Photoelectric Study of
EE Aquarii and AE Phoenicis
By
RICHARD MICHAEL WILLIAMON
A DISSERTAlTION PRESENTED TO THE ADUTEG COUNTTCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FUL'ILTLMT .ENT OF TIE REQUIF.EMEENTS FOR THE DEGREE OF
DOC ORP, O PHILOSOPHY
UNIVERSITY OF FLORIDA
1972
ACKNOWLEDGEMENTS
The author wishes to express his sincere appreciation
to Drs. KY Chen and J. E. Merrill for their advice and
assistance in reducing and analyzing the data and for their
many helpful suggestions for improving this manuscript.
Especial thanks are given to Dr. KY Chen for serving as
chairman of the supervisory comrittee and for his co ntiinuing
encouragement. Appreciation is also given to Drs. A. G.
Smith, Y. B. Wood, T. L. Bailey, and J. K. Gleira who, in
addition to Drs. KY Chen and J. E. Merrill, served on the
supervisory committee.
Appreciation is also given to the INational Science
Foundation for the author's financial support during part
of his stay at the University of Florida, and to Dr. S. S.
Eallard and the University of Florida for the remainder of
the author's financial support, including the funds for the
trip to Cerro Tololo Intmeamerican Observatory. The author's
"gratitude is also extended to Dr. V. Blanco and J. Graham
and the other members of the Cerro Tololo Interamerican
Observatory staff for their assistance during the author's
visit.
The skill of Mr. W. W. Richardson in preparing the
figures, the helpful advice of Mr. J. Whalen in preparing
the computing programs, and the skill of Mr. R. Simons in
preparing the photographs are all very deeply appreciated.
The author also wishes to express his appreciation to
Mrs. L. Honea for her untiring efforts in the typing of
this manuscript.
A special note of thanks is also due the author's wife
for her continuing understanding and encouragement as well
as her help. It is to her that this dissertation is
dedicated.
iii
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS . . .
LIST OF TABLES . . . .
LIST OF FIGURES . . .
ABSTRACT .
CHAPTER
I INTRODUCTION .
II INSTRUMENTATION
. . xi
. . . . . . . 1
. . . . . . . 5
Rosemary Hill Observatory
Cerro Tololo Observatory
OBSERVATIONS . . . . . . .
Rosemary Hill Observatory . . .
Cerro Tololo Observatory ..
REDUCTION . . . . . . .
Extinction and Magnitude Difference
Light Travel Correction . . .
Period Study . . . . . .
Light Curves . . . . . .
Color Curves . . . . . .
Models and Rectification ..
Ellipticity Effect . . . .
Reflection Effect . . . . .
Complications and Perturbations .
Intensity Rectification . . .
Phase Rectification . . . .
V SOLUTIONS . . . . . .
Solution from the X Functions .
Nomographs . . . . .
Solution from the i Functions .
III
S. 26
28
S 30
S. 32
S. 46
TABLE OF CONTENTS (continued)
CHAPTER Page
VI EE AQUARII . . . . . . . . .56
History . . . . . . . . 56
Comparison and Check Stars . . .. 59
Extinction . . . . . . . 62
Period Study . . . . . . 64
Light Curves . . . . . . 72
Color Curves . . . . . . 79
Rectification . . . . . . 84
Solution . . . . . . . 85
Conclusions . . . . . . . 99
VII AE PHOENICIS . . . . .. . . 101
History . . . . . . . . 101
Comparison and Check Stars . . .. .104
Extinction . . . . . . . 107
Period Study . . . . . . 109
Light Curves . . . . . . 117
Color Curves . . . . . . 124
Rectification . . . . . . 126
Solution . . . . . . . 132
Conclusions . . . . . . . 142
APPENDIX . . . . . . . . . . . 146
LIST OF REFERENCES . . . . . . . . . 183
BIOGRAPHICAL SKETCH . . . . . . . .. 185
LIST OF TABLES
Table Page
1 Filter characteristics for the UBV System . . 28
2 EE Aquarii, Comparison and Check Stars ... .62
3 Atmospheric extinction coefficients and
color extinction coefficients for EE Aquarii 63
4 Observed times of primary eclipse for
EE Aquarii . . . . . . . . . 66
5 Photographically determined times of primary
minima used in the period study of EE Aquarii 67
6 Standard star observations and zero point
corrections for the color indices of
EE Aquarii . . . . . . . . ... 81
7 Rectification coefficients for EE Aquarii . 86
8 Elements for the solution of EE Aquarii . . 96
9 Statistical study of EE Aquarii . . . ... 98
10 AE Phoenicis, Comparison and Check Stars . . 107
11 Atmospheric extinction coefficients and color
extinction coefficients for AE Phoenicis . .108
12 Observed times of minima for AE Phoenicis . .. 110
13 Photographically determined times of minima
used in the period study for AE Phoenicis . ill
14 Standard star observations and zero point
corrections for the color indices of
AE Phoenicis . .. . . . . . . . 125
15 Rectification coefficients for AE Phoenicis . 130
16 Elements for the solution of AE Phoenicis . . 141
17 A statistical study of AE Phoenicis ....... 143
 
18 EE Aquarii observations in yellow ..... . 147
19 EE Aauarii observations in blue . . . .
20 EE Aquarii observations in ultraviolet . .
21 Check star observations for EE Aquarii . .
22 Average of every five differences between the
observed rectified observations and the
calculated theoretical curve for EE Aquarii
in yellow light . . . . . . .
23 Average of every five differences between the
observed rectified observations and the
calculated theoretical curve for EE Aquarii
in blue light . . . . . . . .
24 Average of every five differences between the
observed rectified observations and the
calculated theoretical curve for EE Aquarii
in ultraviolet light . . . . . .
25 Average of every five differences between the
observed intensity observations and the
derectified curve for EE Aquarii in yellow
light . . . . . . . . . .
26 Average of every five differences between
observed intensity observations and the
derectified curve for EE Aquarii in blue
light . . . . . . . . ...
27 Average of every five differences between
observed intensity observations and the
derectified curve for EE Aquarii in
ultraviolet light . . . . . .
28 AE Phoenicis observations in yellow . .
29 AE Phoenicis observations in blue ..
30 AE Phoenicis observations in ultraviolet
31 Check star observations for AE Phoenicis
the
. . 165
the
166
. . 167
170
. . 173
. 176
32 Average of every five differences between the
observed rectified observations and the
calculated theoretical curve for AE Phoenicis
in yellow light . . .. . . . . .
177
vii
. 151
. 155
. 159
S. 161
S. 162
S 163
S. 164
33 Average of every five differences between the
observed rectified observations and the
calculated theoretical curve for AE Phoenicis
in blue light . . . . . . . ... 178
34 Average of every five differences between the
observed rectified observations and the
calculated theoretical curve for AE Phoenicis
in ultraviolet light . . . . . ... 179
35 Average of every five differences between the
observed intensity observations and the
derectified curve for AE Phoenicis in yellow
light . . . . . . . . . . . 180
36 Average of every five differences between the
observed intensity observations and the
derectified curve for AE Phoenicis in blue
light . . . . . . . . . . . 181
37 Average of every five differences between the
observed intensity observations and the
derectified curve for AE Phoenicis in
ultraviolet light . . . . . . ... 182
viii
LIST OF FIGURES
Figure Page
1. Thirtyinch telescope building at the Rosemary
Hill Observatory . . . . . . . . 7
2. The dual channel photoelectric photometer
attached to the Rosemary Hill Observatory
thirtyinch reflecting telescope . . . . 9
3. A sample of a strip chart record of EE Aquarii
obtained at Rosemary Hill Observatory with the
thirtyinch reflecting telescope . . .. . 14
4. Sixteeninch telescope building at the Cerro
Tololo Interamerican Observatory . . . . 17
5. The number one, sixteeninch reflecting telescope
at Cerro Tololo Interamerican Observatory . . 19
6. A sample of a strip chart record of AE Phoenicis
obtained at Cerro Tololo Interamerican Observatory
with the number one, sixteeninch telescope . .23
7. A block diagram of the electronics used to
obtain data photoelectrically, at Rosemary Hill
and Cerro Tololo Observatories . . . ... 25
8. A photographic light curve for EE Aquarii . . 58
9. A finding chart for EE Aquarii . . . . . 61
10. Results of the period study of EE Aquarii showing
(OC)'s from photographically and uhotoelectrically
determined times of minima . . .. . . . 69
11. Results of the period study of EE Aquarii showing
(OC)'s from photoelectrically determined times
of minima . . . . . . . . . .. .71
12. EE Aquarii light curve from yellow observations 74
13. EE Aquarii light curve front blue observations 76
14. EE Aquarii light curve from ultraviolet
observations . . . . . . . .
15. EE Aquarii color curves . . . . .
16. Theoretical light curve for the primary of
EE Aquarii in yellow light . . . . .
17. Theoretical light curve for the primary of
EE Aquarii in blue light . . . . .
18. Theoretical light curve for the primary of
EE Aquarii in ultraviolet light . . .
19. A photographic light curve for AE Phoenicis
20. A finding chart for AE Phoenicis . . .
21. Results of the period study of AE Phoenicis
showing (OC)'s from both photographically
and photoelectrically determined times of
minima . . . . . . . . .
22. Results of the period study of AE Phoenicis
showing (OC)'s from photoelectrically
determined times of minima . . . . .
23. AE Phoenicis light curve from yellow
observations . . . . . . . .
24. AE Phoenicis light curve from blue
observations . . . . . . . .
25. AE Phoenicis light curve from ultraviolet
observations . . . . . . . .
26. AE Phoenicis color curves . . . . .
27. Theoretical light curves for AE Phoenicis in
yellow light . . . . . . . .
28. Theoretical light curves for AE Phoenicis in
blue light . . . . . . . . .
29. Theoretical light curves for AE Phoenicis in
ultraviolet light . . . . . . .
. . 78
. . 83
S . 90
S . 92
. . 94
. .103
. .106
. .114
. .116
. .119
. .121
S .123
. .128
. .136
S .138
. .140
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
PHOTOELECTRIC STUDY OF EE AQUARII AND AE PHOENICIS
by
Richard Michael Williamon
June, 1972
Chairman: Dr. KwanYu Chen
Major Department: Astronomy
Photoelectrically observed light curves of the
eclipsing binary star systems EE Aquarii and AE Phoenicis
were obtained using the thirtyinch reflecting telescope
at the University of Florida's Rosemary Hill Observatory
and the number one, sixteeninch telescope at Cerro Tololo
Interamerican Observatory near LaSerena, Chile, respectively.
The observational data from EE Aquarii was obtained on
eleven nights in August, September, and October, 1970, and
on three nights in August and September, 1971. A total of
309, 302, and 299 usable observations were obtained in
yellow, blue, and ultraviolet light for EE Aquarii, and
242 observations were obtained in each of the colors yellow,
blue, and ultraviolet for AE Phoenicis. In both cases, the
colors corresponded to the colors of the (UBV) system of
Johnson and Morgan (Ap. J. 117, 313, 1953).
The Russell model was assumed, and the light curves
were constructed and rectified with the techniques of
Russell and Merrill (Contr. Prin. Obs. No. 26, 1952).
Solutions were obtained with the aid of the tables and
nomographs of Merrill (Contr. Prin. Obs. Nos. 23, 1950, and
24, 1953). For EE Aquarii, a ratio of the radii of 0.69, a
radius of the larger star of 0.468, a luminosity of the
larger star of 0.915, and an inclination of the orbit of
68947 were found. For AE Phoenicis, a ratio of the radii
of 0.574, a radius of the larger star of 0.485, a luminosity
of the larger star of 0.739, and an inclination of the orbit
of 85953 were found.
The eclipsing binary EE Aquarii has an amplitude of
light variations of 065 for primary and 020 for secondary.
The period is 0950899558, and the eclipses are partial with
primary a transit and secondary an occultation. AE Phoenicis
is a W Ursae Majoris type eclipsing binary with a period of
0936237456. The primary eclipse is a complete occultation
and the secondary is a transit. Complications were
encountered in the case of AE Phoenicis in that a rather
large term proportional to the cosine of the phase angle
was present in yellow and ultraviolet light, an asymmetry
proportional to the sine of the phase angle was present for
all colors, and a small amount of orbital eccentricity was
also possibly present.
xii
CHAPTER I
INTRODUCTION
Ancient astronomers, forced to rely solely on
observations with their unaided eyes, probably never
conceived of two stars revolving about a common center of
mass. Astronomers now realize that star systems with two
or more components are not uncommon. Based on the
observation that one hundred and twentyseven of the
nearest two hundred and fiftyfour stars are members of
multiple star systems (Motz and Duveen, 1966), estimates
of the total number of stars in multiple systems range
up to fifty per cent.
There are three different types of two star or binary
star systems which may be detected. Visual binaries
consist of two components which can be seen as individual
stars with available optical telescopes. Spectroscopic
binaries reveal a binary nature due to the periodic
displacement of their spectral lines resulting from the
Doppler shift. Eclipsing binaries are systems in hich
the plane of the orbit is very nearly edgewise to us. As
a result, the light received is diminished periodically
as one component passes in front of the other.
The theory that two stars might revolve about a common
center of mass was probably not seriously considered until
late in the eighteenth century. One of the early proponents
of the theory, following his attempts at parallax measure
ments of stars, was Sir William Herschel (Pannekoek, 1961).
Since direct measurements of stellar positions had errors
too large to determine parallaxes, Herschel proposed to
measure repeatedly the position of a bright star relative
to a fainter star. He mistakenly assumed that the magnitude
of a star was a direct indicator of distance and that any
displacement would be due to parallax of the brighter star
alone. In his early reports on the results of his parallax
studies, Herschel only briefly mentioned the possible
existence of binary star systems. In his reports of 1802
and 1803, however, Herschel described how the position
angle for about fifty of his starpairs had changed by
amounts between 50 and 510. He eliminated the possibility
.f the change being due to the motion of the sun and adopted
the explanation of orbital motion of both stars around a
common center of mass.
Visual binaries, as found by Sir William Herschel,
were, however, probably not the first binary star systems
observed. As mentioned before, an eclipsing binary star,
whose components are not resolved visually and therefore
apear as one point of light, periodically diminishes in
light intensity from our vantage point here on earth. One
such system, Beta Persei, fades almost one and onehalf
magnitudes at intervals of 2 days 20 hours and 49 minutes
for a time of two hours. Beta Persei is also known as
Algol, which probably comes from a name meaning demon or
devil given to it by ancient Arab astronomers (Glasby,
1968). As the name suggests, Algol had been seen to vary
in intensity since antiquity and, indeed, was possibly the
first variable star of any type to be observed. It was not
until the year 1783, however, that John Goodricke correctly
interpreted the light variations on the basis of an
eclipsing binary system (Goodricke, 1783).
Since the time of William Herschel, thousands of
binary stars have been discovered and cataloged. The
discovery of such systems has been aided greatly by the
development of the blink comparator. This instrument
allows the operator to search two photographic plates of
the same star field for a change in magnitude of any star
on the plates. The operator views first one plate and
then the other in such a way that a variable star will
appear to blink. The two eclipsing binary star systems
analyzed in this work were both discovered with the aid of
a blink comparator.
One begins the analysis of an eclipsing binary system
by constructing a light curve. This is a relationship
between the light received from the star as a function of
the orientation of the two stars in their orbit about one
another. The usual appearance of a light curve is a
rather smooth curve with two dips corresponding to the two
eclipses. The deeper of the two eclipses is referred to as
the primary eclipse and occurs when the star with greater
surface brightness is eclipsed by the star with lower
surface brightness. The secondary eclipse, usually located
a half cycle later, occurs when this condition is reversed.
Eclipsing binary systems are valuable sources of
information of properties of stars. Careful analysis of
the light curve will provide the degree of darkening at the
limb of each star, the inclination of the plane of the
orbit with respect to the plane of the sky, the apparent
luminosity received from each star, the ratio of the radii
of the two stars, the ratio of the radii relative to their
separation, and the ratio of the surface brightnesses of
the two stars. The purpose of the present work is to
construct light curves for the eclipsing binaries
EE Aquarii and AE Phoenicis and to analyze these light
curves and determine the properties and orbital parameters
of both systems.
CHAPTER II
INSTRUMENTATION
Rosemary Hill Observatory
The observational data for EE Aquarii was obtained on
eleven nights in August, September, and October, 1970, and
on three nights in August and September, 1971. All
observations were made with the thirtyinch reflecting
telescope owned by the University of Florida and located at
Rosemary Hill Observatory. The site of Rosemary Hill
Observatory is some twentyfive miles southwest of
Gainesville, Florida, and some five miles south of Bronson,
Florida. The thirtyinch telescope, housed in the building
illustrated in Figure 1, was designed and built by Tinsley,
Inc., of California, and has been in operation since 1967.
A dual channel photoelectric photometer, designed and
bu.lt by Astro Mechanics, Inc., of Austin, Texas, was
mounted at the Cassegrain focus of the telescope as shown
in Figure 2. The photometer, which housed the light
enrsi tive photomultiplier tubes, also contained a Fabry
field lens which controlled the size of the area on the
photocathode illuminated by the stellar image; a filter
wheel which allowed the insertion of one of a possible six
Fig. 1. Thirtyinch telescope building at the
Rosemary Hill Observatory
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filter selections into the light path; an aperture
selection wheel which allowed the choice of one of eight
possible apertures for incoming starlight to pass through;
and a narrow field eyepiece located behind the aperture
selection wheel which allowed the observed object to be
centered in the selected aperture. A wide field Erfle eye
piece was also available and located before the photometer.
Provisions were provided for simultaneous use of two
photomultiplier tubes by means of a three position slide.
In the first position, light passed directly through an
opening to one of the photomultiplier tubes. The second
position contained a mirror which routed the light towards
a second photomultiplier tube. The third position contained
a dichroic filter which transmitted eighty per cent of the
impinging light with a wavelength longer than six thousand,
five hundred Angstroms to the first photomultiplier tube,
and reflected ninetyfive per cent of the light with a
wavelength between three thousand, five hundred Angstroms
and six thousand Angstroms to the second photomultiplier
tube. For observations of EE Aquarii, only one channel
corresponding to the second position of the slide was used.
Most of the work on EE ?quarii was accomplished by
using the fifth smallest aperture, which measured 1.98
millimeters and corresponded to 32.5 seconds of arc in the
sky. During the time when the moon was near full phase,
however, the bright sky background became a significant
portion of the total signal received. This background was
reduced by using a smaller aperture which measured 0.93
millimeters and corresponded to 15.2 seconds of arc in the
sky.
The observations collected in 1970 were obtained
using an EMI 6256B photomultiplier tube. A constant
potential difference of one thousand, five hundred volts
was applied to the tube for all of the observations. The
1971 observations were obtained with an EMI 6256S photo
multiplier tube to which a constant potential difference of
one thousand volts was applied. The purpose of the
photomultiplier tube was to convert the received light from
the star into an electron current by the photoelectric
process. This electron current was then further amplified
at various stages by the secondary emission process,
accomplished by applying the above mentioned potential
differences to the phototubes by means of a regulated high
voltage power supply.
The output signal from the photomultiplier tube was
amplified by means of a direct current amplifier. The
armplifier used in 1970 was equipped with coarse and fine
gain controls with steps of 2.5 magnitudes and 0.25
magnitudes respectively. The zero point of the system was
adjustable, although on no occasion was the zero point
changed after having been initially set at the beginning
of the night. The amplifier used in 1971 was equipped
with coarse and fine gain controls with steps of 5.0 and
0.5 magnitudes respectively.
Both amplifiers were equipped to average the input
signal over a specified length of time referred to as the
time constant. This averaging technique was necessary
since the atmosphere of the earth is continuously under
going extremely rapid variations in its transparency.
This results in the rapid fluctuation of the received
light intensity which, when used without a signal averaging
device (zero time constant), leads to a strip chart
recording with high frequency, high amplitude peaks. A
time constant of onehalf second, which was sufficient to
reduce the frequency and amplitude of the variations so
that accurate measurements could be made, was used for all
observations of EE Aquarii.
EE Aquarii, the comparison star, and the check star
were all bright enough so that only moderate amplification
was used. Because of this moderate amplification, the
dark current, electron current caused by thermal motion of
electrons, was extremely low and was of no consequence.
It was, therefore, considered unnecessary to try to reduce
the dark current further by refrigerating the photo
muitiplier tube with dry ice.
The output signal of the direct current amplifier was
fed directly into a Brown strip chart recorder. The
deflection of the recorder, which was linearly proportional
to the amplifier output, measured the intensity of the
light received. A sample of a strip chart record obtained
fIcoom Rosemacy Hill Observatcry is shown in Figure 3. The
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WWV radio time signal and was set to run at the rate of one
inch every minute. The accuracy of the chart drive speed
was checked periodically during the night by means of WWV
time signals and corrected if necessary.
Cerro Tololo Observatory
The observational data of AE Phoenicis was obtained on
four nights in'September and October, 1970, at Cerro Tololo
Interamerican Observatory* near LaSerena, Chile, South
America. All observations were made with the sixteeninch,
number one, reflecting telescope made by Boller and Chivens
Corporation, South Pasadena, California. The building for
the sixteeninch telescope and the telescope itself are
illustrated in Figure 4 and Figure 5, respectively.
A single channel photoelectric photometer was mounted
at the Cassegrain focus of the telescope. A Fabry field
lens, as was the case with the Rosemary Hill Observatory
photometer, was employed to distribute the stellar image
onto the surface of the cathode of the photomultiplier
tube. The photometer also contained an aperture slide,
which allowed the choice of five different aperture sizes
through which incoming light could pass; a filter slide,
*Operated by the Association of Universities for Research
in Astronomy, Inc., under contract with the National
Science Foundation.
Fig. 4. Sixteeninch telescope building at the
Cerro Tololo Interamerican Observatory
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Fig. 5. The number one, sixteeninch reflecting
telescope at Cerro Tololo Interamerican Observatory
a l
which allowed the insertion of one of a possible six
filter selections into the path of the incoming light; a
wide field Erfle eyepiece located before the aperture
selection slide; and a narrow field eyepiece located after
the aperture selection slide which was used to center the
stellar image in the selected aperture. The aperture
selected and used for all observations of AE Phoenicis was
the second smallest, which corresponded to about one
minute of arc in the sky.
All observations were made using an RCA 1P21 photo
multiplier tube, which was refrigerated with dry ice to
effectively reduce dark current to an insignificant level.
A constant potential difference of eight hundred volts,
corresponding to a potential difference of eighty volts
between each of the ten stages, was applied by means of a
regulated high voltage power supply. The potential
difference served to amplify the electron current
originating at the photocathode.
As previously discussed, rapid fluctuations in
atmospheric transparency lead.to similar variations in
received light intensity. This problem is eliminated not
by averaging the signal as was the case with the Rosemary
Hill system, but instead by integrating the signal from
the photomultiplier tube over a specified interval of time.
The total charge of the integrated signal, which is
proportional to the intensity of the light impinging on
the photocathode, was then displayed on a Brown chart
recorder. The integration time for all observations was
chosen to be ten seconds, and the display time on the
strip chart tracing was chosen to be two seconds. An
integration could be started by push buttons located
either at the telescope or at the chart recorder. A
sample of a chart record obtained at Cerro Tololo is
illustrated in Figure 6.
The integrator was equipped with a coarse gain control
of 2.5 magnitudes and a fine gain control of 0.5 magnitudes.
A standard source was supplied by the staff so that the
coarse gain steps could be calibrated before and after each
observing period. The fine gain steps were assumed to be
exact and were not calibrated.
The Brown chart recorder was set to drive the chart at
a rate of one inch per minute. The time was initially set
by WWV time signals broadcast by the National Bureau of
Standards, and monitored at frequent intervals throughout
the night. The accuracy of the chart drive made it
necessary to reset the chart several times each night.
A summary of the equipment used is shown in Figure 7.
This block diagram refers to both Rosemary Hill and Cerro
Tololo Observatories, with the amplifier used in the case
of the former and the integrator in the case of the latter.
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Fig. 7. A block diagram of the electronics used
to obtain data photoelectrically at Rosemary Hill and
Cerro Tololo Observatories
~N~UILII  i. IC
L~~aTYII~BR1nw~u r~ ~saMlr~ ~LUn~ls~rI
CHAPTER III
OBSERVATIONS
Observations of an eclipsing binary star are made
relative to a source of constant light output. Because
of this, differences in magnitude between the variable
star and a comparison star, the source of constant light
output, instead of an absolute determination of the
apparent magnitude of the variable star, are actually
sought. The comparison source should be a nearby star, and
similar to the variable in both magnitude and spectral
classification. The proximity criterion is made mainly to
reduce the effects of differential atmospheric extinction
(Chapter IV), but also the closeness of the two stars
expedites the positioning of the telescopes. The magnitude
restriction would eliminate excessive amplifier gain
changes which might introduce calibration errors. The
spectral classification criterion is imposed so that
similar differences in magnitude in any wavelength region
would result. This would eliminate the need for considera
tion of correction due to the spectral response of the
photomultiplier tube and correction due to differential
color atmospheric extinction (Hardie, 1962).
In addition to a comparison star, a check star was
also observed. The check star, also assumed to produce a
constant light output, afforded a check on the assumption
that the comparison star was not also a variable. This
was accomplished by occasionally substituting a check star
observation for a variable star observation, and conse
quently calculating the magnitude difference of the check
star and the comparison. The same proximity, magnitude,
and spectral classification restrictions used in selecting
a comparison star were also used in the selection of a
check star.
Both EE Aquarii and AE Phoenicis were observed in
three different wavelength regions. This was accomplished
by placing appropriate filters in three of the six slots
provided in the filter wheel in the case of Rosemary Hill
Observatory, or in the filter slide in the case of Cerro
Tololo Interamerican Observatory. The filters chosen
allowed wavelengths corresponding to yellow (y), blue (b),
and ultraviolet (u) light to reach the photocathode. In
both cases, the filters used resulted in a (u,b,y) color
system which very closely matched the standard threecolor
(U,B,V) photometric system set up by Johnson and Morgan
(1953). The effective wavelengths and bandwidths of the
JohnsonMorgan (U,B,V) system are discussed in Mahalas
(1968) and are listed in Table 1.
TABLE 1
Filter characteristics for the UBV System
Approximate
HalfIntensity
Color Effective Wavelength Width
V (visual) 5480 A 700 A
B (blue) 4400 A 900 A
U (ultraviolet) 3650 A 700 A
Equations to transform the observations from the
instrumental system (u,b,y) to the standard system (U,B,V)
have been derived (Hardie, 1962). The observations were,
however, left on the natural systems of the telescopes,
hereafter referred to as (uby).
Rosemary Hill Observatory
Nights in north central Florida during late summer
and early fall are typically warm and humid. The
relatively low altitude of the observatory, resulting in
a large body of atmosphere through which one must look,
and the high moisture content in the atmosphere, required
frequent observations of the comparison in order to detect
and analyze any atmospheric variations. A set of
comparison star observations was obtained on an average
of about cnce every twelve minutes.
A sequence of measurements for the comparison star
consisted of observations of the star in each of the
colors y, b, and u, followed by observations in each
color, but in reverse order, of a point in the sky adjacent
to the star. A sequence of measurements for the variable
star was similar to that of the comparison star except
that, following the sky measurements, a second set of star
observations was made. In all cases, measurements of a
star and the corresponding sky with a particular filter
were made using identical amplifier gain settings. The
procedure of alternately observing the comparison star and
its adjacent sky and the variable star and its adjacent
sky was occasionally interrupted by the substitution of
the check star for the variable star. A sample of the
Rosemary Hill data is reproduced in Figure 3.
Numbers were obtained from the tracings with the aid
of a clear piece of plastic with a thin, dark line drawn
across it so that the line could be visually centered.
After a "best fit" was obtained, a number was read to
three places corresponding to the units in which the chart
was calibrated. The sky reading was then subtracted from
the star reading, with the difference being proportioned
to the light received just from the star alone.
With the chart being transported at the rate of one
inch per minute, time could easily be read from the chart
to an accuracy of 5 seconds. The Greenwich mean time of
the observation, the filter, the deflection due to the
star alone, whether the star was a variable, check, or
comparison (coded 1, 2, and 3, respectively), and the
amplifier gain setting were all punched onto IBM computer
cards for reduction as described later (Chapter IV).
Cerro Tololo Observatory
The different environmental conditions found at Cerro
Tololo Interamerican Observatory enabled the use of
slightly different observing techniques. Cerro Tololo is
located in the Andes mountains, with the result that less
atmosphere is encountered while observing an object. Also,
the humidity at Cerro Tololo is quite low, which, when
combined with the high altitude of the observatory, results
in almost ideal observing conditions. Because of this,
comparison star measurements could be made at slightly
larger time intervals than before with no loss of accuracy.
The following sequence of observations was applied:
comparison, comparison sky, variable, variable sky,
variable, variable sky, variable, comparison, comparison
sky, etc. With this sequence, the comparison star was
observed every fourteen or fifteen minutes. During a time
of rapid sky background change, such as found during
moonrise or moonset, one variable star set and one variable
star sky set of measurements were eliminated from the
previous sequence.
A sample of a Cerro Tololo chart is shown in Figure 6.
Because of the well defined starting and ending points of
such integration, the time of each observation was read to
an accuracy of one second. Usually three integration for
each color were made unless significant scatter was present,
in which case a fourth or possibly a fifth integration
would be made to increase the weight of the average.
Numbers were obtained from the chart by reading the
height of each peak, in units of the chart calibration, and
averaging over all of the integration in that color. Sky
measurements were subtracted from the corresponding star
measurements, so that the difference was proportional to
the light intensity of the star alone. As described before,
IBM computer cards were punched for the reduction procedures
described in Chapter IV.
CHAPTER IV
REDUCTION
The observations of EE Aquarii and AE Phoenicis
punched on IBM computing cards, as described in Chapter III,
were reduced with the aid of an IBM model 360/65 computer
located at the University of Florida. The computing
programs, originally written at the University of
Pennsylvania, had been modified by the staff at the
University of Florida and again by the author in order that
the programs be compatible with the data. The following is
a chronological account of the data reduction as accom
plished by the computing programs.
Extinction and Magnitude Difference
The atmosphere of the earth can greatly affect the
light which passes through by the processes of absorbing
and reddening. The transparency of the atmosphere, which
varies from night to night because of natural causes, is
greater for long wavelengths (red) and decreases with
decreasing wavelengths (blue). A quantitative measure of
the atmospheric transparency is obtained by determining an
extinction coefficient, the amount of light in terms of
stellar magnitudes which is absorbed by the atmosphere per
unit air mass (one air mass is located at the zenith of
the observer).
Extinction coefficients were found from consideration
of the comparison star observations alone. For each
comparison star observation, the air mass (denoted by X)
in units of the air mass at the zenith of the observer was
calculated by means of the following equation (Hardie,
1962)
X = secz 0.001816(secz 1) 0.002875(secz 1)2
0.0008083(secz 1)3 (1)
where z is the zenith distance.
The magnitude of the comparison star was then calcu
lated by the equation:
m = Sc 2.5 log(dc) (2)
where Sc is a zero point magnitude which corresponds to a
particular gain setting and dc is the chart deflection due
to the comparison star.
The extinction coefficient, k, was found to be the
slope of a plot of magnitude versus air mass. A computing
program evaluated the extinction coefficients by making a
linear least squares fit to the relation
mo = m kX (3)
where mo is the outer atmosphere (zero air mass) magnitude.
The variable star observations were then corrected for
differential atmospheric extinction by subtracting the
quantity k(Xv Xc) from the difference in magnitude
(mv mc) between the variable star and the comparison star.
The expression for the difference in magnitude which was
used by the computing programs was
Am = mv mc = 02.5 log(Lv/Lc) k(Xv Xc) (4)
where Lv and Lc are the apparent luminosities of the
variable and comparison stars respectively. The values of
Lc were interpolated to the time of the variable star
observations in all calculations.
Light Travel Correction
As the earth travels around the sun, the distance from
the binary system to the earth and hence the time required
for light to travel from the binary system to the earth,
will change slightly. In order to correct for light travel
time, the recorded geocentric time was converted into
heliocentric time by adding the increment of time t given
by the following equation (Binnendijk, 1960)
t = 0.005775{(cos6cosa)X + (tancsin6 + cos6sina)Y} (5)
where 6 is the declination of the star, a is the right
ascension of the star, X and Y are the rectangular
Cartesian coordinates of the sun (at the midpoint of the
observation) and E is the mean obliquity of the ecliptic.
Period Study
The determination of the period of revolution (P) and
of the mean epoch (T.) to which all observations were
referred was accomplished by a period study. It was
necessary to first determine times of the center of the
minima and to determine the number of cycles which had
elapsed since the mean epoch.
If both the ascending and descending branches of a
minimum had been observed, then the method of Hertzsprung
(1928) was used to obtain the time of the center of the
minimum. If both branches of the minimum were not observed,
then the Hertzsprung method was useless and a method involv
ing tracing paper was necessary. A representation of a
minimum was obtained by aligning and superimposing on
tracing paper the plots of magnitude versus time for those
minima in which the Hertzsprung method had been employed.
Other times of minima were then graphically obtained from
partially observed minima by visually "best fitting" the
tracing paper plot to a plot of each minimum. Lower weight
was given to times of minima determined by the "tracing
paper method" than to those determined by the Hertzsprung
method.
Times of minima determined by photographic means were
combined, with lower weight, with the photoelectrically
determined times of minima, in a linear least squares fit
to the relation
T = T EP (6)
where T is the time of minimum of any date, and E is the
epoch of the observation. The consistency of the period
and the mean epoch derived in the above way were checked
by plotting the difference of the observed time of minimum
and the calculated time of minimum (0 C) versus time.
Light Curves
Using the light elements found in the period study, a
phase, based on the center of the primary eclipse as being
zero phase, was assigned to each observation. A plot of
the difference in magnitude between the variable star and
the comparison versus the phase of the observation was made
for all observations in each color. The three plots,
corresponding to the y, b, and u filters are hereafter
referred to as the yellow, blue, and ultraviolet light
curves, respectively.
Color Curves
The color index of a star refers to the difference in
magnitude obtained from two different wavelength regions.
The two color indices investigated, corresponding to the
standard color indices of the JohnsonMorgan (U,B,V)
system, were obtained by finding the magnitude difference
of the variable star in blue light and yellow light (by)
and the magnitude difference in ultraviolet light and blue
light (ub). The color indices were corrected for atmos
pheric extinction by using the color extinction coefficients
kby and kub to obtain the outer atmosphere color indices.
The outer atmosphere color indices were then transformed,
by the addition of a zero point correction obtained from
standard stars, to a system approximating the Johnson
Morgan standard system. The color indices (BV)' and
(UB)o, obtained in the above manner, were plotted versus
phase to obtain the color curves.
The magnitude of a stellar system in a particular
wavelength region is dependent upon the temperature (or
spectral classification) of the system. Analysis of the
color curve reveals temperature variations throughout the
cycle of the binary system. If the color indices have been
corrected as described above, then information about the
spectral classification of the components can be obtained.
Models and Rectification
The components of an eclipsing binary system by
necessity must be relatively close to one another. When
the components of an eclipsing system are separated by
less than eight or ten radii, the stars will be distorted
by mutual gravitation effects and will vary in brightness
over their surfaces. Exact representation of the light
curve of such a system is not possible with any simple
model. A reasonable approximation to the actual system
was, however, proposed by Russell (1912a, 1912b). The
Russell model assumes that the stars have been distorted
into similar prolate ellipsoids of revolution with the long
axes of the two ellipsoids aligned. It is further assumed
that the two components revolve about each other in a
circular orbit.
Methods to obtain a solution (Chapter V) of the light
curve based on the spherical model have been developed
(Russell and Merrill, 1952). The spherical model is a
binary system in which the components are spherical, the
components revolve about a common center of mass with a
circular orbit, and the components are darkened at the limb
according to
J = Jc(1 x + xcosy) (7)
where y is the angle between the radius and the line of
sight, x is the coefficient of limb darkening, J is the
surface brightness of the star at any point on the surface,
and Jc is the surface brightness at the center of the star.
Definite equations have been developed by Russell and
Merrill to transform the observed light curve in both light
intensity and phase to light curves which would be produced
by spherical stars satisfying the above requirements. The
process of going from the Russell model to the spherical
model is known as rectification.
The rectification formulas involve sinusoidal terms
determined from a truncated Fourier analysis of the outside
of eclipse region of the light curves. The Fourier series
was given by
m m
I A + A, cosne + B sinne (8)
n=l n=l
where I is the unrectified intensity and m is an integer
through which term the Fourier analysis is carried. After
an estimate of the limits of the eclipses had been made, a
computing program calculated the Fourier coefficients (A ,
A, B ) and the corresponding probable errors.
Ellipticity Effect
One of the effects removed through the rectification
process is the ellipticity effect. Because of their
proximity to one another, the stars will be tidally
distorted. The Russell model assumes that both stars are
distorted into similar ellipsoids in such a way that as
the two components revolve around each other, the observer
sees a maximum of light when a maximum area is observed
which occurs at phase 0.25 and phase 0.75.
Rectification for ellipticity must be performed on
both light intensity and phase. Since the ellipticity
effect has a maximum effect on the light intensity at the
quarter points (phase 0.25 and 0.75) and a minimum effect
during the eclipses (phase 0.0 and 0.5), then the first
order Fourier term is proportional to cos26. Since this
term is proportional to the brightness, first order
ellipticity effects are removed by division.
In addition to a term proportional to cos26, a second
order term proportional to cos39, due to a real difference
in shape between the two ellipsoids, may be present.
Merrill (1970) has shown the importance of including cos36
terms in the rectification process in systems in which the
components are as close as the ones considered in this
dissertation. The term proportional to cos38 was considered
as a complication and not part of the formal rectification
for ellipticity.
Reflection Effect
The rather misleading term "reflection" refers to the
result of the heating of the side of each component which
faces the other by the radiation of the latter. The
received energy is absorbed in the outer layers by each
star and subsequently reemitted with no effect on the rate
of escape of energy from the deep interior (Russell and
Merrill, 1952). The total amount of radiation into space
by the system is not altered since the loss of light during
an eclipse in one direction is compensated for by an
increase of radiation in other directions. The amount of
radiation from the regions being irradiated is, however,
greatly enhanced which alters the shape of the light curve
somewhat. Since the cooler star is heated proportionately
more than the hotter star, it is during the time when the
irradiated hemisphere of the cooler star is observed
(secondary eclipse) that this effect is maximum. The effect
upon the light curve is, therefore, to raise the shoulders
of secondary eclipse.
The reflection effect is proportional, to first order,
to the cosine of the phase angle since the minimum effect
is observed at phase 0.0 and the maximum effect at phase
0.5. A second order effect proportional to cos26 is also
present and is considered in the formal rectification of
reflection. Rectification for reflection involves adding
an amount of light to the outer hemisphere of each component
equal to the "reflected" light of the inner hemispheres.
The amount of light added is given by
Aref = CO + C1 cos6 + C2 cos20 (9)
where CO, C, and C2 are the portions of the Fourier
coefficients AO, Al, and A2 from equation (8) due to
reflection alone.
Both the ellipticity and reflection effects are
proportional to a cos28 term. In order to evaluate C2, the
two effects have been separated by Russell and Merrill
(1952) using the assumption that the stars are radiating
as graybodies.
If Gc and Gn are defined as
Gc = Ihr2 (Ec/Eh) (10)
Gh = cr (Eh/Ec)
where Ih and I are the light intensities, rh and rc are
the stellar radii, and Eh and Ec are the luminous efficien
cies of the hotter and cooler components, respectively.
After substituting
Jh h/rh 1
Jc = Ih/r
the ratio
Gc/Gh = Jh E cEh2 (12)
could be calculated. The ratio of the depths of the
eclipses, rectified for ellipticity, is equal to the ratio
Jh/Jc. In order to calculate (Ec/Eh)2, knowledge of the
spectral type (and hence temperature) of the hotter compo
nent must be available. This allows a computation of
Jh/Eh2 from graybody theory represented graphically by
Russell and Merrill (1952), which, when combined with
Jh/Jc, gives the value of Tc and finally Jc/E2.
The reflection coefficients are then expressed as
C1 = A1
C = (0.75 0.25 cos2i) Gc + Gh A cosec i (13)
Gc Gh
C 0.25 Gc + Gh A sin i
2 GC G n
where i is the inclination of the orbit with respect to the
plane of the sky.
The above method for finding the reflection coeffi
cients failed as Gc and Gh approached the same value, as
would occur if the depths of the two eclipses approached
each other. In such a case, reflection coefficients may
be found from
(Gc + G) (0.30 0.10 cos2i + 0.10 sin2i cos28) (14)
+ 0.40(Gc Gh) sini cos6
= C + C1 cosO + C2 cos26
by equating coefficients. The quantity Gc + Gh may be
calculated from
G + Gb
G + G Gc G (I ) (r r ) (15)
c h (G Gh ) 2 ch c
and G Gh follows.
Complications and Perturbations
A theory has not yet been developed which can account
for the presence of any sine terms found in the Fourier
expansion (8). Cos 38 terms and cos 40 terms should,
according to the theory presented, be very small. The
presence of a sizeable cos 39 or cos 40 term, or, for that
matter, the presence of an extraordinary cos0 term, cannot
be explained by theory. These terms may be regarded as
perturbations if the cause is known or complications if
unexplained by theory, and in either case are rectified by
subtracting these terms from the observed light intensity.
Perturbations arise from true and known residue of,
for example, cosO or cos 30 due to a second order difference
of the ellipsoidal shapes. A complication could be due,
for example, to a superluminous or a subluminous region
on one of the components not due to conventional gravity
or irradiation effects. Eccentricity of orbit may also
cause complications which are not predictable with present
theory.
Intensity Rectification
The rectification for reflection is by addition and
the rectification for the complications and perturbations
is by subtraction so that the two types may be combined
into the relation
I" = I + Co + C1 cos8 + C2 cos26 A3 cos38 A4 cos48
B1 sin9 B2 sin28 B3 sin39 B4 sin40 (16)
where the C's are found from either (13) or (14) and the
A and B terms are found from (8). In practice, (8) may be
truncated with 48, 38, or even 20 terms depending upon the
significance of the 48 and 30 terms. If the coefficient C1
in equation (16) is found from (14), then a cosO term
proportional to (A1 Cl) will remain. Instead of removing
this term by separately subtracting (A1 Cl) cosO, one in
practice sets C1 in equation (16) equal to A1 which
effectively removes all of the cosO term at one time.
The ellipticity rectification is accomplished by
dividing the observed intensity by
(Ao + C ) + (A2 + C2) cos20 (17)
due to ellipticity alone. In practice the intensity I'
rectified for reflection and complications is used instead
of the observed intensity. The final relation used to
rectify the observed intensity is given by
I" = 1 1 (18)
(Ao ' Co) + (A2 + C2) cos20
where I' is given by (16).
Phase Rectification
Rectification for phase is carried out using the
relation
sin2 sin2 (19)
1 z cos2
where 0 is the phase angle of the original observation, 0
is the rectified phase angle, and z is the ellipticity
coefficient defined by
z = e2 sin2 i (20)
where e is defined as the eccentricity of the equatorial
section of the component (Binnendijk, 1970). The numerical
value of z is actually obtained from the relation
Nz = 4(A C) (21
(Ao Co A2 + C2)
where N is the ratio of the "photometric ellipticity" to
the geometrical ellipticity and is represented by the
approximation
N = (15 1 x) (1 + y) (22)
15 5x
where x is the coefficient of limb darkening and y is the
coefficient of gravity darkening. Because of the uncer
tainty of the actual variation of N with x and y, it is
customary to adopt N equal to 2.2, 2.6, or 3.2 when x is
assumed to be 0.4, 0.6, or 0.8, respectively.
CHAPTER V
SOLUTIONS
After the light curves had been rectified to the
spherical model, solutions based on methods originally
developed by Russell (1912a, 1912b) and Russell and Shapley
(1912a, 1912b) and summarized by Russell and Merrill (1952)
were used. Other methods developed by Kitamura (1965) and
Kopal (1959) were not considered in this dissertation.
A solution of a light curve is obtained when the
observations comprising the light curves are reasonably
well represented by a theoretical curve defined by a set
of eight parameters. These eight parameters (or elements)
are
P Period of revolution
To Epoch of primary minimum
xs Limb darkening coefficient for the smaller star
Xg Limb darkening coefficient for the larger star
rs Radius of the smaller star
rg Radius of the larger star
i Inclination of the plane of the orbit
L,. Luminosity of the greater star
where the radii rs and rg are expressed in terms of the
distance between the centers of the components.
The period of revolution P and the epoch of primary
minimum To, often referred to as the "light elements," were
satisfactorily determined by the period study previously
described. Of the six remaining elements, the darkening
coefficients Xc and xs must be initially assumed. Merrill
(1950) has compiled tables for solving light curves based
on limb darkening coefficients equal to 0.0, 0.2, 0.4, 0.6,
0.8, and 1.0. It is sufficient with present theory to find
the elements of a system based on tabular values of x, and
to choose the darkening coefficient and the other parameters
of the most reasonable "fit" as the solution of the light
curve. A preliminary value for the darkening coefficient
of the hotter comp.;nent may be obtained from theoretical
considerations if the spectral classification of the system
is known. Since present knowledge of the statistical
relationships between limb darkening coefficients and
spectral types is not perfect, solutions based on darkening
coefficients other than that initially assumed must be
per fo rmed.
Of the four _remaj.inng parameters (rg, rg. i, Lg), g
will be known in the case of a cropl.te eclipse. During
the total phrase of a complete occultation eclipse, at which
time the larger star ecli.ses the smaller star, only the
light from the larger star will be received and Lg, the
luminosity of the large star, is obtained directly from the
rectified light cnrve. Since the light outside of eclipse
is normalized to unity, then Lt, the luminosity of the
smaller star, is given by 1 Lg. For occultation eclipses
which are partial, Lg cannot be obtained from an inspection
of the rectified light curve but must be found from
relations presented shortly. Another instance in which Lg
cannot be directly determined even in the case of complete
eclipses occurs when the eclipsing system is a physical
member of a higher order multiple system. Such a case,
found with neither EE Aquarii nor AE Phoenicis, requires
the removal of the "third light" before a solution can be
obtained.
In order to facilitate the calculation of rs, rg, i,
and Lg (or Ls) in the case of a partial eclipse, the
following quantities have been defined:
k = rs (23a)
rg
p = (6 r) (23b)
rs
where 6 is the apparent distance between the centers of the
two components and is given by
62 = cos2i + sin2i sin2 (24)
Combining equations (23a), (23b), and (24) gives
rg2(1 + kp)2 = cos2i + sin2i sin28 (25)
which is valid for any phase 8. At external contact,
defined by the starting or ending of an eclipse, equation
(25) educess to
rg2(l + k)2 = cos2i + sin2i sin20e (26)
where 0e is the phase angle at external contact and p has
been set to unity since 6 = rg + rs at external contact.
At mideclipse, p = po and 0 = 0 so that
r 2(1 + kpo)2 = cos2i (27)
Both the Merrill tables (1950) and the Merrill nomo
graphs (1953), discussed later in this chapter, are designed
to yield the values of k, Po, and Oe. Equations (26) and
(27) are then used to obtain rg and i.
In order to obtain general expressions for the
luminosities Lg and Ls, it is convenient to define the
quantities a(x,k,p) and T(x,k). The quantity a is the ratio
of the light lost at any phase during an eclipse to the
light lost at internal tangency, the position when the disk
of the smaller star first appears to be entirely either in
front of (transit) or behind (occultation) the disk of the
larger star. The quantity T is the ratio of the light of
the larger star lost at internal tangency to the total
light of the larger star. For a transit eclipse (smaller
star in front) the value of T must be less than unity, and
for an occultation the value of T remains unity since no
light is lost from the larger star.
If Ltr and Zoc are defined to be the apparent light
intensity at any phase Eor the transit and occultation
eclipse, respectively, then 1 tr and 1 oc are the
corresponding light losses. From the above definitions of
a and T, the loss of light for an occultaticn is given by
1 Zoc = Lsaoc(xs,k,p) (28)
and for a transit is given by
1 Ztr = LgT(xg k)atr(xg,kp) (29)
Equations (28) and (29) are in practice evaluated at mid
eclipse and solved for Ls and Lq to obtain
1 toc
Ls = (30)
a, (xs,k,pO)
and
L = 1 tr
9 T(xglk)arr(xgk,Po)
g T (Xg/k ^0iXg/k~po)
The values of T and a are known as functions of the
parameters k,x, and po determined from the adopted solution,
and 1 PO and 1 Ztr are the depths of the rectified
occultation and transit eclipses, respectively. For a
system with complete eclipse, aoc is unity and L =
1 oc (and Lg c)
o g 0
The luminosity of a star is related to the surface
brightness J by the relation
L = 7!'r (31)
Since the radii are known in terms of the distance between
the centers of the components, the ratio of the surface
brightnesses may be expressed as
2
= 5 23 (32)
Js Ls rg
Solution from the XFunctions
A solution based on information derived from the
depths of the two eclipses and from the shape of one of the
eclipses may be obtained for eclipses which are either
partial or complete. In order to define the shape function
X, it is first necessary to scale the eclipse in n such
that n is zero at external contact and unity at mideclipse.
The light lost at any phase of the eclipse will be given by
1 L = n(l Lo) (33)
where 1 .o is the light lost at mideclipse and n for an
occultation eclipse is given by noc = Ooc/aoc and for a
transit eclipse by ntr = atr/atr. For each value of n, the
light curve defines a corresponding value of 0 so that 8(n)
may be read directly from the light curve.
From equation (25) for an arbitrary 6(n), the
following may be written
cos2i + sin2i sin28(n) = rg2{l + kp(x,k,nao)} (34)
At mideclipse
cos2i + rg2{l + kp(x,k,ao)}2 (35)
where 8 has been set to zero and n has been set to one.
The shape function is then defined by
sin28(n) 1 + kp(x,k,nac)2 1 + kp(x,k,an)2
sin(TT) 1 + kp(x,k,ao)2 1 + kp(x,k,ao)2 (36)
= X(x,k,ao,n)
For specified values of x and n, X may be tabulated as a
function of k and ao.
The solution cannot, however, be obtained until
information from the depths of the two eclipses is included.
This is done conveniently by the introduction of a function
q defined by
q(x ,xs,p,k) = T(x ,k)atr(Xgp,k) (37)
g soC xs,p,k)
When equation (37) is combined with equations (28) and (29),
the expression for q becomes
1 ttr (38)
q t= _s (38)
1 LU L
By equation (38), q is physically interpreted to be the
ratio of the light of the large star observed during any
phase of the transit eclipse to the fraction of the light
of the small star observed at the same geometrical phase
during the occultation eclipse.
The function k(xg,xs,aoc,q) has been tabulated for all
combinations of tabular values of x that are likely to
occur. Since Ls + Lg = 1, then equation (38) becomes
oc 1 Lc + (1 + gtr) (39)
qo
so that by varying qo a value of a will be obtained.
This permits a "depth" curve to be plotted with the
coordinates k and aoc. The shape function X(x,k, o,n) may
also be plotted with the coordinates k and a, for a
selected value of n so that a "shape" curve is obtained.
The intersection of the two nonlinear curves yields a
solution to the light curve.
Nomographs
Merrill (1953) has devised a graphical method using
the X functions which is considerably quicker than the
procedure just described. Large scale plots nomographss)
of the shape function X evaluated at n = 0.8 have been
constructed using coordinate scales which allow the "depth"
curve to be represented as a straight line. Four nomo
graphs, based on limb darkening coefficients of 0.2, 0.4,
0.6, and 0.8, are each divided into three sections
corresponding to partial eclipses, almost complete eclipses,
and complete eclipses.
The boundary coordinates of the depth line may be
found from
a = (1 Ztr) + (1 ~oc)
b = 1tr 5 1 o (40)
zucO + to'; (40)
o
S1 tr
c = L _
S0
where a and c are used for a partial eclipse, b and c are
used for nearly complete eclipses, and c is used for
complete eclipses. A piece of clear plastic with a thin,
inked line was positioned, and the intersection of the
depth line with the X0.8 contour represented a solution of
the system.
If both minima are sufficiently deep so that a X0.8
can be obtained from both, then the solution should be
given by the intersection of the depth line and Xtr the
0.8'
intersection of the depth line and Xoc or by the inter
0.8 o
section of Xtr and X oc A theoretical light curve can be
0.8 x0.8.
generated from the nomographic parameters by finding the
shape functions from the tables for other values of n. The
fit of the theoretical curve may be improved by altering
X0.8 and finding another set of nomcgraphic elements. The
solution that "best fits" all of the observed points is
adopted for the solution to the light curve.
Solution from the 1 Functions
If the eclipses are complete or if the eclipses are
partial and the observations are very accurate, it is
recommended that the solution be derived using the p
functions tabulated by Merrill (1950). From equation (25)
it follows that
sin20 sin2 0 (1 + kp)2 (1 + kp2)2
sin22 sin2 (1 + kp2)2 (1 + kp3)2 (41)
= i(x,k,a,a2,a3)
where 032',2,2, and 63'P31a3 are known fixed quantities.
By choosing a2 as 0.6 and a3 as 0.9 and defining the
constants
A = sin2 (0.6)
(42)
B = sin20(0.6) sin2 (0.9) (42)
then equation (41) is reduced to
O(x,k,a) sin26 A (43)
B
or
sin20 = A + Bp(x,k,a) (44)
The above equations require that the values of a
corresponding to a = 0.6 and 0.9 are known and fixed. It
is also necessary to require that the light curve pass
through a third fixed point in order to be uniquely defined.
The midpoint of the eclipse corresponding to 6 = 0 is the
third fixed point. Once these three points have been fixed,
a series of theoretical curves can be generated from the
tables by varying k. The k corresponding to the "best
fitting" theoretical curve is then adopted, and the remain
ing elements can be calculated by finding p, from a table
of ao(k,po) and using equations (26) and (27).
The process of finding the remaining elements has been
abbreviated greatly by the use of the functions ((x,k) and
(2(x,k) which are defined such that
1 (x,k){((x,k,O) {(x, k,l)} = 4k
(45)
Y2(x,k){((l k) f(x,k,O) (1 + k)2i(x,k,l)} = 4k
where 4(x,k,0) is the value of t at external tangency and
p(x,k,l) is the value of i at a = 1. The inclination i and
the radius of the larger star rg may be found from
r2 cosec2i =
g9 l (x,k)
(46)
cot2i =
(2 (x,k) A
CHAPTER VI
EE AQUARII
History
The light variation of EE Aquarii was discovered from
photographic patrol plates reduced at the RemeisBamberg
Observatory, Bamberg, Germany. The discovery was announced
in 1960 by Strohmeier and Knigge (1960), and the system was
given a provisional designation of BV 320.
Ten times of primary minima were obtained photograph
ically by Filatov (1961). Strohmeier, Knigge, and Ott
(1962) published six more photographicallydetermined times
of primary minima which, when combined with the times of
minima obtained by Filatov, gave the light elements
JD 2429881.310 + Od5089951 E
Strohmeier, Knigge and Ott also published a light curve
from their photographic observations which is reproduced in
Figure 8. The system was thought to have a light curve
similar to that of Algol with a magnitude range of from a
maximum of 8T35 to a minimum of 9T10. No secondary minimum
was detected from the photographic plates.
The author of this dissertation was unable to find any
photoelectric work or orbital solutions for EE Aquarii.
oi
0
44
rd
0
4<
0
O4
0
U
CO
O
*l
I Ii  I5I I  I 
0
o
0
0
0
0O
0
o
0 0
0 0
OO
0
8
0co
0 0
o8 o
o O
O 0
) C 0)C 0)
Comparison and Check Stars
Selection of a comparison and a check star for EE
Aquarii was a relatively straightforward process. Many of
the stars in the vicinity of EE Aquarii were of similar
magnitude to EE Aquarii, and were listed in the Bonner
Durchmusterung (BD) catalog, the Henry Draper (HD) catalog,
and the Smithsonian Astrophysical Observatory (SAO) catalog.
An investigation of the spectral classifications (obtained
from the SAO catalog) revealed, however, that the spectral
classifications were, except for the stars ultimately chosen
to be the comparison and check stars, quite different from
that of EE Aquarii.
A summary of relevant information about EE Aquarii,
the comparison star, and the check star is included in
Table 2. The coordinates of each star are processed to
epoch 1970.5 from 1950.0 coordinates given by the SAO
catalog. A finding chart reproduced from the Atlas
Eclipticalis (Bevair, 1964a), with EE Aquarii, the compari
son star, and the check star labeled, to the right of the
respective star, is shown in Figure 9.
Fig. 9. A finding chart for EE Aquarii
oO o oo c
O o o
o 0 o a0
00
0 o
0
ocomp
o
0 "check
0
0
0 0
O EE Aqr
00
0 O
O v
1
0 o
1 "
0
0 0
22h 34m 22h 30n
RIGHT ASCENSION
0 
0
130
190
 0
0 20
H
,<
21
22
22h 38m
22h 26m
91 i I '~~I~
TABLE 2
EE AQUARII, COMPARISON AND CHECK STARS
R.A. (1970.5)
Dec. (1970.5)
BD Catalog
HD Catalog
SAO Catalog
Spectral Class
Magnitude(m )
EE AQUARII
22h 33m 06 0
20 00' 44.6
2006454
213863
191236
FO
8.0
COMPARISON
22h 32m 48s6
190 12' 17'.4
1906300
213791
165165
F8
8.5
CHECK
22h 31m 31s8
190 32' 36"3
2006446
213623
165157
AO
9.1
Extinction
Extinction coefficients for each of the three filters
and also for the colors (bv) and (ub) were determined for
each night from comparison star observations as described
in Chapter IV. The coefficients obtained are listed along
with the mean extinction coefficients in Table 3. The mean
coefficients have no real significance except to reveal the
order of magnitude of the extinction coefficients one might
encounter during a typical night in the late summer at
Rosemary Hill Observatory.
The extinction coefficients used in later calculations
are listed in Table 3. The extinction coefficients actually
found for September 17, 1970, were considerably less than
the coefficients found for other nights, due, at least in
TABLE 3
ATMOSPHERIC EXTINCTION COEFFICIENTS
AND COLOR EXTINCTION COEFFICIENTS FOR EE AQUARII
DATE ky kb ku kby ku_b
82970 0.5444 0.6787 1.0823 0.1369 0.4052
83170 0.3353 0.5037 0.8617 0.1724 0.3680
9470 0.4905 0.6969 1.0668 0.1989 0.3567
9870 0.5943 0.7617 1.4150 0.1683 0.6550
91270 0.6076 0.8348 0.9983 0.2309 0.1721
91470 0.3844 0.5671 0.9252 0.1798 0.3622
91670 0.5056 0.7075 1.1853 0.2048 0.4752
91770 0.4653* 0.6395* 1.0446* 0.1744* 0.4051*
91870 0.3073 0.4401 0.8296 0.1294 0.3695
10170 0.2910 0.4455 0.8470 0.1583 0.4015
10370 0.5933 0.7593 1.2355 0.1679 0.4747
71871 0.5377 0.7710 0.9818 0.2346 0.2131
81771 0.5473 0.6411 1.0806 0.0935 0.4403
83071 0.3540 0.5321 0.9434 0.1779 0.4144
Mean 0.4686 0.6415 1.0348 0.1733 0.3929
*91770 not included in mean.
part, to a rather short range of airmass to which the least
squares fit described in Chapter IV was applied. A more
realistic set of coefficients for each filter was obtained
by averaging the coefficients from the 1970 observations.
The color extinction coefficients (bv) and (ub) for
September 17, 1970, were obtained by finding the differences
between the extinction coefficients in blue and yellow and
in ultraviolet and blue, respectively.
Period Study
The primary minimum of EE Aquarii was observed by the
author on five different nights. The times of primary
minima were found using the Hertzsprung method for three of
the nights and the tracing paper method for the remaining
two nights. Both of these methods were discussed in
Chapter IV. Since observations in each of the three colors
yield essentially independent information, a time of
central minimum was found for each color on each of the
five nights.
In addition to the times of minima obtained by the
author, the ten epochs of minima determined photographically
by Filatov (1961) and the six photographically determined
times of minima by Strohmeier, Knigge, and Ott (1962) were
also available. These photographically determined epochs
were combined with the photoelectrically determined epochs
to determine a period and reference epoch by a linear least
squares fit to equation (6). In this calculation, each
time of minimum determined photographically was given a
weight of unity, while each time of minimum in each color
found by the Hertzsprung method was given a weight of four
and by the tracing paper method, a weight of two.
The light elements
JD Heliocentric = 2440828.7809 + 050899558 E
.0006 .00000009
were found from the least squares fit, where the probable
errors are given below the elements. The results of this
period study are summarized in Table 4, which contains
information relevant to the photoelectrically determined
times of minimum obtained by the author, and in Table 5,
which contains information relevant to photographically
determined times of minimum obtained from other sources.
In Tables 4 and 5, epoch refers to the number of cycles of
the orbital revolution counted from the reference epoch
JD 2440828.7809 and (OC) refers to the difference between
the observed and calculated times of minima.
A plot of the residuals (OC) for all the observed
minima is shown in Figure 10. Because of the small scale
of Figure 10, the photoelectric residuals are not clearly
shown. Figure 11, therefore, shows just the (OC)'s for
the minima observed photoelectrically. From Figure 10 one
may conclude that linear light elements give a satisfaction
fit to the data. A conclusion concerning the constancy of
the period of EE Aquarii cannot be made from the present
period study; although no variation is indicated.
0. 0 l
000
000
000
oo
00
00
00
c o
+ 1
r r
CO
CO 00
r r'
ncm
o1 r
00CO
Coo.D
c0c0CO
o00 00 Co
cococ
>I, Q :: 3 >1a Q :5^ : >
o rI 0 r c
(N m r4
C00o O' Om oh
ON r
00
00
00
I I
OLO
00
00
00
+ 1
00 ,3T
00
00
00
00
I I
O
+
0
U :
U
ro
co
o co
cq
a) 0
o0 o) oN
in U) in
Mo
S0
pOo
o00 c
00
n n
oo
oo o
TABLE 5
PHOTOGRAPHICALLY DETERMINED TIMES OF PRIMARY MINIMA
USED IN THE PERIOD STUDY OF EE AQUARII
JD HELIOCENTRIC = 2440828.7809 + 0.5099558 E
JD HEL
(OBSERVED)
2400000+
29881.297
29902.185
29904.212
30200.426
31375.215
32744.388
32771.383
33187.235
33897.287
36080.348
36844.375
36868.306
36822.465
36845.375
36846.396
36893.242
EPOCH
21508.0
21467.0
21463.0
20881.0
18573.0
15883.0
15830.0
15013.0
13618.0
 9329.0
 7828.0
 7781.0
 7871.0
 7826.0
 7824.0
 7737.0
OC
0.007
+0.012
+0.003
0.01.8
+0.009
0.016
+0.002
+0.005
+0.008
0.013
+0.012
+0.020
0.012
0.007
0.004
+0.015
SOURCE
Filatov (1961)
II
I;
II
II
II
11
ii
I!
I1
II
Strohmeier (1962)
II
i;
II
U) U
0 H
*H 4J
E 4J 0
44 , ,
U 0
Ul , 4 4 J>
0 0 0
U O
o
I 0
0 4
*4 0
:? c
0 nr 0
,.4 A
*U) E 34
O '0
C U 0
S0 (0
O 'B 4
M H I
0 10
C 0
0 0u 41>
SrA 0
Q 0 r 0
0 Q) Cr
0
0 1 > o
4 a 4J
0 0 0
C)
(U U c)
r0 0
0 0 U)
n* 'U3
0 O
C)4 H C)
H 0 (U
0 42 I
.c^ r.)
i '4; 0
I IX
)( )(
0
>
o 
l 
00
L OL
00
I I
00
I I
CL CL
x
x
'
0 0 0
d d i
I I
15I I _I _I a~amay y~l I
0
0
0
0
0
0
0
0
I
0
0
a
0
0
o
N
o
0
0
0
0
0
0
0
N
0
0
0
'O
1d
.IL
0
U)
44
0
0
I
rn
*H
4H
0
>1
r) o
4)
ra
o
Q4 0
w w
4i
4
0
ri
4)
o
0
H '
O
Q4
0
 I I I
*0
0
L0
Ln
to
0
O
lo
Q
<
3
o
ro
'C" S
0O
0O
o o
O 0 O
0 0
o 0o
I>
00a
U)
z
0
w
m
0
Cr
ILl
I
0
I
0
0L
0
0
O0
o
a.
0d
'Y~~  I~ L P~U~pBL~q~
_I
Light Curves
There were 309 usable observationsin yellow light,
302 usable observations in blue light, and 299 usable
observations in ultraviolet light of EE Aquarii. The phase
for each of these observations was calculated as described
in Chapter IV using the period (P) and reference epoch (T,)
discussed in the previous section. The time of each obser
vation given in terms of the heliocentric Julian date
(Julian day number and decimal), the phase of each observa
tion, and the magnitude difference between the variable and
the comparison star (V C) are given in Tables 18 20 in
the Appendix.
The light curves of EE Aquarii were obtained by
plotting the magnitude difference between EE Aquarii and
the comparison star versus phase for all the observations
in each of the colors. The light curves of EE Aquarii in
yellow, blue, and ultraviolet light are shown in Figures
12 14.
The light curves appear regular with no vertical
displacements due to variations in the light from the
comparison star. Further proof of the constancy of the
comparison star light output was provided by check star
observations. The heliocentric Julian date and the
difference in magnitude beLween the check star and the
comparison star are recorded in Table 21 in the Appendix.
The scatter of the observations could have been due to
0
tI
a)
0
r)
,4
3
>
,
3
1
rl
0
4C
H
SI I I I
I
O@
*
.*
*
* **
S***
*e
oo
03
..
0
0**
0** *
'I"
/** '
****
,.so
0
p**
0
0
> og
ge
ci I
I I I
I i I I
c O ( cJ
< I I I
<1
1~111~   q_e  I U  II~
U)
0
*,
r
4)
0
0
H
4U
0
0
4)
4
rO
*,
n,
r
n3l
&4
 I i I I I T
0
V.
Jo
o
84:
0%
00
0,,
Ad,
* 10
0
go
a.
0s0
o a
1, .
$0
gO
0*~__
L. L _LLI ilL _
CO
d
Oi
O
0
01
~I~PC  
7 1 I r
I~Z~L~CN
U)
0
4)
rd
U)
0
,Q
0
11
rd
4
C)
0
*l
n3
4)
14
I
*0
669
**0
10
**
0.O
o (D 0 0
b *
*0
'o
d *
*..
6**
*6** 0
** 0
*6
". *~
8 0 0
: O
0
,O 0 0
E 0 d d o O
**3 .
actual short term lignt variations from the system, but was
probably due mostly to random fluctuations in the transpar
ency of the atmosphere. It is noted that the scatter
during secondary eclipse was somewhat larger than that in
other portions of the light curve. It is also noted that
the nights during which the secondary eclipse was observed
were less transparent than average as evidenced by the
larger extinction coefficients obtained for those nights.
Color Curves
The color indices (by) and (ub) found from the
observations of EE Aquarii alone were transferred to values
for outside the atmosphere by application of the color
extinction coefficients given in Table 3. A study of
standard stars observed on seven nights yielded "zero point
corrections" for these nights which, when added to the out
side the atmosphere values of the color indices, gave the
color indices (BV)o and (UB) approximating the Johnson
Morgan standard system. These calculations are represented
by
(3V)6 = (by) kb yX + Aby
(47)
(UB)o = (ub) kubX 1 Aub
where X is the airmass, kby and kub are the color
extinction coefficients for (by) and (ub). respectively,
and Aby and Aub are the zero point corrections (in units
of stellar magnitude) for (by) and (ub), respectively.
The standard star observations and the zero point
corrections for each night are located in Table 6. Zero
point corrections for the nights with no standard star
observations were obtained by shifting the data for each
night until agreement with the seven nights with standard
star observations was achieved.
It is emphasized that the color indices (BV)o aid
(UB)o were not transformed to the JohnsonMorgan standard
star system. In finding outside the atmosphere values of
the color indices, second order color extinction
coefficients were assumed to be zero. In transforming from
the natural systems of the telescope to one approximating
that of Johnson and Morgan, the transformation coefficients
(V and i as defined by Hardie (1962)) were assumed to be
unity.
The color curves were obtained by plotting (BV)' and
(UB)o versus phase, and are shown in Figure 15. Inspec
tion of Figure 15 reveals no evident color change during
any part of the cycle. The lack of color change is
possibly due to the partial nature of the eclipses (shown
in a later section). The color indices (BV)o and (UB)0
found for EE Aquarii agree reasonably well with the Johnson
Morgan standard system color indices of an FO main sequence
star.
n oC LU) inL
XNooo
M 0
0
m
O
wm
"noM
HHH+
H
OH
H
0 u
a W
U)
0
O
EO
oa
ri P
W a
w o
0
O>
O >
M 1
M J
CNi r
I I
I I
00
00
I~I
o
00
IN o m0
II03
0 rq D
III
'..0
NNP
0'0
++
0000
++++
0000
0000
+>NN
W In L I
HHOH
IIII.
Ln
Ln
C
0
0000
I I I I
(^JH (1)
Q 0 0rC
* I I
+1+
or 00
+ +
oo
CC) IC
++ +
co o
0000
I I I
M 0
I I o
I I I I
I I I
I I
K K
o,
OH
I I
co
I ri 
U)
0 1
OH
41 >
o,c
m 0
4J U)
fo
r
U) 0
MK
I c
o 0o
1 i
0OOO
r+ +
+ +,
0
0
un
0
4
0
in
83
S't * *
0 0
0 .
:00* 0 0, O
*0 0
* **
40e
* * :* .
:t. .
0*% * *
* C
or "
S*. *a ".
o 0
00 9
0 0 0
0
* 
* .,* *
**
* o 0
6 0
* *
S "" %*
* ** .+ d
* 9 0
**4* .
0 00.. 0
o 0 0o
Rectification
The light curves of EE Aquarii were rectified in both
intensity and phase according to the procedures outlined in
Chapter IV. The first step of the rectification procedure
is to express the extraeclipse regions of the light curve
in terms of a Fourier expansion as given by equation (8).
It is first necessary, however, to find what portion of the
light curve is excluded from the eclipse regions by finding
the phase angle of external contact (6e). By removing a
considerable portion of the cos28 term due to ellipticity,
the angle of external contact appeared to be about 450.
The Fourier expansion was then applied to the regions
between 450 and 1350 and between 2250 and 3150 by the
method of least squares.
The reflection coefficients (CO, C1, C2) were calcu
lated from equation (13) according to the procedure
outlined in Chapter IV. A spectral classification for the
hotter component of FO was assumed in order to evaluate
the reflection coefficients. As a result of this
calculation, it was also found that the cooler component
should be approximately of spectral type F5.
Different sets of Fourier coefficients were found by
truncating equation (8) with 46, 36, and 26 terms. In the
expansion which was truncated with 48 terms, however, it
was found that the coefficients of the sin48 and the cos46
terms were comparable in size to their associated probable
errors. For this reason, the set of Fourier coefficients
corresponding to the expansion of equation (8) through the
39 terms was finally adopted for the process of rectifica
tion.
As will be shown in the next section, the yellow and
the blue curves were solved under the assumption that the
coefficient of limb darkening (x) was 0.4. The ultraviolet
was, however, solved with the assumption that x was 0.6.
The ellipticity coefficient (z) used in equation (19) to
rectify the phase angle (8) was then found for yellow and
blue with the assumption that x 0.4 and for ultraviolet
with the assumption that x 0.6. The values of the
Fourier coefficients, the reflection coefficients, and the
ellipticity coefficients used are given in Table 7.
Solution
The process of rectification made the secondary
eclipses extremely shallow, primarily through the removal
of the cos20 terms. For this reason, the depths but not
the shapes of the rectified secondary eclipses were used in
the solution.
The rectified intensities and phase angles within a
range of 550 of midprimary eclipse were plotted to large
scale. The points on the descending branch were reflected
onto the ascending branch (for example, a point at 3550
would be ploited at 50). Tn order to more easily fit a
TABLE 7
RECTIFICATION COEFFICIENTS FOR EE AQUARII
Fourier Coefficients
0.88580
233
0.89469
219
0.87967
304
B1
y 0.00124*
138
b 0.00158*
126
uv 0.00508
176
0.03596
817
0.02966
731
0.03630
1058
B2
0.00769
148
0.00472
134
+0.00417
 185
0.07057
318
0.08133
291
0.07183
399
0.02199
448
0.01926
390
0.01164
561
0.00046*
189
0.00199*
172
0.00121*
236
Reflection and Ellipticity
Coefficients
0.12991
0.16576
uv 0.23673
*Set equal to zero in rectification.
0.03596
0.02966
0.03630
0.04330
0.05525
0.07891
0.23036
0.28691
0.29219
 
mean curve through the points, normals (averages) were
taken for intervals of every two and a half degrees,
without regard as to whether the observation was originally
positioned on the ascending or the descending branch. A
smooth curve was then drawn with regard to the normal
points and, to a lesser extent, the individual observations.
From this smooth curve, the intensity (Lo) at phase
angle 090 was read and the corresponding depth (1  t) was
obtained. The depth of the primary eclipse, combined with
the depth of secondary eclipse, allowed the determination
of the boundary coordinates (40) to be used on the nomo
graphs. The eclipse was also scaled according to the
fraction of the depth (n) as described in Chapter V, and
values of the rectified phase angle 0 corresponding to
values of n were read from the smooth curve. The shape
function XO.8 was obtained from equation (36).
The appropriate nomographs were entered, and no inter
section of the depth line and the X0.8 contour occurred
under the assumption that the primary eclipse was an
occultation. Under the assumption that the primary eclipse
was a transit, two solutions corresponding to a k of about
0.7 and a k of about 0.5 were found in the partial eclipse
region of the nomographs for each color. The k = 0.5
solution was discarded because the computed curves resulting
from this assumption were too wide at the top of the eclipse
in each color. The solution corresponding to k = 0.7, with
a darkening coefficient of 0.4 for yellow and blue and of
0.6 in ultraviolet, was adopted and refined for each color
individually.
The values of k and po were read from the nomograph at
the point of intersection of the depth line and the shape
contour. Values of or aoc, and T were obtained from
tables, and Ls and L were calculated from equation (30).
The parameters were checked for consistency by the relation
Ls + Lg = 1, and a theoretical curve was plotted with the
aid of the Xtr(k, a,n) tables. The theoretical curve was
compared to the observations and improvement was obtained
by adjusting the value of Xtr until a "best fitting" curve
was obtained. The final theoretical curves adopted for EE
Aquarii are shown in Figures 16 18.
Using the parameters of the best fitting curve for
each color, other parameters of the spherical model were
found. The values of r and i were calculated from ecua
q
tions (26) and (27), and rs followed from the definition
of k. The ratio of the surface brightnesses (Jg/Js) was
calculated from equation (32).
A derectification procedure was applied to translate
these elements into elements representing the Russell model
for the unrectified light curves. A derectified angle of
inclination j was obtained from
2
cos j = (1 z) cos2i (48)
where z is the ellipticity coefficient. It was assumed,
for the purpose of rectification, that the radii (rs and
rg) of the spherical stars could be set equal to the semi
