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Photoelectric study of EE Aquarii and AE Phoenicis

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Title:
Photoelectric study of EE Aquarii and AE Phoenicis
Creator:
Williamon, Richard Michael, 1946-
Publication Date:
Copyright Date:
1972
Language:
English
Physical Description:
xii, 185 leaves. : illus. ; 28 cm.

Subjects

Subjects / Keywords:
Astronomical magnitude ( jstor )
Comparison stars ( jstor )
Eclipses ( jstor )
Ellipticity ( jstor )
Light curves ( jstor )
Observational astronomy ( jstor )
Rectifiable curves ( jstor )
Space observatories ( jstor )
Ultraviolet observatories ( jstor )
Variable stars ( jstor )
Astronomical photometry ( lcsh )
Astronomy thesis Ph. D ( lcsh )
Dissertations, Academic -- Astronomy -- UF ( lcsh )
Double stars ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Thesis -- University of Florida.
Bibliography:
Bibliography: leaves 183-184.
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Also available on World Wide Web
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Typescript.
General Note:
Vita.

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University of Florida
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University of Florida
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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.
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13980687 ( OCLC )
ADA5137 ( NOTIS )

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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. K-Y 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. K-Y Chen for serving as

chairman of the supervisory comr-ittee 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. K-Y 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
de-rectified curve for EE Aquarii in yellow
light . . . . . . . . . .


26 Average of every five differences between
observed intensity observations and the
de-rectified curve for EE Aquarii in blue
light . . . . . . . . ...

27 Average of every five differences between
observed intensity observations and the
de-rectified 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
de-rectified curve for AE Phoenicis in yellow
light . . . . . . . . . . . 180

36 Average of every five differences between the
observed intensity observations and the
de-rectified curve for AE Phoenicis in blue
light . . . . . . . . . . . 181

37 Average of every five differences between the
observed intensity observations and the
de-rectified curve for AE Phoenicis in
ultraviolet light . . . . . . ... 182


viii













LIST OF FIGURES


Figure Page

1. Thirty-inch telescope building at the Rosemary
Hill Observatory . . . . . . . . 7

2. The dual channel photoelectric photometer
attached to the Rosemary Hill Observatory
thirty-inch reflecting telescope . . . . 9

3. A sample of a strip chart record of EE Aquarii
obtained at Rosemary Hill Observatory with the
thirty-inch reflecting telescope . . .. . 14

4. Sixteen-inch telescope building at the Cerro
Tololo Interamerican Observatory . . . . 17

5. The number one, sixteen-inch 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, sixteen-inch 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
(O-C)'s from photographically and uhotoelectrically
determined times of minima . . .. . . . 69

11. Results of the period study of EE Aquarii showing
(O-C)'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 (O-C)'s from both photographically
and photoelectrically determined times of
minima . . . . . . . . .

22. Results of the period study of AE Phoenicis
showing (O-C)'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. Kwan-Yu Chen
Major Department: Astronomy

Photoelectrically observed light curves of the

eclipsing binary star systems EE Aquarii and AE Phoenicis

were obtained using the thirty-inch reflecting telescope

at the University of Florida's Rosemary Hill Observatory

and the number one, sixteen-inch 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 twenty-seven of the

nearest two hundred and fifty-four 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 star-pairs 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

ap-ear 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 one-half








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 thirty-inch reflecting

telescope owned by the University of Florida and located at

Rosemary Hill Observatory. The site of Rosemary Hill

Observatory is some twenty-five miles southwest of

Gainesville, Florida, and some five miles south of Bronson,

Florida. The thirty-inch 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. Thirty-inch 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 ninety-five 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 one-half 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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chart was originally set by a National Bureau of Standards

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 sixteen-inch,

number one, reflecting telescope made by Boller and Chivens

Corporation, South Pasadena, California. The building for

the sixteen-inch 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. Sixteen-inch telescope building at the

Cerro Tololo Interamerican Observatory





























I I I l .


.. I


.... ... il .....
































Fig. 5. The number one, sixteen-inch 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.



























-0 -
rQ






U Q


a)
,A-





0
C








4- U
SC













0 .
0
-r, -)








-I
4
O-
U (








04 >i
M OC





O4 0


0 -0




0
o .0










0



r)-
0(

*H a
--0














0
a)












01 ^-
0)













.5 -


t-- 9


-- _


(r


PiOSI 4Wd

I -- -





-
-4 oi
J7XL-7--
- ItJ _l _-L


-I-.--


-Ii
.i I-Ii~S --

,. _L- i I t1- -


wr o


-I 4
-I-
7 -:


6t"


'lo











rs


t-D ^<


5 '


9q

I-


7- 19


_5-3 ",w


I


..--,---


--~i


r r
































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~BR1n-w--~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 three-color

(U,B,V) photometric system set up by Johnson and Morgan

(1953). The effective wavelengths and bandwidths of the

Johnson-Morgan (U,B,V) system are discussed in Mahalas

(1968) and are listed in Table 1.








TABLE 1

Filter characteristics for the UBV System

Approximate
Half-Intensity
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 Johnson-Morgan (U,B,V)

system, were obtained by finding the magnitude difference

of the variable star in blue light and yellow light (b-y)

and the magnitude difference in ultraviolet light and blue

light (u-b). The color indices were corrected for atmos-

pheric extinction by using the color extinction coefficients

kb-y 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 (B-V)' and

(U-B)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

intensi-ty 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 re-emitted 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 gray-bodies.

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 gray-body 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 super-luminous or a sub-luminous 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 mid-eclipse, 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 -2-3 (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 mid-eclipse.

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 mid-eclipse 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 mid-eclipse

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 = 1-tr 5 1 o (40)
zucO + to'; (40)
-o
S1 tr
c = L- _
S-0

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 Remeis-Bamberg

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
4-4
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


9--1 --------------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 (b-v) and (u-b) 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 kb-y ku_b

8-29-70 0.5444 0.6787 1.0823 0.1369 0.4052

8-31-70 0.3353 0.5037 0.8617 0.1724 0.3680

9-4-70 0.4905 0.6969 1.0668 0.1989 0.3567

9-8-70 0.5943 0.7617 1.4150 0.1683 0.6550

9-12-70 0.6076 0.8348 0.9983 0.2309 0.1721

9-14-70 0.3844 0.5671 0.9252 0.1798 0.3622

9-16-70 0.5056 0.7075 1.1853 0.2048 0.4752

9-17-70 0.4653* 0.6395* 1.0446* 0.1744* 0.4051*

9-18-70 0.3073 0.4401 0.8296 0.1294 0.3695

10-1-70 0.2910 0.4455 0.8470 0.1583 0.4015

10-3-70 0.5933 0.7593 1.2355 0.1679 0.4747

7-18-71 0.5377 0.7710 0.9818 0.2346 0.2131

8-17-71 0.5473 0.6411 1.0806 0.0935 0.4403

8-30-71 0.3540 0.5321 0.9434 0.1779 0.4144

Mean 0.4686 0.6415 1.0348 0.1733 0.3929


*9-17-70 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 (b-v) and (u-b) 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 (O-C) refers to the difference between

the observed and calculated times of minima.

A plot of the residuals (O-C) 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 (O-C)'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 r-I 0 r- c
(N m r-4

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


r-o
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


O-C


-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
4-4 ,- ,-

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 4-J





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


-15--I -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)
4-4





0





0
I








rn
*H







4-H
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
t-I






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
1-1











rd

-4
C)



0
*l
n3








4)




























1-4












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 (b-y) and (u-b) 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 (B-V)o and (U-B) approximating the Johnson-

Morgan standard system. These calculations are represented

by

(3-V)6 = (b-y) kb yX + Aby
(47)
(U-B)o = (u-b) kubX 1 Aub

where X is the airmass, kb-y and kub are the color

extinction coefficients for (b-y) and (u-b). respectively,

and Ab-y and Au-b are the zero point corrections (in units

of stellar magnitude) for (b-y) and (u-b), 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 (B-V)o aid

(U-B)o were not transformed to the Johnson-Morgan 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 (B-V)' and

(U-B)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 (B-V)o and (U-B)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 r--q 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 r--i -


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



* *

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* ** .+ d
* 9 0
**4* .


0 0-0.. 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 extra-eclipse 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 mid-primary 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 de-rectification procedure was applied to translate

these elements into elements representing the Russell model

for the unrectified light curves. A de-rectified 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-




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Photoelectric Study of EE Aquarii and AE Phoenicis By RICHARD MICH;^i;L WILLIAMON A DISSERTATION T'S.'ZSEll'VET) TO THE :1B--\DUATE COL^TCIL 01 TI-IE ITIUVSRSITY OF FLORIDA IN PARTIAL FULl-ILLi^'P.NT OF THE REQUIREMEHTS FOR THE DEGREE OF DOC TOP Or PHILOSOPHY UNIVERSITY or FLORIDA 197 2

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ACKNCWLEDGEMENTS The author wishes to express his sincere appreciation to Drs . K~Y Chen and J. E. Merrill for their advice and assistance in reducing and an^.lyzing the data and for their raany helpful suggestions for improving this manuscript. Especial tb.anks are given to Dr. K-Y Chen for serving as chairman of the supervisory comriittee and for his continuing encouragerrent. Appreciation is also given to Drs . _A. G. Smith, F. B. Wood, T. L. Bailey, and J. K. Gleira v.'ho, in addition ro Drs. K-Y Chen and J. E. Merrill, served on the supervisory committee. Appreciation is also given to the I'ational Science Foundation for the author's fiuanci.al .'iupport during part of his st.ay at the University of Florida, and to Dr. o. S. Ealla.rd and the University of Florida for the remainder of the author's financial support, including the funds for the trip to Cerro Tololo Interam.erican Observatory. The author's gratitude is also extended to Dc. V. Blanco and J. Grahan\ and the other members of the Cerro Tololo Interamericnn 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

PAGE 3

the computing prograriis , 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. :.i:

PAGE 4

TABLE OF CONTENTS Page ACKNOWLEDGEIIENTS ii LIST OF TABLES vi LIST OF FIGURES ix ABSTRACT xi CHAPTER I INTRODUCTION _ . . 1 II INSTRUMENTATION 5 Rosemary Hill Observatory 5 Cerro Tololo Observatory 15 III OBSERVATIONS 26 Rosemary Hill Observatory 2 3 Cerro Tololo Observatory 30 IV REDUCTION 32 Extinction and Magnitude Difference . . 32 Light Travel Correction 34 Period Study 34 Light Curves 36 Color Curves 36 Models and Rectification 37 Ellipticity Effect 39 Reflection Effect 40 Complications and Perturbations .... 43 Intensity Rectification 4 4 Phase Rectification 45 V SOLUTIONS 46 Solution from the x Functions 50 Nomographs 52 Solution from the w Functions 54 iv

PAGE 5

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 ?^ 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

PAGE 6

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 extincuion coefficients for EE Aquarii . . 63 4 Observed times of primary eclipse for EE Aquarii 66 5 Photographically determined times of oriraary minima used in the period study of E?: Aquarii . . 67 6 Standard star observations and zero point corrections for the color indices of EE Aquarii 81 7 Rectification coefficients for EE Aquarii .... 36 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 coefficiencs 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 VI

PAGE 7

18 EE Aquarii observations in yellow 147 19 EE Aquarii observations in blue 151 20 EE Aquarii observations in ultraviolet 155 21 Check star observations for EE Aquarii 159 22 Averaae of every five differences betv/een the observed rectified observations and the calculated theoretical curve for EE Aquarii in yellow light 1^1 23 Average of every five differences betv/een the observed rectified observations and the calculated theoretical curve for EE Aquarii in blue light ......... 162 24 Average of every five differences between the observed rectified observations and the calculated theoretical curve for EE Aquarii in ultraviolet light 163 25 Average of every five differences between the observed intensity observations and the de-rectified curve for EE Aquarii in yellow light 16'* '1 26 Average of every five differences betv/een the observed intensity observations and the de-rectified curve for EE Aquarii in blue light i65 27 Average of every five differences betv/een the observed intensity observations and the de-rectified curve for EE Aquarii in ultraviolet light 166 2 3 AX Phoenicia observations in yellow 167 29 AE Phcenicis observati.ons in blue 170 30 AE Phoenicis observations in ultraviolet .... 173 31 Check star observations for AE Phoenicis .... 176 32 Average of every five differences between the observed rectified observations and the calculated theoretical curve for AE Phoenicis in yellow light 177 vxi

PAGE 8

33 Average of every five differences between the observed rectified observations and the calculated theoretical curve for 7\E 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 betv/een the observed intensity observations and the de-rectified curve for AE Phoenicis in yellow light 180 36 Average of every five differences between the observed intensity observations and the de-rectified curve for AE Phoenicis in blue light 181 37 Average of every five differences between the observed intensity observations and the de-rectified curve for AE Phoenicis in ultraviolet light 182 vii:

PAGE 9

LIST OF FIGURES Page . 7 Figure 1. Thirty-inch telescope building at the Rosemary Hill Observatory • 2. The dual channel photoelectric photometer attached to the Rosemary Hill Observatory thirty-inch reflecting telescope .... 9 3. A sample of a strip chart record of EE Aquarii obtained at Rosemary Hill Observatory with the thirty-inch reflecting telescope l*^4. Sixteen-inch telescope building at the Cerro Tololo Interamerican Observatory . 17 5. The number one, sixteen-inch reflecting telescope at Cerro Tololo Interam.erican Observatory .... 19 6. A sam.ple of a strip chart record of AF Phoenicis ' obtained at Cerro Tololo Interamerican Observatory v/ith the number one, sixteen-inch telescope ... 23 7. A block diaciram of the electronics used to obtain data" photoelectrically , at Rosemary Hill and Cerro Tololo Observatories 2!} 8. A piiotographic light curve for EE Aquarii 58 9. A finding chart for EE Aquarii 61 IC. Results of the r-eriod study of EE Aquarii showing (O-C)'s from photographically and ahotoelectrically determined times of minima ^9 11. Results of the oariod study of EE Aauarii showing (0-C;'s from photoelectrically determined times of minima 12. EE Aquarii light curve from yellow observations 13. EE Aquarii light curve from blue observations 71 74 76 IX

PAGE 10

14. EE Aquarii light curve from ultraviolet observations 78 15. EE Aquarii color curves 83 16. Theoretical light curve for the primary of EE Aquarii in yellow light 90 17. Theoretical light curve for the primary of EE Aquarii in blue light 92 13. Theoretical light curve for the primary of EE Aquarii in ultraviolet light 94 19. A photographic light curve for AE Phoenicis . . .103 20. A finding chart for AE Phoenicis 106 21. Results of the period study of AE Phoenicis showing (O-C)'s from both photographically and photoelectrically determined times of minima 114 22. Results of the period study of AE Phoenicis showing (O-C)'s from photoelectrically determined times of minima 116 23. AE Phoenicis light curve from yellow observations 119 24. AE Phoenicis light curve from blue observations 121 25. AE Phoenicis light curve from ultraviolet observations 12 3 26. AE Phoenicis color curves 12 8 27. Theoretical light curves for AE Phoenicis in yellov; light 136 28. Theoretical light i;urves for AE Phoenicis in blue light 138 29. Theoretical light curves for AE Phoenicis in ultraviolet light 140

PAGE 11

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, 197 2 Chairman: Dr. Kwan-Yu Chen Major Department: Astronomy Photoelectrically observed light curves of the eclipsing binary star systems EE Aquarii and AE Phoenicis were obtained using the thirty-inch reflecting telescope at the University of Florida's Rosem.ary Hill Observatory and the number one, sixteen-inch telescope at Cerro Tololo Interamerican Observatory near LaSerena, Chile, respectively, The observational data from EE Aquarii v;as obtained on eleven nights in August, September, and October, 1970, and on three nights in August and September, 19 71. A total of 309, 302, ajid 299 usable observations v/ere obtained in yellow, blue, and ultraviolet light for EE Aquarii, and 242 observations v;ere obtained in each of the colors yellov;, 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). XI

PAGE 12

The Russell model v;as assumed, and the light curves were constructed and rectifj.ed with the techniques of Russell and Merrill (Contr. Prin. Obs. No. 26, 1952). Solutions v/ere obtained v.'ith 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 68?47 v^ere 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 8 5953 v;ere found. The eclipsing binary EE Aquarii has an amplitude of light variations of 01*6 5 for primary and 0^72 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 Ma j oris type eclipsing binary' with a period of 0936237456. The primary eclipse is a complete occultation and the secondary is a transit. Complications were ercountered in the case of AE Phoenicis in that a rather large term proportional to the cosine of the phase angle v;as present in yellov; and ultraviolet light, an asymmetry proportional to the sine of the phase angle was present for all colors, ar.d a small amount of orbital eccentricity was also possibly present. Xil

PAGE 13

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 com.ponents are not uncommon. Based on the observation that one hundred and twenty-seven of the nearest two hundred and fifty-four stars are members of multiple star systems (Motz and Duveen , 1966), estimates of the total number of stars in multiple system.s range up to fifty per cent. Tliere are three different types of tv/o star or binary star systems v;hich may be detected. Visual binaries consist of two coiriponents which can be seen as individual stars with availableoptical 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 which 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 tlie other.

PAGE 14

Ths theory that two stars might revolve about a conunon center of raass was probably not seriously considered until late in the eighteenth century. One of the early proponents of the theory, following his attempts at parallax measurem.ents 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 star-pairs had changed by araouats between 5*^ and 51°. He eliminated the possibility of 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, v/are, however, probably not the first binary star systems observed. As mentioned before, an eclipsing binary star, whose components are not resolved visually and therefore appear 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 one-half

PAGE 15

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 com.es from a name meaning demon or devil given to it by ancient Arab astronomers (Glasby, 19G8) . 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 CTOodricke 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. Tnis instrument allows the operator to search two photographic plates of the sam.e 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 ro 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 tv/o stars in their orbit about one another. The usual appearance cf a light curve is a rather smooth curve with two dips corresponding to the two

PAGE 16

4 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 r.wo stars, the ratio of the radii relative to their separation, and the ratio of the surface brightnesses of the tv/o 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.

PAGE 17

CHAPTER II INSTRUMENTATION Rosemary Kill Observator;;^ 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 thirty -inch reflecting telescope owned by the University of Florida and located at Rosemary Hill Observatory. The site of Rosemary Hill Observatory is seme twenty-five miles southwest of Gainesville, Florida, and some five miles south of Bronson, Florida. The thirty-inch telescope, housed in the building illus .traced in Figure 1, was designed and built by Tinsley, Inc., of California, and has been in operation since 1967. A dual channel photoelectric photometer, desiv-ned and bu.ilt by Astro Mechanics, Inc., of Austin, Texab, '^}as mounted at the Cassegrain focus of the telescope as shown in Figure 2. The photometer, which housed the light sensitive photomul tiplier tubes, also contained a Fabry field lens which controlled the size of the area on the photocalhode illum..inated "ny the stellar image; a filter wheel which allov/e-i the insertion of one of a posoible six

PAGE 18

Fig. 1. Thirty-inch telescope building at the Rosemary Hill Observatory

PAGE 20

>1

PAGE 22

10 filter selections into the light path; an aperture selection v:heel 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 eyepiece 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 tov.'ards a second photomultiplier tube. The third position contained a dichroic filter v;hich transmitted eighty per cent of the impinging light witli a wavelength longer than six thousand, five hundred Angstroms to the first photomultiplier tube, and reflected ninety-five per cent of the light v/ith a wavelength between three thousand, five hundred Jmgstroms and six thousand Angstroms to the second photonmltiplier 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 i^quarii was accomplished by using the fifth smallest aperture, v/hich measured 1.98 miiJimeters and corresponded to 32.5 seconds of arc in the .sky. During the time when the moon was near full phase, hov/ever, the bright sky background became a significant portion of the total signal received. This background was

PAGE 23

11 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 photomultiplier 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 photomviltiplier tube was amplified by means of a direct current amplifier. The amplifier used in 1970 was equipped with coarse and fine gain controls with steps of 2.5 magnitudes and 0.2 5 magnitudes respectively. The zero point of the system was adjustable, although on no occasion was the zero point cViancied after having been initially set at the beginning of the night. The am.olifier used in 1971 was equipped with coarse and fine gain controls with steps of 5.0 and 0.5 miagnitudes respectively.

PAGE 24

12 Both amplifiers were equipped to average the input signal over a specified length of time referred ho as the time constant. This averaging technique was necessary since the atmosphere of the earth is continuously undergoing 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 tim.e constant), leads to a strip chart recording with high frequency, high amplitude peaks. A time constant of one-half second, which was sufficient to reduce the frequency and am.plitude 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 v/as of no consequence. It v/as, therefore, considered unnecessary to try to reduce the dark current further by refrigerating the photorauitipli^r tube with dry ice. The output signal of the direct current am.plifier was fed directly into a Brown strip chart recorder. The deflection of the recorder, which was linearly proportional to the aniplifier output, ineasured the intensity of the light received. A sample of a strip chart record obtained from Rosemacy Hill Observatory is shown in Pigur^e 3. The

PAGE 25

+J

PAGE 26

14

PAGE 27

15 chart was originally set by a National Bureau of Standards W/TV radio time signal and v/as 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 necessai-y. Cerro Tololo Observatory The observational data of AE Phoenicis was obtained on four nighcs in September and October, 1970, at Cerro Tololo Interam.erican Observatory* near LaSerena, Chile, South America. All observations were made with the sixteen -inch , number one, reflecting telescope made by Boiler and Chivens Corporation, South Pasadena, California. The building for the sixteen-inch 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 photomei:er, 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 al lov.^ed the choice of five different aperture sizes through whicli incoming light could pass; a filter slide. •*Operated by the Associatioii of Universities for Research in Astronomy, Inc., under contract with the National Science Foundation.

PAGE 28

Fig. 4. Sixceen-inch telescope building at the Cerro Tololo Interamerican Observatory

PAGE 29

17

PAGE 30

Fig. 5. The nunber one, sixteen -inch reflecting telescope at Cerro Tclolc Interamerican Observatory

PAGE 31

19

PAGE 32

20 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 photomultiplier tube, which v/as 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, v/as applied by means of a regulated high voltage power supply. The potentj.al difference served to amplify the electron current originating at the photocathode . As previously discussed, rapid fluctuations in atm.ospheric 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

PAGE 33

21 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 Brov;n chart recorder was set to drive the chart at a rate of one inch per minute. The time was initially set by VJWV 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 shovm 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.

PAGE 34

-u

PAGE 35

* %\r

PAGE 36

Fig. 7. A block diagram of the electronics used to obtain data photoelectrically at Rosemary Hill and Cerro Tololo Observatories

PAGE 37

25 0, >*

PAGE 38

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 proxim.ity criterion is made mainly to reduce the effects of differential atmospheric extinction (Chapter IV) , but also the closeness of the two stars expedites t'ne positioning of the telescopes. The m.agnitude re.'-triction 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 v;ould eliminate the need for consideration of correction due to the spectral response of the photomultiplier tube and correctJ.cn due to differential color atmospheric extinction (Hardie, 1962). In addition to a comparison star, a check star v/as 26

PAGE 39

27 also observed. The check star, also assumed to produce a coustant light output, afforded a check on the assumption that the comparison star was not also a variable. This v/as accomplished by occasionally substituting a check star observation for a variable star observation, and consequently 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 three-color (Q,B,V) photometric system set up by Johnson and Morgan (1953) . The effective wavelengths and bandwidths of the Johnson-Morgan (U,B,V) system are discussed in Mahalas (1968) and are listed in Table 1.

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28 TABLE 1 Filter characteristics for the UBV System Approximate Half-Intensity Colo r Effective Wavelength Width V (visual) 5480 A 700 A B (blue) 4400 f. 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 O bservatory 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 once every twelve minutes. A sequence of measurem.ents for the com.parison star consisted of observations of the star in each of the colors y, b, and u, followed by observations in each

PAGE 41

29 color, bat 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 v;ere 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. Num.bers 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 tc tJie 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 ail accuracy of 5 seconds. The Greenwich mean time of the observation, the fi. Iter, the deflection due to the star alone, whether the star was a variable, check, or comparison (coded 1, 2, and 3, respectively), and the

PAGE 42

30 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 techniqiies . 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 v/ith 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 m.oonrise or moonset, one variable star set and one variable star sky set of measurements v/ere 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 v/as read to

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31 an accuracy of one second. Usually three integrations for each color were made unless significant scatter v/as present, in v;hich 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 ail of the integrations in that color. Sky measurements v;ere 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 v/ere punched foi the reduction procedures described in Chapter IV.

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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 36 0/65 computer located at the University of Florida. The computing programs, originally v/ritten 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 accomplished by the computing programs. Extinction and Magnitude Differenc e The atmosphere of the earth can greatly affect the light v/hich passes through by the processes of absorbing and reddening. The transparency of the atmosphere, \/hich 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 32

PAGE 45

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) ^ -0.0008083 (secz -1) ^ (1) where z is the zenith distance. The magnitude of the comparison star was then calculated by the equation: m = S^ 2.5 log(d^) (2) where S^^ is a zero point magnitude which corresponds to a particular gain setting and d^ 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 ].east squares fit to the relation m^ = m kX (3) where m^ is the outer atmosphere (zero air mass) magnitude. The variable star observations were then corrected for differential atmospheric extinction by subtracting the quantity k(X^^ X^) from the difference in magnitude

PAGE 46

34 (in^ m^^) between the variable star and the comparison star. The expression for the difference in magnitude v/hich was used by the computing programs was Am = m^ m^ = 02.5 log(LyL<.) k (X^ X^) (4) where L^ and L^ are the apparent luminosities of the variable and comparison stars respectively. The values of L^ 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 foi" light travel time, the recorded geocentric time v/as converted into heliocentric time by adding the increment of time t given by the following equation (Binnendijk, I960) t O.C05775{ (cos6cosa)X + (tanesin6 + cos6sina)Y} (5) where 6 is the declination of the star, a is the right ciscension of the star, X and Y are the rectangular Cartesian coordinates of the sun (at the midpoint of the observation) and £ is the mean obliquity of the ecliptic. Period Study The determination of the period of revolution (F) and of the mean epoch (T ) to which all observations v/ere

PAGE 47

35 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 m.ean 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 involving tracing paper was necessary. A representation of a minimum v/as obtained by aligning and superimposing on tracing paper the plots of magnitude versus time for those minima in v;hich the Hertzsprung m^ethod 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 mininia determined by photographic means v/ere coiT^Dined, with lower v/eight, with the photoolectrically determined times of minima, in a linear least squares fit to the relation T =• T >EP (6) o where T is the time of m.inimum 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 v/ere checked

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36 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 Johnson-Morgan (U,B,V) system, v/ere obtained by finding the magnitude difference of the variable star in blue light and yellow light (b-y) and the magnitude difference in ultraviolet light and blue light (u-b) . The color indices were corrected for atmospheric extinction by using the color extinction coefficients kj^_ and k^_]^ to obtain the outer atmosphere color indices. The outer atmosphere color indices were then transformed.

PAGE 49

37 by the addition of a zero point correction obtained from standard stars, to a system approximating the JohnsonMorgan standard system. The color indices (B-V) ' and (U-B) ', obtained in the above manner, v;ere 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 Rectificat ion The components of an eclipsing binary system by necessity must be relatively close to one another. Vihen the components of an eclipsing system are separated by less than eight or ten radii, the stars v;ill be distorted by mutual gravitation effects and v/ill vary in brightness over theii" 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, ho\;ever, 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

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38 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 = J^(l X + xcosy) (7) where y i^ ''-"^^ 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 J^ 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 intensi-ty 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 knov/n as rectification. The rectification formulas involve sinusoidal terms determined from a truncated Fourier analysis of the outside cf eclipse region of the light curves. The Fourier series was given by m m 1 = A^ -r T A„ cosno + F B^ sinne (8) o ' n '• n n-1 n=l where I is the unrectified intensity and ra is an integer

PAGE 51

39 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 assum.es that both stars are distorted into similar ellipsoids in such a way that as the tv/o com.ponents revolve around each other, the observer sees a maximum of light when a maximum area is observed v/hich occurs at phase 0.25 and phase 0.75. Rectification for ellipticity miust 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 co a term, proportional to cos26, a second order term proportional to cos30, due to a real difference in shape between the two ellipsoids, may be present. Werrill (1970) has shown the importance of including cos39

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40 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 cos30 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 re-emitted v/ith 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, hov/ever, 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 heni.sphere 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. Tlie reflection effect is proportional, to first order.

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41 to the cosine of the phase angle since the minimum affect 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 am.ount 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 A^ef = Cq + C-L cose + C2 cos2e (9) v/here Cq , C, and C2 are the portions of the Fourier coefficients Aq , A-j_, and A2 from equation (8) due to reflection alone. Both the ellipticity and reflection effects are proportional to a cos26 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 gray-bodies. If G(-. and G^ are defined as Gc = Ihri (EcAh) (10) Gh = I^rg (Eh/E^) where i-^ and I^ are the light intensities, rj.^ and r are the stellar radii, and Ej^ and E^ are the luminous efficiencies of the hotter and cooler components, respectively. After SLibs ti tuting Jh Ih/^h '11) ^c = hr'4

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42 tha ratio could be calculated. The ratio of the depths of the eclipses, rectified for ellipticity, is equal to the ratio Jj^/JcIn order to calculate (E^/E-^)^, knov.'ledge of the spectral type (and hence temperature) of the hotter component must be available. This allows a computation of J^/Ej^ from gray-body theory represented graphically by Russell and Merrill (1952) , which, when combined with J;,-^/J^, gives the value of T^. and finally J^/E^^ _ The reflection coefficients are then expressed as ^1 = -^1 C = -(0.75 0.25 cos^i) ^R_l_2ll A, cosec i (13) C„ -0.25 ^c "^ ^h ^ in i 2 G, G, 1 c h where i is the inclination of the orbit with respect to the plane of the sky. The above method for finding the reflection coefficients failed as G and Gj^ 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 (G^ -IGj^)(0.30 0.10 cos^i + 0.10 sin'^i cos26) (14) + 0.40 (G^ G, ) sini cosB c n = C + C, cosO + C., cos29 o 1 2 by equating coefficients. The quantity G + G. may be calculated from

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43 G + G^ = -^_ II (I I, )^ (r r^,) (15) c h (G G^)'^ c h c h c h and G -G, follows. c h Complications and Perturbations A theory has not yet been developed v;hich can account for the presence of any sine terms found in the Fourier expansion (8) . Cos 3 terms and cos 4 9 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 cos9 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 knov/n residue of, for example, cos9 or cos 39 due to a second order difference of the ellipsoidal shapes. A complication could bs due, for example, to a super-luminous or a sub-li,minous 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.

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44 Intensity Rectification The rectification for reflection i3 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 + Cq + C-i_ cose + C2 cos29 A3 cosSe A^ cos4e B^ sine B2 sin2e B^ sin3e B^ sin4e (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 4e, 39, or even 26 terms depending upon the significance of the 46 and 36 terms. If the coefficient C-, in equation (16) is found from (14) , then a cosG lierm proportional to (A^^ C]^) v/ill remain. Instead of removing this term by separately subtracting (A^^ Cj^) cosG, one in practice sets C-^ in equation (16) equal to A-, which effectively removes all of the cosG term at one time. The ellipticity rectification is accomplished by dividing the observed intensity by (Aq + Cq) + (A2 + C2) cos29 (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'^ = ^ (18) (Aq •:Cq) + (A2 + C2) cos 2 6 v/here I' is given by (16) .

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45 Phase Rectification Rectification for phase is carried out using the relation sin^e EHL_i_^.. (19) 1 z cos^O v/here 6 is the phase angle of the original observation, is the rectified phase angle, and z is the ellipticity coefficient defined by z = e'^ sm"^ 1 (20) where e is defined as the eccentricity of the equatorial section of the component (Binnendijk, 1970). The nunierical value of z is actually obtained from the relation Nz = -4(A^ _C^ ^ (21) (Aq Co A2 + C2) where N is the ratio of the "photometric ellipticity" to the geometrical ellipticity and is represented by the approximation N (15 f x) (1 f y) ,^,2S 15 5x ' ' where x is the coefficient of limb darkening and y is the coefficient of gravity darkening. Because of the uncertainty of the actual variation of N with x and y, it is custoip.ary to adopt N equal to 2.2, 2.6, or 3.2 when x is assxxmed to be 0.4, O.G, or 0.8, respectively.

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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 v;ell represented by a theoretical curve defined by a set of eight parameters. These eight parameters (or elements) are P Period of revolution Tq Epoch of primary minimum Xg Limb darkening coefficient for the sm.aller star Xg Limb darkening coefficient tor the larger star rg 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 v.'here the radii r^ and r are expressed in terms of the distance between the centers of the components. 46

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47 The period of revolution F and the epoch of primary minimum T^ , often referred to as the "lighr elements," were satisfactorily determined by the period study previously described. Of the sj.x remaining elements, the darkening coefficients x^ and x^ must be initially assumed. Merrill (195G) has compiled tables for sciving light cxirves based en 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 elemients of a system based on tabular values of x, and to choose the darkening coefficient and the other parameters of the m.ost reasonable "fit" as the solution of the light curve. A preliminary value for the darliening coefficient of the hotter component may be obcai'ied from theoretical considerations if tl-.e spectral classification of the system is kno'.vn .. Since present knowledge of the statistical rolations'iiips between limb darkening coefficients and spectral types is not perfect, =iolutions based on darkening coef f icier.ts other than that initially assumed must be performed. Of the four rem.aining parameters (rg, rg . i, Lg) , I.g v/ill be known in tlie case of a complete eclipse. During the total phr.oc of a complete occaltation eclipse, at wt.ich time the larger star eclipses the smaller star, only the light from the largei; star will be received and Lg , the lumii";(.-)sity of the larger star, is obtained directly from the rectified IJ.ght carve. Since the light outside of eclipse 3 3 normalized to unity, th(='n L., the liiminosi ty of the

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48 smaller star, is given by 1 Lg. For occialtation 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 rg , rg, i, and Lg (or Lg) in the case of a partial eclipse, the follov/ing quantities have been defined: k = Ls. {23a) ^ (23b) ^s where 5 is the apparent distance between the centers of the tv;o components and is given by 62 = cos^i + sin^i sin^G . (24) Combining equations (23a), (23b), and (24) aives r^^d + kp)2 = cos^i + sin^i sin^e (25) which is valid for any phase 6. At external contact, defined by the starting or endinc of an ecJ.ipse, equation (25) reduces to rg2(l Ik)2 3-. cos^i + sin2i sin^G^ (26) v;here 9g is the phase angle at external contact and p has

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4S been set to unity since 5 = rg + rg at external contact. At mid-eclipse, p = Po and 9 = so that r 2(1 + kpQ)2 = cos2i . (27) Both the Merrill tables (1950) and the Merrill nomographs (1953), discussed later in this chapter, are designed to yield the values of k, Po, and Bq. Equations (26) and (27) are then used to obtain rg and i. In order to obtain general expressions for the luminosities Lg and L3 , 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 x 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 x must be less than unity, and for an occultation the value of x remains unity since no light is lost from the larger star. If £tr and £°*=^ are defined to be the apparent light intensity at any phase for the transit and occultation eclipse, respectively, then 1 -C^r and 1 l^'^ are the corresponding light losses. From the above definitions of a and x, the loss of light for an occultation is given by 1 IOC = LsaO'^(Xs,k,p) (23)

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50 and for a transit is givezi by 1 l^"^ = LgT(Xg,k)atr(Xg,k,p) . (29) Equations (28) and (29) are in practice evaluated at mideclipse and solved for L^ and L^ to obtain 1 _ /OC a^^(Xs,k,poT and 1 l^^ 5 T (Xg,k)a§-^"(Xg,k,po) The values of t and a are known as functions of the parameters k,x, and p^ determined from the adopted solution, and 1 1°^ and 1 l^^ are the depths of the rectified occultation and transit eclipses, respectively. For a system with complete eclipse, a^^ is unity and L^ = 1 £g^ (and Lg ^3^) . The luminosity of a star is related to the surface brightness J by the relation L = T-r'^J . (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 ^ = ^ • ^"3^ . (32) "^s ^s ^g Sol ution from the x ^m"i ctions A solvation 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 eitheir

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51 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 mid-eclipse. The light lost at any phase of the eclipse will be given by 1 £ n(l Iq) (33) where 1 Z^ is the light lost at iriid-eclipse and n for an occultation eclipse is given by n°'^ = a°*^/a^^ and for a transit eclipse by n^^ = a^''^/a^^ . For each value of n, the light curve defines a corresponding value of 9 so that 9 (n) may be read directly from the light curve. From equation (25) for an arbitrary 9 (n) , the following may be written cos^i + sin^i sin^O (n) = rg^{l + kp(x,k,naQ)} . (34) At mid-eclipse cos^i + rg2{l + kp(x,k,ao)}2 (35) where has been set to zero and n has been set to one. The shape function is then defined by sin^ e (n) _ 1 + kp(x,k,na^)^ 1 + kp(x,k,a ^)^ sin^'elTjy 1 + kp(x,k,J2ao) ^ 1 + kp(x,k,aoy^ (36) = x(x,k,aQ,n) For specified values of x and n, x ni^Y be tabulated as a function of k and a^. The solution cannot, however, be obtained until informtation from the depths of the two eclipses is included. This is done conveniently by the introduction of a function q defined by q(x„,x.,p,k) T jx k)a^ ^(Xg,p,J<_)_ . (37) ^ g' ^^ aOctxs,p,k)^

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52 When equation (37) is combined with equations (28) and (29) , the expression for q becomes = 1-1^^ L, 1 £^^ • r q = ^_ji4;; . ^ . (38) g By equation (38) , q is physically interpreted to be the ratio of the light of the large star observed during any phase of the transd.t eclipse to the fraction of the light of the small star observed at the same geometrical phase during the occultation eclipse. The function k (x,-, Xg ,rx°'-^ ,q) has been tabulated for all combinations of tabular values of x that are likely to occur. Since Lg + Lq = 1 , then equation (38) becomes a°^ = 1_:_^1 ^ t (1 ^ Ji^± (39) ° ^-o so that by varying qQ a value of a^ will be obtained. This permits a "depth" curve to be plotted with the coordinates k and o.^^ . The shape function xi'^r^r q/^) ^^Y 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 Me5:rill (1953) has devised a graphical method using the X functions which is considerably quicker than the procedure just described. Large scale plots (nomographs) of the shape function x evaluated at n 0.8 have been

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53 constructed using coordinate scales v/hich allow the "depth" curve to be represented as a straight line. Four nomographs, 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 £tr) + (1 „ £OC) b = 1 -l^"" + 1 L.^J^ (40) ^o o 1 l^^ 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 Xn 8 contour represented a solution of the system. If both minima are sufficiently deep so that a Xq 8 can be obtained from both, then the solution should be given by the intersection of the depth line and Xn o ' the intersection of the depth line and Xn^o/ or by the intersection of Xo"o ^'^^ Xa o' ^ theoretical light curve can be J . o U . o 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 Xq p SJ^d finding another set of nomographic elements. The

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54 solution that "best fits" all of the observed points is adopted for the solution to the light curve. S olution from the '|j 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 i|) functions tabulated by Merrill (1950) . From equation (25) it follov/s that sin^ e sin^e2 ^ (1 + kp)^„(1 + kpp)2 sin2~92 sin^63 (1 + kp2)2 (1 + kp3) ^ (41) = ^ (x,k,a,a2 ,a3) v/here Q2 rP2''^2' ^^^^ ^ 2 '^"^ ''^3 ^'-^ knov;n fixed quantities. By choosing 0-2 as 0.6 and a3 as 0.9 and defining the constants A -= sin^e (0.6) B = sin-9(0.6) sin2e(0.9) (42) then equation (41) is reduced to iij(x,k,a) _ sin^e A (43) B or sin^e = A + B(|j(x,k/Ci) . (44) The above equations require that the Vdlaes of 6 corresponding to ex = 0.6 ;?n.d 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

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55 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 remaining elements can be calculated by finding Pq from a table of aQ(k,pQ) 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 c})2(x,k) which are defined such that 6, (x,k) {;|j(x,k,0) i|'(x,k,l)} = 4k (j52(x,k) {-^(1 k)^^(x,k,0) (1 + k)^;J;(x,k,l) } = 4k v/here i|i(x,k,0) is the value of '> at external tangency and 'ji(x,k,l) is the value of '^ at a = 1 . The inclination i and the radius of the larger star r^ may be found from r^ cosec 1 ° 5 4'i(x,k) (46) COt^i rB )2(X/k) -

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CHAPTER VI EE AQUARII History The light variation of EE Aquarii v;as discovered from photographic patrol plates reduced at the Remeis-Bamberg 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 photographically by Filatov (1961). Strohmeier, Knigge, and Ott (1962) published six more photographically determined times of primary minima which, when combined with the times of minima obtained by Filatov, gave the light elements JD 2429881.310 + 0^5089951 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 simi. lar to that of Algol v;ith a magnitude range of from a maxim.um of 8™35 to a minimum, of 9'?10. No secondary minimum was detected f?:om the photographic plates. The author of this dissertation was unable to find any photoelectric vi/ork or orbital solutions for EE Aquarii. 56

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-H H U m «: w O > o +J H o •H 5^ tr. O -P O Xi a 00 en -H

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58 i I I o o a o o o o o s o o 8 o o 8 q O o 8 o T > 1 r~T § o oQ CNJ (X) p ^" 00 03 CO CD* <7> Cr> 00 d CD d < X O vM d o o

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59 CorriDarison and Check Stars Selection of a comparison and a check star for EE Aquarii \/as 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 precessed to epoch 1970.5 from 1950.0 coordinates given by the SAO catalog. A finding chart reproduced from the Atlas Eclipticalis (Becvaf, 1964a), with EE Aquarii, the comparison star, and the check star labeled, to the right of the respective star, is shown in Figure 9.

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'ig. 9. A finding chart for EE Aquarii

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61 -13^ o o 19 o o H < IH ^^ O w Q -20 o -21^ -O O o .oO Q -O ocorr.p o o -0^check 1 3 -oO EE Aqr oo -22 o o o O ^'^ -o-oo 1 JL O Oj -O22^ 38^22^ 3 4"^22^ 3 0'" RIGHT ASCENSION O o o o o i 22^ 26"'

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62 TABLE 2 EE AQUARII, COMPARISON AND CHECK STARS EE AQUARII COMPARISON CHECK R. A. (1970.5) 22^ 33"^ 06?0 22^ 32"^ 48?6 22^ 3l'^ 31?8 Dec, (1970.5) -20° 00' 44V6 -19° 12' 17V4 -19° 32' 36'.'3 BD Catalog -20°6454 -19°6300 -20°6446 HD Catalog 213863 213791 213623 SAO Catalog 191236 165165 165157 Spectral Class FO F8 AG Magnitude (m^) 8.0 8.5 9.1 Extinction Extinction coefficients for each of the three filters and also for the colors (b-v) and (u-b) 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 coefficient? used in later calculations are listed in Table 3. The extinccion coefficients actually found for September 17, 1970, \/ere considerably less than the coefficients found for other nights, due, at least in

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63 TABLE 3 ATMOSPHERIC EXTINCTION COEFFICIENTS AND COLOR EXTINCTION COEFFICIENTS FOR EE AQUARII DATE

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64 part, to a rather short range of aiririass to which the least squares fit described in Chapter IV v/as applied. A more realistic set of coefficients for each filter was obtained by averaging the coefficients from the 1970 observations. The color extinction coefficients (b-v) and (u-b) 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 St udy 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 v/ere 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 (19 62) were also available. These photographically determined epochs were con±)ined v/ith the photoelectrically determ.ined epochs to determine a period and reference epoch by a linear least

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65 squares fit to equation (6) . In this calculation, each time of minimum determined photographically was given a weight of unity, v^hile each tim.e 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 + O'^SOBggSSS E +.0006 +.00000009 were found from the least squares fit, v/here the probable errors are given below the elements. The results of this period study are summarized in Table 4, v/hich contains information relevant to the photoelectrically determined times of m.inimum 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 (0-C) refers to the difference between the observed and calculated times of minima. A plot of the residuals (0-C) for all the observed minima is shown in Figure 10.Because of the small scale of Figure 10, the photoelectric residuals are not clearly shovm. Figure 11, therefore, shovv-s just the (O-C)'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.

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66 2:

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67 TABLE 5 PHOTOGPAPHICALLY DETERMINED TIFJES OF PRirL\RY MINIMA USED IN THE PERIOD STUDY OF EE AQUARII JD HELIOCENTRIC 2440828.7809 + 0.5099558 E JD HEL (OBSERVED) EPOCH 0-C SOURCE 2400000+ 29881.297 -21508.0 -0.007 Filatov (1961) 29902.185 -21467.0 +0.012 " 29904.212 -21463.0 +0.003 " 30200.426 -20881.0 -0.018 " 31375.215 -18573.0 +0.009 " 32744.388 -15883.0 -0.016 32771.383 -15830.0 +0.002 33187.235 -15013.0 +0.005 33897.287 -13618.0 +0.008 36080.348 9329.0 -0.013 36844.375 7828.0 +0.012 Strohmeier (1962) 36868.306 7781=0 +0.020 36822.465 7871.0 -0.012 36845.375 7826.0 -0.007 36846.396 7324.0 -0.004 36893.242 7737.0 +0.015

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69 < ~3 O o IT) C\J CM O o o o CM O o iO ro CM O O o IT) CM O o lO CM ro CM O o o o fO CM •
PAGE 82

g o u u 1

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71 -{O Q

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72 Light Curve s There were 309 usable observations in 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 (Tq) discussed in the previous section. The time of each observation given in terms of the heliocentric Julian date (Julian day number and decimal) , the phase of each observation, and the magnitude difference betv/een 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 iii yellov/, blue, and ultraviolet light are shov/n 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 v;as provided by check star observations. The heliocentric Julian date and the difference in magnitude beLween the check star and the comparison star are x-eccrded in Table 21 in the Appendix. The scatter of the observations could have been due to

PAGE 85

CO a o H 4J > U . e O u > O P x: en •rH •H •H < W CN •H fa

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74

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O -H -P > U u o +> H -H w -H

PAGE 88

76

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CO o -H -p > M (U (0 Xi O 4J 0) -I o •H > (0 4J g o (U > o H -H •H •H

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78

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79 actual short term lignt variations from the system, but was probably due mostly to random fluctuations in the transparency 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. C olor Curves The color indices (b-y) and (u-b) found from the observations of EE Aquairii alone ware 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 outside the atmosphere values of the color indices, gave the color indices (B-V)q and (U-E)q, approximating the JohnsonMorgan standard system. These calculations are represented by (3-V)6 = (b-y) k^ X 4. Aj^^_ "^ ^ "^ ^ (47) (U-B)i (u-b) k^_j^X HA^_j3 v.'here X is the airmass, k)3_Y and k^.j^ are the color extinction coefficients for (b-y) and (u-b) .. respectively, and Aj^_„ and ^,_,_]:j ore the zero point corrections (in units of stellar magnitude) for (b-y) and (u-b), respectively.

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80 The standard star observations and the zero point corrections for each night are located in Table 6 . Zero point corrections for the nights v/ith 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 (B-V)' aid (U"B)q were not transformed to the Johnson-Morgan standard star system. In finding outside the atmosphere values of the cclor indices, second order color extinction coefficients were assum.ed to be zero. In transforming from the natural systems of the telescope to one approxi'm.ating that of Johnson and Morgan, the transformation coefficients (y and v as defined by Hardie (1962)) were assumed to be unity. The color curves were obtained by plotting (B-V) ' and (U-B) Q versus phase, and are shown in Figure 15. Inspection 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 n^\ture of the eclipses (shown in a later section). The color indices (B--V)q and (U-B)^ found for EE Aquarii agree reasonably well v/ith the JohnsonMorgan standard system, color indices of an FO main sequence star.

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1

PAGE 94

> u u u o r-i o o •H -H rO in H

PAGE 95

r 5« • !• •• %3* • • • I •I 15 •• 4 * • •rj V « q e «*» o « « A 9 — L o d r~i — r • •• • •• o I ,• • • • • •• . V • • »• ", • • • • •• • e .V o ro d — L J. >^ o I o ID d d LJ id< a. q 00 d

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84 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 extra-eclipse 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 (9^). By rem.oving a considerable portion of the cos29 term due to ellipticity, the angle of external contact appeared to be about 45°. The Fourier expansion was then applied to the regions between 45^ and 135° and between 225° and 315° by the method of least squares. The reflection coefficients '\Cq , C^, C2) were calculated 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 v/ere found by truncating equation (8) with 4G, 36, and 2G terms. In the expansion which was truncated with 4 9 terms, however, it was found that the coefficients of the sin4 9 and the cos 4 6 terms were comparable in size to their associated probable

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85 errors. Fcr 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 rectification. As will be shown in the next section, the yellow and the blue curves v/ere solved under the assumption that the coefficient of limb darkening (x) v/as 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 (6) was then found for yellow and blue with the assum.ption 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 extrem.ely 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 55^ of mid-primary eclipse were plotted to large scale. Tlie points on the descending branch v/ere reflected onto the ascending branch (for exam.pl e, a point at 355 would be plotted at 5'-^) . Tn order to more easily fit a

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86 TABLE 7 RECTIFICATION COEFFICIENTS FOR EE AQUARII Fourier Coefficients uv 0.88580 ± 233 0.894G9 ± 219 0.87967 ± 304 -0.03596 ± 817 -0.02966 ± 731 -0.03630 ± 1058 0.07057 ± 313 0.08133 ± 291 0.07183 ± 399 -0.02199 + 448 -0.01926 ± 390 -0.01164 ± 561 Bn B. y

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87 mean curve through the points, normals (averages) were taken for intervals of every tv70 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 (Iq) at phase angle 090 was read and the corresponding depth (1 l^) v;as 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 nomographs. The eclipse v/as also scaled according to the fraction of the depth (n) as described in Chapter V, and values of the rectified phase angle corresponding to values of n were read from the smooth curve. The shape function Xq 8 "'^^ obtained from equation (36) . The appropriate nomographs were entered, and no intersection of the depth line and the Xq.S contour occurred under the assumption that the primary eclipse was an occultation. Under the assumption that the primary eclipse v/as 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 v;as 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 darkenina coefficient of 0.4 for yellow and blue and of

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88 0.6 in ultraviolet, v/as adopted and refined for each color individually. The values of k and p^ were read from the nomograph at the point of intersection of the depth line and the shape contour. Values of a^T ct^g, and x were obtained from tables, and L_ and L^ v.'ere calculated from eauation (30). The parameters were checked for consistency by the relation Lg + L„ = 1 , and a theoretical curve was plotted with the aid of the x^''^(k, OL^,n) tables. The theoretical cur^^e was compared to the observations and improvement v/as obtained by adjusting the value of Xn''^o until a "best fitting" curve U . o was obtained. The final theoretical curves adopted for EE Aquarii are shov;n in Figures 16 18. Using the parameters of the best fitting curve for each color, ocher parameters of the sphericeil model v;ere found. The values of r and i were calculated from equations (26) and (27) , and r^ followed from the definition of k. The ratio of the surface brightnesses (J^/Jg) was calculated from equation (32). A de-rectification procedure v;as applied to translate these elements into elements representing the Russell m.odel for the unrectified light curves. A d'3-rectif ied angle of inclination j was obtained from cos"^j = (1 z) cos^i (48) where z is the ellipticity coefficient. It was assumed, for the purpose of rectification, that the radii (r^ and r^) of the spherical stars could be set equal to the semi-

PAGE 102

90 o (D 00 o o lO o o ro UJ Q © o CM -O

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92 "T _ X •o » o O 0> d 00 o o in o o ro Q © O cvj O O

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4J

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94 o to .Jo CD O o o ro Q © o CM O

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95 major axes {a^ and a ) of the two similar prolate ellipsoids approximating the shapes of the stars of the actual system. The semi-minor axis (b) was then obtained for each ellipsoid by b •= a(tani/tanj) . (49) The oblateness of the ellipsoid v.'as obtained from £ = ^ k . (50) a The final elements in each color for EE Aquarii are included in Table 8. A problem was encountered with EE Aquarii in that, for the region of the nom.ograph in v.'hich the solution was defined, a very small change in x produced a rather large change in all the elements of the system. For this reason the final parameters obtained in this investigation are certainly subject to refinement by further investigation. Because of the inherent uncertainty of the final elements, a set of "mean" elements, obtained by averaging the values of k, j, and z from each color and calculating a consistent set of elements from these, would probably be quite sufficient to represent the system at this time. Such a set of "mean" elements was calculated and has been so designated in Table 8. In order to check the agreement of the theoretical curve and the rectified observations, residuals consisting of the difference in intensity units between eac:h observed point and the theoretical curves were obtained. Average residuals for (^very five points were calculated and are

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96 TABLE 8 ELEMENTS FOR THE SOLUTION OF EE AQUARII

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97 tabulated as a function of rectified phase for each color in Tables 22 24 in the Appendix. A further check on the consistency of the fit was obtained by calculating root mean square (RxMS) residuals for different parts of the curve. The results of these calculations are listed in Table 9. A measure was sought of how well the Russell model, as adopted, represented the observations of EE Aquarii. The adopted theoretical curves were de-rectified using only terms associated with the Russell model, and the de-rectified curve was compared to the unrectified intensity observations. The theoretical curves were de-rectified according to I = I^'{(Aq + Cq) + (A2 + C2) COS23} (51) Cq C-jCOsB C2COs2e + A3CCS3O Tables 25 27 in the Appendix give the average of every five differences between the observed unrectified intensity points. and the de-rectified theoretical curve, as defined above, as a function of unrectified phase. Root mean square (RMS) residuals were calculated for different portions of the curve, and the results for each color are located in Table 9. It is noted that the Pi4S residuals in both of the above investigations were largest for the secondary eclipse, and that the average residuals were also largest in secondary. These effects were not unexpected since, as noted before, the scatter of the observations was most pronounced during secondary eclipse.

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98 TABLE 9 STATISTICAL STUDY OF EE AQUARII RMS residuals of the observed rectified points from the theoretical liglit curves COLOR Yellow Blue Ultraviolet PORTION OF CURVE Primary First Maximum Secondai"y Second Maximum Primary First Maximum Secondary Second Maximum Primary First I'laximum Secondary Second Maximum P^S OF

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99 Conclusions The following conclusions and opinions follow from the present photoelectric study of EE Aquarii: 1) The eclipses are partial and, after rectification, very shallow. The primary eclipse is a transit and the secondary eclipse is an occultation. 2) The observational scatter during the secondary eclipse was larger than for any other part of the curves. Future investigations should reveal whether or not this scatter is in part due to an intrinsic variation in one of the components or totally to atmospheric fluctuations. It is the opinion of the author that the latter are chiefly responsible. 3) The two components of EE Aquarii are very different in luminosity. It is the opinion of the author that the possibility of obtaining a double line spectrum from this system is rather remobe. 4) The light curves of EE Aquarii show an amplitude of Although no attempt is made to classify the system, EE Aquarii should not be classified as an Algol type eclipsing variable as v/as previously done (Strohmeier, Knigge, Ott, 1962) . 5) From a period study from timias of minima observed during the last thirty years, one cannot fdnd any indication of a variation in the orbital period. The limited number »f variability of 0?65 for primary and O'I'ZO for secondary.

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100 of times of minima and the short time scale over which these have been gathered would prohibit any conclusions regarding the constancy of the period at this tim.e . Several other references (Kuklin, 1962; Locker, 1971a, 1971b) , listed in the card catalog of eclipsing binary stars at the University of Florida, contain times of minima not used in the period study included in this dissertation. It is the intention of the author to include these times of minima in another period study of EE Aquarii. Although the light elements should be better determined, a definitive conclusion about the constancy of the period will remain impossible because of the short time scale (about 30 years). The orbital parameters should not be changed by the revised light elements.

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CHAPTER VII AE PHOENICIS History The light variability of AE Phoenicis was discovered frora photographic patrol plates reduced at Remeis-Baipberg Observatory, Bamberg, Germany. The discovery was announced in 1964 by Strohmeier, Xnigge , and Ott (1964), and the system was given a provisional designation of BV 483. The system was believed tc have a magnitude of 7'r'9 at maximum light, and an amplitude of variation of 0^6. Avery and Sievers (1968) published a list of thirtyfive times of minimum from v/hich an orbital period of 0,362378 days was derived. The same authors also published a light cvrve from their photographic observations which is reproduced in Figure i9. The system was thought Lo have a light curve similar to that of W Ursae Majoris with an amplitude of light varieition of 0?'^5. The card catalog of eclipsing binary stars, maintained by tlie Astronomy Departm.ent of the University of Florida, gave no indication of any photoelectric observations or orbital solutions for AE Phoenicis prior to this investigation. 101

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CO •H U •H C 0) o w < u o •4-1 > u -p x; •H u •H x: O -P O x:
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103 i 1 1

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104 Comparison and Check Stars Selection of a comparison and a check star for AE Phoenicis v;as a straightforward process. Many of the stars in the vicinity of AE Phoenicis were of similar magnitude to AE Phoenicis, and were listed in the Smithsonian Astrophysical Observatory (SAO) catalog. An investigation of the spectral classifications of these nearby stars revealed a very limited number of suitable possibilities fr-om which a comparison and a check star could be selected. A summary of information concerning the comparison star, the check star, and AE Phoenicis is given in Table 10. The spectral classifications in Table 10 are taken from tlie Henry Draper (HD) catalog. The coordinates of each star were processed to epoch 1970.8 from 1950.0 coordinates given by the SAO catalog. A finding chart taken from the Atlas Australia (Be5var, 1964b), with AE Phoenicis, the comparison star, and the check star labeled, is shown in Figure 20.

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Fig, A finding chart for AE Phoenicis

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106 ^T -4 8 o 49 o o M M U w Q O —O O p, «> « o ^o -50 o "> O o o o o -51 o 9> q, o O' O comp O h ocm OAE Phe O check o o o O 3_h 3Qm j_h 25m o o O o o o 1^ 20^^ RIGHT ASCENSION

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107 TABLE 10 AE PHOENICIS, COMPARISON AND CHECK STARS AE PHOENICIS COMPARISON CHECK R. A. (1970.8) 1^31^-21715 1^27^^ 4-?58 1^^31^25?46 Dec. (1970.3) -49° 46' 54'.'9 -49° 15' 46V1 -49° 58' 52V8 HD Catalog 9528 9067 9544 SAO Catalog 215545 215517 2155.47 Spectral Class GO GO F5 Magnitude (my) 7.9 7.1 6.4 E xtinction Extinction coefficients for each of the three filters and also for the colors (b-y) and (u-b) were determined from observations of the comparison star. The coefficients obtained and the mean values of the coefficients are listed in Table 11. The means were calculated primarily to provide an indication of the atmospheric transparency at Cerro Tololo. The observations on September 30, 19 70, v;ere treated differently than those of the other three nights. The values of the coefficients derived from the least squares fit described in Chapter IV were erroneous due to the rather short range of airmass to which the fit was made. The coefficients from, the other three nights were averaged to obtain the coefficients actually used for the night SeptemlDer 30, 19 70.

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108

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109 Period Study Times of four primary minima and two secondary minima were determined for each color by the author. Nine of the times of minima were determined by the Kertzsprung method and the remaining nine by the tracing paper method. Both of ttiese methods were discussed in Chapter IV. In addition to the times of minimum obtained by the author, thirty-five times of minimum determined by Avery and Sievers (1968) were available. As was done with EE Ziquarii, the photographically determined times of minimum were combined with the photoelectrically determined times of minimum to determine a period and reference epoch by a linear least squares fit to equation (6) . In this calculation, each time of minimum determ.ined photographically was given a weight of unity, while each time of minimum in each color found by the Ilertzsprung method was given a weight of four and by the tracing paper method, a weight of two. The light elements JD Heliocentric 2440857.3151 + 0^36237456 E ±.0006 +.00000019 were obtained from the least squares fit, \7here the probable errors of the elements are given directly below the elements. The results of this period study are contained in Table 12, which sumiaarizes the infcrm.ation relevant to the photoelectrically determined times of minima, and in Table 13, \v'hich summarizes the information pertinent to the photographically determined times of minima.

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110 is

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Ill TABLE 13 PHOTOGRAPHICALLY DETEP^IINED TIMES OF MINIMA USED IN THE PERIOD STUDY FOR AE PHOENICIS JD HELIOCENTRIC = 2440857.8151 + 0.36237456 E JD HEL (OBSERVED) EPOCH O-C 2400000+ 38257.591 -7175.5 -0.005 38263.579 -7159.0 +0.003 38283.496 -7104.0 -0.010 38297.452 -7065.5 -0.006 38314.456 -7018.5 -0.033 38315.414 -7016.0 +0.019 38316.460 -7013.0 -0.022 38317.416 -7010.5 +0.028 38319.373 -7005.0 -0.008 38340.383 -6947.0 -0.016 38353.281 -6911.5 +0.018 38354.347 -6908.5 -0.004 38358.324 -6897.5 -0.013 30380.273 -6837.0 +0.013 38641.540 -6116.0 +0.008 38643.544 -6110.5 +0.019 38697,492 -5961.5 -0.027 38701.340 -5951.0 +0.016 38709.304 -5929.0 +0.008 38711.307 -5923.5 +0.018 38723.269 -5890.5 +0.021 38724.317 -5887.5 -0.018 38726.315 -5882.0 -0.013 38728.319 -5875.5 -0.002 38738.279 -5849.0 -0.007 38740.303 -5843.5 +0.024 39206.285 -4333.5 +0.007 39361.579 -4129.0 +0.009 39383.507 -4068.5 +0.013 39389.458 -4052.0 -0.035 39435.307 -3925.5 -0.007 39444.359 -3900.5 -0.014 39761.445 -3025.5 -0.006 39767.436 -3009.0 +0.006 39771.418 -2998.0 +0.002

PAGE 124

1.12 All the times of mininaim in Table 13 are due to Avery and Sievers (1968) . In Tables 12 13, (0-C) refers to the difference between the observed and calculated times of minima and epoch refers to the num±)er of cycles of the orbital revolution from the reference epoch JD 2440857.8151. A plot of the (0-C) 's for all the observed mi.nima is shown in Figure 21. Because of the small scale of Figure 21, the lO-C) 's for the photoelectrically observed, times of minima are not clearly shown. Figure 22, therefore, shows the (0-C) 's for each photoelectrically observed minim.um. A study of Figure 22 shows that the (0-C) 's from the primary minima are predominantly negative, while those from the secondary mini.m.a are all positive. This indicates that the centers of the primary and secondary do not occur exactly at phase 0.0 and 0.5, respectively. The possible causes and treatment of the displacement of the primary to earlier phase and the secondary to later phase are discussed later. This displacement was not considered in the period study for AE Phoenicis although the (0--C) 's of the photographically observed minima revealed a shift similar to the photoelectrically determ.inod (0-C) 's. From consideration of the photoelectrically determined (O-C)'s for primary, it is suggested that the reference epoch JD 2440857.8140 be used with the previously stated period to compute times of central primary minim.a.

PAGE 126

114 I< D < -I -3 O o Ol 1

PAGE 127

e o u I

PAGE 128

CD CO CD CD CO CD UJOJ < O < -J -i CD CO O
PAGE 129

117 Light Curve s There were 242 usable observations in each of the colors yellow, blue, and ultraviolet for AE Phoenicis. The phase for each of these observations was calculated as described in Chapter IV using the period (P) and the reference epoch (T ) discussed in the previous section. The tines for each observation given in terms of the heliocentric Julian date, the phase of each observation, and the difference in magnitude v/ith the corrparison star (V C) are given in Tables 2 8 30 in the Appendix. The light curves of AE Phoenicis, shown in Figures 23 25, v/ere obtained by plotting the difference in magnitude v/ith the comparison star versus phase for each observation in each color. Inspection of Figures 23 25 reveals that the maximram around phase 0.25 is, in each color, considerably higher than the maximum around phase 0.75. This asymmetry, treated as a complication in the rectification procedures, was partially responsible for the shift of the primary minima to earlier phase and the secondary minima to later phase. As v/ill be shown later, this displacement of the minima could eilso in part be due to a very small eccentricity of orbit. Inspection of Figures 23 25 also reveals a region of constant light output in the middle of each primary eclipse indicating that tlie eclipse is total. The duration of eclipse is about two and one-half hours while the duration of cotalit'.' is about one-half hour.

PAGE 130

c o •H -P > u Q) W Xi O o 0) >1 o u > u u -p t/5 •H O •H a) o w rt3 [in

PAGE 131

119

PAGE 132

0) a o +j > u O" 0) P e o u m (1) > o +> •H H U] •H O •H fl Q) O Xi Ai W en

PAGE 133

121

PAGE 134

w c o •H 4J > (U U) O -P O •H > 4-) o > o +1 en •H m •H o •H a (1) o Xi Pa in (N

PAGE 135

123 'T — T" T — r — r 4 CO d y *««» ^ I d 1 •£3 '% ^^ .•• i» / 1% d 2 Q. O * o r— o £ d

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124 The scatter in the observations appears to be the largest in the region including egress primary and the maximvm around phase 0.25. By inspecting the plots for the individual nights in this region, it was found that there were slight vertical shifts in the light curve for the different nights. The reason for this is not known and this effect v/as ignored for all of the calculations which followed. Aside from the region mentioned above, the scatter in the observations was found to be less than O'^OIS in each color. No indication of light variation in the comparison star is present from an inspection of the light curves. Further proof of the constancy of the comparison star light output was provided by check star observations. The heliocentric Julian dates of the observations and the differences in magnitude between the check star and the comparison star are recorded in Table 31 in the Appendix. Color Curves The color indices (b-y) and (u-b) were corrected for atmospheric extinction by using the first order color extinction coefficients listed in Table 11. Zero point corrections were added as given by equation (47) to obtain the color indices (B-V) ' and (U-B) ', approximating the Johnson-Morgan standard star system. Zero point corrections for each night, as well as the standard star observations from v.'hich they were obtained, are listed in Table 14.

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125 I 3 CN pq < EH O H U U « CO (^ H o u U M H H 2 O H K o a. (X w o <; w fa Nl o C/3 w u H Q H o M < (« > o w o to U M O W <: CO Q (X < Q Eh CO o ic Q W O >^ W I CO ;3 o >1 I * Q W > w CO j3 o I CO iH U3 ^ r~ cr\ v£) o rO CO CTi ro o v^ r^ r^ vD r^ o ro^ ro o O iH + + + + + rH rH + + iH iH r-H .H rH O ++++++ TRUE (U-B)

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126 The color curves, obtained by plotting the color indices (B-V)q and (U-B)^ versus phase, are shown in Figure 26. Inspection of both color curves shows that during both minima, the radiation received was of longer wavelengths. The shift towards longer wavelengths, often referred to as "reddening," was not unexpected and, according to Binnendijk (1970) , is due to a combination of the reflection and gravity effects. Those portions of the ellipsoidal shaped stars which, along the line joining their centers, do not face each other are cooler and redder than any other portions, and it is these portions which present themselves during the minima. The color indices obtained for AE Phoenicis agree closely v/ith the color indices of a GO star of luminosity class V (main sequence) . The spectral classification of GO was listed for the system in the Henry Draper catalog and was adopted for the rectification procedure. Rectification The light curves of AE Phoenicis were rectified in both intensity and phase according to the procedures outlined in Chapter IV. As \\?as done in the case of EE Aquarii, the phase angle of external contact (Sg) v/as determined to be about 45"-^ by removing some of the term proportional to cos2G from the light curve. The Fourier expansion as given by equation (8) was hhen applied to the regions outside of eclipse by the method of least squares.

PAGE 139

> u u o rH o o o •H C 0) o x: H CM

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128 f y \ •'" I ••.r. 4f •I . > •.A . \ • • v.. •.%< I>9 « a o » > • •• ..%• ..'• •v. . ^ • • •I, \ . A* ••I ' . 4. *• e»< '•« to d d CVJ d CO < CL ^» ••• 1 » • ••

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129 The reflection coefficients could not be determined in the same manner as for EE Aquarii because the primary and secondary minima of AE Phoenicis v;ere of nearly equal depth. The coefficients Cg and C2 were determined from equation (14) . Different sets of Fourier coefficients were found by truncating equation (8) to 46, 36, and 2 8 terms. When the light curves were rectified using the equation including the 46 terms, both minima in each color were shifted to earlier phase, and the primary eclipse in each color was found to be considerably shallov/er than when rectified using the coefficients resulting from truncating the Fourier expansion to 39 terms. The Fourier coefficients, obtained by truncating equation (8) with the 36 terms, v;ere adopted for use in the rectification of AE Phoenicis. As will be shown in the next section, theoretical curves with a limb darkening coefficient of 0.4 best represented the observed points in primary eclipse, while theoretical curves with a darkening coefficient of 0.6 best fit the secondary eclipse. Ellipticity coefficients for each color and for the darkening coefficients 0.4 and 0.6 were derived from equation (21) . The values of the Fourier coefficients and their associated probable errors, the reflection coefficients, and the ellipticity coefficients are listed in Table 15. After rectification, both the primary and secondary minima in each color had been shifted to an earlier phase.

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130 TABLE 15 RECTIFICATION COEFFICIENTS FOR AE PHOENICIS Fourier Coefficients Aq a^^ A2 A3 y 0.85186 -0.01024 -0.11527 0.00260 ± 107 ± 362 ± 150 + 193 b 0.84595 -0.00008 -0.11623 0.00598 + 130 ± 441 ± 181 ± 235 u 0.83471 -0.01879 -0.12507 -0.00006* ± 135 + 460 ± 188 ± 244 ^1 ^2 ^3 y 0.01880 -0.00826 0.00125* ±61 ±64 ±89 b 0.02502 -0.00577 0.00408 ±74 ±77 + 108 u 0.02599 -0.00672 0.00172 ±77 ±79 ± 112 Reflections and Ellioticity Coefficients C-,^ C2 z(0.4) z(O.G) 0.01024 0.01037 0.24137 0.20424 0.00008 O.OIOGI 0.24509 0.20738 0.01879 0.01064 0.26291 0.22247

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131 After shifting the primary minima so that the center of eclipse v;as at phase angle 0.0, it was found that secondary minimum occurred 1?2 after mid-period for the yellov; curve, 0?6 for the blue curve, and 1?2 for the ultraviolet curve. For an average discrepancy of 1?0 and an assumed inclination of 90*^, a value of (e coscL))/(l-e ) of 0.007 was obtained from tables presented by Irwin (19 62) , v.iiere ca is the longitude of periastron of the star eclipsed at primary minimum and e is the eccentricity of the orbit. In order to gain an order of magnitude estimate of e , a value of 45 was assumed for oj which resulted in an eccentricity of 0.01 . An eccentricity of this size should not invalidate the v.'orking hypothesis of a circular orbit. In yellow light and ultraviolet light the rectified primary minim.a became slial lower than the secondary minima. The physical cause for this is not definitely known, but would px'obably depend on such considerations as relative sizes of the tv/o components or relative surface brightnesses. The small amount of orbital eccentricity discussed above would cause the distance betv/een the two components to vary which, in turn, would cause a change in the reflectioii coef f :Lcient3 . It is the opinion of the author that the small am.ount of eccentricity present is large enough to account for the observed behavior of the depths of the eclipses .

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132 Solution Each primary eclipse was shifted in phase, as described in the last section, so that the center of the minimum coincided with a phase angle of 0°. Large scale plots of the rectified intensities and phase angles within a range of 55'^ of the adjusted mid-primary eclipse were constructed. As in the case of EE Aquarii, the descending branch of each minimum was reflected onto the ascending branch, and norm.al points were obtained for every five degrees of phase. This procedure was repeated for each secondary eclipse, and smooth curves were fitted to each of the six eclipses. As v/as noted earlier, the prim.ary eclipse shov/ed an interval of constant light. It was, therefore, evident that the primary was a total eclipse and that the light output, found by averaging the individual points in the region of totality, represented the light of the back side of the larger star. The depth of secondary minimum, found from the smooth curve, when combined with the depth of primary minimum, allowed the determination of the depth f uncti on. . After evaluating the shape function, the nomographs were entered and possible solutions v/ere found for each value of limb darkening. Each of these possibilities was investigated, and the darkening coefficient corresponding to the best fitting theoretical curve was adopted in each case. Instead of generating theoretical curv<^s by adjusting

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133 the shape function as was done with EE Aquarii, the nomographic solutions were improved and the theoretical curves were generated by means of the ijj(k,aQ) tables supplied by Merrill (1950) . The light of the larger star (L ) was obtained directly from the light during the total portion of the primary eclipse, and the light of the smaller star followed from L = 1 L . The quantity a°^ was known to be unity s g for the primary eclipse, and the values of A and B were obtained from equation (41) . A theoretical curve was calculated from equation (43) and compared to the observations and normal points in the primary eclipse. Other theoretical curves were calculated in an effort to improve the fit by slightly changing k or by varying the values of A and B. The value of k adopted for each color corresponded to the value which yielded the best fitting theoretical curve in that color. For all three colors, a limb darkening coefficient of 0.4 gave the best fitting theoretical curve for the primary eclipse. After satisfactory solutions for the prim.ary eclipse had been obtained, a value of a^'" corresponding to the value of k for each color v/as found from equation (30) . Theoretical curves v/ere then generated for the secondary eclipses using the elements found from the corresponding primary eclipse. It was found that the theoretical curves with a darkening coefficient of . 6 fit the observed points better than curves wi;:h a darkening coefficient of 0.4.

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134 The value of 0.6 was therefore adopted as the limb darkening of the larger star, in all three colors. The theoretical curves for both primary and secondary rninima are shown in Figures 27 -29. The values of r and i were calculated from equation (45) , and the value of the ratio of the luminosities from equation (31) . The elements were then de-rectified, as was done v.'ith EE Aquarii, using equations (48), (49), and (50). A summary of the elements found for AE Phoenicis is given in Table 16. It is noted that the fit of the theoretical curves to the ultraviolet primary and secondary is not as close as that of the yellow and blue curves. It is also noted that the phase angle of internal contact is somewhat larger in ultraviolet light than in yellow and blue light. This could be explained by the larger star being physically larger . in ultraviolet light than in yellov'i? and blue light. Final conclusions regarding the system as observed in ultraviolet light should not be made until more observations, especially around the internal tangency points of the secondary eclipse, are available. As was done in the case of EE Aquarii, a measure was sought of hov; well the Russell m.odel, as adopted, represented the observations. Residuals consisting of the difference, in units of intensity, between each rectified observation and the theoretical curves were obtained. Average residuals for every five points were calculated

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136

PAGE 150

138 O O T o o o 00 d o o to 5: O If) o o O ® O O O O o o o o CO o o d amamta/w—etoiMmm

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4-)

PAGE 152

140 o o

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141 TABLE ] 6 ELEiMENTS FOR THE SOLUTION OF AE PHOENICIS YELLOW BLUE ULTRAVIOLET k Sg (prim) Gg (sec) Gj_ (prim) 6^ (sec) ^s i Jq/Jc zIO.4) J ag N Po ,oc a tr 1 _ /"oc 1 l"^^ 573 4829 2767 7543 2457 49915 48980 10975 10968 0.6 0.4 84990 1.0081 0.2414 85956 0.4829 2767 4195 2404 1312 4242 •1 V tr 1.0000 1.0725 0.2457 0.2490 1.7342 579 4891 2832 7175 2825 50933 48933 10960 10917 0.6 0.4 84960 0.8513 0.2451 85932 0.4891 2832 4239 2454 1333 3927 1 1.0000 1.0688 0.2825 0.2500 1.7001 570 4823 2749 7446 2554 49903 49908 10995 10988 0.6 0.4 85900 0.9475 0.2629 85971 -1 4823 2749 4128 2353 14 41 4377 1.0000 1.0742 0.2554 0.2845 1.7478

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142 and are tabulated as a function of rectified phase angle for each color in Tables 32 34 in the Appendix. A further check on the consistency of the fit was obtained by calculating root mean square (RMS) residuals for different parts of the curve. The results of these calculations are listed in Table 17. The theoretical curves were then de-rectified by using only terms associated with the Russell model according to equation (51). Differences between the observed intensity observations and the de-rectified curves were obtained. Averages of every five differences were calculated and are listed as a function of unrectified phase angle in Tables 35 37 in the Appendix. Root mean square (RI-!S) residuals were calculated for different portions of the curves, and the results are tabulated in Table 17. It is noted that the P_MS residuals of the difference between the observed intensities and the de-rectified curves were considerably larger than the RMS residuals of the difference between the rectified observations and the theoretical curves. This was not unexpected since the rather significant sine terms were not included in the de-rectification procedure. Conclusions The follov/ing conclusions and opinions follov; from the present photoelectric study of AE Phoenicis.

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143 TABLE 17 STATISTICAL STUDY OF AE PHOENICIS RMS Residuals of the observed rectified points from the theoretical light curves COLOR Yellow Blue Ultraviolet PORTION OF CURVE Primary First Maximum Secondary Second Maximum Primary First Maximum Secondary Second Maximum Primary First Maximum Secondary Second Maximum RMS OF

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144 1) The eclipses are complete, with the primary an occultation and the secondary a transit. 2) The light curve of AE Phoenicis, with nearly equal depths of minima and highly curved maxima, is classified as a W Ursae Majoris type eclipsing binary. The presence of highly curved maxima is indicative of the large tidal forces distorting the stars from a sph.erical shape. 3) The orbit of AE Phoenicis is believed to contain a small amount of orbital eccentricity. A value of 0.01 v.'as deduced for the eccentricity from a one degree displacement of the center of the secondary eclipse. Further evidence for possible eccentricity is provided by the unusual relative depths of the primary and secondary minima, perhaps caused partly by orbital eccentricity through the reflection effect. 4) Complications exist in the light curve of AE Phoenicis. This is evidenced by a ratlier large asymmetry proportional to the sine of the phase angle, by displacement of the primary to an earlier phase and the secondary to a later phase in the unrectified light curve, and by a slight vertical shift in the light from night to niglit in one region of the light curve. 5) The components of AE Phoenicis are intrinsically faint, yellow stars. AE Phoenicis is one of the brighter W Ursae Majoris systems with complete eclipses, and should therefore be considered for future spectroscopic as well as photoelectric observations.

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145 6) Although no indication is given by the present period study that the period of the orbital revolution has changed during the last eight years, no conclusions rr.ay be made at this time because of the insufficient time interval over which times of minimum are available.

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APPENDIX

PAGE 159

147 TABLE 18 JD HEL 2440000+ 0828.6444 .6574 .6704 .6880 .7022 .7138 . 7251 .7585 .7687 .7792 .7896 .8203* .8295 .8410 .8524 0830.7050 .7104 .7155 .7212 .7440 .7504 .7565 .7617 .7694 .7743 .7796 .7342 .7901 .7948 .8017 .8066 .8138 .8189 .8257 .8316 .8378 .8436 .8503 .8559 0834.6629 .6634 .6760 .6807 .6937 .7011 EE AQUARII OBSERVATIONS IN YELLOW PHASE V-C JD HEL PHASE JD HEL 2440000+ V-C .7318 .7573 .7830 .8174 .8454 .8682 .8903 .9560 .9760 .9966 .0171 .0774 .0954 .1181 .1404 .7802 .7908 .8027 .8119 .8569 .8694 .8814 .8916 .9067 .9164 .9268 .9359 .9473 .9566 .9701 .9798 .9940 .0040 .0174 .0290 .0410 .0524 .0656 .0767 .5562 .5669 .5318 .5910 .6165 .6312 -0 -0 -0 -0 -0 -0 -0, -0 --0 -0, -0, -0, -0, -0. -0, -0, -0, -0, -0, -0, -0, -0. -0. -0. -0. -0. -0, -0. -0. -0. -0. ~0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. 0. -0. 672 668 675 684 618 599 559 221 099 069 111 450 493 577 605 671 678 666 657 613 593 579 531 483 443 393 350 269 227 169 097 058 057 061 115 187 267 358 389 517 541 544 568 602 629 0834 0838 0842 7310 7389 7430 7492 7545 7608 7656 7739 7798 7869 3134 8278 6173 6239 6296 6482 6542 6610 6665 6779 6842 5842 5898 6016 6036 6100 6133 6195 6230 6313 6379 6430 6554 6617 6678 6745 6798 7149 7210 7266 7313 7371 7429 7500 7550 .6899 .7054 .7135 .7256 .7360 .7483 .7578 .7741 .7858 .7997 .8518 .8801 .3250 .3381 .3494 ,3858 ,3977 ,4109 ,4217 ,4441 ,4566 .1187 ,1298 ,1529 ,1568 ,1695 ,1759 ,1881 ,1949 ,2113 ,2242 ,2343 ,2586 2709 2830 2961 3065 3755 3875 3986 4078 4190 4305 4445 4543 -0 -0 -0 -0 -0 -0 -0 -0 -0, -0 -0 -0, -0, -0, -0, -0, -0, -0, -0, -0, -0, -0, -0, -0, -0. -0, -0, -0. -0, -0, -0, -0, -0, -0. -0. 0. 0. 0. 0, -0, 0. 0. 0. 0. •0. 604 655 580 650 667 659 675 710 645 661 612 579 644 615 593 565 577 578 522 533 496 571 569 590 635 632 651 633 665 703 642 648 690 670 625 648 637 570 580 593 565 514 500 476 451

PAGE 160

148 TABLE 18 CONTINUED JD HEL

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149 TABLE 18 CONTINUED JD HEL PHASE V-C JD HEL PHASE V-C 2440000+ 2440000+ 0848.5793 .8969 -0.528 0861.6905 .6559 -0.615 .5859* .9100 -0.516 .7123 .6987 --0.633 .5921 .9221 -0.423 .7249 .7236 -0.655 .5978 .9333 -0.364 .7404 .7539 -0.651 .6041 .9457 -0.286 .7528 .7784 -0.666 .6095 .9563 -0.237 .7719 .8159 -0.629 .6152 .9675 -0.149 .7819 .8355 -0.631 .6204 .9777 -0.116 0863.5562 .3215 -0.652 .6266 .9899 -0.056 .5614 .3317 -0.625 .6316 .9997 -0.051 .5736 .3556 -0.640 .6366 .0095 -0.048 .5798 .3677 -0.589 .6418 .0198 -0.099 .5901 .3880 -0.552 .6474 .0308 -0.162 .5953 .3983 -0.569 .6533* .0423 -0.260 .6057 .4186 -0.553 .6586 .0527 -0.264 .6132 .4333 -0.526 .6638 .0629 -0.320 .6296 .4656 -0.503 .6690 .0732 -0.391 .6426 .4912 -0.503 .6741 .0833 -0.449 .6598 .5250 -0.506 .6794 .0936 -0.484 .6808 .5663 -0.539 .6845 .1037 -0.497 .6860 .5764 -0.541 .6902 .1149 -0.559 .6960 .5960 -0.561 .6966 .1275 -0.554 .7025 .6089 -0.581 .7134 .1604 -0.613 .7132 .6298 -0.595 .7186 .1707 -0.610 .7208 .6448 -0.635 .7241 .1814 -0.640 .7374 .6773 -0.647 .7297 .1924 -0.643 .7458 .6939 -0.665 .7380 .2088 -0.657 1151.7537 .5296 -0.477 .7426 .2179 -0.675 .7572 .5366 -0.492 .7477 .2279 -0.661 .7677 .5572 -0.532 .7529 .2380 -0.622 .7710 .5637 -0.535 .7601 .2521 -0.620 .7767 .5748 -0.529 .7654 .2627 -0.621 .7794 .5803 -0.528 .7715 .2745 -0.671 .7852 .5916 -0.566 .7782 ,2877 -0.668 .7882 .5974 -0.570 .7848 .3008 -0.655 .7940 .6089 -0,567 .7916 .3141 -0.633 .7973 .6153 -0.581 .7987 .3279 -0.638 1181.7103 .3841 -0.626 .8050 .3403 -0.602 .7176 .3983 -0.564 0861.5837 .4461 -0.505 .7245 .4119 -0.571 .5919 .4623 -0.470 .7311 .4248 -0.532 .6016 .4813 -0.473 .7382 .4388 -0.530 .6132 .5042 -0.491 .7456 .4534 -0.508 .6275 .5322 -0.476 .7536 .4690 -0.471 .6390 .5549 -0.517 .7603 .4822 -0.524 .6492 .5748 -0.537 .7761 .5132 -0.519 .6671 .6100 -0.558 .7830 .5268 -0.509 .6773 .6300 -0.583 .7898 .5402 -0.504

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150 TABLE 18 CONTINUED JD HEL 2440000+ 1181, 1194 7967 8040 3110 8177 8242 8312 8379 8430 8511 8764 8822 8902 8958 6112 6148 6207 PPIASE .5537 .5680 .5819 .5950 .6077 .6216 .6347 .6448 .6606 .7104 .7216 .7374 .7485 .7297 .7369 .7484 V-C 0.523 0.513 •0.549 •0.543 •0.537 •0.589 0.570 0.602 •0.611 0.659 0.665 0.681 0.719 0.668 •0.680 -0.692 JD HEL

PAGE 163

151 TABLE 19 EE AQUARII OBSERVATIONS IN BLUE JD HEL PHASE V-C JD HEL PHASE V-C 2440000+ 2440000+ 0828.6454 .7337 -0.784 0834.7319 .6917 -0.812 ,6583 .7591 -0.809 .7378 .7033 -0.801 .6715 .7850 -0.809 .7424 .7123 -0.818 .6888 .8191 -0.788 .7498 .7267 -0.756 .7032 .8473 -0.740 .7539 .7349 -0.815 .7146 .8697 -0.715 .7613 .7494 -0.788 .7256 .8914 -0.658 .7649 .7565 -0.811 .7594 .9577 -0.339 .7792 .7845 -0.788 .7698 .9782 -0.213 .7878 .8015 -0.778 .7798 .9978 -0.154 .7923 .8104 -0.773 .7905 .0188 -0.223 .8143 .8536 -0.734 .8211 .0790 -0.584 .8285 .8813 -0.704 .8302 .0969 -0.633 .8326 .8895 -0.699 .8420 .1201 -0.709 0838.6168 .3241 -0.745 .8532 .1421 -0.714 .6247 .3397 -0.699 0830.7057 .7815 -0.799 .6290 .3481 -0.723 .7097 .7894 -0.795 .6489 .3872 -0.728 .7172 .8041 -0.792 .6534 .3961 -0.715 .7205 .8106 -0.784 .6616 .4121 -0.711 .7448 .8583 -0.743 .6656 .4200 -0.670 .7498 .8682 -0.730 .6849 .4578 -0.617 .7573 .8830 -0.692 .7163 .5196 -0.638 .7611 .8904 -0.672 0842.5849 .1201 -0.714 .7702 .9083 -0.621 .5890 .1282 -0.668 ,7739 .9155 -0.582 .6009 .1515 -0.709 .7802 .9279 -0.532 .6026 .1549 -0.703 .7835 .9344 -0.486 .6107 .1708 -0.749 .7908* .9487 -0.357 . .6126 .1746 -0.779 .7942 .9554 -0.327 .6205 .1899 -0.802 .8024 .9716 -0.282 .6385 .2253 -0.818 .8059 ,9785 -0.221 .6424 .2330 -0.817 .8144 .9951 -0.171 .6548 .2574 -0.831 .8179 .0020 -0.179 .6624 .2722 -0.804 .8266 .0191 -0.198 .6667 .2808 -0.791 .8307 .0272 -0.222 .6751 .2974 -0.764 .8385 .0425 -0.368 .6790 .3049 -0.752 .8429 .0510 -0.407 .7154 .3765 -0.683 .8511 .0672 -0.505 .7202 .3859 -0.683 .8553 .0754 -0.547 .7272 .3997 -0.714 0834.6638 .5578 -0.614 .7308 .4067 -0.723 .6678 .5657 -0.612 .7379 .4206 -0.660 .6766 .5830 -0.633 .7422 .4291 -0.674 .6801 .5898 -0.677 .7506 .4456 -0.626 .6930 .6152 -0.727 .,7544 .4531 -0.595 .7005 .6300 -0.764 .7628 .4695 -0.617

PAGE 164

152 TABLE 19 CONTINUED JD HEL

PAGE 165

153 TABLE 19 CONTINUED JD HEL

PAGE 166

154 TABLE 19 CONTINUED JD HEL 2440000+ PHASE V-C JD HEL 2440000+ PHASE V-C .181, 1194 8773 8831 8894 8950 6027 G058 6120 6154 6214 6242 6294 6319 7121 7236 7359 7469 7131 7193 7315 7381 7498 7553 7656 7705 -0 -0 -0 -0 -0 -0 -0, -0, -0, -0 -0 -0, 799 808 811 808 749 752 802 800 821 802 793 797 1194 6378 6404 6458 6486 6544 6571 6638 6671 6725 6751 6801 6827 .7820 .7872 .7979 .8032 .8147 .8199 .8332 .8396 .8502 .8553 .8653 .8703 •0.825 •0.826 •0.826 •0.807 •0.773 •0.785 •0.779 •0.772 •0.757 •0.755 •0.720 0.736 *Oraitted from final solution

PAGE 167

.55 TABLE 2 EE AQUARII OBSERVATIONS IN ULTRAVIOLET JD IIEL 2440000+ 0828.6336 .6464 .6593 .6723 .6900 .7041 .7155 .7264 .7600* .7709 .7807 .7914 .8218 .8313 .8553 0330.7063 .7088 .7178 .7198 .7467 .7490 .7532 .7603 .7709 .7732 .7309 .7828 .7914* .7934 .8031 .8053 .8151 .8172 .8279 .8301 .8394 .8419 .8520 .8545 0834.6645 .6671 .6795 .6890 .6922 .7369 PHASE 7105 7358 7611 7866 8214 8490 8714 8929 9590 9803 ,9996 0206 0803 ,0990 ,1461 ,7827 ,7876 ,8052 ,8092 ,8621 , 8666 .8847 .8889 .9096 .9142 .9294 .9330 .9499 .9539 .9729 .9772 .9965 .0006 .0216 .0260 .0442 .0491 .0690 .0739 .5593 .5643 .5887 .6074 .6136 .7015 V-C JD HEL 2440000+ -0. -0. -0. -0. -0. -0. -0. -0. ~0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0. -0, -0, -0, -0, -0, -0, -0, -0, -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 821 784 801 792 745 715 703 641 372 168 123 204 553 611 725 773 775 777 768 725 713 679 654 598 556 500 474 306 301 237 201 132 141 202 227 350 384 503 532 623 662 ,627 ,638 ,718 ,814 0834 0838 0842 0844 7414 7530 7641 7756 7783 7888 7915 3053 8153 3291 8318 6017 6282 6501 ,6527 6623 ,6648 ,6739 ,6764 ,6806 ,6890 ,6982 ,7003 ,7154 ,6393 .6416 .6538 .6631 .6659 .6758 .7190 .7280 .7302 .7388 .7414 .7514 .7537 .7645 .7672 .7790 .7904 .7927 .8037 .8064 .5934 PHASE .7104 .7331 .7549 .7774 .7828 .8033 .8083 .3353 .8554 .8826 .8879 .2945 .3465 .3895 .3946 .4135 .4185 .4363 .4412 .4494 .4659 .4841 .4892 .5179 .2269 .2314 .2554 .2737 .2792 .2987 .3835 .4013 .4056 .4224 .4276 .4472 .4538 .4729 .4781 .5014 .5237 .5283 .5500 .5552 .0660 V-C 0. 0. •0. 0. 0. •0. 0. -0. -0. -0. 0. -0. -0. -0. -0. -0, -0, -0, -0, -0, -0. -0, -0, -0, -0 -0, -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 749 760 794 835 780 767 754 817 670 657 653 774 730 726 713 674 665 639 576 596 602 637 649 635 800 813 795 806 799 723 690 670 668 599 620 626 574 624 ,605 ,574 ,573 ,575 ,6 40 ,630 ,470

PAGE 168

156 TABLE 2 CONTINUED JD HEL

PAGE 169

157 TABLE 2 CONTINUED JD HEL 2440000+ 0848 0861 0863 6304 6378 6404 6439 6518 6601* 6625* 6706* 6728* 6807 6832 6923 6947 7150 7172 7257 7282 7393 7414 7490 7516 7617 7641 7731 7761 7866 7896 8002 8030 5852 5935 6031 6157 6407 6511 6 6 89 6737 7157 7276 7233 7419 7535 5580 5819 5922 5973 ,6087 PHASE .9973 .0119 .0170 .0337 .0395 .0558 .0605 .0765 .0807 .0962 .1011 .1190 .1236 .1635 .1680 .1847 .1896 .2113 .2155 .2304 .2355 .2554 .2600 .2778 .2837 .3043 .3102 .3310 .3365 .4490 .4654 .4843 .5089 .5582 .5785 .6135 .6329 .7055 .7288 .7303 .7570 .7796 .3249 .3718 .3921 .4022 .4246 •0, •0. 0. •0. •0, 0, 0, 0. •0, 0, 0, 0, •0, 0, 0, 0, •0. •0, 0, 0, 0, 0, 0, -0, 0. 0, •0, -0, •0, •0, 0 0 0 -0 •0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 V--C 152 168 161 257 299 349 413 488 501 589 602 679 667 723 752 776 796 789 798 787 797 744 759 744 755 776 777 765 797 633 620 597 608 593 610 683 710 786 807 761 769 794 791 751 676 676 668 JD HEL 2440000+ 0863.6164 .6323 .6332 .6416 .6622 .6697 .6877 .6981 .7045 .7163 .7228 .7426 .7479 1151.7487 .7553 .7694 .7724 .7779 .7812 .7866 .7895 .7954 .7979 1181.7119 .7191 .7259 .7325 .7397 .7471 .7550 .7617 .7775 .7844 .7910 .7982 .8055 .8126 .3193 .8257 .8327 .8392 .8445 .8593 .8654 .8731 .8838 .8887 PHASE .4397 .4709 .4727 .4893 .5298 .5444 .5798 .6002 .6128 .6359 .6487 .6877 .6980 .5199 .5323 .5606 .5664 .5773 .5836 .5943 .6000 .6115 .6165 .3871 .4013 .4147 .4276 .4413 .4564 .4717 .4849 .5160 .5296 .5426 .5568 .5710 .5850 .5982 .6108 .6245 .6373 .6476 .6768 .6888 .7137 .7249 .7345 V-C 612 666 646 642 ,642 ,664 ,666 ,705 ,740 ,755 ,777 ,808 ,800 ,653 ,655 ,674 ,663 ,641 ,638 .677 ,682 ,669 ,696 .724 .656 .664 .592 .636 .661 .614 .642 .630 .626 .605 .618 .673 .690 .682 .722 .699 .712 .791 .786 .794 .789 .794 .799

PAGE 170

158 /. TABLE 2 JD HEL PHASE V-C 440000+ 1181.8942 .7453 -0.807 .9021 .7609 -0.824 1194.6034 .7144 -0.839 .6065 .7206 -0.851 .6129 .7332 -0.792 .6160 .7392 -0.814 .6221 .7512 -0.830 .6248 .7565 -0.805 .6299 .7666 -0.786 .6326 .7718 -0.798 .6383 .7831 -0.820 i\ X X IN U IJ U JD HEL

PAGE 171

159 TABLE 21 CHECK STAR OBSERVATIONS FOR EE AQUARII Yellow DATE jD HEL 2440000+ ch c 8-29-70 828.7355 8-31-70 830.7279 9-4-70 834.70965 9-8-70 838.6364 9-12-70 842.6865 9-14-70 • 844.6910 9-16--70 846.6924 9-17-70 847.7067 9-18-70 848.7020 Blue +0.4211 +0.3891 +0.4342 +0.4100 +0.4107 +0.3966 +0.4132 +0.4394 +0.3739 DATE jn jiFT 2440000+ ^^ c t'il'ia 828.7367 +0.0084 g 4^;^° 830.7287 +0.0046 r^i°834.7105 +0.0279 9"f? ?n 838.6372 +0.0222 9 l4'7n ^f -^^^2 +0.0150 l]c n. 844.6910 +0.0080 9'}?~7n 846.6932 +0.0117 9-l8"70 o:^3-'°'^ +^^-0061 9 18-70 848.7027 +0.0085

PAGE 172

160 TABLE 21 CONTINUED Ultraviolet DATE JD HEL m ^ m^ 2440000+ 8-29-70 828.7380 +0.1065 8-31-70 830.7300 +0.0939 9-4-70 834.7114 +0.1120 9-8-70 838.6380 +0.0543 9-12-70 842.6880 +0.0585 9-14-70 844.6924 +0.1045 9-16-70 846.6941 +0.1128 9-17-70 847.7094 +0.0952 9-18-70 848.7035 +0.1166

PAGE 173

161 TABLE 2 2 AVERAGE OF EVERY FIVE DIFFERENCES EETVJEEN THE OBSERVED RECTIFIED OBSERVATIONS AND THE CALCULATED THEORETICAL CURVE FOR EE AQUARII IN YELLOW LIGHT 0-C 0-C 4.654

PAGE 174

162 TABLE 2 3 A^TERAGE OF EVERY FIVE DIFFERENCES BEIWEEN THE OBSERVED RECTIFIED OBSERVATIONS AND THE CALCULATED THEORETICAL CURVE FOR EE AQUARII IN BLUE LIGHT 0-C 0-C 4.836

PAGE 175

163 TABLE 2 4 A\^RAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED RECTIFIED OBSERVATIONS AND THE CALCULATED THEORETICAL CURVE FOR EE AQUARII IN ULTRAVIOLET LIGHT 0-C 0~C 4.688

PAGE 176

164 TABLE 2 5 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED INTENSITY OBSERVATIONS AND THE DE-RECTIFIED CURVE FOR EE AQUARII IN YELLOW LIGHT 0-C 0-C 4.065

PAGE 177

165 TABLE 2 6 AVEPAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED INTENSITY OBSERVATIONS AND THE DE-RECTIFIED CURVE FOR EE AQUARII IN BLUE LIGHT O-C O-C 4.088

PAGE 178

166 TABLE 2 7 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED INTENSITY OBSERVATIONS AND THE DE-RECTIFIED CURVE FOR EE AQUARII IN ULTRAVIOLET LIGHT O-C 0-C 3,950

PAGE 179

167 TABLE 2 8 AE PHOENICIS OBSERVATIONS IN YELLOW JD HEL

PAGE 180

168

PAGE 181

169 TABLE 2 8 CONTINUED JD HEL

PAGE 182

170 TABLE 29 AE PHOENICIS OBSERVATIONS IN BLUE JD HEL

PAGE 183

171 TABLE 2 9 CONTINUED JD HEL PHASE V-C 2440000+ 0858.6343 .2605 0.463 .6383 .2716 0.467 .6409 .2790 0.473 .6433 .2855 0.484 .6448 .2897 0.480 .6508 .3060 0.493 .6531 .3124 0.506 .6554 .3189 0.509 .6571 .3236 0.511 .6595 .3301 0.525 0850.7193 .0143 1.156 .7209 .0189 1.153 .7233 .0254 1.151 .7253 .0310 1.124 .7279 .0380 1.078 .7303 .0447 1.031 .7367 .0624 0.909 .7387 .0680 0.882 .7414 .0752 0.832 -7437 .0818 0.791 .7465 .0893 0.755 .7493 .0971 0.712 ,7553* .1138 0.663 .7575* .1198 0.647 .7606 .1282 0.620 .7623 .1329 0.611 .7653 .1413 0.608 .7672 .1465 0.591 .7735 .1639 0.564 .7765 .1721 0.538 .7795 .1804 0.524 .7817 .1865 0.518 .7861 .1988 0.501 .7886 .2056 0.496 .7953 .2240 0.479 .7965 .2274 0.476 .7999 .2368 0.484 .30].5 .2411 0.478 0867.5123 .7601 0.531 .5158 .7698 0.519 .5192 .7792 0.502 .5210 .7842 0.522 .5272 .3014 0.542 .5291 .8064 0.560 .5316 .8133 0.549 .5337 .8191 0.551 .5364 .8265 0.559 JD HEL

PAGE 184

172 TABLE 2 9 CONTINUED JD HEL PHASE V-C 440000+

PAGE 185

173 TABLE 3 AE PHOENICIS OBSERVATIONS IN ULTRAVIOLET JD HEL

PAGE 186

174 TABLE 30 CONTINUED JD HEL

PAGE 187

175 TABLE 3 CONTINUED JD HEL

PAGE 188

176 TABLE 31 CHECK STAR OBSERVATIONS FOR AE PHOENICIS Yellow DATE JD HEL m^|-^ ra„ 2440000+ 9-27-70 857.7156 -0.701 9-28-70 858.5441 -0.697 10-7-70 867.6528 -0.690 lG-7-70 867.7207 -0.693 Blue DATE JD HEL m^^j-^ ra,-. 2440000+ 9-27-70 857.7163 -1.001 9-28-70 858.5447 -1.011 10-7-70 867.6534 -1.002 10-7-70 867.7215 -1.009 Ultraviolet DATE JD HEL m^l^ ~ m^ 2440000+ 9-27-70 857.7140 -1.341 9-28-70 858.5454 -1.353 10-7-70 867.6541 -1.351 10--7-70 867.7218 -1.346

PAGE 189

177 TABLE 32 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED RECTIFIED OBSERVATIONS AND THE CALCULATED THEORETICAL CURVE FOR AE PHOENICIS IN YELLOW LIGHT G (0-C) (0-C) 7.041

PAGE 190

17S TABLE 33 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED RECTIFIED OBSERVATIONS AND THE CALCULATED THEORETICAL CURVE FOR AE PHOENICIS IN BLUE LIGHT (0-C) e (O-C) 6.262

PAGE 191

179 TABLE 34 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED RECTIFIED OBSERVATIONS AND THE CALCULATED THEORETICAL CURVE FOR AE PHOENICIS IN ULTRAVIOLET LIGHT (0-C) e (o-c) 6.375

PAGE 192

180 TABLE 3 5 AVEPAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED INTENSITY OBSERVATIONS AND THE DE-RFCTIFIED CURVE FOR AE PHOENICIS IN YELLOW LIGHT 0-C 0-C 4.658

PAGE 193

181 TABLE 3 6 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED INTENSITY OBSERVATIONS AI^D THE DE-RECTIFIED CURVE FOR AE PHOENICIS IN BLUE LIGHT 0-C 0-C 4.752

PAGE 194

182 TABLE 37 AVERAGE OF EVERY FIVE DIFFERENCES BETWEEN THE OBSERVED INTENSITY OBSERVATIONS AND THE DE-RECTIFIED CURVE FOR AE PHOENICIS IN ULTRAVIOLET LIGHT 0-C 0-C 4.880

PAGE 195

18: LIST OF REFERENCES Avery, R. and Sievers, J. 19C8, Verof f entlichungen de r Rer.e is-Sternv/arte Ban berg , VII (76). Beovar, A. 1964a, A tlas Eclipticalis (CarpiDridge : Sky Publishing Company) . Becvar, A. 196 4b,, Atlas Ausbralis (Cambridge: Sky Publishing Company) . Binnendijk, L. 1960, Properties of Double Stars (Philadelphia: University of Pennsylvania Press) . Binnendijk, L. 1970, "The Orbital Elements of W Ursae I"lajoris Systems," Vistas In Astro n opy , Volume 12, :;dited by A. Beer (New York: Pergar^on Press). Filatov, G. S. 1961, Astronomical Circular , Stat e Univer s ity of Kazan , 223 , 24. Glasby, J. S. 1968, Variable Stars (London: Consrable and Company, Ltd. ) . Gocvlricke , J. 17 83, Philosophical Transactions of the f^Yi^ Society of Lon don , 73 , 4 74. Hardio, P. H. 1962, "Photoelectric Reductions," 7ist rononical Tech niques, edited by VI. A. Hiltner (Chicago: The University of Chicago Press). ITert^^sprung , E . 19 2 8, Bull eti n of th e Astron omical 1 -'''S ti tutes o f th e < e therland s , 4 , 17 8. Irv;in, J. 3. 1962, "Orbit-al Determ.ination of Eclipsing Binaries," Astr onom.ical T echn iques , edited by V7. A. Hiltner (Chicago: The University of Chicago Press). Johnson, H. L. and rlorga-, W. W. 1953, T he As trochysical J ournal , 117 , 313. Kitaniura, M. 1S65, "Determination of the Elements of Eclipsing Variables from the Fourier Transforms of Their Light Curves," Advances in Astronom y an d Astr o phy s ics , Vol. 3, edited by Z. Kopai (Mev/ York: Academic Press) . Kopal, Z. 195 9, Close Binary Stars (Nev; York: John Wiley and Sons , Inc > ) Kuklin, G, P. 1962, A stronomical Circu lar, State Universitv of Kazan, 223, 25.

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184 Locker, K. 1971a, Orion , 126 , 2. Locker, K. 1971b, Orion , 127 , 2. Mahalas, D. 196 8, Galactic Astronomy (San Francisco: W. H. Freeman and Company). Merrill, J. E. 1950, "Tables for Solution of Light Curves of Eclipsing Binaries," Contribution from the Princeton University Observatory , No. 23. Nerriil, J. E. 1953, "Nom.ographs for the Solution of Light Curves of Eclipsing Binaries," Contributions from the Princeton University Observatory , No . 24 . Merrill, J. E. 1970, "Rectification of Light Curves of W Ursae Majoris-Type Systems on the Russell Model," , Vi stas In Astronomy , Vol. 12, edited by A. Beer (Kev/ YorJ:: Pergam.on Press). Motz, L. and Duveen , A. 196 6, Essentials of T^stronom.y (Belmiont, California: Wadsv/orth Publishing Company, Inc. ) . Pannekock, A. 1961, A History of Astronomy (London: George Allen & Unv/in, Ltd.) . Russell, H. N. 1912a, The Astrophysical Jou rnal, 35_, 349. Russell, H. N. 1912b, The Astrophysical Journal , 36 , 54. Russell, H, U. and Merrill, J. E. 1952, "The Determination of the Elements of Eclipsing Binaries," Contr i butions fro m the Princeton Universit y Obs ervatory , Mo. 26. Russell, H. N. and Shapley, H. 1912a, The Astrophysical Journal , 3_6 , 2 3 9. Russell, H. N. and Shapley, H. 1912b, The Astrophysical vTourn al , 36 , 385. Strohmeier, W. and F^nigge , P.. 1960, \'c r o f f en 1 1 i chungen de r Remeis-S ternwarte Bamberg , V ( 5 ) , 2 . Strohmeier, W. , Kuigge, R. and Ott, H. 1962, Vero f fentlichungen der Remeis-Stern;var te Bamberg , V ( I IT , 8 . Strohmeier, W . , Knigge, P. and Ott, H. 1964, Co mjnission 27 of the International Astro nomi cal Union information Bulletin on Variable Stars, No~. 70 .

PAGE 197

185 BIOGPAPHICAL SKETCH Richard Michael Williamon, the son of Mr. and Mrs. Paul S . Williamon, was born on January 26, 1946, in Clemson, South Carolina. He attended local elementary and secondary schools, and graduated from D. W. Daniel High School in June, 1964. In the fall of 1964, he enrolled at Clemson University and received a Bachelor of Science degree in physics in May, 196 8. After teaching at the Tri-County Technical Education Center near Clemson, he enrolled as a graduate student of astronomy at the University of Florida in the fall of 1968. Since October, 1971, he has been employed by Fernbank Science Center in Atlanta, Georgia. Mr. Williamon is a member of the Phi Kappa Phi Society, the Sigma Pi Sigma Society, the Astronomical Society of the Paci.fic, and the .American Association for the Advancement of Science. His v;ife is the former Barbara Kaye Allen, v/hom he married December 17, 1967.

PAGE 198

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. yiuj"? )^c7^i\*<-' vLa/>^4 A'\ Kwan-Yu CheriT Chairman Associate Professor of Astronomy and Physical Science 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. lAin ^ ^^w^^ ^ohn Ellsworth Merriil Visiting Professor of Ast nomy 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 scop? and quality, as a dissertation for the degree o£-^Doc^or/,of Pjsilosophy. Alexander Goudy Smith Professor of Physics and Astronomy Chairman for Astronomy

PAGE 199

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 to.e degree of Doctor of Philosophy. {jmXiIj^U/ km Frank Bradshav/ Wood Professor of Astronomy Director of Optical Observatories 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. Thomas Lee Bailey Professor of Physics and Electrical Engineering This dissertation was submitted to the Department of Astronomy in the College of Arts and Sciences and to the Graduate Council, and v/as accepted as partial fulfillm.ent of the requirements for the degree of Doctor of Philosophy. June, 1972 Dean of Graduate School

PAGE 200

33 I .V H&-^.^^-_l 7. 1.17. \\%