x-rays are actually detected in the y2 Vel system.
Some of the theories which have been proposed to accommodate
such a result include the following: Moffat and Seggewiss
(1979) state that if the density of material in the wind is
sufficiently large, then the hard x-rays may be re-absorbed.
Sunyaev (1978) explains that hard x-rays may not be detected
if accretion occurs at supercritical rates so that the
accretion disk becomes opaque to this radiation.

171
If a neutron-star is present in the y2 Vel system,
perturbations in the radial velocity measurements, should,
in principle, be detected. Moffat et al. (1986) carried out
a spectroscopic study to investigate this possibility and to
define the orbital parameters more precisely. These authors
tried to fit sine waves with periods between 1^675 and
infinity, with arbitrary phases, to the deviations which
result from an orbital fit to the data. They did not detect
any variations with an amplitude larger than 10km/sec. This
low amplitude cannot represent a real physical situation.
According to the model presented by Jeffers et al . ,
the amplitude of variation of the deviations in radial
velocity measurements would have to be larger that 21km/sec
to accommodate an orbital period of 5^4. The suggested
orbital period between 13 and 19 days which was proposed by
Moffat (1977) would indicate a lower limit on the amplitude
of the radial velocities of 15km/sec. In either case, a
10km/sec variation almost certainly excludes the possibility
of a neutron-star companion.
Variations in the strengths and shapes of the emission
lines of y2 Vel may be explained in the context of the
theoretical studies which have been discussed in some detail
above. Several mathematical models have been computed that
describe the variations in the strength of the Hell emission
line. The amplitudes of these fluctuations amount to a few
percent of the total intensity, and are in good agreement
with variations reported by other authors.

172
Table 5-1 lists all of the periods and corresponding
frequencies found in the analyses of the narrowband
photometry used in this study. With the exceptions of the
0.07 hour period (Data Set I) and the 0.57 hour period (Data
Set II), the shortest periods in all of these data are
P = 1.14 hours (Data Set II), and P = 1.26 hours (Data Set
Via). It is interesting to note that many of the periods
listed in column three of the table are near-harmonics or
subharmonics of one of these two periods. An average period
of 1.20 hours is used to compute the multiplicity of each of
the periods in this table. These results are listed in
column five. The two periods found to satisfy Data Set III
are not simple harmonics of the 1.20 hour fundamental
period. However, as stated in the discussion of this data
set in Chapter 4, relatively large intervals when y2 Vel was
not observed, exist, and make it difficult to determine a
reliable period. In addition, Data Set VIb has been
described with a period of 2.64 hours and a period of 8.03
hours. The longer of these periods is seven times the 1.20
hour period, but, the 2.64 hour period cannot be related to
the assumed fundamental period. A look at Figures 4-60
(P = 8.03 hours) and 4-61 (P = 2.64 hours) indicates that
the longer period that is found for these data is a better
representation of the variations.
At this point, it is interesting to link these results
with the results of theoretical modeling of Wolf-Rayet stars

173
Summary
Table 5-1
of Periodicities Found
in the
Data Set
Phase
Hell Emission Line
Period Frequency
Multiplicity
Interval
(hours) (d1)
(P = 1.20)
I
0 .
. 302 â– 
-0 .
. 303
0
.07
327
. 3
P/16
2
. 40
10
. 0
2 P
II
0 ,
.671-
-0 .
.675
0 ,
. 57
42 ,
. 1
P/2
1
. 14
21
. 1
~P
111
0 .
.688-
-0 .
.691
2 .
. 00
12 ,
. 0
—
2
. 75
8
. 7
IVa
0 .
.698-
-0 .
, 703
5 .
. 79
4 .
. 1
5 P
IVb
0 ,
. 704-
-0 .
. 708
6
. 31
3
. 8
11 P/2
2 .
.23
10 .
, 8
2 P
V
0 ,
. 782 â– 
-0 .
. 789
—
Via
0 .
.863-
-0 .
866
1 .
. 26
19 .
, 0
~P
VIb
0 .
.866-
-0 .
,872
2 .
. 64
9 .
. 1
-2 P
8 .
03
3 .
0
7 P
VII
0 .
, 387-
-0 .
.395
3 .
. 66
6 .
. 6
3 P
1 .
72
14 .
0
3P/2

174
accomplished by Cox and Cahn (1988). They have computed the
evolution of an 8 5 star which has undergone core hydrogen
burning and is left with a hydrogen - burning shell. The
first-overtone period for the case of non-adiabatic radial
mode pulsations is 1.23 hours. This period is in excellent
agreement with the results of the observational study
presented in this dissertation which indicates a period of
1.20 hours.
In addition, the fundamental period which Cox and Cahn
derive for this particular model is 1.98 hours. Perhaps the
2.00 hour period which describes the changes in intensity of
the helium emission line in Data Set III, actually
represents the fundamental mode of vibration.
An attempt was also made to mathematically model the
line intensity fluctuations exhibited by the carbon emission
line of y2 Vel at 5696Á. This 1 ine is considerably weaker
than the helium line at 4686Á and variations are much more
difficult to model. In almost every case the amplitude of
fluctuation amounts to about only 1% of the total intensity.
Models have been computed for the two cases when the
variation exceeded a 1% fluctuation. Data Set I exhibited
the largest percentage of variations in both the carbon and
helium emission line strengths. The carbon line was modeled
with a period of 1.10 hours and the amplitude of the
variation was approximately 2% of the carbon line intensity.
Data Set VIb exhibited fluctuations with P = 2.38 hours, and

175
an amplitude of fluctuation of nearly 1.5% of the intensity.
It is interesting to realize that in each of these cases,
the period of variation is slightly smaller than that which
was derived for the corresponding helium line data.
Although the sample is small, this may provide evidence of
where the lines are formed in the extended wind. That is,
if these variations are a result of pulsational
instabilities produced by mass loss, material closer to the
surface of the star may exhibit more rapid oscillations than
more distant material which may be damped by underlying
layers of the wind.
In any case, analyses of narrowband photometry of
72 Vel that we have presented in this study may well provide
considerable support for recent theoretical developments of
Wolf-Rayet stars.

REFERENCES
Austin, R.R.,
No. 2552
Bahng, J.D.R.
Bahng, J.D.R.
Bahng, J.D.R.
Banachiewicz,
Schneider, W.H., Wood, F.B. 1973,
1973, Bull. Am. Astron. Soc . . 5_,
1974, Bull. Am. Astron. Soc.. 6.,
1975, As trophvs. J. . 200. 128.
T. 1942, Astron. J. . 5 0. 38.
IAU
412 .
455 .
C i r c
Bappu, M.K.V. 1973, Wolf-Ravet and High Temperature Stars
IAU Svmp. No. 49. eds.: M.K.V. Bappu and J. Sahade,
Dordrecht: Reidel.
Barlow, M.J., Hummer, D.G. 1982, Wolf-Ravet Stars:
Observations. Physics, and Evolution. IAU Svmn. No.
9 9 . eds.: C.W.H. de Loore and A.J. Willis, Dordrecht:
Reidel, 387.
Bisiacchi, G. , Firmani, C. , de Lara, E. 1982, Wolf-Rave t
Stars: Observations. Physics and Evolution. IAU Svmp.
No , 9 9. eds.: C.W.H. de Loore and A.J. Willis,
Dordrecht: Reidel, 583.
Brown, R.H., Davis, F., Herbison - Evans, D., Allen, L.R.
1970, Mon. Not. R. Astron. Soc.. 148 . 103.
Cherepashchuck, A.M., Eaton, J.A., Khaliullin, K. F. 1984,
As trophvs. J. . 2 81. 774.
Conti, P.S., Smith, L.F. 1972, Astrophvs. J., 172. 623.
Cox, A.N., Cahn, J.H. 1988, Astrophvs. J.. 326. 804.
Deeming, T.J. 1975, Astrophvs. Space Sci. Rev.. 3 6. 137.
Esper, J. 1986, Master's Thesis, Univ. of Florida.
Ganesh, K.S., Bappu, M.K.V. 1967, Kodaikanal Obs. Bull.
Ser. A. No. 183.
Gaposchkin, S. 1959, Astron. J.. 64. 127.
176

177
Haefner, R., Metz, K., Schoembs, R. 1977, Astron.
Astroohvs. . 5 5. 5.
Henden, A.A., Kaitchuck, R.H. 1982, Astronomical Photometry.
New York, Van Nostrand Reinhold Co.
Hógbom, J.A. 1974, Astron. Astrophvs.. Supnl. Ser.. 15. 417.
Jeffers, S., Stiff, T., Weller, W.G. 1985, Astron. J. . 9 0.
1852 .
Jeffers, S., Weller, W., Sanyal, A. 1973a, IAU Circ. . No.
2495 .
Jeffers, S., Weller, W., Sanyal, A. 1973b, IAU Circ.. No.
2531 .
Kondo, Y., Feibelman, W.A., West, D.K. 1982, As trophvs. J . .
252. 208.
Lindgren, H., Lundstrom, I., Stenholm, B. 1975, Astron.
Astrophvs.. 44. 219.
Maeder, A. 1985, Astron. Astrophvs.. 147. 300.
Moffat, A.F.J. 1977, Astron. Astrophvs. . 5 7 . 151.
Moffat, A.F.J. , Haupt, W. 1974, Astron. Astrophvs.. 3 2. 435.
Moffat, A.F.J., Seggewiss, W. 1979, Astron. Astrophvs. . 7 7.
128 .
Moffat, A.F.J., Shara, M.M. 1986, Astron. J . . 92., 952.
Moffat, A.F.J. , Vogt, N. , Paquin, G. , Lamontagne, R. ,
Barrera, L.H. 1986, Astron. J.. 91. 6, 1386.
Niemela, V.S., Sahade, J. 1980, Astrophvs. J.. 238. 244.
Noels,
A. ,
, Gabriel,
M.
1981 ,
Astron
. Astroohvs..
Noels,
A. ,
, Gabriel,
M .
1984 ,
Proc .
25th Liéee Int
Astrophvs. Astrophvs. Colloq.. 5 9 .
Noels, A., Scuflaire, R. 1987, in Instabilities in Luminous
Early Type Stars, eds.: H.J.G.L.M. Lamers and
C.W.H. de Loore, Dordrecht: Reidel, 213.
Palomar Observatory, 1954. National Geographic - Palomar
Observatory Skv Survey. Pasadena: California Institute
of Technology.
Perrine, C.D. 1918, Astrophvs. J.. 47. 52.

178
Roberts, R.H., Lehár, J., Dreher, J.W. 1987, Astron. J. . 9 3.
968 .
Rublev, S.V. 1972, Izv. Soets . Astrofiz. Obs . . 4_, 3.
Smith, L.F. 1973, Wolf-Ravet and High Temperature Stars. IAU
Svmp. No. 49. eds.: M.K.V. Bappu and J. Sahade,
Dordrecht: Reidel.
Sunyaev, R.A. 1978, Proc. of the Int. School of Phvs..
"Enrico Fermi," 697, New York, North-Holland.
Underhill, A.B. 1959, Pub. Pom. Astrophvs. Obs.. 11. 209.
Underhill, A.B. 1982, Wolf-Ravet Stars: Observations.
Physics, and Evolution. IAU Svmp. No. 99. eds.:
C.W.H. de Loore and A.J. Willis, Dordrecht: Reidel.
Vreux, J.-M. 1987, Instabilities in Luminous Early Type
Stars. eds.: H.J.G.L.M. Lamers and C.y.H. de Loore,
Dordrecht: Reidel, 81.
West, P.K. 1972, NASA SP - 310. 441.
Willis, A.J. 1980, in Symposium: The 2nd European IUE Conf..
eds. : B. Battrick and J. Mort, ESA SP. 157.
Willis, A.J. 1987, 0. J. R. Astron. Soc. . 2 8. 217.
Willis, A. J . , Wilson, R. 1976, Astron. Astrophvs.. 47. 429.
Wood, F.B. 1941, Contrib. Princeton Univ, Obs. No. 21.

BIOGRAPHICAL SKETCH
MaryJane was born in Lake Worth, Florida, on 15 March
1961, to Peter Oates and Constance Ann Taylor. MaryJane's
father played a very major role in her life, supplying
support and encouragement along the way. At age five she
assisted her father in building an 8-inch Newtonian
telescope, and then proceeded to assemble her own 6-inch
reflector at age six.
In 1968, MaryJane became the youngest member of the
American Association of Variable Star Observers (AAVSO) as
an active contributor, submitting thousands of sunspot and
variable star observations. She continues to use her
original 6-inch telescope for solar observations, and is one
of about 100 international contributors to the American
Sunspot Program.
MaryJane's father offered her an early exposure to
mathematics by teaching her first and second year algebra
while she was in third-grade. She skipped the fourth-grade,
and in 1974, published her first astronomical paper in the
Journal of the American Association of Variable Star
Observers. Since that time, she has published a number of
additional astronomical papers on variable star astronomy.
179

180
MaryJane attended Lake Worth High School and was
dually and triply enrolled at Palm Beach Junior College
and/or Florida Atlantic University during her junior and
senior years. During this period, she acquired a background
in upper-level mathematics and enrolled in a number of
computer classes.
During the summers of 1974 and 1975, MaryJane
participated as a research assistant at the Maria Mitchell
Observatory, under the direction of Dr. Dorrit Hoffleit,
long associated with Yale University. Although designed for
college students, she was accepted to the program during the
summers following her freshmen and sophomore years in high
school. Also, in 1975, MaryJane worked in the Harvard Patrol
Plate stacks for the AAVSO to resolve a long-term
misidentification of the variable stars, SX and Cl Librae,
resulting in the official cataloging of Cl Librae as a newly
discovered variable star.
During her high school career, MaryJane was an active
participant in science fairs, winning numerous awards on the
local and state level. In 1976, MaryJane was awarded first
place in the Earth and Space Sciences Division of the 1976
State Science and Engineering Fair and received an award for
best overall project in the Physical Sciences Division. In
addition, MaryJane won first place in the State Science
Talent Search for presentation of a portion of her work at
Maria Mitchell Observatory, "Two Variables in Sagittarius."

181
MaryJane was graduated from high school in June 1978
and became seriously ill in August. The illness pre-empted
her college career, and so it was not until August 1980 that
she was able to leave South Florida. She was graduated with
a Bachelor of Arts degree from Agnes Scott College in 1983
with majors in physics/astronomy and in mathematics.
In August 1983, MaryJane became a graduate student in
the Astronomy Department at the University of Florida.
During her career as a graduate student, MaryJane has been a
teaching assistant in the Mathematics Department where she
conducted lectures in Fundamental Mathematics, Survey
Calculus I and II, and Science and Engineering Calculus I.
In addition, she has had research positions in the Astronomy
Department with grants from National Science Foundation and
the Division of Sponsored Research from the University of
Florida.
In January 1988, MaryJane had the opportunity to
travel to the South Pole as a member of the South Pole
Optical Telescope research project team. Her duties
included installation and testing of software upgrades,
training winter-over personnel, and preparing
hardware/software/user documentation.

I certify that I have read this study and that in ray
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
in Parker Oliver, Chairman
issociate Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Kwan- Yu Ghen
Professor of Astronomy
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
JVpir/l-
Frank Bradshaw Wood
Professor of Astronomy

I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality, as
a dissertation for the degree of Doctor of Philosophy.
Bruce T. Edwards
Associate Professor of Mathematics
This dissertation was submitted to the Graduate
Faculty of the Department of Astronomy in the College of
Liberal Arts and Sciences and to the Graduate School and was
accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
December, 1988
Dean, Graduate School





PAGE 1

$1$/<6,6 2) \ 9(/2580 3+2720(75< )520 7+( 6287+ 32/( %< 0$5<-$1( 7$
PAGE 2

$&.12:/('*(0(176 $W WKLV WLPH ZRXOG OLNH WR H[SUHVV P\ JUDWLWXGH WR WKRVH SHRSOH ZLWKRXW ZKRVH FRQWULEXWLRQV WKLV ZRUN ZRXOG KDYH EHHQ IDU PRUH IUXVWUDWLQJ )LUVW ZRXOG OLNH WR WKDQN WKH FKDLUPDQ RI P\ 3K' FRPPLWWHH 'U -RKQ 3 2OLYHU IRU KLV GHGLFDWLRQ DQG H[SHUWLVH LQ WKH FRQVWUXFWLRQ RI WKH 6RXWK 3ROH 2SWLFDO 7HOHVFRSH 6327f DQG IRU KLV NQRZOHGJH LQ WKH GHYHORSPHQW RI WKH VRIWZDUH QHFHVVDU\ WR DXWRPDWH WKH 6327 V\VWHP WKDQN KLP IRU WKH PDQ\ KRXUV WKDW KH KDV GHGLFDWHG WR WKLV SURMHFW DQG IRU WKH HQGOHVV KHDGDFKHV LQFXUUHG HVSHFLDOO\ QHDU WKH EHJLQQLQJ RI HDFK REVHUYLQJ VHDVRQ ZRXOG DOVR OLNH WR H[SUHVV P\ VLQFHUH DSSUHFLDWLRQ WR 'U .ZDQ
PAGE 3

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

PAGE 4

WZHQW\RQH \HDUV KDYH PDGH PH DOZD\V VWULYH IRU WKH EHVW RZH P\ ORYH RI DVWURQRP\ WR KLP ,Q DGGLWLRQ KH LV WR EH WKDQNHG IRU WKH ORQJ KRXUV GHYRWHG WR WKH WHGLRXV SURFHVV RI IRUPDWWLQJ WKLV GLVVHUWDWLRQ LQ DFFRUGDQFH ZLWK *UDGXDWH 6FKRRO UHJXODWLRQV DQG IRU WKH XVH RI KLV +HZOHWW3DFNDUG SULQWHU 7KH 6WHOOHU 3KRWRPHWU\ 3URJUDP ZDV VXSSRUWHG E\ JUDQWV IURP WKH 'LYLVLRQ RI 3RODU 3URJUDPV RI WKH 1DWLRQDO 6FLHQFH )RXQGDWLRQ ,9

PAGE 5

7$%/( 2) &217(176 $&.12:/('*(0(176 LL $%675$&7 YL &+$37(56 %$&.*5281' ,167580(17$7,21 $1' '$7$ $&48,6,7,21 6<67(0 ,QVWUXPHQWDWLRQ 'DWD $FTXLVLWLRQ '$7$ 5('8&7,21 7(&+1,48(6 '$7$ $1$/<6,6 6SHFWUDO 3KRWRPHWU\ RI WKH +HOO DQG &,,, (PLVVLRQ /LQHV 'DWD 6HWO 'DWD6HWOO 'DWD 6HW ,,, 'DWD6HW,9 'DWD6HW9 'DWD 6HW 9, 'DWD 6HW 9,, % DQG 9 3KRWRPHWU\ ',6&866,21 $1' &21&/86,216 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ Y

PAGE 6

$EVWUDFW RI 'LVVHUWDWLRQ 3UHVHQWHG WR WKH *UDGXDWH 6FKRRO RI WKH 8QLYHUVLW\ RI )ORULGD LQ 3DUWLDO )XOILOOPHQW RI WKH 5HTXLUHPHQWV IRU WKH 'HJUHH RI 'RFWRU RI 3KLORVRSK\ $1$/<6,6 2) \ 9(/2580 3+2720(75< )520 7+( 6287+ 32/( %< 0$5<-$1( 7$
PAGE 7

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

PAGE 8

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

PAGE 9

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

PAGE 10

6SHFWUDO REVHUYDWLRQV KDYH OHG DVWURQRPHUV WR UHDOL]H WKDW WKHUH DUH WZR PDLQ VHTXHQFHV RI :ROI5D\HW VWDUV 7KH :1 W\SH :ROI5D\HW VWDU KDV DQ RSWLFDO VSHFWUXP GRPLQDWHG E\ QLWURJHQ DQG KHOLXP LRQV 6HYHUDO RI WKH :1 VXEW\SHV H[KLELW WUDFHV RI FDUERQ LQ SDUWLFXODU WKH &,9 OLQHV DW ƒ DQG ƒ LQ WKH RSWLFDO UHJLRQ RI WKH VSHFWUXP DQG DW ƒ LQ WKH XOWUDYLROHW UHJLRQ 7KH HQKDQFHG DEXQGDQFHV RI KHOLXP DQG QLWURJHQ DV ZHOO DV WKH ORZHU WKDQ QRUPDO DEXQGDQFHV RI FDUERQ DQG R[\JHQ PD\ EH H[SODLQHG E\ WKH IDFW WKDW WKHVH VWDUV H[SRVH PDWHULDO ZKLFK KDV EHHQ SURFHVVHG LQ WKH &12 F\FOH 7KH VHFRQG VHTXHQFH RI :ROI 5D\HW VWDUV DUH WKH :& W\SHV $V WKHLU QRPHQFODWXUH LQGLFDWHV WKHVH VWDUV KDYH VSHFWUD ZKLFK DUH DOPRVW HQWLUHO\ GRPLQDWHG E\ FDUERQ LRQV EXW ZKLFK PD\ DOVR GLVSOD\ KHOLXP DQG R[\JHQ OLQHV DV ZHOO 8QOLNH WKH :1 W\SHV WKH SUHVHQFH RI QLWURJHQ LQ :& VWDUV LV YHU\ ZHDN 7KH SRVVLEOH EOHQGV RI 1,,, LRQV ZLWK VWURQJ FDUERQ IHDWXUHV KDYH EHHQ LGHQWLILHG 8QGHUKLOO %DSSX f DW RSWLFDO ZDYHOHQJWKV DQG EOHQGV RI 1,9 DQG 19 LRQV ZLWK RWKHU FDUERQ HPLVVLRQV PD\ EH SUHVHQW DOVR DW XOWUDYLROHW ZDYHOHQJWKV :LOOLV f 7KH :&f FODVV RI :ROI5D\HW VWDUV H[SRVHV PDWHULDO ZKLFK KDV SUREDEO\ EHHQ SURFHVVHG E\ D KHOLXPn EXUQLQJ FRQYHFWLYH FRUH ,Q WKLV FDVH KRZHYHU WKH SUHVHQFH RI FDUERQ DQG R[\JHQ LV HQKDQFHG DW WKH H[SHQVH RI QLWURJHQ DQG KHOLXP 5HFHQWO\ %DUORZ DQG +XPPHU f KDYH LGHQWLILHG D WKLUG FODVV RI :ROI5D\HW VWDUV :2 7KH

PAGE 11

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f DQ DQDO\VLV FDUULHG RXW LQ WKLV PDQQHU UHVXOWHG LQ 1+f1+Hf IRU ODWH :1 :ROI5D\HW VWDUV :1/f QDPHO\ :1 DQG :1 6KH IRXQG FRQVLGHUDEO\ ORZHU UDWLRV 1+f1+Hf IRU HDUOLHU :1 W\SHV :1(f 7KHVH UHVXOWV LQGLFDWH WKDW VWDUV ZKLFK H[KLELW WKH :ROI5D\HW SKHQRPHQRQ PD\ LQGHHG EH FKHPLFDOO\ HYROYHG VWDUV ZLWK :1( W\SHV PRUH HYROYHG WKDQ :1/ W\SHV $ VLPLODU VWXG\ ZDV FDUULHG RXW IRU :& VWDUV E\ 5XEOHY f ZKR FRQFOXGHG 1+f1+Hf 7KLV UHVXOW PXVW EH UHJDUGHG ZLWK D OLWWOH PRUH FDXWLRQ WKDQ WKRVH UHVXOWV IRU :1 W\SHV EHFDXVH RI WKH VHYHUH EOHQGLQJ HIIHFWV RI WKH +HOO 3LFNHULQJ OLQHV

PAGE 12

ZLWK VHYHUDO FDUERQ DQG R[\JHQ WUDQVLWLRQV +RZHYHU LW LV WKH DSSDUHQW SRVLWLRQ RI WKLV FODVV RI VWDUV RQ WKH +HUW]VSUXQJ5XVVH GLDJUDP ZKLFK FRQWUDGLFWV WKH REVHUYDWLRQDO HYLGHQFH IRU ORZ K\GURJHQ DEXQGDQFHV ,Q IDFW WKLV LV WKH SULPDU\ UHDVRQ WKDW 8QGHUKLOO f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

PAGE 13

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rn f 7KH \ 9HO V\VWHP LV WKH EULJKWHVW :ROI5D\HW VWDU LQ WKH VN\ ZLWK D YLVXDO PDJQLWXGH RI 7KH VWDU LV DQ LQWULJXLQJ V\VWHP IRU ERWK REVHUYDWLRQDO DQG WKHRUHWLFDO DVWURQRPHUV ,W KDV EHHQ WKH REMHFW RI PDQ\ LQYHVWLJDWLRQV LQ WKH ODVW WZHQW\ \HDUV RU VR DQG LV WKH VXEMHFW RI WKLV GLVVHUWDWLRQ

PAGE 14

7KH \ 9HO V\VWHP ZDV GLVFRYHUHG WR EH D VSHFWURVFRSLF ELQDU\ E\ 6DKDGH LQ 7KH V\VWHP FRQVLVWV RI D :ROI 5D\HW VWDU RI VSHFWUDO FODVVLILFDWLRQ :& DQG D KRW VXSHUJLDQW FRPSRQHQW RI VSHFWUDO W\SH &RQWL DQG 6PLWK f *DQHVK DQG %DSSX f FDUULHG RXW WKH ILUVW VSHFWURVFRSLF VWXG\ LQ RUGHU WR GHWHUPLQH WKH RUELWDO SDUDPHWHUV RI WKH \ 9HO V\VWHP 8VLQJ WKH &,,,,9 EOHQG DW ƒ ZKLFK LV DVVXPHG WR EH IRUPHG LQ WKH HQYHORSH RI WKH :ROI5D\HW FRPSRQHQW DQG WKH K\GURJHQ DEVRUSWLRQ OLQH RI WKH VWDU DW ƒ WKH\ IRXQG DQ RUELWDO SHULRG RI A $ PRUH DFFXUDWH HVWLPDWH RI WKH SHULRG ZDV DFFRPSOLVKHG E\ 1LHPHOD DQG 6DKDGH f 7KH\ GHWHUPLQHG D UDGLDO YHORFLW\ FXUYH XVLQJ WKH VWURQJ YLROHWVKLIWHG DEVRUSWLRQ OLQH +HO DW ƒ 7KH SRVLWLRQ RI WKLV IHDWXUH ZDV PHDVXUHG UHODWLYH WR WKH + DEVRUSWLRQ OLQH RI WKH VXSHUJLDQW 7KH\ GHWHUPLQHG D SHULRG RI UHYROXWLRQ RI A ZLWK DQ HFFHQWULFLW\ RI 0RIIDW HW DO f FDUULHG RXW DQRWKHU UDGLDO YHORFLW\ DQDO\VLV RI WKH \ 9HO V\VWHP LQ DQ DWWHPSW WR GHILQH WKH RUELWDO HOHPHQWV RI WKH V\VWHP PRUH SUHFLVHO\ 7KH\ XVHG UDGLDO YHORFLW\ FXUYHV REWDLQHG IRU VHYHUDO HPLVVLRQ IHDWXUHV EXW ZHUH XQDEOH WR LPSURYH XSRQ WKH A SHULRG RU WKH HFFHQWULFLW\ +RZHYHU WKHLU UHILQHG HSKHPHULV FRXSOHG ZLWK WKH SHULRG RI 1LHPHOD DQG 6DKDGH LV JLYHQ E\ WKH IROORZLQJ HTXDWLRQ ( -' A '

PAGE 15

:LWK WKH DLG RI HTXDWLRQ f WKH VWDU LV IRXQG WR EH LQ IURQW RI WKH :& VWDU DW SKDVH DQG EHKLQG WKH :ROI 5D\HW FRPSRQHQW QHDU SKDVH $FFRUGLQJ WR WKH VWXG\ E\ 0RIIDW HW DO f WKH HSRFK RI SHULDVWURQ SDVVDJH LV GHWHUPLQHG WR EH -' $OWKRXJK DQ HFOLSVH KDV EHHQ GHWHFWHG LQ WKH XOWUDYLROHW UHJLRQ RI WKH VSHFWUXP :LOOLV DQG :LOVRQ f DW SKDVH QR GHILQLWH HFOLSVH HIIHFWV KDYH EHHQ UHFRUGHG DW RSWLFDO ZDYHOHQJWKV *DSRVFKNLQ f UHSRUWHG D G SHULRG ZLWK DQ DPSOLWXGH RI YDULDWLRQ UDQJLQJ EHWZHHQ DQG XVLQJ YLVXDO DQG SKRWRJUDSKLF PHDVXUHPHQWV EXW WR GDWH QR RWKHU ZRUN KDV EHHQ DEOH WR FRQILUP WKHVH UHVXOWV 7KH PDVVHV RI WKH LQGLYLGXDO FRPSRQHQWV FDQ EH GHWHUPLQHG IURP WKH VLPSOH UHODWLRQ ADEV AHP P:U 0 f ZKHUH .DEV DQG .HP DUH WKH DPSOLWXGHV RI WKH DEVRUSWLRQ DQG HPLVVLRQ UDGLDO YHORFLW\ FXUYHV DQG 0\J04J UHSUHVHQWV WKH UDWLR RI WKH PDVV RI WKH :& VWDU WR WKH VXSHUJLDQW $FFRUGLQJ WR WKH VWXG\ E\ 0RIIDW HW DO WKH EHVW YDOXHV IRU .DEV DQG .HP DUH JLYHQ E\ .DEV s NPVHF DQG .HP s NPVHF

PAGE 16

7KHVH YDOXHV DUH LQ JRRG DJUHHPHQW ZLWK LQGHSHQGHQWO\ GHWHUPLQHG UHVXOWV E\ 1LHPHOD DQG 6DKDGH EXW KDYH VPDOOHU XQFHUWDLQWLHV 7KH\ LQGLFDWH WKDW WKH PDVV RI WKH :ROI 5D\HW FRPSDQLRQ LV DSSUR[LPDWHO\ WLPHV WKDW RI WKH VWDU ZLWK PLQLPXP PDVVHV RI 0 IRU WKH :& FRPSRQHQW DQG 0T IRU WKH VXSHUJLDQW ,Q DQ DWWHPSW WR PHDVXUH WKH UDGLXV RI WKH LQGLYLGXDO FRPSRQHQWV RI \ 9HO DV ZHOO DV WKH VHSDUDWLRQ GLVWDQFH RI WKLV ELQDU\ V\VWHP %URZQ HW DO f REWDLQHG LQWHUIHURPHWU\ PHDVXUHPHQWV ZLWK WKH VWHOODU LQWHQVLW\ LQWHUIHURPHWHU DW WKH 1DUUDEUL 2EVHUYDWRU\ 7KHLU PHDVXUHPHQWV LQ WKH FRQWLQXXP DW ƒ LQFOXGHG WKH HIIHFWV RI ERWK \ 9HO DQG 9HO ZKLFK LV ORFDWHG DZD\ IURP \ +RZHYHU WKH DQDO\VLV LV VLPSOLILHG E\ WKH IDFW WKDW WKH DVVXPHG FRQWULEXWLRQ IURP 9HO LV QHJOLJLEOH 7KHVH GDWD LQGLFDWH DQ DQJXODU GLDPHWHU RI WKH :ROI5D\HW FRPSRQHQW RI nn s [ n DQG DQ DQJXODU VHPLPDMRU D[LV RI WKH RUELW RI s [ 8VLQJ VSHFWURVFRSLF REVHUYDWLRQV RI *DQHVK DQG %DSSX f WRJHWKHU ZLWK WKHVH DQJXODU PHDVXUHPHQWV WKH GLVWDQFH PRGXOXV IRU WKH 9HO V\VWHP LV s RU DERXW SF $ VLPSOH WULJRQRPHWULF FRPSXWDWLRQ \LHOGV D UDGLXV IRU WKH :& FRPSRQHQW RI 54 $ PRUH FRQYHQWLRQDO PHWKRG RI PHDVXULQJ +% LQGLFHV RI VHYHUDO VWDUV LQ WKH YLFLQLW\ RI 9HO UHVXOWV LQ D GLVWDQFH RI SF DQG KHQFH D UDGLXV IRU WKH :ROI5D\HW VWDU RI 54

PAGE 17

6LQFH DQ HFOLSVH LQ WKH RSWLFDO UHJLRQ RI WKH VSHFWUXP KDV QRW EHHQ GHWHFWHG WKH LQFOLQDWLRQ RI WKH V\VWHP FDQ EH DVVXPHG WR EH DSSUR[LPDWHO\ r 8VLQJ WKH DQJXODU VHPLn PDMRU D[LV RI WKH RUELW REWDLQHG E\ %URZQ HW DO WKH SURMHFWHG VHPLPDMRU D[LV LV s [ FP RU DVVXPLQJ L r [ FP 7KHVH PHDVXUHPHQWV LPSO\ D VHSDUDWLRQ GLVWDQFH RI DSSUR[LPDWHO\ 54 %URZQ HW DO DOVR REWDLQHG PHDVXUHPHQWV RI WKH &,,,,9 HPLVVLRQ OLQH RI \ 9HO DW ƒ 7KHVH GDWD LQFOXGH HIIHFWV RI ERWK FRPSRQHQWV RI \ 9HO LQ DGGLWLRQ WR WKH HIIHFWV IURP WKH PDWHULDO VXUURXQGLQJ WKH :ROI5D\HW FRPSRQHQW 7KH LQWHUIHURPHWU\ PHDVXUHPHQWV JLYH DQ DQJXODU GLDPHWHU RI WKH UHJLRQ HPLWWLQJ DW WKH &,,,,9 IUHTXHQF\ RI f s [ n ,I RQH DVVXPHV WKH SF GLVWDQFH HVWDEOLVKHG E\ %URZQ HW DO WKHVH GDWD LQGLFDWH WKDW WKH &,,,,9 EOHQG LV IRUPHG LQ D UHJLRQ DURXQG WKH :& FRPSRQHQW DQG DW D GLVWDQFH RI s 5T IURP WKDW VWDU :RRG f GHULYHG D UHODWLRQVKLS EHWZHHQ WKH GLPHQVLRQV RI 5RFKH HTXLSRWHQWLDO VXUIDFHV UHODWLYH WR WKH VHPLPDMRU D[LV DV D IXQFWLRQ RI WKH PDVV UDWLR *LYHQ WKH GHVFULEHG GLPHQVLRQV IRU \ 9HO WKH UDGLXV RI WKH 5RFKH OREH LV 54 +HQFH HYLGHQFH LQGLFDWHV WKDW WKLV HPLVVLRQ IHDWXUH DW ƒ LV IRUPHG LQ WKH RXWHU UHJLRQV RI WKH FLUFXPVWHOODU PDWHULDO DQG LQ IDFW LQ PDWHULDO ZKLFK QHDUO\ ILOOV WKH 5RFKH OREH RI WKH 9HO ELQDU\ V\VWHP 7KHVH DQG RWKHU SURSHUWLHV RI WKLV LQWULJXLQJ V\VWHP DUH VXPPDUL]HG LQ 7DEOH ,Q OLJKW RI

PAGE 18

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f :HVW f ZDV WKH ILUVW DXWKRU WR DVVHUW WKDW YDULDWLRQ LQ OLQH LQWHQVLWLHV PLJKW DOVR H[LVW 7KH UHVXOWV RI KRXUV RI REVHUYDWLRQ ZLWK WKH 2UELWLQJ $VWURQRPLFDO 2EVHUYDWRU\ $ LQGLFDWHG QR VKRUW WHUP YDULDWLRQV LQ WKH &,,, HPLVVLRQ OLQH DW ƒ 6LQFH WKHQ KRZHYHU VHYHUDO DXWKRUV KDYH SUHVHQWHG HYLGHQFH RI YDULDWLRQV LQ ERWK OLQH SURILOH DQG OLQH VWUHQJWK -HIIHUV HW DO Df ILUVW UHSRUWHG D PLQXWH SHULRGLFLW\ LQ ERWK OLQH LQWHQVLW\ DQG OLQH VKDSH XVLQJ SKRWRPHWULF PHDVXUHPHQWV RI WKH +HOO ƒf DQG &,,,,9 ƒf IHDWXUHV $ ODWHU UHDQDO\VLV RI WKHLU GDWD Ef UHYHDOHG HYHQ PRUH UDSLG IOXFWXDWLRQV RQ WKH RUGHU RI s VHFRQGV ZLWK DQ DPSOLWXGH RI RQO\ DERXW b

PAGE 19

7DEOH 3URSHUWLHV RI WKH 9HORUXP 6\VWHP 6SHFWURVFRSLF ELQDU\ :& Df 3 A Ef 9 f F f PZU r VLQL 0 2 Gf 0 r VLQL 0 2 H f T I f UZU s 5 Jf 5 s 25 2 Kf D r VLQLf 5 Lf ++H

PAGE 20

$XVWLQ 6FKQHLGHU DQG :RRG f FDUULHG RXW D YHU\ H[WHQVLYH SKRWRPHWULF VWXG\ RI \ 9HO DW WKH 0W -RKQ 2EVHUYDWRU\ LQ 1HZ =HDODQG 7KH\ XVHG VL[ QDUURZEDQG ILOWHUV ZLWK $$ ƒ 7KUHH RI WKHVH ILOWHUV LVRODWHG WKH HPLVVLRQ OLQHV LGHQWLILHG DV +HOO DW ƒ &,,, DW ƒ DQG &,9 DW ƒ WKH RWKHU WKUHH ILOWHUV ZHUH FHQWHUHG RQ WKH FRQWLQXXP DW ƒ ƒ DQG ƒ 7KHVH DXWKRUV IRXQG WKDW WKH FRQWLQXXP UHPDLQHG HVVHQWLDOO\ FRQVWDQW DV GLG WKH WULSO\LRQL]HG FDUERQ IHDWXUH DW OHDVW LQ WKH WLPH LQWHUYDO VSDQQHG E\ WKHLU REVHUYDWLRQV $XVWLQ HW DO GHWHFWHG GHILQLWH QLJKWWR QLJKW YDULDWLRQV LQ WKH VWUHQJWKV RI WKH &,,, DQG +HOO IHDWXUHV 7KH DPSOLWXGH RI YDULDWLRQ IRU WKH GRXEO\LRQL]HG FDUERQ OLQH DPRXQWHG WR DERXW ," )OXFWXDWLRQV RI DOVR RFFXUUHG LQ D WLPH SHULRG RI OHVV WKDQ KRXUV 7KH VLQJO\ LRQL]HG KHOLXP IHDWXUH H[KLELWHG DQ LQFUHDVH LQ EULJKWQHVV RI r1 LQ OHVV WKDQ PLQXWHV EXW $XVWLQ HW DO DOVR QRWH WKDW WKLV OLQH DSSHDUV WR UHPDLQ TXLWH VWDEOH LH ZLWKLQ Af IRU UHODWLYHO\ ORQJ WLPH LQWHUYDOV $XVWLQ HW DO FDUULHG RXW D PRUH LQWHQVLYH \ 9HO REVHUYLQJ SURJUDP FRQFHQWUDWLQJ RQO\ RQ YDULDWLRQV LQ WKH +HOO OLQH 7KHLU UHVXOWV ZHUH LQFRQFOXVLYH \ 9HO H[KLELWHG QR GHWHFWDEOH YDULDWLRQV LQ D PLQXWH WLPH SHULRG RQ WKH ILUVW QLJKW RI REVHUYDWLRQ 2Q WKH VHFRQG QLJKW KRZHYHU WKH LRQL]HG +HOO OLQH LQFUHDVHG LQ EULJKWQHVV E\ DV PXFK DV 2f3O2 LQ PLQXWHV DQG DJDLQ ODWHU WKDW QLJKW EULJKWHQHG E\ WKH VDPH DPRXQW LQ RQO\ PLQXWHV ,Q VXPPDU\ $XVWLQ

PAGE 21

HW DO FRQILUPHG SUHYLRXV UHVXOWV WKDW UDSLG YDULDWLRQV FDQ RFFXU DOWKRXJK QRW ZLWK WKH UDSLGLW\ RI WKH GHJUHH UHSRUWHG E\ -HIIHUV HW DO 7KH\ IRXQG WKHVH YDULDWLRQV WR EH WHPSRUDO DQG QRW SHULRGLF $QRWKHU SKRWRPHWULF VWXG\ FDUULHG RXW E\ /LQGJUHQ HW DO f GLG QRW SURYLGH HYLGHQFH IRU VWDEOH YDULDWLRQV EHWZHHQ DQG PLQXWHV LQ OHQJWK LQ HLWKHU WKH &,,,,9 OLQH DW ƒ RU WKH +HOO OLQH DW ƒ +RZHYHU /LQGJUHQ HW DO GLG FRQILUP QLJKWO\ YDULDWLRQV LQ HPLVVLRQ OLQH VWUHQJWK LQ SDUWLFXODU D A WR FKDQJH LQ WKH ƒ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ƒ WR ƒ ,Q %DKQJ UHSRUWHG RQ UHVXOWV RI VSHFWUXP VFDQV ZKLFK ZHUH DFTXLUHG RQ IRXU GLIIHUHQW QLJKWV 7KH HPLVVLRQ OLQHV RI SDUWLFXODU LQWHUHVW ZHUH &,,,,9 ƒf DQG +HOO ƒf $OWKRXJK D SRZHU VSHFWUXP DQDO\VLV GLG QRW \LHOG HYLGHQFH RI D SHULRGLF SKHQRPHQRQ %DKQJ GLG ILQG YDULDWLRQV LQ WKH HTXLYDOHQW ZLGWKV RI WKHVH OLQHV 7KHVH FKDQJHV DPRXQWHG WR

PAGE 22

D b YDULDWLRQ IRU &,,,&,9 DQG D b YDULDWLRQ IRU +HOO DQG RFFXUUHG RQ WLPH VFDOHV RI WR PLQXWHV /DWHU %DWLQJ f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nL2O ZLWK D SHULRGLFLW\ RI VHFRQGV ,Q DGGLWLRQ D VLJQLILFDQW SHDN DW PLQXWHV ZDV DOVR SUHVHQW LQ WKH GDWD :H KDYH SUHVHQWHG RQO\ D EULHI VXPPDU\ RI VRPH RI WKH PRUH QRWDEOH REVHUYDWLRQDO VWXGLHV RI \ 9HO ZKLFK KDYH DSSHDUHG LQ WKH OLWHUDWXUH RYHU WKH ODVW IHZ \HDUV $V SRLQWHG RXW E\ +DHIQHU HW DO f XOWUDVKRUW SHULRG YDULDWLRQV LH WKRVH RQ WKH RUGHU RI PLQXWHVf VHHP WR EH

PAGE 23

GHWHFWHG RQO\ LQ REVHUYDWLRQV ZKLFK GR QRW H[FHHG PXFK PRUH WKDQ PLQXWHV LQ OHQJWK )OXFWXDWLRQV RI WKLV QDWXUH DUH QRW JHQHUDOO\ GHWHFWHG LQ ORQJHU UXQV RI FRQWLQXRXV GDWD EXW YDULDWLRQV LQ ERWK HPLVVLRQ OLQH VWUHQJWK DQG SURILOH DUH DOPRVW DOZD\V GHWHFWHG RQ D QLJKWO\ EDVLV

PAGE 24

&+$37(5 ,167580(17$7,21 $1' '$7$ $&48,6,7,21 6<67(0 ,QVWUXPHQWDWLRQ 7KH GDWD XVHG IRU WKLV UHVHDUFK ZHUH REWDLQHG ZLWK WKH DXWRPDWHG RSWLFDO WHOHVFRSH ORFDWHG DW WKH $PXQGVHQ6FRWW 6RXWK 3ROH 6WDWLRQ RQ $QWDUFWLFD 7KLV LQVWUXPHQW LV D WZR PLUURU VLGHURVWDW ]HQLWK WHOHVFRSH ZLWK DQ I FP DFKURPDWLF OHQV )LJXUH f $V RQH ZRXOG H[SHFW FHUWDLQ VSHFLDO GHVLJQ FRQVLGHUDWLRQV DUH QHFHVVDU\ WR HQVXUH QRUPDO RSHUDWLRQ LQ WKH H[WUHPHO\ RZWHPSHUDWXUH KDUVK HQYLURQPHQW ZKLFK SUHYDLOV DW WKH 6RXWK 3ROH $ VSHFLDOO\ LQVXODWHG EXLOGLQJ PHDVXULQJ n [ n [ n ZDV FRQVWUXFWHG RQ VLWH E\ FDUSHQWHUV RI ,77 $QWDUFWLF 6HUYLFHV GXULQJ WKH DXVWUDO VXPPHU 7KH WHOHVFRSH RFFXSLHV DQ n [ n VHFWLRQ RI WKH EXLOGLQJ ZKLOH WKH FRPSXWHU V\VWHP DQG FRQWURO HOHFWURQLFV DUH KRXVHG LQ DQ DGMDFHQW URRP PHDVXULQJ n [ f 7KLV VPDOOHU URRP FDQ EH FRPSOHWHO\ LVRODWHG IURP WKH WHOHVFRSH URRP DQG KHDWHG ZKHQ QHFHVVDU\ 8QGHU QRUPDO FLUFXPVWDQFHV WKH WHOHVFRSH URRP WHPSHUDWXUH LV PDLQWDLQHG DW DSSUR[LPDWHO\ r& 7KH RSWLFDO KHDG RI WKH WHOHVFRSH FRQWDLQV WZR RSWLFDOO\I DW IURQWVXUIDFHG PLUURUV ZKLFK DFW WR UHGLUHFW

PAGE 25

)LJXUH 6RXWK 3ROH 2SWLFDO 7HOHVFRSH

PAGE 26

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f WKURXJK D F\OLQGULFDO WXEH FRQWDLQLQJ D GHVLFFDQW DQG LQWR WKH RSWLFDO KHDG RI WKH WHOHVFRSH +HUH WKH DLU LV KHDWHG VR WKDW D WHPSHUDWXUH JUDGLHQW RI DERXW r& LV PDLQWDLQHG EHWZHHQ WKH DLU LQ WKH RSWLFDO KHDG DQG WKH DLU LPPHGLDWHO\ RXWVLGH RI WKH HQWUDQFH ZLQGRZ ZLWK WKH LQVLGH DLU EHLQJ WKH ZDUPHU 2I WKH HQWLUH GHVLJQ RI WKH WHOHVFRSH WKH RQO\ PRYLQJ SDUW ZKLFK LV GLUHFWO\ H[SRVHG WR DPELHQW FRQGLWLRQV LV WKH HOHYDWLRQ D[OH 7KH JHDU V\VWHP ZKLFK PRYHV WKH WHOHVFRSH LQ WKH HDVW WR ZHVW GLUHFWLRQ DOVR PRYHV WKH HQWLUH WHOHVFRSH WXEH LQ ERWK D]LPXWKDO GLUHFWLRQV 7KH D]LPXWKPRWLRQ UHVLGHV ZHOO ZLWKLQ WKH WHOHVFRSH URRP DQG D ZRUP JHDU

PAGE 27

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f LV VWUDLJKW IRUZDUG ,QFRPLQJ OLJKW UD\V VWULNH WKH REMHFWLYH DIWHU EHLQJ GHIOHFWHG E\ WKH WZR IURQW VXUIDFHG PLUURUV LQ WKH RSWLFDO KHDG HDFK LV SRVLWLRQHG DW DERXW r ZLWK UHVSHFW WR WKH QRUPDO $V WKH SKRWRQV HPHUJH IURP WKH OHQV WKH\ SDVV WKURXJK D GLDSKUDJP D ILOWHU DQG D IDEU\ OHQV ZKLFK LPDJHV WKH REMHFWLYH RI WKH WHOHVFRSH RQWR WKH SKRWRPXOWLSOLHU ,W LV WKH SKRWRPHWHU ZKLFK PHDVXUHV WKH VLJQDO DQG FRQYHUWV LW LQWR DQ HTXLYDOHQW QXPEHU RI SKRWRQV 7KH 6RXWK 3ROH 2SWLFDO 7HOHVFRSH 6327f LV HTXLSSHG ZLWK VHYHUDO GLDSKUDJPV DQG ILOWHUV DOORZLQJ WKH VHOHFWLRQ RI DQ\ RQH RI D QXPEHU RI FRPELQDWLRQV GHSHQGLQJ XSRQ WKH EULJKWQHVV RI WKH VRXUFH WKH WUDFNLQJ UDWH RI WKH WHOHVFRSH DQG WKH W\SH RI REVHUYDWLRQ GHVLUHG 7KH FXUUHQW V\VWHP KDV GLDSKUDJPV ZKLFK UHVWULFW WKH ILHOG RI YLHZ WR r n n RU n UHJLRQV RI WKH VN\ 7KH ILHOG RI YLHZ

PAGE 28

LV XVHG RQO\ LQ WKH LQLWLDO VWDU SRLQWLQJ SURFHGXUH ZKLOH WKH n n DQG n GLDSKUDJPV DUH XVHG LQ WKH VWDU FHQWHULQJ SURFHGXUH DQG IRU WKH DFWXDO PHDVXUHPHQW RI WKH REMHFWnV EULJKWQHVV WKH VPDOOHU WKH GLDSKUDJP WKH VPDOOHU WKH FRQWULEXWLRQ IURP VN\ EDFNJURXQG DQG WKH IDLQWHU WKH OLPLWLQJ PDJQLWXGH RI WKH WHOHVFRSH 7KH ILOWHUV ZKLFK DUH FXUUHQWO\ PRXQWHG LQ WKH ILOWHU ZKHHO LQFOXGH WKH -RKQVRQ VWDQGDUG % DQG 9 ILOWHUV D QHXWUDO GHQVLW\ ILOWHU DQG IRXU QDUURZEDQG ILOWHUV 7KH QHXWUDO GHQVLW\ ILOWHU LV D 1' DQG LV XVHG LQ WKH VWDU ILQG SURFHGXUH 7KLV ILOWHUnV VROH SXUSRVH LV WR SUHYHQW WKH WHOHVFRSH IURP REVHUYLQJ D VRXUFH ZKLFK LV WRR EULJKW WKDW LV RQH ZKLFK FRXOG VDWXUDWH DQG VXEVHTXHQWO\ GDPDJH WKH SKRWRPXOWLSOLHU 2I WKH QDUURZEDQG ILOWHUV RQH LV XVHG WR PRQLWRU DXURUDO DQGRU VN\ YDULDWLRQV 7KLV ILOWHU LV FHQWHUHG RQ WKH YHU\ VWURQJ HPLVVLRQ OLQH DW ƒ 7KH UHPDLQLQJ WKUHH QDUURZEDQG ILOWHUV ZHUH FKRVHQ VSHFLILFDOO\ IRU RXU UHVHDUFK RQ WKH :ROI5D\HW VWDU 9HO 7ZR RI WKHVH ILOWHUV LVRODWH HPLVVLRQ IHDWXUHV LQ WKH VSHFWUXP RI 9HO WKH +HOO HPLVVLRQ OLQH DW ƒ DQG WKH &,,, HPLVVLRQ OLQH DW ƒ 7KH WKLUG ILOWHU LVRODWHV WKH FRQWLQXXP UHJLRQ DW ƒ 0RUH VSHFLILF GHWDLOV RI HDFK RI WKHVH ILOWHUV DUH SUHVHQWHG LQ 7DEOH $OVR WKH UHVSRQVH FXUYHV IRU VL[ RI WKH ILOWHUV DUH SUHVHQWHG LQ )LJXUHV WKURXJK ,Q DGGLWLRQ WR WKH ILOWHUV RQH SRVLWLRQ RQ WKH ILOWHU ZKHHO FRQWDLQV D PLUURU ZKLFK GLUHFWV WKH OLJKW EHDP SHUSHQGLFXODU

PAGE 29

7DEOH )LOWHU 6SHFLILFDWLRQV 3HDN :DYHOHQJWK $f +DOI3RZHU EDQGZLGWK $f ,QWHJUDWLRQ WLPH VHFVf 3XUSR VH % 9 +HOO FP &RQWLQXXP

PAGE 30

7UDQVPLWWDQFH R :DYHOHQJWK $ f )LJXUH 5HVSRQVH FXUYHV IRU WKH 8 2f % $f DQG 9 f ILOWHUV &2

PAGE 31

7UDQVPLWWDQFH 7 R R R R R R f§ R R R R :DYHOHQJWK $ f R )LJXUH 5HVSRQVH FXUYH IRU WKH KHOLXP HPLVVLRQ ILOWHU ‘)

PAGE 32

7UDQVPLWWDQFH R R R R R R / L R :DYHOHQJWK $ f R R R L R )LJXUH 5HVSRQVH FXUYH IRU WKH FDUERQ HPLVVLRQ ILOWHU UR /Q

PAGE 33

7UDQVPLWWDQFH R f§ f§ f§ R R R R R R R R R R R f§ R R R R R R R B R :DYHOHQJWK $ f )LJXUH 5HVSRQVH FXUYH IRU WKH R[\JHQ HPLVVLRQ ILOWHU A

PAGE 34

7UDQVPLWWDQFH R :DYHOHQJWK $ f )LJXUH 5HVSRQVH FXUYH IRU WKH FRQWLQXXP ILOWHU UR

PAGE 35

WR WKH RSWLFDO D[LV ZKHUH LW FDQ EH YLHZHG WKURXJK DQ H\HSLHFH 7KLV VHFWLRQ RI WKH WHOHVFRSH LV PDLQWDLQHG DW D WHPSHUDWXUH EHORZ r& LQ DQ HIIRUW WR UHGXFH WKH GDUN FXUUHQW LQ WKH V\VWHP 'XULQJ WKH WKUHH \HDUV LQ ZKLFK WKH WHOHVFRSH KDV EHHQ RSHUDWLQJ WKLV OHYHO KDV UHPDLQHG QHDUO\ FRQVWDQW DW FRXQWVVHF 2WKHU UHJLRQV RI WKH WHOHVFRSH DUH DOVR PDLQWDLQHG DW RSWLPXP WHPSHUDWXUHV DFFRUGLQJ WR D WKHUPDO GHVLJQ DQDO\VLV FDUULHG RXW E\ (VSHU f 7KHVH WHPSHUDWXUHV DUH DFKLHYHG ZLWK WKHUPDO LQVXODWLRQ KHDWHUV DQG UHVLGXDO KHDW JHQHUDWHG E\ VWHSSLQJ PRWRUV DQG RWKHU HOHFWURQLF FRPSRQHQWV 7KH HQWLUH WKHUPDO GHVLJQ RI WKH WHOHVFRSH ZRUNV LQ FRQFHUW ZLWK DPELHQW WHPSHUDWXUHV UDQJLQJ IURP r& WR r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

PAGE 36

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

PAGE 37

FRPSXWHU LQ 0DODEDU )ORULGD YLD WKH $76 VDWHOOLWH FRPPXQLFDWLRQ OLQN 7KHVH GDWD DUH UHWULHYHG IURP WKH 9$; V\VWHP LQ 0DODEDU DQG VWRUHG RQ WKH $769$; DW WKH 8QLYHUVLW\ RI )ORULGD ZKHUH WKH UHGXFWLRQ RI WKH GDWD SURFHHGV

PAGE 38

&+$37(5 '$7$ 5('8&7,21 7(&+1,48(6 'XULQJ WKH DQG REVHUYLQJ VHDVRQV 6327 REWDLQHG GDWD IRU VHYHUDO GLIIHUHQW UHVHDUFK SURMHFWV ,Q RUGHU WR RSWLPL]H WHOHVFRSH WLPH DQG WR PDNH WKH EHVW XVH RI FOHDU VNLHV WKH GDWD IRU PRUH WKDQ RQH UHVHDUFK SURMHFW ZHUH XVXDOO\ DFTXLUHG ZLWKLQ WKH VDPH REVHUYLQJ SURJUDP 7KH UHVHDUFK SURMHFW WKDW LV WKH VXEMHFW RI WKLV GLVVHUWDWLRQ KDV WZR PDMRU REMHFWLYHV f WKH VHDUFK IRU YDULDWLRQV LQ HPLVVLRQ OLQH VWUHQJWKV XVLQJ WKH +HOO &,,, DQG FRQWLQXXP ILOWHUV DQG f WKH VHDUFK IRU HFOLSVH HIIHFWV UHTXLULQJ REVHUYDWLRQV LQ WKH % DQG 9 ILOWHUV DV ZHOO DV LQWHJUDWLRQV RQ WKH FRPSDULVRQ VWDU +5 ,Q VRPH REVHUYLQJ SURJUDPV ERWK \ 9HO DQG +5 ZHUH REVHUYHG XVLQJ DOO DYDLODEOH ILOWHUV ,Q RWKHU FDVHV DQ LQFUHDVH LQ WKH WLPHUHVROXWLRQ RI WKH GDWD ZDV DFFRPSOLVKHG E\ UHPRYLQJ +5 IURP WKH REVHUYLQJ SURJUDP DQG RPLWWLQJ WKH % DQG 9 ILOWHU REVHUYDWLRQV 7KHUHIRUH VHYHUDO GLIIHUHQW REVHUYLQJ VHTXHQFHV ZHUH UHTXLUHG WR REWDLQ WKH GDWD QHFHVVDU\ IRU WKLV LQYHVWLJDWLRQ 6SHFLILF GHWDLOV RI WKHVH SURJUDPV ZLOO EH SUHVHQWHG LQ &KDSWHU ZKHUH LQGLYLGXDO GDWD VHWV DUH GLVFXVVHG ,Q JHQHUDO KRZHYHU WKH GDWD DFTXLUHG IRU WKLV SURMHFW ZHUH REWDLQHG ZLWK WKH XVH RI WKH n GLDSKUDJP LQ

PAGE 39

FRQMXQFWLRQ ZLWK WKH % 9 +HOO &,,, DQG FRQWLQXXP ILOWHUV $V RQH ZRXOG H[SHFW WKH SKRWRPHWULF LQWHJUDWLRQ WLPHV QHFHVVDU\ WR REWDLQ DGHTXDWH VLJQDOV IRU D JLYHQ VWDU DQG ILOWHU YDULHV IURP VWDU WR VWDU DQG IURP ILOWHU WR ILOWHU 7KH UHTXLUHG LQWHJUDWLRQ OHQJWK GHSHQGV RQ D SDUWLFXODU VWDUn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

PAGE 40

7DEOH 2SWLPXP ,QWHJUDWLRQ 7LPHV IRU \ 9HO DQG +5 6WDU )LOWHU ,QWHJUDWLRQ WLPH VHFVf 9HO % 9 +HOO &O &RQWLQXXP +5 % 9

PAGE 41

WHOHVFRSH EXW ZKLFK LV UHDVRQDEO\ FORVH WR WKH VWDU XQGHU REVHUYDWLRQ ,Q DQ DXWRPDWHG WHOHVFRSH V\VWHP LW LV QHFHVVDU\ WR PDNH FHUWDLQ WKDW ERWK RI WKHVH FULWHULD DUH VDWLVILHG $ VHDUFK RI WKH 3DORPDU 6N\ 6XUYH\ f SULQWV LQGLFDWHV WKDW D GLVSODFHPHQW RI n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n WLPH WKH GDWD ZHUH LQLWLDOO\ SDVVHG WKURXJK D SURJUDP ZKLFK XQSDFNHG WKH GDWD IURP KH[DGHFLPDO WR GHFLPDO IRUPDW DQG GLYLGHG WKH GDWD LQWR WZR ILOHV DQ HQJLQHHULQJ ILOH FRQWDLQLQJ LQIRUPDWLRQ UHODWLQJ WR WKH GHVLJQ DQG RSHUDWLRQ RI WKH WHOHVFRSH DQG D VHSDUDWH SKRWRPHWULF ILOH FRQWDLQLQJ WKH GDWD UHODWLQJ WR WKH VWHOODU GHIOHFWLRQV 7KH UDZ VWHOODU GHIOHFWLRQV ZHUH WKHQ UHGXFHG WR SKRWRPHWULF PDJQLWXGHV XVLQJ VWDQGDUG UHGXFWLRQ WHFKQLTXHV +HQGHQ DQG .DLWFKXFN f 6LQFH HDFK GHIOHFWLRQ LV UHFRUGHG DV D FRXQWVSHULPH TXDQWLW\ DQG VLQFH DQ LQWHJUDWLRQ WLPH YDULHV IURP ILOWHU WR ILOWHU LW LV QHFHVVDU\ WR FRQYHUW DOO PHDVXUHPHQWV WR WKH VDPH XQLW D FRXQWV SHU VHFRQG TXDQWLW\

PAGE 42

2QFH WKLV LV GRQH WKH QH[W VWHS LV WR UHPRYH WKH VN\ EDFNJURXQG IURP HDFK VWHOODU PHDVXUHPHQW 7KLV LV DFFRPSOLVKHG WKURXJK D VLPSOH OLQHDU LQWHUSRODWLRQ RI WKH VN\ UHDGLQJV WR WKH WLPH RI WKH VWDU UHDGLQJ 7KH QXPEHU RI FRXQWV SHU VHFRQGf GXH WR WKH EDFNJURXQG DUH WKHQ VXEWUDFWHG IURP WKH WRWDO FRXQWV SHU VHFRQGf GXH WR WKH VWDU ,Q WKH FDVH RI WKH % DQG 9 GDWD +5 ZDV FKRVHQ DV WKH FRPSDULVRQ VWDU +5 LV YHU\ FORVH LQ FRORU WR \ 9HO ZLWK D VSHFWUDO FODVVLILFDWLRQ RI %9 DQG LV ORFDWHG RQO\ n IURP WKH YDULDEOH )RU WKHVH UHDVRQV DQG WKH IDFW WKDW +5 KDV QRW EHHQ UHSRUWHG WR EH YDULDEOH LW PHHWV WKH FULWHULD IRU D UHDVRQDEO\ JRRG FRPSDULVRQ VWDU $ VHFRQG OLQHDU LQWHUSRODWLRQ VFKHPH LV WKHQ XVHG WR GHWHUPLQH WKH LQWHQVLW\ RI WKH FRPSDULVRQ VWDU +5 DW WKH WLPH RI WKH YDULDEOH VWDU UHDGLQJ )LQDOO\ HTXDWLRQ f LV XVHG WR JLYH WKH GLIIHUHQWLDO SKRWRPHWULF PDJQLWXGH $P r ORJ f ZKHUH $P LV WKH GLIIHUHQWLDO PDJQLWXGH EHWZHHQ \ 9HO DQG +5 DQG 'Y DQG 'F DUH WKH GHIOHFWLRQV LQ FRXQWV SHU VHFRQGf RI \ 9HO DQG RI +5 UHVSHFWLYHO\ $ PDMRU SRUWLRQ RI WKLV UHVHDUFK LV WKH VHDUFK IRU SHULRGLF LQWHQVLW\ YDULDWLRQV LQ WKH +HOO DQG &,,, HPLVVLRQ OLQHV RI \ 9HO 2QH WHFKQLTXH ZKLFK LV FRPPRQO\ HPSOR\HG LQ

PAGE 43

SHULRG GHWHUPLQDWLRQ DQDO\VHV XVHV WKH PHWKRG RI )RXULHU WUDQVIRUPDWLRQ RI WLPH DQG LQWHQVLW\ PHDVXUHPHQWV ,Q SDUWLFXODU 'HHPLQJnV PHWKRG f RI )DVW )RXULHU WUDQVIRUPV LV ZHOO VXLWHG IRU GDWD ZKLFK DUH XQHTXDOO\ VSDFHG LQ WLPH ,Q JHQHUDO )RXULHU WUDQVIRUP DQDO\VLV LV XVHG WR WUDQVIRUP GDWD IURP WKH WLPH GRPDLQ LQWR WKH IUHTXHQF\ GRPDLQ RU YLFHYHUVD )RU LQVWDQFH D FRQWLQXRXV IXQFWLRQ RI WLPH IWf FDQ EH WUDQVIHUUHG LQWR IUHTXHQF\ VSDFH DFFRUGLQJ WR WKH IRUPXOD )Mf )7 > I @ IWf r H[S QLLWf GW f 7KH LQYHUVH )RXULHU WUDQVIRUP LV JLYHQ E\ IWf )7 > ) @ )Yf r H[S ZLLWf GX f ,Q HTXDWLRQ f )Wf UHSUHVHQWV D VSHFWUXP RI IUHTXHQFLHV (DFK SHDN LQ )Yf FRUUHVSRQGV WR WKH UHODWLYH VWUHQJWK RI HDFK IUHTXHQF\ LQ WKH GDWD IWf ,Q PRVW FDVHV DQG LQ WKLV GDWD IWf LV QRW D FRQWLQXRXV IXQFWLRQ EXW UDWKHU D VDPSOHGVHW GHWHUPLQHG DW GLVFUHWH WLPHV WA +HQFH HTXDWLRQ f FDQ EH UHZULWWHQ DV D GLVFUHWH WUDQVI RUP 1 )Yf ( >IWf r H[S }ULLWf@ f

PAGE 44

ZLWK 1 HTXDO WR WKH QXPEHU RI GDWD SRLQWV LQ IWf 7KH SHULRGLF H[SRQHQWLDO IXQFWLRQ LQ HTXDWLRQ f LQGLFDWHV WKDW WKH )RXULHU WUDQVIRUP RI IWf KDV ERWK UHDO DQG LPDJLQDU\ FRPSRQHQWV 6LQFH LW LV WKH SRZHU DV D IXQFWLRQ RI IUHTXHQF\ UDWKHU WKDQ WKH )RXULHU WUDQVIRUP WKDW LV RI LQWHUHVW WKH GHULYDWLRQ PXVW EH FDUULHG RQH VWHS IXUWKHU ,I IWf LV DVVXPHG WR EH UHDO WKHQ )Yf PXVW VDWLVI\ WKH UHODWLRQ )mf )r Lf f ZKHUH )r Xf LV WKH FRPSOH[ FRQMXJDWH RI )\f 7KHQ DFFRUGLQJ WR 5D\OHLJKnV WKHRUHP ZKLFK VWDWHV IWf GW f ZH FDQ GHILQH WKH SRZHU 3Lf DV WKH SURGXFW RI )Lf DQG LWV FRPSOH[ FRQMXJDWH ZKLFK OHDYHV RQO\ WKH UHDO TXDQWLW\ 3 Lf )LAf & Yf 6 Lf f &Lf DQG 6\f LQ HTXDWLRQ f DUH WKH FRVLQH DQG VLQH FRPSRQHQWV UHVSHFWLYHO\ DQG DUH GHILQHG DV &ff A 6 >IWf r FRV MQWf@ DQG f

PAGE 45

6Lf = >I Wf r VLQ QYWf @ f 6LQFH WKH GDWD XVHG LQ WKLV LQYHVWLJDWLRQ DUH DFTXLUHG RYHU D ILQLWH VSDQ RI WLPH DQG DUH QRW VDPSOHG DW HTXDO WLPH LQWHUYDOV WKH )RXULHU WUDQVIRUP LV FRQWDPLQDWHG ZLWK WKH VDPSOLQJ IXQFWLRQ 7KLV VSHFWUDO ZLQGRZ RU EHDP DV LW LV IUHTXHQWO\ UHIHUUHG WR LV JLYHQ E\ WKH UHODWLRQ :Lf Yf + Mf f ,Q f *Lf DQG +Lf DUH GHILQHG LQ WKH IROORZLQJ PDQQHU *\f 6 >FRV LULWf@ DQG f +Af 6 >VLQ LULWf @ f $FFRUGLQJ WR HTXDWLRQV f WR f WKH VSHFWUDO ZLQGRZ IXQFWLRQ LV QRUPDOL]HG VR WKDW :f 7KH )RXULHU FRQYROXWLRQ WKHRUHP VWDWHV WKDW WKH )RXULHU WUDQVIRUP RI WKH VLJQDO )7>I@ LV WKH FRQYROXWLRQ RI WKH GDWD ZLWK WKH VDPSOLQJ IXQFWLRQ LH EHDPf 7KH IDFW WKDW WKH GDWD DUH XQHTXDOO\ VSDFHG RYHU D ILQLWH WLPH LQWHUYDO KDV VHYHUDO FRQVHTXHQFHV 7KH VDPSOLQJ SHULRG RI WKH GDWD SXWV D FRQVWUDLQW RQ WKH ORZHVW IUHTXHQF\ ZKLFK LV UHVROYDEOH LQ WKH GDWD 7KH IUHTXHQF\ UHVROXWLRQ LQ D GLVFUHWH GDWD VHW FRUUHVSRQGV WR WKH ZLGWK RI WKH EHDP :f DW X DQG LV

PAGE 46

UHODWHG WR WKH VDPSOLQJ SHULRG E\ HTXDWLRQ f 6X m 7 f ZKHUH 7 LV WKH OHQJWK RI WKH GDWD VHW 7KHUHIRUH WKH PLQLPXP IUHTXHQF\ ZKLFK LV UHWULHYDEOH IURP D JLYHQ GDWD VHW LV JLYHQ E\ WKH 1\TXLVW IUHTXHQF\ fnPLQ f $7f f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f fnPD[ f O$Wf f ,Q DGGLWLRQ WR XSSHU DQG ORZHU OLPLWV EHLQJ SODFHG RQ WKH IUHTXHQFLHV ZKLFK FDQ EH UHFRYHUHG IURP WKH GDWD WKH GDWD DUH DOVR QRW UHSUHVHQWHG E\ D FRQWLQXRXV IXQFWLRQ RI WLPH 7KH GLVFUHWH QDWXUH RI WKH GDWD FRQWULEXWHV VHYHUDO

PAGE 47

IHDWXUHV WR WKH VSHFWUDO ZLQGRZ EHDPf ZKLFK DGGV FRQVLGHUDEOH FRPSOH[LW\ WR WKH QDWXUH RI WKH VSHFWUXP 6LQFH WKH )RXULHU WUDQVIRUP RI WKH GDWD LV D FRQYROXWLRQ RI WKH VSHFWUXP ZLWK WKH ZLQGRZ WKH UHVXOWLQJ GLUW\ VSHFWUXP PD\ EH VXEVHTXHQWO\ FRQWDPLQDWHG ZLWK VSXULRXV VDPSOLQJ IHDWXUHV 7KLV DOLDVLQJ FDQ EH PDVNHG DV HLWKHU D GDPSLQJ RI UHDO IHDWXUHV RU DQ HQKDQFHPHQW RI IDOVH IHDWXUHV PDNLQJ LQWHUSUHWDWLRQ GLIILFXOW )RU WKLV UHDVRQ D RQHn GLPHQVLRQDO GHFRQYROXWLRQ &/($1f DOJRULWKP +RJERP f KDV EHHQ DGRSWHG IRU XVH LQ WKLV VWXG\ 7KH &/($1 DOJRULWKP LV WKDW XVHG E\ 5REHUWV /HK£U DQG 'UHKDU f ZKLFK GHFRQYROYHV WKH VDPSOLQJ IXQFWLRQ IURP WKH GLUW\ VSHFWUXP WR JLYH D EHWWHU UHSUHVHQWDWLRQ RI WKH WUXH VSHFWUXP 7KLV GHFRQYROXWLRQ LV DFFRPSOLVKHG LQ WKH IROORZLQJ PDQQHU )LUVW WKH ODUJHVW SHDN LQ WKH GLUW\ VSHFWUXP LV ORFDWHG DQG VWRUHG 7KH VSHFWUDO ZLQGRZ EHDPf LV WKHQ VXSHULPSRVHG RQWR WKH GLUW\ VSHFWUXP VR WKDW WKH PDLQ SHDN LQ WKH EHDP FRLQFLGHV ZLWK WKH ODUJHVW SHDN LQ WKH GLUW\ VSHFWUXP $ JLYHQ SHUFHQWDJH RI WKH VSHFWUDO ZLQGRZ LV VXEWUDFWHG IURP WKH FRUUHVSRQGLQJ GLUW\ VSHFWUXP 7KH SHDN RI WKH VSHFWUXP ZKLFK LV UHPRYHG IURP WKH GDWD VSHFWUXP LV VWRUHG DV D &/($1 FRPSRQHQW 7KH VSHFWUXP ZKLFK UHVXOWV DIWHU VXEWUDFWLQJ WKH VSHFWUDO ZLQGRZ LV DOVR VWRUHG LQ D UHVLGXDO ILOH 7KLV SURFHGXUH LV UHSHDWHG IRU D JLYHQ QXPEHU RI GHFRQYROXWLRQV SHDNVf $ JDXVVLDQ UHSUHVHQWDWLRQ RI WKH VSHFWUDO ZLQGRZ LV WKHQ FRQYROYHG ZLWK HDFK &/($1 VSHFWUD )LQDOO\ LQ

PAGE 48

RUGHU WR SUHVHUYH WKH QRLVH ZKLFK H[LVWHG LQ WKH RULJLQDO VSHFWUXP WKH UHVLGXDO YDOXHV DUH DGGHG EDFN WR WKH FRQYROYHG &/($1 FRPSRQHQWV JLYLQJ WKH &/($1 VSHFWUXP 7KH LQSXW SDUDPHWHUV IRU WKH &/($1 DOJRULWKP LQFOXGH WKH QXPEHU RI &/($16 WR EH SHUIRUPHG SHDNVf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f LQ )LJXUH GHFUHDVHV LQ VWUHQJWK DV WKH QXPEHU RI &/($16 LQFUHDVHV ZKLOH WKH SHDN DW L G JHWV SURJUHVVLYHO\ VWURQJHU ,Q IDFW LQ )LJXUH WKLV LV WKH RQO\ SHDN ZKLFK UHPDLQV LQ WKH VSHFWUXP ZLWK FRQVLGHUDEOH DPSOLWXGH ,W LV OLNHO\ WKDW D PLVLQWHUSUHWDWLRQ RI WKH GDWD ZRXOG UHVXOW LQ WKLV W\SH RI RYHUDSSOLFDWLRQ ,Q WKLV VWXG\ FDUH KDV EHHQ

PAGE 49

5HODWLYH 3RZHU )UHTXHQF\ G f )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU WKH WHVW GDWD VHW

PAGE 50

5HODWLYH 3RZHU )UHTXHQF\ G f )LJXUH 3RZHU VSHFWUXP IRU WKH WHVW GDWD VHW EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 51

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH WHVW GDWD VHW DIWHU LWHUDWLRQV RI WKH FOHDQ DOJRULWKP

PAGE 52

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH WHVW GDWD VHW DIWHU LWHUDWLRQV RI WKH FOHDQ DOJRULWKP

PAGE 53

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH WHVW GDWD VHW DIWHU LWHUDWLRQV RI WKH FOHDQ DOJRULWKP

PAGE 54

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH WHVW GDWD VHW DIWHU LWHUDWLRQV RI WKH FOHDQ DOJRULWKP

PAGE 55

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f $ $W r VLQ 3 f

PAGE 56

,Q HTXDWLRQ f 'WAf UHSUHVHQWV WKH FDOFXODWHG GHIOHFWLRQ DW WLPH WA $ LV WKH EDVHOLQH LQ LQWHQVLW\ XQLWV $M LV WKH DPSOLWXGH RI WKH VLQH ZDYH H[SUHVVHG LQ LQWHQVLW\ XQLWV W LV WKH SKDVH RU WLPH RI PD[LPXP DQG 3 LV WKH SHULRG ,W VKRXOG EH QRWHG DW WKLV SRLQW WKDW 3 LV H[SUHVVHG LQ GD\V GLYLGHG E\ ,Q UDGLDQV D UHSUHVHQWDWLRQ WKDW ZDV FKRVHQ VR WKDW ZKHQ RQH FRPSOHWH F\FOH KDG HODSVHG WKDW LV W W 3 WKHQ FRV W W4f3 ,Q WKH HDVWVTXDUHV SURFHGXUH IRXU SDUDPHWHUV DUH DGMXVWHG 7KH\ DUH $ $W W DQG 3 $ ILUVWRUGHU 7D\ORU H[SDQVLRQ LV FRPSXWHG XVLQJ WKH SDUWLDO GHULYDWLYHV RI HTXDWLRQ f ZLWK UHVSHFW WR HDFK RI WKHVH YDULDEOHV 7KH URXWLQH XVHG LQ WKLV FRQWH[W LV WKDW RI %DQDFKLHZLF] f DQG XVHV &UDFRYLDQ FDOFXOXV ,I WKH HUURUV ZKLFK ZHUH FDOFXODWHG LQ WKH HDVW VTXDUHV URXWLQH H[FHHG WKH FRQYHUJHQFH FULWHULD FRUUHFWLRQV DUH FRPSXWHG DQG DGGHG WR $ $W W DQG 3 WR JLYH WKH QHZ TXDQWLWLHV $R $$ $ $$! W $W DQG 3 $3 7KH FRQYHUJHQFH FULWHULD DUH WKH IROORZLQJ

PAGE 57

($f [ n? ($Wf [ n (Wf [ 2n DQG ( 3f [ n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f LQ WKH SKDVH SDUDPHWHU D VLPLODU DSSURDFK LV DGRSWHG 7KDW LV D GHVLUHG LQFUHPHQW LQ WKH SKDVH LV HQWHUHG DV DQ LQSXW SDUDPHWHU ,Q HYHU\ FDVH WKH WLPH RI WKH ILUVW REVHUYDWLRQ ZDV XVHG DV DQ LQLWLDO JXHVV RI WKH YDOXH IRU W 7KH SDUDPHWHU ZDV LQFUHPHQWHG IURP WKLV LQLWLDO YDOXH WR WKH LQLWLDO YDOXH SOXV Q

PAGE 58

:HDWKHU UHSRUWV DUH UHFRUGHG DW WKH 6RXWK 3ROH HYHU\ VL[ KRXUV 3KRWRPHWULF GDWD UHFRUGHG EHWZHHQ WZR FRQVHFXWLYH SHULRGV RI UHSRUWHG FOHDU VNLHV ZHUH XVHG LQ WKLV LQYHVWLJDWLRQ RI a 9HO ,Q DGGLWLRQ VLQFH WKH WHOHVFRSH GRHV QRW KDYH WKH FDSDELOLW\ WR PRQLWRU WKH VWDU DQG QHDUE\ VN\ VLPXOWDQHRXVO\ GDWD UHFRUGHG GXULQJ WLPHV RI DXURUDO DFWLYLW\ ZHUH RPLWWHG IURP WKH VWXG\ 7KHVH RPLVVLRQ FULWHULD UHVXOWHG LQ WKH VHOHFWLRQ RI VHYHQ LQGLYLGXDO GDWD VHWV REWDLQHG GXULQJ WKH DXVWUDO ZLQWHU ,QGLYLGXDO REVHUYDWLRQV LQ HDFK GDWD VHW ZHUH RPLWWHG LI WKH REVHUYDWLRQ LQ WKH FRQWLQXXP ILOWHU DQG WKH REVHUYDWLRQ LQ RQH RU PRUH RI WKH HPLVVLRQ OLQH ILOWHUV GHYLDWHG E\ PRUH WKDQ b 7KH DQDO\VLV RI WKH GDWD VHWV WKDW UHVXOWHG LV SUHVHQWHG LQ WKH IROORZLQJ FKDSWHU

PAGE 59

&+$37(5 '$7$ $1$/<6,6 6SHFWUDO 3KRWRPHWU\ RI WKH +HOO DQG &,,, (PLVVLRQ /LQHV 'DWD 6HW 7KLV GDWD VHW LQFOXGHV REVHUYDWLRQV RI \ 9HO WKDW H[WHQG IURP -' WKURXJK -' 7KHVH f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nV PHWKRG RI )DVW )RXULHU 7UDQVIRUPV DQG DVVXPLQJ D PLQLPXP RI WZR VDPSOHV SHU F\FOH *LYHQ WKH GDWD VSDFLQJ RI A DQG WKH KRXU WLPH LQWHUYDO VSDQQHG E\ WKLV GDWD WKH PLQLPXP DQG PD[LPXP IUHTXHQFLHV ZKLFK FDQ EH UHWULHYHG IURP WKH GDWD VHW ZLWK D UHDVRQDEOH DPRXQW RI FRQILGHQFH DUH LPAQ OLGn DQG PD[ AOAGn )LJXUH UHSUHVHQWV WKH VDPSOLQJ

PAGE 60

7DEOH 2EVHUYLQJ 3URJUDP 6WDU % 9 +H & &r &6 &r 9r 9r 9r 9r YV Yr Yr Yr Yr Fr &6 Fr 9r Yr Yr 9r Yr Yr 9r Yr 9r Yr Yr Yr ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;;; ;;; ;;; ;;; ;;; ;;; ;;; ;;; ;;; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; &r FRPSDULVRQ VWDU &6 FRPSDULVRQ VN\ 9r YDULDEOH VWDU 96 YDULDEOH VN\ &RQWLQXXP ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

PAGE 61

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW ,

PAGE 62

IXQFWLRQ IRU ERWK WKH KHOLXP DQG WKH FDUERQ REVHUYDWLRQV 7KH SHDNV DW X DQG Gn DUH WKH RGG KDUPRQLFV LS 9 DQG YM DQG WKH SHDN DW Gn LV WKH HYHQ KDUPRQLF LJ 7KH FRQYROXWLRQ RI WKH KHOLXP GDWD ZLWK WKH VSHFWUDO ZLQGRZ IXQFWLRQ JLYHV WKH VSHFWUXP LQ )LJXUH 7KH QRLVH OHYHO LV GHQRWHG LQ WKH ORZHU OHIWKDQG SRUWLRQ RI WKH SORW DW [ n $OO IHDWXUHV WR WKH ULJKW RI WKH DUURZ VDWLVI\ WKH 1\TXLVW FULWHULRQ RI D PLQLPXP RI WZR VDPSOHV SHU F\FOH 7KH SHDNV DW Y DQG GB UHSUHVHQW VWDWLVWLFDOO\ VLJQLILFDQW IHDWXUHV DW WKH FWAM OHYHO $ FORVHU ORRN DW WKH VSHFWUXP UHYHDOV WKDW WKH SHDN DW G LV SDUW DOLDV ZLWK WKDW DW Y Gn DQG WKH SHDN DW Y GO LV SDUW DOLDV ZLWK D SHDN DW Y GB 7KLV ODVW IHDWXUH LV VKRZQ LQ )LJXUH EXW VLQFH LW GRHV QRW VDWLVI\ WKH 1\TXLVW FULWHULRQ LW LV GLVFDUGHG LQ WKLV VWXG\ 6SXULRXV IHDWXUHV DW RU DERYH WKH DA OHYHO DUH UHPRYHG IURP WKLV VSHFWUXP ZLWK D GHFRQYROXWLRQ SURFHVV WKH UHVXOWLQJ SRZHU VSHFWUXP LV GHSLFWHG LQ )LJXUH +HUH ZH VHH WKDW WKH SHDN DW Y Gn LV WKH RQO\ IHDWXUH ZKLFK UHPDLQV DERYH WKH FWA OHYHO DW DA 7KLV IUHTXHQF\ FRUUHVSRQGV WR D SHULRG RI PLQXWHV $FFRUGLQJ WR WKH OHDVW VTXDUHV URXWLQH WKH EHVW ILW WR WKH GDWD ZLWKLQ WKH OLPLWDWLRQV RI WKH 1\TXLVW FULWHULRQ RFFXUV IRU 3 PLQXWHV DQG LV GHVFULEHG E\

PAGE 63

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 64

5HODWLYH 3RZHU )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 65

'WWf r VLQ U Ws r? f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f r VLQ U [ f LV JUDSKLFDOO\ SUHVHQWHG LQ )LJXUH 7KH DPSOLWXGH RI WKH YDULDWLRQ RI WKLV PRGHO LV RQO\ VOLJKWO\ ORZHU WKDQ WKDW IRU WKH PRUH UDSLG IOXFWXDWLRQV GHSLFWHG LQ )LJXUH 7KH SRZHU VSHFWUXP IRU WKH FDUERQ GDWD REWDLQHG LQ WKLV WLPH LQWHUYDO LV VKRZQ LQ )LJXUH $JDLQ WKH DUURZ LQ WKH ORZHU OHIW RI WKH SORW UHSUHVHQWV WKH QRLVH OHYHO DW [ nr ,W VKRXOG EH QRWHG WKDW WKH SHDN DW Y GB LV SDUW DOLDV ZLWK WKDW DW Y Gn DQG VKRXOG LQ SULQFLSOH EH UHPRYHG ZLWK WKH &/($1 DOJRULWKP )LJXUH f

PAGE 66

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO RI WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW ZLWK 3 PLQXWHV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO /Q

PAGE 67

+HOO &RQWLQXXP )LJXUH 0RGHO RI WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D r" LQWHUYDO &7V 2

PAGE 68

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKUD

PAGE 69

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP DV

PAGE 70

7KLV LV LQGHHG WKH FDVH DQG WKH RQO\ SHDN ZKLFK VXUYLYHV WKH &/($1LQJ SURFHVV LV WKDW DW Y Gf FRUUHVSRQGLQJ WR A KRXUVf 7KH VLJQDOR QRLVH UDWLR IRU WKLV SHDN LV TXLWH ODUJH DW FWMM 7KH H D V W V TXDU H V URXWLQH FRQYHUJHV WR D VHW RI TXDQWLWHV ZLWK D VLPLODU SHULRG 3 A KRXUVf DFFRUGLQJ WR WKH IROORZLQJ UHODWLRQ 'WWf r VLQ U W/ / f 7KH ILW WR WKH GDWD XVLQJ HTXDWLRQ f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

PAGE 71

&OOO &RQWLQXXP )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH FDUERQ GDWD IRU GDWD VHW ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D LQWHUYDO

PAGE 72

'DWD 6HW ,, 2QHKXQGUHG DQG HLJKW\QLQH REVHUYDWLRQV RI 9HO ZHUH REWDLQHG LQ WKH KHOLXP DQG FDUERQ ILOWHUV XVLQJ WKH REVHUYLQJ VHTXHQFH SUHVHQWHG LQ 7DEOH 7KH WLPH LQWHUYDO VSDQQHG E\ WKHVH GDWD H[WHQGV IURP -' WKURXJK -' 7KH DYHUDJH VSDFLQJ EHWZHHQ VXFFHVVLYH REVHUYDWLRQV IRU D JLYHQ ILOWHU LV W 7KH GHFUHDVH LQ WLPHUHVROXWLRQ RI WKHVH GDWD UHODWLYH WR WKH WLPHUHVROXWLRQ RI WKH GDWD GLVFXVVHG LQ WKH SUHYLRXV VHFWLRQ LV DQ DUWLIDFW RI WKH QHZ REVHUYLQJ SURJUDP $V FDQ EH VHHQ IURP 7DEOH VHYHUDO QHZ REMHFWV KDYH EHHQ DGGHG WR WKH RULJLQDO REVHUYLQJ SURJUDP 7DEOH f 7KLV XOWLPDWHO\ OLPLWV WKH KLJK IUHTXHQF\ FRPSRQHQWV ZKLFK DUH UHFRYHUDEOH LQ WKH GDWD DQDO\VLV 7KH VSHFWUDO ZLQGRZ IRU WKHVH GDWD LV JLYHQ LQ )LJXUH DQG LQFOXGHV ILUVW VHFRQG WKLUG DQG IRXUWK KDUPRQLFV DW Y DQG Gn UHVSHFWLYHO\ ,Q JHQHUDO DV LQ WKLV FDVH VSHFWUDO ZLQGRZV IRU XQHTXDOO\ VDPSOHG GDWD WHQG WR EH PRUH FRPSOH[ DW KLJK IUHTXHQFLHV 7KH SRZHU VSHFWUXP RI WKH KHOLXP GDWD LV FRPSXWHG IRU IUHTXHQFLHV EHWZHHQ LPAQ Gr DQG LPD[ Gn )LJXUH f 7KH QRLVH OHYHO LQ WKLV ILJXUH LV GHQRWHG LQ WKH ORZHU OHIWKDQG SRUWLRQ RI WKLV SORW DW [ n *LYHQ WKLV OHYHO RI QRLVH WKH SHDNV DW Y DQG GB PLJKW EH FODVVLILHG DV VWDWLVWLFDOO\ VLJQLILFDQW +RZHYHU ZKHQ RQH VXSHULPSRVHV WKH VSHFWUDO ZLQGRZ RQ WKH SRZHU VSHFWUXP LW

PAGE 73

7DEOH 2EVHUYLQJ 3URJUDP 6WDU % 9 +H & &RQWLQXXP 9r ; ; 9r ; ; 9r ; ; 9r ; ; 9r 9r 9r 9r 96 ; ; 9r Yr Yr Yr 9r ; ; 9r ; ; 9r ; ; 9r ; ; &r ; ; &6 ; ; &r ; ; 9r ; ; &r ; ; &6 ; ; &r ; ; 9r ; ; 9r ; ; 9r ; ; 9r ; ; 9r 9r Yr Yr 96 ; ; 9r Yr Yr Yr ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;n ; ; ; ; ;

PAGE 74

7DEOH FRQWLQXHG 6WDU % 9 +H & &RQWLQXXP 9r ; ; 9r ; ; 9r ; ; 9r ; ; &r ; ; &6 ; ; &r ; ; 6N\ % ; ; D *UX ; ; D *UX 6 ; ; D *UX ; ; IL *UX ; ; % *UX 6n ; ; % *UX ; ; 6N\ $ ; ; ; ; ; ; ; ; ; ;; ; ;; ; ;; ; ;; ; ; ; ; ; ; ; 9r YDULDEOH VWDU 96 YDULDEOH VN\ &r FRPSDULVRQ VWDU &6 FRPSDULVRQ VN\ 6N\ % r UHJLRQ RI VN\ T *UX 6 D *UX VN\ % *UX 6 % *UX VN\ 6N\ $ r UHJLRQ RI VN\ r IURP YDULDEOH VWDU r IURP 6N\ %

PAGE 75

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW ,, D? FR

PAGE 76

5HODWLYH 3RZHU ( ( (f§ ‘ ( f )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW ,, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 77

LV DSSDUHQW WKDW WKH SHDNV DW X GB DQG Y G DUH SDUW DOLDVHV RI WKH FRQVLGHUDEO\ VWURQJHU SHDN DW Y G O 7KH &/($1 DOJRULWKP KDV EHHQ XVHG LQ DQ DWWHPSW WR HOLPLQDWH WKHVH VSXULRXV IHDWXUHV DQG WKH UHVXOWV DUH VKRZQ LQ )LJXUH 1RZ RQO\ WZR SHDNV KDYH VXUYLYHG WKH &/($1LQJ SURFHVV ZKLOH UHPDLQLQJ VWDWLVWLFDOO\ VLJQLILFDQW 7KH\ LQFOXGH WKH IHDWXUH DW X GB 3 A f ZLWK D VLJQDO WRQRLVH UDWLR RI FUMT DQG WKH IHDWXUH DW Y G 3 Af ZKLFK KDV D ZHDNHU VLJQDO RI DA $OWKRXJK WKH OHDVWVTXDUHV URXWLQH FRQYHUJHV IRU SHULRGV FRUUHVSRQGLQJ WR HDFK RI WKHVH IUHTXHQFLHV WKH ODUJHVW DPSOLWXGH RI YDULDWLRQ DQG WKH ORZHVW REVHUYHG PLQXV FDOFXODWHG &f UHVLGXDOV RFFXUV IRU 3 f§ A c Gnf ,W LV LQWHUHVWLQJ WR QRWH WKDW WKLV SHULRG LV H[DFWO\ WZLFH WKH SHULRG FRQFOXGHG IURP WKH SRZHU VSHFWUXP DQDO\VLV DQG SHUKDSV WKLV LV DQ LQGLFDWLRQ RI WKH SUHVHQFH RI KDUPRQLFV RI D IXQGDPHQWDO SHULRG ,Q RUGHU WR GHWHUPLQH ZKHWKHU RU QRW RQH RI WKHVH SHULRGV ILWV WKH GDWD PRUH VDWLVIDFWRULO\ WKDQ DQRWKHU PRGHOV DUH FRPSXWHG IRU HDFK 5HVXOWV LQGLFDWH WKDW WKHVH GDWD DUH PRVW DGHTXDWHO\ UHSUHVHQWHG E\ 3 A )LJXUH f DFFRUGLQJ WR WKH HTXDWLRQ 'W/f r VLQ U A f 3HUKDSV WKLV SHULRG UHIOHFWV WKH IXQGDPHQWDO PRGH RI

PAGE 78

5HODWLYH 3RZHU )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW ,, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP -

PAGE 79

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW ,, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 80

YDULDWLRQ ZKLOH WKH SURPLQHQW IHDWXUH LQ WKH SRZHU VSHFWUXP PD\ UHSUHVHQW WKH ILUVW KDUPRQLF 7KH FDUERQ GDWD KDYH DOVR EHHQ DQDO\]HG LQ D PDQQHU FRPSDUDEOH WR WKDW XVHG LQ WKH DQDO\VLV RI WKH KHOLXP GDWD 7KH GLUW\ DQG FOHDQ VSHFWUXPV DUH SUHVHQWHG LQ )LJXUHV DQG UHVSHFWLYHO\ $OWKRXJK WKH IHDWXUH DW Y Gn/ LV FRQVLGHUDEO\ ZHDNHU WKDQ WKH FRUUHVSRQGLQJ SHDN LQ WKH SRZHU VSHFWUXP IRU WKH KHOLXP GDWD LW VWLOO UHPDLQV WKH VWURQJHVW IHDWXUH LQ WKH SORW DW FMIM ZKHUH DMT [ n 7KUHH RWKHU SHDNV ULVH MXVW DERYH WKH DAM OHYHO WKH\ RFFXU DW c DQG Gn 7KH HDVWVTXDUHV URXWLQH LV H[HFXWHG IRU SHULRGV UDQJLQJ EHWZHHQ A Y Gf DQG A Y Gnf 6ROXWLRQV ZLWK SHULRGV FRUUHVSRQGLQJ WR HDFK RI WKH IUHTXHQFLHV IRXQG IURP WKH SRZHU VSHFWUXP DQDO\VLV UHVXOW 2QFH DJDLQ WKH VWURQJHVW FRQYHUJHQFH RFFXUV IRU D SHULRG RI A Y Gf 2QH PXVW UHDOL]H WKDW LQ DOO RI WKHVH PRGHOV WKH DPSOLWXGH RI YDULDWLRQ RI WKH &,,, HPLVVLRQ OLQH LV VR VPDOO WKDW UHVXOWV FDQQRW EH FRQVLGHUHG WR UHSUHVHQW VLJQLILFDQW LQWHQVLW\ FKDQJHV 7KH ODUJHVW YDULDWLRQ LQ WKLV FDVH LV RQO\ b RI WKH DYHUDJH GHIOHFWLRQ ,W LV IRU WKLV UHDVRQ WKDW QR VSHFLILF PRGHO LV VXJJHVWHG IRU WKLV GDWD ,Q VXPPDU\ SRZHU VSHFWUXP DQDO\VLV JLYHV D SHULRG RI A IRU ERWK KHOLXP DQG FDUERQ GDWD 7KH HDVWVTXDUHV URXWLQH DOWKRXJK VWLOO FRQYHUJLQJ IRU SHULRGV RQ WKLV

PAGE 81

5HODWLYH 3RZHU ( ( ( )UHTXHQF\ G af )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW ,, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJR ULWKP

PAGE 82

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW ,, DIWHU H[HFXWLRQ RI WKH FOHDQ D JR ULWKP /Q

PAGE 83

RUGHU JLYH D EHVWILW ZLWK D SHULRG HTXDO WR WZLFH WKLV SHULRG RU 3 A 'DWD 6HW ,,, 7KLV GDWD VHW VSDQV WKH WLPH LQWHUYDO WKDW H[WHQGV IURP -' WKURXJK -' 7KH VHW LQFOXGHV REVHUYDWLRQV RI 9HO IRU WKH KHOLXP FDUERQ DQG FRQWLQXXP ILOWHUV 7KH REVHUYLQJ VHTXHQFH XVHG WR REWDLQ WKHVH GDWD LV WKDW VKRZQ LQ 7DEOH ZLWK $W A $VVXPLQJ D PLQLPXP RI WZR VDPSOHV SHU F\FOH DQG D PLQLPXP RI WZR F\FOHV SHU GDWD VHW IUHTXHQFLHV IURP LAPAQ O2Gn WR YPD[ Gn FDQ EH WHVWHG )LJXUH VKRZV WKH VDPSOLQJ IXQFWLRQ ZKLFK UHVXOWV IURP WKH DFTXLVLWLRQ RI WKHVH GDWD 7KH SHDN DW Y G KDV FRQVLGHUDEOH SRZHU DPRXQWLQJ WR DERXW RQH WKLUG WKDW RI WKH PDLQ SHDN DW L G O 7KH VHFRQG KDUPRQLF DW Y Gn DQG WKH WKLUG KDUPRQLF DW Y GaO DUH DOVR LGHQWLILHG ,Q DGGLWLRQ D EURDG IHDWXUH FHQWHUHG DW L G KDV D VHFRQG KDUPRQLF DW X Gn 7KH FRQYROXWLRQ RI WKH KHOLXP GDWD ZLWK WKH EHDP LV GHWHUPLQHG DQG SUHVHQWHG LQ )LJXUH LQ WHUPV RI UHODWLYH SRZHU 7KH QRLVH OHYHO RI WKHVH GDWD LV LQGLFDWHG E\ DQ DUURZ DW WKH ERWWRP OHIW RI WKH SORW DW [ n 7KH IHDWXUHV DW X DQG G ULVH DERYH WKH FWMA OHYHO EXW WKH SHDN DW Y G LV SDUW DOLDV ZLWK WKH VWURQJHU IHDWXUH DW Y OLGn 7KH &/($1LQJ SURFHVV OHDYHV WZR RI WKH WKUHH SHDNV PHQWLRQHG DERYH DV VWDWLVWLFDOO\ VLJQLILFDQW )LJXUH f 7KH

PAGE 84

5HODWLYH 3RZHU )UHTXHQF\ G f )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW ,,,

PAGE 85

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW ,,, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 86

5HODWLYH 3RZHU 2 )UHTXHQF\ Gff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW ,,, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 87

IHDWXUH DW Y OLGn KDV D UHODWLYH SRZHU RI DA ZKLOH WKH SHDN DW Gf KDV WKH VOLJKWO\ ORZHU V L JQDO W R QR L V H UDWLR RI FUA DQG UHSUHVHQWV PRUHUDSLG IOXFWXDWLRQV LQ LQWHQVLW\ 7KH HDVWVTXDUHV URXWLQH FRQYHUJHV WR WZR VHWV RI SDUDPHWHUV ZLWK DOPRVW WKH VDPH DPSOLWXGH LQ HDFK FDVH 7KH ILUVW RI WKHVH PRGHOV KDV D SHULRG RI KRXUV DQG FRUUHVSRQGV WR WKH SHDN DW X OLGn LQ )LJXUH 7KH PRGHO IRU WKLV SHULRG LV VKRZQ LQ )LJXUH DQG KDV EHHQ FRPSXWHG ZLWK WKH UHODWLRQ 'WLf r VLQ U Ws f 7KH VHFRQG VHW RI SDUDPHWHUV ZKLFK UHVXOWV IURP WKH OHDVW VTXDUHV URXWLQH KDV D SHULRG RI KRXUV DQG LV GHVFULEHG E\ HTXDWLRQ f '&WLf r VLQ U WW / f (YHQ WKRXJK WKLV SHULRG GRHV QRW FRUUHVSRQG WR DQ\ SURPLQHQW IHDWXUH LQ WKH SRZHU VSHFWUXP WKH PRGHO SLFWXUHG LQ )LJXUH VHHPV WR ILW WKH GDWD DV ZHOO DV WKDW GHVFULEHG E\ HTXDWLRQ f (YHQ WKRXJK WKH & UHVLGXDOV FRPSXWHG IURP HTXDWLRQ f DUH VOLJKWO\ VPDOOHU WKDQ WKH UHVLGXDOV GHWHUPLQHG ZLWK HTXDWLRQ f JDSV LQ WKH GDWD PDNH LW

PAGE 88

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW ,,, ZLWK 3 KRXUV HDFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO RR

PAGE 89

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW ,,, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO .!

PAGE 90

GLIILFXOW WR GHWHUPLQH ZKLFK ILW LV EHWWHU 6LQFH WKH ILUVW RI WKH WZR SHULRGV LV VXSSRUWHG E\ ERWK PHWKRGV RI DQDO\VLV 3 KRXUV PD\ EH WKH PRUH DSSURSULDWH SHULRG 7KH )RXULHU WUDQVIRUP RI WKH FDUERQ GDWD LV FRQYHUWHG WR SRZHU DQG )LJXUHV DQG VKRZ WKH UHVXOWV EHIRUH DQG DIWHU WKH UHPRYDO RI VSXULRXV IHDWXUHV 7KH OD QRLVH OHYHO RI WKHVH GDWD LV DW [ n 6LQFH WKH FDUERQ HPLVVLRQ IHDWXUH LV FRQVLGHUDEO\ ZHDNHU WKDQ WKH KHOLXP OLQH LW LV PRUH GLIILFXOW WR GLVFHUQ LQWHQVLW\ YDULDWLRQV LQ WKH FDUERQ IHDWXUH IURP UDQGRP QRLVH ,Q WKLV SDUWLFXODU FDVH WKH &/($1LQJ SURFHVV UHGXFHV WKH VWURQJHVW SHDNV LQ )LJXUH EHORZ WKH DAc OHYHO 7KH HDVW VTXDUHV URXWLQH KDV WKH VWURQJHVW FRQYHUJHQFH IRU D SHULRG RI PLQXWHV +RZHYHU VLQFH WKH DPSOLWXGH RI YDULDWLRQ LV RQO\ D YHU\ VPDOO IUDFWLRQ RI WKH WRWDO GHIOHFWLRQ LQ WKDW ILOWHU RQH FDQQRW UHJDUG WKLV UHVXOW ZLWK PXFK FRQILGHQFH ,Q VXPPDU\ WKH LQWHQVLW\ YDULDWLRQV RI WKH KHOLXP HPLVVLRQ OLQH DUH SUREDEO\ EHVW GHVFULEHG ZLWK D SHULRG RI KRXUV 7KLV SHULRG ZDV VHOHFWHG E\ ERWK SRZHU VSHFWUXP DQG HDVWVTXDUHV DQDO\WLF WHFKQLTXHV ,Q DGGLWLRQ WKH GDWD KDYH EHHQ PRGHOHG ZLWK D SHULRG RI KRXUV ZKLFK ZDV D VROXWLRQ RI WKH HDVWVTXDUHV PHWKRG RQO\ 5HODWLYHO\ ODUJH JDSV LQ WKH GDWD PDNH LW GLIILFXOW WR GHWHUPLQH ZKLFK SHULRG LI HLWKHU LV WKH PRUH DSSURSULDWH 7KH SRZHU VSHFWUXP RI WKH FDUERQ GDWD LQGLFDWHV QR VWURQJ SHULRGLFLW\ $ HDVW VTXDUHV VROXWLRQ ZLWK D SHULRG RI PLQXWHV LV

PAGE 91

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW ,,, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 92

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW ,,, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 93

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n DQG X GB DUH VHSDUDWHG E\ WKH VDPH DPRXQW DV WKH SHDNV DW X GB DQG Y GB 7KH FRQYROXWLRQ RI WKH LQWHQVLW\ PHDVXUHPHQWV RI WKH KHOLXP HPLVVLRQ OLQH ZLWK WKLV VRPHZKDW PRUH FRPSOLFDWHG VDPSOLQJ IXQFWLRQ JLYHV WKH SRZHU VSHFWUXP LQ )LJXUH 7KH OD1 QRLVH OHYHO DW [ nn LQGLFDWHV WKDW SHDNV DW L DQG G DUH VWDWLVWLFDOO\ VLJQLILFDQW IHDWXUHV $ VWURQJ IHDWXUH DOVR RFFXUV DW L Gn[ EXW WKLV IUHTXHQF\ GRHV QRW VDWLVI\ WKH 1\TXLVW FULWHULRQ RI WZR VDPSOHV SHU F\FOH 7KH SHDN DW Y GO LV SDUW DOLDV ZLWK WKDW DW Y O6G DQG VKRXOG LQ SULQFLSOH EH UHGXFHG LQ VWUHQJWK E\ WKH &/($1 DOJRULWKP )LJXUH VKRZV WKH SRZHU VSHFWUXP WKDW UHVXOWV DIWHU WKH VSXULRXV IHDWXUHV DUH UHPRYHG IURP WKH GLUW\ VSHFWUXP 7KH RQO\ IHDWXUH ZKLFK

PAGE 94

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW ,9

PAGE 95

5HODWLYH 3RZHU )UHTXHQF\ Gff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW ,9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 96

5HODWLYH 3RZHU )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW ,9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 97

UHPDLQV VWDWLVWLFDOO\ VLJQLILFDQW LV WKH SHDN DW Y GfO EXW DV PHQWLRQHG SUHYLRXVO\ WKH SHULRG ZKLFK FRUUHVSRQGV WR WKLV IUHTXHQF\ GRHV QRW VDWLVI\ WKH 1\TXLVW FULWHULRQ DQG WKXV FDQQRW EH FRQVLGHUHG LQ WKH FRQWH[W RI WKLV UHVHDUFK $OWKRXJK QR SHDNV VXUYLYH WKH &/($1LQJ SURFHVV WKH OHDVW VTXDUHV URXWLQH LV VWLOO H[HFXWHG &RQYHUJHQFH WR D VHW RI SDUDPHWHUV ZLWK 3 A KRXUVf LV JLYHQ E\ '&WLf r VLQ U Ws / f 7KLV ILW GHSLFWHG LQ )LJXUH KDV D YHU\ ORZ DPSOLWXGH RI YDULDWLRQ DQG LV FHUWDLQO\ QRW D YHU\ JRRG UHSUHVHQWDWLRQ RI WKH FKDQJHV LQ LQWHQVLW\ 7KLV XQVDWLVIDFWRU\ ILW LV FRQILUPHG E\ WKH ORZ SRZHU LQ WKH SHDNV RI WKH &/($1HG SRZHU VSHFWUXP )LJXUH f $Q DWWHPSW ZDV PDGH WR LPSURYH WKH UHVXOWV REWDLQHG ZLWK WKHVH GDWD E\ GLYLGLQJ WKH ORQJ GDWD VHW LQWR WZR VXEVHWV 7KH ILUVW VXEVHW FRQWDLQV REVHUYDWLRQV PDGH EHWZHHQ -' DQG -' DQG KDV D WLPH UHVROXWLRQ ZKLFK LV H[DFWO\ WKH VDPH DV WKDW IRU WKH ZKROH GDWD VHW 7KH VKRUWHU VSDQ RI WKLV VXEVHW DOORZV IRU IUHTXHQFLHV EHWZHHQ G DQG APD[ Gn WR EH WHVWHG 7KH VSHFWUDO ZLQGRZ IXQFWLRQ IRU WKHVH GDWD GHSLFWHG LQ )LJXUH VKRZV VLPLODU EXW VOLJKWO\ VWURQJHU SHDNV DV WKRVH IURP )LJXUH 'LUW\ DQG FOHDQ VSHFWUD DUH VKRZQ LQ

PAGE 98

+HOO &RQWLQXXP -XOLDQ GDWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD RI GDWD VHW ,9 ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO 92

PAGE 99

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU VXEVHW RI GDWD VHW ,9 UR

PAGE 100

)LJXUHV DQG 7KH QRLVH OHYHO LQ HDFK RI WKHVH SORWV LV GHQRWHG LQ WKH ORZHU OHIWKDQG SRUWLRQ RI WKH ILJXUHV DW [ n 7KH &/($1 DOJRULWKP UHPRYHV DOO VSXULRXV IHDWXUHV DW RU EHORZ WKH FWA OHYHO $V FDQ EH VHHQ IURP )LJXUH WKLV SURFHVV DJDLQ OHDYHV QR VWDWLVWLFDOO\ VLJQLILFDQW IHDWXUHV 7KH HDVWVTXDUHV VROXWLRQ IRU WKLV GDWD VHW JLYHV D SHULRG RI KRXUV FRUUHVSRQGLQJ LQ IUHTXHQF\ WR WKH VWURQJHVW SHDN LQ )LJXUHV DQG HYHQ WKRXJK WKHVH SHDNV GR QRW PHHW WKH VWDWLVWLFDO VLJQLILFDQFH FULWHULRQ 7KLV PRGHO LV GHVFULEHG E\ '&WLf r VLQ Ws f DQG LV VKRZQ LQ )LJXUH 7KH & UHVLGXDOV FRPSXWHG ZLWK WKLV HTXDWLRQ DUH PDUJLQDOO\ EHWWHU WKDQ WKRVH GHWHUPLQHG ZLWK HTXDWLRQ f 7KH VHFRQG VXEVHW RI GDWD VHW ,9 FRYHUV WKH WLPH UDQJH WKDW H[WHQGV IURP -' WKURXJK -' 7KH PLQLPXP DQG PD[LPXP IUHTXHQFLHV UHWULHYDEOH IURP WKLV VXEVHW DUH WKH VDPH DV IRU VXEVHW $JDLQ WKH VSHFWUDO ZLQGRZ )LJXUH f LV TXDOLWDWLYHO\ VLPLODU WR WKH EHDP IRU WKH HQWLUH GDWD VHW EXW LV TXDQWLWDWLYHO\ GLIIHUHQW )LJXUHV DQG UHSUHVHQW WKH SRZHU VSHFWUD EHIRUH DQG DIWHU &/($1LQJ WKH GDWD 7KH RQO\ SHDNV ZKLFK UHPDLQ DERYH WKH DA OHYHO DIWHU WKH &/($1LQJ SURFHVV DUH WKRVH DW

PAGE 101

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW RI GDWD VHW ,9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 102

5HODWLYH 3RZHU )UHTXHQF\ G af )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW RI GDWD VHW ,9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 103

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD RI VXEVHW RI GDWD VHW ,9 ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 104

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU VXEVHW ,, RI GDWD VHW ,9 Y2

PAGE 105

5HODWLYH 3RZHU (f§ ( ( ( )UHTXHQF\ Gaf )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW ,9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP 92 22

PAGE 106

5HODWLYH 3RZHU (f§ ( ( ( )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW ,9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP 92 92

PAGE 107

Y O2G DQG X G 7KH ILUVW RI WKHVH IHDWXUHV FRUUHVSRQGV WR 3 KRXUVf DQG KDV D UHODWLYH SRZHU RI DERXW FWA ZKHUH [ n 7KH VHFRQG SHDN ZKLFK ULVHV DERYH WKH FWA OHYHO LQ )LJXUH UHSUHVHQWV UDSLG IOXFWXDWLRQV ZLWK 3 PLQXWHV DQG KDV D VLJQDOWRQRLVH UDWLR RI $FFRUGLQJ WR WKH OHDVW VTXDUHV DQDO\VLV RI VXEVHW D SHULRG RI A KRXUVf EHVW VDWLVILHV WKH YDULDWLRQV SUHVHQW LQ WKHVH GDWD 7KLV SHULRG GRHV QRW DJUHH ZLWK HLWKHU RI WKH SHULRGV IRXQG WKURXJK SRZHU VSHFWUXP DQDO\VLV +RZHYHU EHFDXVH LW FRUUHVSRQGV WR D IUHTXHQF\ ZKLFK LV OHVV WKDQ APLQ WKLV LV QRW D VHULRXV GLVFUHSDQF\ 7KH PRGHO ZKLFK LV FRPSXWHG ZLWK WKH SDUDPHWHUV FKRVHQ E\ WKH OHDVW VTXDUHV URXWLQH QDPHO\ 'WWf r VLQ U WMB / f LV VKRZQ ZLWK DYHUDJHG SRLQWV LQ )LJXUH ,Q DGGLWLRQ WR WKLV VROXWLRQ D VKRUWHU SHULRG ZDV DOVR IRXQG WKURXJK WKH HDVWVTXDUHV PHWKRG RI DQDO\VLV 7KLV VROXWLRQ KDV D VPDOOHU DPSOLWXGH RI YDULDWLRQ EXW WKH SHULRG RI KRXUV DJUHHV ZHOO ZLWK WKH SHDN LQ )LJXUH DW Y O2G 7KH PDWKHPDWLFDO UHSUHVHQWDWLRQ RI WKLV PRGHO LV GHVFULEHG E\ U Ws 'W$f r VLQ f

PAGE 108

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD RI VXEVHW ,, RI GDWD VHW ,9 ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO R

PAGE 109

DQG LV GHSLFWHG LQ )LJXUH 7KH WKH 2& UHVLGXDOV GHWHUPLQHG IURP WKLV PRGHO DUH ODUJHU WKDQ WKRVH GHWHUPLQHG IURP HTXDWLRQ f ,Q DGGLWLRQ WKH DPSOLWXGH RI IOXFWXDWLRQ LQ HTXDWLRQ f UHSUHVHQWV D VPDOOHU SHUFHQWDJH RI WKH WRWDO LQWHQVLW\ RI WKH KHOLXP OLQH WKDQ GRHV HTXDWLRQ f 7KHVH FRQVLGHUDWLRQV OHDG RQH WR EHOLHYH WKDW HTXDWLRQ f LV D EHWWHU UHSUHVHQWDWLRQ RI WKH FKDQJHV LQ LQWHQVLW\ RI WKLV VXEVHW $QDO\VLV RI WKH FDUERQ GDWD DUH UHDOO\ TXLWH LQFRQFOXVLYH 7KH SRZHU VSHFWUD IRU WKH HQWLUH GDWD VHW EHIRUH DQG DIWHU H[HFXWLRQ RI WKH &/($1 DOJRULWKP DUH VKRZQ LQ )LJXUHV DQG UHVSHFWLYHO\ 7KH QRLVH OHYHO RI WKHVH GDWD LQ WHUPV RI UHODWLYH SRZHU LV GHQRWHG LQ WKH ORZHU OHIWKDQG SRUWLRQ RI WKH ILJXUHV DW [ n 1RQH RI WKH IHDWXUHV LQ )LJXUH DUH VWDWLVWLFDOO\ VLJQLILFDQW DW WKH DA OHYHO ,Q DQ DWWHPSW WR LPSURYH WKH LQWHUSUHWDWLRQ RI WKHVH GDWD WKH GDWD VHW ZDV DJDLQ GLYLGHG LQWR WZR VXEVHWV ZKLFK FRUUHVSRQG LQ WLPH WR VXEVHWV DQG ,, RI WKH KHOLXP GDWD 7KH GLUW\ DQG FOHDQ SRZHU VSHFWUD ZKLFK UHVXOW DIWHU WDNLQJ WKH )RXULHU WUDQVIRUP RI WKH ILUVW VXEVHW RI WKHVH GDWD DUH VKRZQ LQ )LJXUHV DQG 7KH QRLVH OHYHO LQ HDFK RI WKHVH SORWV LV DW [ n $V VHHQ LQ WKHVH ILJXUHV WKUHH KLJKIUHTXHQF\ FRPSRQHQWV H[FHHG WKH FWA OHYHO E\ D VPDOO DPRXQW )LJXUHV DQG DUH VLPLODU SORWV IRU WKH VHFRQG VXEVHW ,Q WKLV FDVH WKH SHDN DW Y GB 3 Af KDV D ODUJHU

PAGE 110

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD RI VXEVHW ,, RI GDWD VHW ,9 ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO r R &2

PAGE 111

5HODWLYH 3RZHU )UHTXHQF\ G f )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW ,9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 112

5HODWLYH 3RZHU ( 7 )UHTXHQF\ GB f )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW ,9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 113

5HODWLYH 3RZHU )UHTXHQF\ F0f )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW RI GDWD VHW ,9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 114

5HODWLYH 3RZHU ( ( ( )UHTXHQF\ Gff )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW RI GDWD VHW ,9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP R

PAGE 115

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW ,9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 116

5HODWLYH 3RZHU ( ( ( (f§ )UHTXHQF\ G f )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW ,9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 117

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f GDWD SRLQWV FRYHU WKH WLPH LQWHUYDO H[WHQGLQJ IURP -' WR -' 7KH REVHUYLQJ SURJUDP XVHG WR VHFXUH WKHVH GDWD LV WKDW GHVFULEHG LQ 7DEOH 6XFFHVVLYH REVHUYDWLRQV LQ DQ\ VLQJOH ILOWHU DUH VHSDUDWHG E\ r

PAGE 118

,OO $FFRUGLQJ WR WKH 1\TXLVW WKHRUHP SHDNV LQ WKH SRZHU VSHFWUXP DW IUHTXHQFLHV EHWZHHQ GB DQG fnPD[ G FDQ EH UHJDUGHG ZLWK FRQILGHQFH 7KH VDPSOLQJ IXQFWLRQ IRU WKLV GDWD VHW LV FRPSXWHG IRU WKLV UDQJH RI IUHTXHQFLHV DQG LV VKRZQ LQ )LJXUH ,Q WKH GLDJUDP ZH QRWH WKDW WKH FRPSOH[LW\ RI WKH ZLQGRZ LQFUHDVHV WRZDUG KLJK IUHTXHQFLHV EXW WKH ILUVW WKUHH RGG KDUPRQLFV DW DSSUR[LPDWHO\ X Gn[ L O22Gn DQG A Gnr DUH VWLOO LGHQWLILDEOH 7KH FRQYROXWLRQ RI WKH LQWHQVLWLHV RI WKH KHOLXP HPLVVLRQ OLQH ZLWK WKH EHDP UHVXOWV LQ )LJXUH 7KH QRLVH OHYHO DW [ f LQGLFDWHV WKDW SHDNV DW Y DQG GB DUH VWDWLVWLFDOO\ VLJQLILFDQW 7KH SHDN DW X G FRUUHVSRQGV WR D SHDN LQ WKH VSHFWUDO ZLQGRZ KRZHYHU WKH KHLJKW RI WKH SHDN LQ WKH EHDP LV QRW VXIILFLHQW WR H[FOXGH WKH SRVVLELOLW\ WKDW DQ DGGLWLRQDO FRPSRQHQW FRQWULEXWHV WR WKLV IHDWXUH 7KH FRUUHVSRQGLQJ SHDN LQ WKH EHDP LV RQO\ DERXW b DV VWURQJ DV WKH PDLQ FRPSRQHQW LQ WKDW VSHFWUXP ZKLOH WKH SHDN DW Y Gf VKRZQ LQ )LJXUH LV QHDUO\ b RI WKH PDLQ IHDWXUH DW X G 'HFRQYROYLQJ WKH SRZHU VSHFWUXP )LJXUH f IURP WKH VSHFWUDO ZLQGRZ IXQFWLRQ UHVXOWV LQ WKH &/($1HG VSHFWUXP GHSLFWHG LQ )LJXUH ZKHUH WKH IHDWXUHV DW Y DQG GO UHPDLQ VWDWLVWLFDOO\ VLJQLILFDQW 7KH VWURQJHVW RI WKHVH IHDWXUHV DW Y G KDV D UHODWLYH SRZHU RI DSSUR[LPDWHO\ DA DQG FRUUHVSRQGV WR D SHULRG RI KRXUV 7KH IHDWXUH DW Y Gn[ DOVR

PAGE 119

5HODWLYH 3RZHU )UHTXHQF\ Gaf )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW 9

PAGE 120

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW 9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP L 2-

PAGE 121

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW 9 DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 122

ULVHV VXEVWDQWLDOO\ DERYH WKH FWAM OHYHO DW XAc DQG FRUUHVSRQGV WR 3 KRXUV )LQDOO\ WKH SHDN DW Y GB KDV D UHODWLYH SRZHU HTXDO WR WLPHV WKDW RI WKH ORIO OHYHO DQG UHSUHVHQWV PRUH UDSLG IOXFWXDWLRQV RQ WKH RUGHU RI PLQXWHV 7KH HDVWVTXDUHV URXWLQH FRQYHUJHV WR D SHULRG ZKLFK FRUUHVSRQGV PRVW FORVHO\ WR WKH IHDWXUH DW Y Gn LQ )LJXUH 7KH UHVXOWLQJ PRGHO GHVFULEHG PDWKHPDWLFDOO\ DV 'WLBf r VLQ U W / f LV GHSLFWHG LQ )LJXUH 7KLV ILW VHHPV WR UHSUHVHQW WKH FKDQJHV LQ WKH REVHUYDWLRQV IRU WKH HDUOLHU SRLQWV PXFK PRUH VDWLVIDFWRULO\ WKDQ IRU GDWD SRLQWV QHDU WKH HQG RI WKH GDWD VHW 7KH FKDQJH LQ WKH TXDOLW\ RI WKH ILW PD\ LQGLFDWH WKDW WKH YDULDWLRQV H[KLELWHG LQ WKHVH GDWD PLJKW EHWWHU EH XQGHUVWRRG LI WKH GDWD DUH GLYLGHG LQWR WZR VXEVHWV 7KH VHSDUDWLRQ SRLQW ZDV FKRVHQ WR FRLQFLGH ZLWK D PLQXWH JDS LQ WKH GDWD 8VLQJ WKH PHWKRG RI HDVWVTXDUHV WKH EHVW ILW VLQH ZDYH IRU WKH GDWD REWDLQHG EHWZHHQ -' DQG -' ZDV DFFRPSOLVKHG ZLWK WKH HTXDWLRQ I WL A 'WLf r VLQ f

PAGE 123

+HOO &RQWLQXXP f§ n n f f§ -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW 9 ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 124

ZKHUH 3 KRXUV 7KH GDWD VSDQQLQJ WKH UHPDLQGHU RI WKH GDWD VHW -' WKURXJK -' UHVXOWV LQ D YDULDWLRQ ZKLFK LV VOLJKWO\ ODUJHU WKDQ WKDW IRU WKH ILUVW VXEVHW ZLWK 3 KRXUV 7KH DSSURSULDWH SDUDPHWHUV VHOHFWHG E\ WKH HDVWVTXDUHV URXWLQH UHVXOWV LQ HTXDWLRQ f '&WLf r VLQ U WW f 0RGHOV IRU HDFK RI WKHVH SRUWLRQV RI WKH GDWD DUH VKRZQ LQ )LJXUHV DQG ZKHUH HDFK SORWWHG SRLQW LV D A DYHUDJH 7KH DFFRPSDQ\LQJ HUURUEDUV UHSUHVHQW WKH VWDQGDUG GHYLDWLRQ DVVRFLDWHG ZLWK WKH LQGLYLGXDO SRLQWV ZLWKLQ HDFK DYHUDJH )LJXUHV DQG VKRZ GLUW\ DQG FOHDQ SRZHU VSHFWUD IRU WKH FDUERQ GDWD 6HYHUDO SHDNV LQ HDFK RI WKHVH SORWV KDYH D UHODWLYH SRZHU LQ H[FHVV RI HUA [ n $OWKRXJK LW LV LQWHUHVWLQJ WR QRWH WKDW WKH WKUHH VWURQJHVW SHDNV RFFXU DW IUHTXHQFLHV DURXQG X GfO LW LV QRW SRVVLEOH WR VLQJOH RXW WKH GRPLQDQFH RI DQ\ RQH SHULRG 7KH OHDVW VTXDUHV URXWLQH DOVR KDV GLIILFXOW\ LQ FRQYHUJLQJ WR D EHVW RYHUDOO SHULRG ZKLFK VDWLVILHV WKH YDULDWLRQV ,Q IDFW WKH URXWLQH FRQYHUJHV IRU VHYHUDO SHULRGV ZLWK WKH ODUJHVW DPSOLWXGH RI YDULDWLRQ DPRXQWLQJ WR RQO\ b RI WKH LQWHQVLW\ RI WKH FDUERQ HPLVVLRQ IHDWXUH ,W LV FHUWDLQO\

PAGE 125

+HOO &RQWLQXXP n m f§ -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU VXEVHW RI GDWD VHW 9 ZLWK 3 f§ KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO RR

PAGE 126

+HOO &RQWLQXXP -XOLDQ 'DWH f / )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU VXEVHW ,, RI GDWD VHW 9 ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO c,, 92

PAGE 127

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW 9 EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 128

5HODWLYH 3RZHU ( )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW 9 DIWHU H[HFXWLRQ RI WKH FOHDQ DJRULW KP

PAGE 129

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f FRQVLVWV RQO\ RI GHIOHFWLRQV RI \ 9HO DQG +5 7KH DYHUDJH VSDFLQJ EHWZHHQ UHGXFHG REVHUYDWLRQV LQ HDFK RI WKH KHOLXP DQG FDUERQ ILOWHUV LV A 7KH WLPH UHVROXWLRQ RI WKHVH GDWD WDNHQ WRJHWKHU ZLWK WKH KRXUV VSDQQHG E\ WKH

PAGE 130

+HOO &RQWLQXXP -XOLDQ 'DWH f / )LJXUH 3ORW RI WKH LQWHQVLW\ RI WKH KHOLXP HPLVVLRQ OLQH IRU WKH ODVW VL[ KRXUV RI GDWD VHW 9 UR /2

PAGE 131

&OOO &RQWLQXXP )LJXUH 3ORW RI WKH LQWHQVLW\ RI WKH FDUERQ HPLVVLRQ OLQH IRU WKH ODVW VL[ KRXUV RI GDWD VHW 9

PAGE 132

GDWD VHW DOORZV IRU D ZLGH UDQJH RI IUHTXHQFLHV WR EH DQDO\]HG LPAQ Gn DQG YPD[ G n )LJXUH VKRZV WKH UHVXOWLQJ VSHFWUDO ZLQGRZ IXQFWLRQ IRU WKLV JURXS RI GDWD $OWKRXJK WKH UHODWLYH SRZHU RI WKH SHDNV LV TXLWH ORZ RQH FDQ VWLOO LGHQWLI\ ILUVW DQG VHFRQG KDUPRQLFV $IWHU WKDW WKH EHDP EHFRPHV FRQWDPLQDWHG ZLWK H[WUHPHO\ ORZ DPSOLWXGH QRLVH 7KH VWURQJHVW KDUPRQLF RWKHU WKDQ WKDW DW Y 2G LV ORFDWHG DW DERXW Y G DQG LV RQO\ DERXW b DV VWURQJ DV WKH PDLQ FRPSRQHQW +HQFH RQH ZRXOG H[SHFW WKDW WKH VDPSOLQJ IXQFWLRQ PD\ LQWURGXFH RQO\ VPDOO DPSOLWXGH VSXULRXV IHDWXUHV LQWR WKH SRZHU VSHFWUXP RI WKHVH GDWD )LJXUHV DQG VKRZ WKH )RXULHU WUDQVIRUP RI WKH KHOLXP LQWHQVLW\ PHDVXUHPHQWV LQ WHUPV RI UHODWLYH SRZHU DQG WKH VSHFWUXP ZKLFK UHVXOWV DIWHU FOHDQLQJ DW WKH FWMA OHYHO 7KH QRLVH OHYHO IRU HDFK RI WKHVH SORWV LV DW [ n DQG D QXPEHU RI SHDNV DUH VWDWLVWLFDOO\ VLJQLILFDQW ,Q )LJXUH WKH VWURQJHVW IHDWXUH DW Y Gn 3 KRXUVf RFFXUV DW WKH EHJLQQLQJ RI WKH SORW DQG FRUUHVSRQGV DOPRVW H[DFWO\ WR YPAQ 7KLV IHDWXUH KDV D VLJQDOWRQRLVH UDWLR RI DSSUR[LPDWHO\ 7KH QH[W VWURQJHVW IHDWXUH LV DW Y G 3 KRXUVf DQG KDV D UHODWLYH SRZHU HTXLYDOHQW WR FWA $QRWKHU UHDVRQDEO\ VWURQJ SHDN LV ORFDWHG DW X G VXJJHVWLQJ UDSLG YDULDWLRQV RQ WKH RUGHU RI VHFRQGV 7KLV SHDN KDV D VWUHQJWK ZKLFK H[FHHGV WKH QRLVH OHYHO E\ D IDFWRU RI DQG PXVW EH FRQVLGHUHG ZLWKLQ WKH FRQWH[W RI WKLV DQDO\VLV

PAGE 133

7DEOH 2EVHUYLQJ 3URJUDP 6WDU % 9 +H & &RQWLQXXP &r ; ; &6 ; ; &r ; ; 9r ; ; 9r ; ; 9r ; ; 9r ; ; 9r Yr Yr 9r 96 ; ; 9r 9r 9r Yr 9r ; ; 9r ; ; 9r ; ; 9r ; ; &r ; ; &6 ; ; &r ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; &r FRPSDULVRQ VWDU &6 FRPSDULVRQ VN\ 9r YDULDEOH VWDU 96 YDULDEOH VN\

PAGE 134

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW 9, AVM

PAGE 135

5HODWLYH 3RZHU )UHTXHQF\ F0f )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW 9, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 136

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW 9, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP ?e!

PAGE 137

2QH ODVW IHDWXUH ZKLFK VKRXOG EH PHQWLRQHG LV WKDW DW Y G 3 VHFRQGVf $OWKRXJK WKLV SHDN LV FHUWDLQO\ QRW DV VWURQJ DV WKH WKUHH SUHYLRXV IHDWXUHV ZKLFK KDYH EHHQ PHQWLRQHG LW VWLOO UHPDLQV VWDWLVWLFDOO\ VLJQLILFDQW 7KH OHDVW VTXDUHV URXWLQH GRHV QRW FRQYHUJH IRU D SHULRG WKLV VKRUW EXW WKH SRVVLEOH VLJQLILFDQFH RI WKH SUHVHQFH RI WKLV IHDWXUH PXVW QRW EH RYHUORRNHG 9HU\ UDSLG IOXFWXDWLRQV EHWZHHQ DQG VHFRQGV KDYH EHHQ SUHYLRXVO\ UHSRUWHG -HIIHUV HW DO D Ef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f 'WWf r VLQ U Ws L / f 1HLWKHU WKLV PRGHO QRU WKH PRGHO IRU WKH KRXU SHULRG VHHPV WR EH DQ DGHTXDWH UHSUHVHQWDWLRQ RI WKHVH GDWD

PAGE 138

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD RI GDWD VHW 9, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO f§ /2

PAGE 139

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f DQG WKH FOHDQ )LJXUH f VSHFWUD $OWKRXJK WKUHH SHDNV ULVH DERYH WKH VWDWLVWLFDOO\ VLJQLILFDQW OHYHO DW DMM RQO\ RQH IHDWXUH UHPDLQV VLJQLILFDQW DIWHU HOLPLQDWLQJ VSXULRXV IHDWXUHV GXH WR WKH VDPSOLQJ IXQFWLRQ 7KLV SHDN ORFDWHG DW X G f KDV D UHODWLYH SRZHU RI WLPHV WKH QRLVH OHYHO 7KH OHDVW VTXDUHV URXWLQH FRQYHUJHV WR D SHULRG RI KRXUV DQG KDV RQH RI WKH ODUJHVW DPSOLWXGHV RI YDULDWLRQ IRXQG LQ WKH GDWD VR IDU 7KH PRGHO ZKLFK FRUUHVSRQGV WR WKHVH

PAGE 140

5HODWLYH 3RZHU )UHTXHQF\ GBf )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW RI GDWD VHW 9, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 141

5HODWLYH 3RZHU )UHTXHQF\ Gf )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW RI GDWD VHW 9, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 142

YDOXHV LV VKRZQ LQ )LJXUH DQG LV FRPSXWHG ZLWK HTXDWLRQ f 'Wf r VLQ U f $OWKRXJK WKH HUURUEDUV IURP WKH SORW DUH UDWKHU ODUJH WKH PRGHO VHHPV WR UHSUHVHQW D SHULRGLF YDULDWLRQ RQ WKH RUGHU RI KRXUV $JDLQ DV RQH PLJKW H[SHFW WKH EHDP IRU WKH VHFRQG VXEVHW LV YHU\ VLPLODU WR WKDW FRPSXWHG IRU WKH ZKROH GDWD VHW )LJXUH f 7KH QRLVH OHYHO IRU WKHVH GDWD LV LQ IDFW TXLWH ORZ DW [ n DQG VHYHUDO SHDNV LQ WKH SRZHU VSHFWUXP )LJXUH f PLJKW EH FRQVLGHUHG DV VWDWLVWLFDOO\ VLJQLILFDQW 7KH &/($1 SURFHGXUH UHGXFHV WKH VSXULRXV SHDNV OHDYLQJ RQH GRPLQDQW IHDWXUH DW X G DQG WZR ORZHU DPSOLWXGH IHDWXUHV DW X G DQG Y GO )LJXUH f 7KH IHDWXUH DW Y G LV DFWXDOO\ QRW ZLWKLQ WKH 1\TXLVW IUHTXHQF\ OLPLW IRU WKLV VXEVHW EXW WKH JRRG DJUHHPHQW ZLWK WKH OHDVW VTXDUHV URXWLQH LPSOLHV WKDW WKLV PD\ UHSUHVHQW D SRVVLEOH SHULRGLFLW\ LQ WKHVH GDWD 7KH HTXDWLRQ XVHG WR GHVFULEH WKHVH YDULDWLRQV LV 'WLf r VLQ U f

PAGE 143

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU VXEVHW RI GDWD VHW 9, ZLWK 3 f§ KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 144

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW 9, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 145

5HODWLYH 3RZHU )UHTXHQF\ Gaf )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW 9, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 146

DQG WKH UHVXOWLQJ PRGHO LV VKRZQ LQ )LJXUH DORQJ ZLWK A DYHUDJH LQWHQVLW\ PHDVXUHPHQWV $ VHFRQG VROXWLRQ IRU WKH TXDQWLWLHV $ $W W4 DQG 3 LV DOVR SURGXFHG WKURXJK WKH HDVWVTXDUHV PHWKRG RI DQDO\VLV 7KH SHULRG KRXUV FRUUHVSRQGV WR D IUHTXHQF\ RI X G DQG LV DOVR D SURPLQHQW IHDWXUH LQ WKH SRZHU VSHFWUXP 7KH PRGHO IRU WKLV VROXWLRQ LV GHSLFWHG LQ )LJXUH DQG LV FRPSXWHG ZLWK WKH UHODWLRQ '&WLf r VLQ U W/ / f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

PAGE 147

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU VXEVHW ,, RI GDWD VHW 9, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 148

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU VXEVHW ,, RI GDWD VHW 9, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D LQWHUYDO

PAGE 149

5HODWLYH 3RZHU ( )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW RI GDWD VHW 9, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 150

5HODWLYH 3RZHU )UHTXHQF\ GBf )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW RI GDWD VHW 9, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 151

5HODWLYH 3RZHU ( 7 7 ( ( )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW 9, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 152

5HODWLYH 3RZHU ( ( ( ( )UHTXHQF\ G ff )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI VXEVHW ,, RI GDWD VHW 9, DIWHU H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 153

'Wf r VLQ U Ws nM / f $OWKRXJK WKH PRGHO )LJXUH f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

PAGE 154

&OOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH FDUERQ GDWD IRU GDWD VHW 9, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 155

REVHUYHG IRU KRXUV IURP -' WKURXJK -' EHIRUH EHLQJ LQWHUUXSWHG E\ EDG ZHDWKHU 7KH VDPSOLQJ IXQFWLRQ IRU WKLV GDWD VHW LV UHSUHVHQWHG E\ WKH SRZHU VSHFWUXP SUHVHQWHG LQ )LJXUH +HUH WKH WKUHH REYLRXV SHDNV DUH WKH ILUVW VHFRQG DQG WKLUG KDUPRQLFV DW Y[ Gn X G DQG L G O (YHQ WKRXJK D IHZ DGGLWLRQDO REVHUYDWLRQV RI WKH FDUERQ HPLVVLRQ OLQH ZHUH XVHG WKH VSHFWUDO ZLQGRZ GRHV QRW GLIIHU IURP )LJXUH IRU HLWKHU WKH KHOLXP RU WKH FDUERQ GDWD 5HODWLYH SRZHU FRPSXWHG IURP WKH UHDO DQG LPDJLQDU\ FRPSRQHQWV RI WKH )RXULHU WUDQVIRUP RI WKH KHOLXP LQWHQVLW\ PHDVXUHPHQWV LV SORWWHG LQ )LJXUH )HDWXUHV ZKLFK ULVH DERYH WKH VWDWLVWLFDOO\ VLJQLILFDQW OHYHO RI A LQFOXGH SHDNV DW X DQG G ,QGHHG PDQ\ RI WKHVH IHDWXUHV ULVH RQO\ PLQLPDOO\ DERYH WKH FUAc OHYHO DQG VRPH DUH DW OHDVW SDUW DOLDVHV ZLWK VWURQJHU SHDNV 7KH &/($1LQJ IXQFWLRQ )LJXUH f KDV UHGXFHG WKH VWUHQJWK RI HDFK IHDWXUH TXLWH VXEVWDQWLDOO\ +RZHYHU SHDNV DW X DQG GB VWLOO QHHG WR EH FRQVLGHUHG DV SRVVLEOH SHULRGLFLWLHV ZLWKLQ WKHVH GDWD 7KH SHDN DW Y Gn 3 KRXUVf LV FOHDUO\ WKH VWURQJHVW IHDWXUH LQ ERWK )LJXUHV DQG DW FWA OXA [ n 7KH QH[W ODUJHVW SHDN RFFXUV DW Y G KRXUVf ZLWK D VLJQDOR QRLVH UDWLR RQ WKH RUGHU RI 7KH RWKHU WZR IHDWXUHV LQ )LJXUH FRUUHVSRQG WR SHULRGV RI PLQXWHV

PAGE 156

5HODWLYH 3RZHU )LJXUH 6SHFWUDO ZLQGRZ IXQFWLRQ IRU GDWD VHW 9,,

PAGE 157

5HODWLYH 3RZHU )UHTXHQF\ Ga f )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW 9,, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJR ULWKP A R

PAGE 158

5HODWLYH 3RZHU )LJXUH 3RZHU VSHFWUXP IRU WKH KHOLXP REVHUYDWLRQV RI GDWD VHW 9,, DIWHU H[HFXWLRQ RI WKH FOHDQ W! DO JRU L W KP

PAGE 159

DQG PLQXWHV HDFK ZLWK UHODWLYH SRZHUV HTXDO WR FWA ([HFXWLRQ RI WKH HDVWVTXDUHV URXWLQH WR GHWHUPLQH WKH EHVWILW VLQH ZDYH WR WKH GDWD JLYHV D SHULRG RI 3 KRXUV RU LQ WHUPV RI F\FOHV SHU GD\ X Gn %HFDXVH RI WKH DJUHHPHQW EHWZHHQ WKHVH WZR PHWKRGV D PRGHO LV FRPSXWHG DFFRUGLQJ WR WKH HDVWVTXDUHV SDUDPHWHUV ZKLFK DUH H[SUHVVHG E\ WKH UHODWLRQ 'WWf r VLQ U WMB / f 7KH PRGHO LWVHOI LV VKRZQ LQ )LJXUH WRJHWKHU ZLWK WKH VWDQGDUG GHYLDWLRQV DVVRFLDWHG ZLWK A DYHUDJHV $ YDULDWLRQ ZLWK D VOLJKWO\ VPDOOHU DPSOLWXGH LV DOVR IRXQG E\ WKH HDVWVTXDUHV PHWKRG RI DQDO\VLV 7KH VHFRQG FRQYHUJHQFH FRUUHVSRQGV ZHOO ZLWK WKH SHDN DW X Gn LQ WKH SRZHU VSHFWUXP RI )LJXUH 7KH PRGHO VKRZQ LQ )LJXUH LV FRPSXWHG IURP HTXDWLRQ f 'Wf r VLQ U W/ / f %DVHG RQ WKH & UHVLGXDOV FRPSXWHG IURP HTXDWLRQV f DQG f WKH ORQJHU RI WKHVH WZR SHULRGV VHHPV WR EH D EHWWHU UHSUHVHQWDWLRQ IRU WKH YDULDWLRQV 7KH OHDVW VTXDUHV URXWLQH GRHV QRW FRQYHUJH IRU SHULRGV FRUUHVSRQGLQJ

PAGE 160

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH r 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW 9,, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 161

+HOO &RQWLQXXP -XOLDQ 'DWH f )LJXUH 0RGHO IRU WKH YDULDWLRQV RI WKH KHOLXP GDWD IRU GDWD VHW 9,, ZLWK 3 KRXUV (DFK SORWWHG SRLQW UHSUHVHQWV WKH DYHUDJH LQWHQVLW\ ZLWKLQ D A LQWHUYDO

PAGE 162

WR Y GO RU Y Gn DQG VR QR PRGHO IRU WKHVH SHULRGV LV FRQVLGHUHG VXFFHVVIXO 7KH SRZHU VSHFWUXP UHVXOWLQJ IURP D FRQYROXWLRQ RI WKH FDUERQ HPLVVLRQ GDWD ZLWK WKH VSHFWUDO ZLQGRZ )LJXUH f LV JLYHQ LQ )LJXUH 7KH ODA OHYHO LQ WKLV SORW LV DW [ n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f $OWKRXJK *DSRVFKNLQ f UHSRUWHG D SHULRG RI A EDVHG RQ D VHW RI YLVXDO DQG SKRWRJUDSKLF PHDVXUHPHQWV QR RWKHU LQYHVWLJDWRU KDV GHWHFWHG DQ RSWLFDO HFOLSVH RI WKLV V\VWHP

PAGE 163

5HODWLYH 3RZHU )UHTXHQF\ Gaf )LJXUH 3RZHU VSHFWUXP IRU WKH FDUERQ REVHUYDWLRQV RI GDWD VHW 9,, EHIRUH H[HFXWLRQ RI WKH FOHDQ DOJRULWKP

PAGE 164

:LOOLV DQG :LOVRQ f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f 2QH DVSHFW RI WKLV GLVVHUWDWLRQ LQYROYHV WKH DFTXLVLWLRQ RI % DQG 9 SKRWRPHWU\ RI \ 9HO ZLWK WKH 6327 LQ DQ DWWHPSW WR FRQILUP RU GHQ\ WKH H[LVWHQFH RI DQ RSWLFDO HFOLSVH 'XULQJ WKH DQG REVHUYLQJ VHDVRQV DW WKH 6RXWK 3ROH VHYHUDO GLIIHUHQW REVHUYLQJ SURJUDPV ZHUH XVHG WR REWDLQ % DQG 9 SKRWRPHWU\ RI WKLV V\VWHP $ WRWDO RI REVHUYDWLRQV -' WKURXJK -'f ZLWK WKH YLVXDO ILOWHU DQG REVHUYDWLRQV -' WKURXJK f ZLWK WKH EOXH ILOWHU ZHUH UHFRUGHG 7KHVH GDWD ZHUH UHGXFHG WR SKRWRPHWULF PDJQLWXGHV XVLQJ WKH WHFKQLTXHV GHVFULEHG LQ &KDSWHU ZLWK +5 DV WKH FRPSDULVRQ VRXUFH 7KH WLPH RI HDFK REVHUYDWLRQ LV FRQYHUWHG WR SKDVH XVLQJ WKH HSKHPHULV JLYHQ E\ HTXDWLRQ f DQG DYHUDJH PDJQLWXGHV DUH FRPSXWHG DW SKDVH LQWHUYDOV 7DEOH OLVWV WKH DYHUDJH $9 PDJQLWXGHV WKH

PAGE 165

7DEOH $YHUDJH $9 0DJQLWXGHV IRU 9HO 3KD V H 0DJQLWXGH $9f 506 'HYLDWLRQ 1

PAGE 166

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ƒ UDSLG YDULDWLRQV LQ WKH LQWHQVLW\ RI WKH VN\ EDFNJURXQG FDQQRW EH SURSHUO\ DFFRXQWHG IRU 7ZR ZD\V WR PLQLPL]H WKH HIIHFWV RI YDULDWLRQV LQ WKH VN\ EULJKWQHVV ZRXOG EH WR REVHUYH WKH VN\ PRUH IUHTXHQWO\ DQG WR XVH D VPDOOHU ILHOG VWRS +RZHYHU PRVW RI WKH GDWD XVHG LQ WKLV LQYHVWLJDWLRQ ZHUH REWDLQHG GXULQJ WKH

PAGE 167

REVHUYLQJ VHDVRQ $W WKLV SRLQW LQ WKH SURMHFW WKH HIIHFWV RI YDULDWLRQV LQ WKH VN\ EULJKWQHVV ZHUH QRW ZHOO XQGHUVWRRG DQG WKH WHOHVFRSH ZDV QRW HTXLSSHG ZLWK GLDSKUDJPV RWKHU WKDQ WKH n ILHOG VWRS ,Q WKH DFTXLVLWLRQ RI PRVW RI WKHVH GDWD D VN\ REVHUYDWLRQ ZDV REWDLQHG RQO\ RQFH HYHU\ WR PLQXWHV 7KLV LV FHUWDLQO\ QRW DGHTXDWH WR DFFRXQW SURSHUO\ IRU VSDWLDO DQGRU WHPSRUDO FKDQJHV ,Q DGGLWLRQ VLQFH WKHVH GDWD ZHUH REWDLQHG ZLWK D n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

PAGE 168

7DEOH $YHUDJH $% 0DJQLWXGHV IRU 9HO 3KD V H 0DJQLWXGH $%f 506 'HYLDWLRQ 1

PAGE 169

3KDVH )LJXUH 3KDVHPDJQLWXGH GLDJUDP IRU WKH EOXH SKRWRPHWU\ RI \ 9HO (DFK SRLQW UHSUHVHQWV WKH DYHUDJH PDJQLWXGH ZLWKLQ D SKDVH LQWHUYDO 7KH HUURUEDUV VKRZ WKH VWDQGDUG GHYLDWLRQV DVVRFLDWHG ZLWK HDFK DYHUDJH

PAGE 170

3KDVH )LJXUH 3KDVHPDJQLWXGH GLDJUDP IRU WKH YLVXDO SKRWRPHWU\ RI \ 9HO (DFK SRLQW UHSUHVHQWV WKH DYHUDJH PDJQLWXGH ZLWKLQ D SKDVH LQWHUYDO 7KH HUURU EDUV VKRZ WKH VWDQGDUG GHYLDWLRQV DVVRFLDWHG ZLWK HDFK DYHUDJH

PAGE 171

,W LV SUREDEO\ VRPH FRPELQDWLRQ RI WKHVH VLWXDWLRQV WKDW OHDGV WR WKH ODUJH UPV GHYLDWLRQV LQ WKH UHGXFHG GDWD &RPSOHWH SKDVH FRYHUDJH LQ WKH EOXH DQG YLVXDO ILOWHUV KDV QRW \HW EHHQ REWDLQHG )LJXUHV DQG f ,Q IDFW REVHUYDWLRQV DW FHUWDLQ FULWLFDO SKDVH SRVLWLRQV DUH PLVVLQJ SDUWLFXODUO\ WKRVH MXVW EHIRUH SKDVH ,Q DQ\ FDVH ZLWKLQ WKH OLPLWDWLRQV RI WKHVH GDWDf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

PAGE 172

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b $V UHFHQWO\ DV %LVLDFFKL HW DO UHSRUWHG WKDW XS WR b RI WKH :1( W\SH :ROI5D\HW VWDUV PD\ EH PHPEHUV RI ELQDU\ V\VWHPV 2Q WKH RWKHU KDQG D PRUH UHFHQW VWXG\ E\ 0RIIDW f VKRZV WKDW WKH IUHTXHQF\ RI ELQDULW\ DPRQJ WKLV FODVV RI VWDUV KDV SUREDEO\ EHHQ ODUJHO\ RYHUHVWLPDWHG 7RGD\ DSSUR[LPDWHO\ b RI WKH :ROI5D\HW VWDUV LQ WKH 0LON\ :D\ *DOD[\ DUH NQRZQ WR EH PHPEHUV RI ELQDU\ V\VWHPV 7KLV QHZ VWDWLVWLF WDNHQ WRJHWKHU ZLWK WKH IDFW WKDW VSHFWUDO YDULDWLRQV KDYH DOVR EHHQ REVHUYHG LQ :ROI5D\HW

PAGE 173

VWDUV ZKLFK GR QRW KDYH D FRPSDQLRQ LQGLFDWHV WKDW DQRWKHU PHFKDQLVP PXVW H[LVW WR H[SODLQ WKH FKDQJHV LQ WKH VSHFWUD RI WKHVH VWDUV 7KHRUHWLFDO UHVHDUFK QRZ LQGLFDWHV WKDW WKHVH FKDQJHV DUH SUREDEO\ OLQNHG WR WKH KLJK UDWHV RI PDVV ORVV LH a [ n0R \Uf DVVRFLDWHG ZLWK WKLV W\SH RI REMHFW $OWKRXJK SHULRGLF RU TXDVL SHULRGLF YDULDWLRQV UHVXOWLQJ IURP D VWHDG\ ODPLQDU IORZ RI PDWHULDO ZRXOG VLPSOLI\ WKH WKHRUHWLFDO LQWHUSUHWDWLRQ RI WKHVH VWDUV DFFRUGLQJ WR 9UHX[ f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n VSHHG SKRWRPHWU\ LV UHFRUGHG IRU D SHULRG RI DERXW WHQ PLQXWHV 2WKHU LQYHVWLJDWRUV ZKR KDYH UHFRUGHG ORQJHU LQWHUYDOV RI REVHUYDWLRQV UHSRUW YDULDWLRQV RI ORQJHU GXUDWLRQ 6RPH UHSRUW SHULRGLF EHKDYLRU RWKHUV UHSRUW

PAGE 174

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f ZHUH SXEOLVKHG WKHRU\ GLG QRW VXSSRUW WKH H[LVWHQFH RI QRQUDGLDO SXOVDWLRQV LQ KHLXPEXUQLQJ VWDUV +RZHYHU 1RHOV DQG 6FXIODLUH KDYH FRQVLGHUHG WKH HYROXWLRQDU\ VHTXHQFH RI D O220A VWDU DQG KDYH VKRZQ WKDW QRQUDGLDO SXOVDWLRQV FDQ SURGXFH YLEUDWLRQDO LQVWDELOLWLHV LI UHJLRQV RI K\GURJHQ VKH EXUQLQJ SHUVLVW 7KLV WKHRU\ SHUWDLQV PRVW FORVHO\ WR WKH :1 W\SH :ROI5D\HW VWDUV ZKLFK KDYH KLJKHU ++H UDWLRV WKDQ GR WKH :& W\SHV 7KHVH UHVXOWV PD\ OHQG VXSSRUW WR 9UHX[nV VXJJHVWLRQ WKDW QRQUDGLDO

PAGE 175

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f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f DQG UDGLDWLRQ SUHVVXUH DORQH FDQ VXSSRUW VXFK D VWURQJ DQG HIILFLHQW SKHQRPHQRQ &R[ DQG &DKQ f KDYH FDUULHG RXW DQ LQGHSHQGHQW VWXG\ LQYROYLQJ WKH WKHRUHWLFDO PRGHOLQJ RI WKHVH VWDUV DV

PAGE 176

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f &R[ DQG &DKQ FRQFOXGH WKDW LI PDVV ORVV LV WR EH WKH UHVXOW RI SXOVDWLRQV LW PXVW EH RQO\ IURP WKH UDGLDO IXQGDPHQWDO PRGH ZKLFK DSSHDUV LQ K\GURJHQIUHH VWDUV 7KHLU 04 PRGHO H[KLELWV UDGLDO SXOVDWLRQV ZKLFK DUH DJDLQ LQ DJUHHPHQW ZLWK 0DHGHU n V VWXG\ 2QH RI WKH IHZ PRGHOV ZKLFK KDV EHHQ SXW IRUZDUG LQ DQ DWWHPSW WR H[SODLQ VSHFWUDO YDULDWLRQV LQ \ 9HO LQFOXGHV WKH SUHVHQFH RI D WKLUG ERG\ ,Q 0RIIDW VXJJHVWHG WKDW WKLV VWDU PD\ EH RQH PHPEHU RI D WULSOHVWDU V\VWHP $QDO\VLV RI QDUURZEDQG SKRWRHOHFWULF SKRWRPHWU\ UHYHDOHG D SHULRG ,Q DGGLWLRQ UDSLG VSHFWURSKRWRPHWULF VFDQV LQGLFDWHG WKH SUHVHQFH RI D ORQJHU SHULRG RI YDULDWLRQ EHWZHHQ DQG GD\V ,W ZDV QRW XQWLO WKDW -HIIHUV HW DO SXUVXHG 0RIIDWnV VXJJHVWLRQ DQG GHULYHG D PRGHO IRU

PAGE 177

WKLV V\VWHP FRQWDLQLQJ D QHXWURQVWDU FRPSDQLRQ WR WKH :& FRPSRQHQW $FFRUGLQJ WR WKHLU LQYHVWLJDWLRQ WKH :ROI 5D\HW FRPSRQHQW ZRXOG KDYH D PDVV RI 04 DQG D UDGLXV RI s 5 ZKLOH WKH QHXWURQVWDU ZRXOG KDYH D PDVV RI s 0 $VVXPLQJ D FLUFXODU RUELW IRU WKH QHXWURQVWDU DQG EHFDXVH RI WKH ODFN RI RSWLFDO HFOLSVHV DQ LQFOLQDWLRQ RI r WKH PD[LPXP VHSDUDWLRQ GLVWDQFH EHWZHHQ WKH QHXWURQ VWDU DQG WKH :& VWDU ZRXOG EH 5 ,QWHUIHURPHWU\ PHDVXUHPHQWV PDGH E\ %URZQ HW DO f LQGLFDWH WKDW WKH H[WHQGHG ZLQG RI WKH :ROI5D\HW FRPSRQHQW KDV D UDGLXV RI 5 +HQFH WKH QHXWURQVWDU ZRXOG EH FRQWDLQHG ZHOO ZLWKLQ WKLV HQYHORSH RI PDWHULDO 7KH RUELWDO SHULRG ZKLFK UHVXOWV IURP WKLV VLWXDWLRQ LV DERXW r" 6LQFH WKHVH WZR VWDUV DUH UHDVRQDEO\ FORVH WR RQH DQRWKHU RQH ZRXOG H[SHFW WKDW WKH DFFUHWLRQ RI PDWHULDO RQWR WKH QHXWURQVWDU ZRXOG SURGXFH D ODUJH IOX[ RI KDUG [UD\V +RZHYHU REVHUYDWLRQV UHYHDO WKDW YHU\ ORZ DPRXQWV RI KDUG [UD\V DUH DFWXDOO\ GHWHFWHG LQ WKH \ 9HO V\VWHP 6RPH RI WKH WKHRULHV ZKLFK KDYH EHHQ SURSRVHG WR DFFRPPRGDWH VXFK D UHVXOW LQFOXGH WKH IROORZLQJ 0RIIDW DQG 6HJJHZLVV f VWDWH WKDW LI WKH GHQVLW\ RI PDWHULDO LQ WKH ZLQG LV VXIILFLHQWO\ ODUJH WKHQ WKH KDUG [UD\V PD\ EH UHDEVRUEHG 6XQ\DHY f H[SODLQV WKDW KDUG [UD\V PD\ QRW EH GHWHFWHG LI DFFUHWLRQ RFFXUV DW VXSHUFULWLFDO UDWHV VR WKDW WKH DFFUHWLRQ GLVN EHFRPHV RSDTXH WR WKLV UDGLDWLRQ

PAGE 178

,I D QHXWURQVWDU LV SUHVHQW LQ WKH \ 9HO V\VWHP SHUWXUEDWLRQV LQ WKH UDGLDO YHORFLW\ PHDVXUHPHQWV VKRXOG LQ SULQFLSOH EH GHWHFWHG 0RIIDW HW DO f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f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

PAGE 179

7DEOH OLVWV DOO RI WKH SHULRGV DQG FRUUHVSRQGLQJ IUHTXHQFLHV IRXQG LQ WKH DQDO\VHV RI WKH QDUURZEDQG SKRWRPHWU\ XVHG LQ WKLV VWXG\ :LWK WKH H[FHSWLRQV RI WKH KRXU SHULRG 'DWD 6HW ,f DQG WKH KRXU SHULRG 'DWD 6HW ,,f WKH VKRUWHVW SHULRGV LQ DOO RI WKHVH GDWD DUH 3 KRXUV 'DWD 6HW ,,f DQG 3 KRXUV 'DWD 6HW 9LDf ,W LV LQWHUHVWLQJ WR QRWH WKDW PDQ\ RI WKH SHULRGV OLVWHG LQ FROXPQ WKUHH RI WKH WDEOH DUH QHDUKDUPRQLFV RU VXEKDUPRQLFV RI RQH RI WKHVH WZR SHULRGV $Q DYHUDJH SHULRG RI KRXUV LV XVHG WR FRPSXWH WKH PXOWLSOLFLW\ RI HDFK RI WKH SHULRGV LQ WKLV WDEOH 7KHVH UHVXOWV DUH OLVWHG LQ FROXPQ ILYH 7KH WZR SHULRGV IRXQG WR VDWLVI\ 'DWD 6HW ,,, DUH QRW VLPSOH KDUPRQLFV RI WKH KRXU IXQGDPHQWDO SHULRG +RZHYHU DV VWDWHG LQ WKH GLVFXVVLRQ RI WKLV GDWD VHW LQ &KDSWHU UHODWLYHO\ ODUJH LQWHUYDOV ZKHQ \ 9HO ZDV QRW REVHUYHG H[LVW DQG PDNH LW GLIILFXOW WR GHWHUPLQH D UHOLDEOH SHULRG ,Q DGGLWLRQ 'DWD 6HW 9,E KDV EHHQ GHVFULEHG ZLWK D SHULRG RI KRXUV DQG D SHULRG RI KRXUV 7KH ORQJHU RI WKHVH SHULRGV LV VHYHQ WLPHV WKH KRXU SHULRG EXW WKH KRXU SHULRG FDQQRW EH UHODWHG WR WKH DVVXPHG IXQGDPHQWDO SHULRG $ ORRN DW )LJXUHV 3 KRXUVf DQG 3 KRXUVf LQGLFDWHV WKDW WKH ORQJHU SHULRG WKDW LV IRXQG IRU WKHVH GDWD LV D EHWWHU UHSUHVHQWDWLRQ RI WKH YDULDWLRQV $W WKLV SRLQW LW LV LQWHUHVWLQJ WR OLQN WKHVH UHVXOWV ZLWK WKH UHVXOWV RI WKHRUHWLFDO PRGHOLQJ RI :ROI5D\HW VWDUV

PAGE 180

7DEOH 6XPPDU\ RI 3HULRGLFLWLHV )RXQG LQ WKH +HOO (PLVVLRQ /LQH 'DWD 6HW 3KDVH 3HULRG )UHTXHQF\ 0XOWLSOLFLW\ ,QWHUYDO KRXUVf Gf 3 f ‘ 3 3 ,, 3 a3 f§ ,9D 3 ,9E 3 3 9 ‘ f§ 9LD a3 9,E 3 3 9,, 3 3

PAGE 181

DFFRPSOLVKHG E\ &R[ DQG &DKQ f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ƒ 7KLV LQH LV FRQVLGHUDEO\ ZHDNHU WKDQ WKH KHOLXP OLQH DW ƒ DQG YDULDWLRQV DUH PXFK PRUH GLIILFXOW WR PRGHO ,Q DOPRVW HYHU\ FDVH WKH DPSOLWXGH RI IOXFWXDWLRQ DPRXQWV WR DERXW RQO\ b RI WKH WRWDO LQWHQVLW\ 0RGHOV KDYH EHHQ FRPSXWHG IRU WKH WZR FDVHV ZKHQ WKH YDULDWLRQ H[FHHGHG D ,W IOXFWXDWLRQ 'DWD 6HW H[KLELWHG WKH ODUJHVW SHUFHQWDJH RI YDULDWLRQV LQ ERWK WKH FDUERQ DQG KHOLXP HPLVVLRQ OLQH VWUHQJWKV 7KH FDUERQ OLQH ZDV PRGHOHG ZLWK D SHULRG RI KRXUV DQG WKH DPSOLWXGH RI WKH YDULDWLRQ ZDV DSSUR[LPDWHO\ b RI WKH FDUERQ OLQH LQWHQVLW\ 'DWD 6HW 9,E H[KLELWHG IOXFWXDWLRQV ZLWK 3 KRXUV DQG

PAGE 182

DQ DPSOLWXGH RI IOXFWXDWLRQ RI QHDUO\ b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

PAGE 183

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nV 7KHVLV 8QLY RI )ORULGD *DQHVK .6 %DSSX 0.9 .RGDLNDQDO 2EV %XOO 6HU $ 1R *DSRVFKNLQ 6 $VWURQ

PAGE 184

+DHIQHU 5 0HW] 6FKRHPEV 5 $VWURQ $VWURRKYV +HQGHQ $$ .DLWFKXFN 5+ $VWURQRPLFDO 3KRWRPHWU\ 1HZ
PAGE 185

5REHUWV 5+ /HK£U 'UHKHU -: $VWURQ 5XEOHY 69 ,]Y 6RHWV $VWURIL] 2EV NB 6PLWK /) :ROI5DYHW DQG +LJK 7HPSHUDWXUH 6WDUV ,$8 6YPS 1R HGV 0.9 %DSSX DQG 6DKDGH 'RUGUHFKW 5HLGHO 6XQ\DHY 5$ 3URF RI WKH ,QW 6FKRRO RI 3KYV (QULFR )HUPL 1HZ
PAGE 186

%,2*5$3+,&$/ 6.(7&+ 0DU\-DQH ZDV ERUQ LQ /DNH :RUWK )ORULGD RQ 0DUFK WR 3HWHU 2DWHV DQG &RQVWDQFH $QQ 7D\ORU 0DU\-DQHnV IDWKHU SOD\HG D YHU\ PDMRU UROH LQ KHU OLIH VXSSO\LQJ VXSSRUW DQG HQFRXUDJHPHQW DORQJ WKH ZD\ $W DJH ILYH VKH DVVLVWHG KHU IDWKHU LQ EXLOGLQJ DQ LQFK 1HZWRQLDQ WHOHVFRSH DQG WKHQ SURFHHGHG WR DVVHPEOH KHU RZQ LQFK UHIOHFWRU DW DJH VL[ ,Q 0DU\-DQH EHFDPH WKH \RXQJHVW PHPEHU RI WKH $PHULFDQ $VVRFLDWLRQ RI 9DULDEOH 6WDU 2EVHUYHUV $$962f DV DQ DFWLYH FRQWULEXWRU VXEPLWWLQJ WKRXVDQGV RI VXQVSRW DQG YDULDEOH VWDU REVHUYDWLRQV 6KH FRQWLQXHV WR XVH KHU RULJLQDO LQFK WHOHVFRSH IRU VRODU REVHUYDWLRQV DQG LV RQH RI DERXW LQWHUQDWLRQDO FRQWULEXWRUV WR WKH $PHULFDQ 6XQVSRW 3URJUDP 0DU\-DQHnV IDWKHU RIIHUHG KHU DQ HDUO\ H[SRVXUH WR PDWKHPDWLFV E\ WHDFKLQJ KHU ILUVW DQG VHFRQG \HDU DOJHEUD ZKLOH VKH ZDV LQ WKLUGJUDGH 6KH VNLSSHG WKH IRXUWKJUDGH DQG LQ SXEOLVKHG KHU ILUVW DVWURQRPLFDO SDSHU LQ WKH -RXUQDO RI WKH $PHULFDQ $VVRFLDWLRQ RI 9DULDEOH 6WDU 2EVHUYHUV 6LQFH WKDW WLPH VKH KDV SXEOLVKHG D QXPEHU RI DGGLWLRQDO DVWURQRPLFDO SDSHUV RQ YDULDEOH VWDU DVWURQRP\

PAGE 187

0DU\-DQH DWWHQGHG /DNH :RUWK +LJK 6FKRRO DQG ZDV GXDOO\ DQG WULSO\ HQUROOHG DW 3DOP %HDFK -XQLRU &ROOHJH DQGRU )ORULGD $WODQWLF 8QLYHUVLW\ GXULQJ KHU MXQLRU DQG VHQLRU \HDUV 'XULQJ WKLV SHULRG VKH DFTXLUHG D EDFNJURXQG LQ XSSHUOHYHO PDWKHPDWLFV DQG HQUROOHG LQ D QXPEHU RI FRPSXWHU FODVVHV 'XULQJ WKH VXPPHUV RI DQG 0DU\-DQH SDUWLFLSDWHG DV D UHVHDUFK DVVLVWDQW DW WKH 0DULD 0LWFKHOO 2EVHUYDWRU\ XQGHU WKH GLUHFWLRQ RI 'U 'RUULW +RIIOHLW ORQJ DVVRFLDWHG ZLWK
PAGE 188

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

PAGE 189

, FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ LQ 3DUNHU 2OLYHU &KDLUPDQ LVVRFLDWH 3URIHVVRU RI $VWURQRP\ FHUWLI\ WKDW KDYH UHDG WKLV VWXG\ DQG WKDW LQ P\ RSLQLRQ LW FRQIRUPV WR DFFHSWDEOH VWDQGDUGV RI VFKRODUO\ SUHVHQWDWLRQ DQG LV IXOO\ DGHTXDWH LQ VFRSH DQG TXDOLW\ DV D GLVVHUWDWLRQ IRU WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ .ZDQ
PAGE 190

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