Title: Photoelectric study of EE Aquarii and AE Phoenicis
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Title: Photoelectric study of EE Aquarii and AE Phoenicis
Physical Description: xii, 185 leaves. : illus. ; 28 cm.
Language: English
Creator: Williamon, Richard Michael, 1946-
Publication Date: 1972
Copyright Date: 1972
 Subjects
Subject: Double stars   ( lcsh )
Astronomical photometry   ( lcsh )
Astronomy thesis Ph. D   ( lcsh )
Dissertations, Academic -- Astronomy -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 183-184.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097650
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000577442
oclc - 13980687
notis - ADA5137

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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
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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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