• TABLE OF CONTENTS
HIDE
 Title Page
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Instrumentation and observatio...
 Reductions of the data
 AA CETI
 UZ Puppis
 Appendix
 References
 Biographical sketch














Title: Photoelectric investigations of AA Ceti and UZ Puppis
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Permanent Link: http://ufdc.ufl.edu/UF00097563/00001
 Material Information
Title: Photoelectric investigations of AA Ceti and UZ Puppis
Physical Description: xiv, 173 leaves. : illus. ; 28 cm.
Language: English
Creator: Bloomer, Raymond Howard, 1945-
Publication Date: 1973
Copyright Date: 1973
 Subjects
Subject: Eclipsing binaries   ( lcsh )
Physics and Astronomy thesis Ph. D
Dissertations, Academic -- Physics and Astronomy -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 170-172.
Additional Physical Form: Also available on World Wide Web
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Raymond H. Bloomer.
 Record Information
Bibliographic ID: UF00097563
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 - 000582562
oclc - 14125002
notis - ADB0939

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Table of Contents
    Title Page
        Page i
    Acknowledgement
        Page ii
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
        Page vii
        Page viii
    List of Figures
        Page ix
        Page x
        Page xi
    Abstract
        Page xii
        Page xiii
        Page xiv
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
    Instrumentation and observations
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
    Reductions of the data
        Page 20
        Page 21
        Page 22
        Page 23
        Page 24
    AA CETI
        Page 25
        Page 26
        Page 27
        Page 28
        Page 29
        Page 30
        Page 31
        Page 32
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        Page 91
    UZ Puppis
        Page 92
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    Appendix
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    References
        Page 170
        Page 171
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    Biographical sketch
        Page 173
        Page 174
        Page 175
        Page 176
Full Text


















PHOTOELECTRIC INVESTIGATIONS F AA CETI AND UZ FUPPIS


By




RAYMOND H, B3LOOVER, JR.













A DISSERTATION PRINENT1ED DD T'iE GRADUATE CGUi-;;ITT,
OF TME'T C)VRST F ?10RJ)JA IN PA~RTAT. FULVILIATENT 0" TKE Rl !i.!2PT
FOR TH2i DFGRE, CF CC'fOR OF ThILnoSop.y






IJIIIVERSIcT7C OF FLORIDA
1973












ACKOWFI)3IU*: ENTS


In thet years spent in the preprtation of this dissortation a.ny

. pcrrors unselfishly gave their tlms and talents to support the effort.

Pejrhaps nosi- critical, however, Iero the encouragement, patience, and

op
tion necessary to carry on through many nights of observing, through

hour:s of r-ducin3 data, and during the tine spent in analyzing the

dt. as ~n iritiigl the juiIL-.cript.

First f.ong these peorS'ons was Dr. K. Y. Chen who chaired the

Superv.scory Com.mitte'., nad! InnuLerab]c :.'SC.stions about the work,

Cave up nights of use on the telescope, and encouraged the author at

cvcry stage. Perhaps nore important were the love of astronomy, the

colf-reliar.co, and the discipline which resulted directly from his

l~caderchip and ca6.plo.

Dr. F. B. Wo~V served on the Supervisory Committee and was re-

sponsible for the author's work at the Dr. Reneis Sternwarte, Baz;-

berg, Vc.:i. Gerna!ny, through a National Science Foundation grant.

lHo v:3. the first to instruct this author on the proper use of the

telescope ar-l continued to offer help by giving up time on the tolo-

scope which he could have used for his work and by giving advice

b.scd upon his rcany ycarz of experience rith photometric observations.

It i: a ra h.nor to have had the opportunity to pa-ticipate with him

in sovexrA r,-sea-ch efforts not evs.ociaLed with this dissertation






which resulted in contributions which he and this author published

together.

Other members of the Supervisory Committee were Dr. A. G. Smith,

Dr. F. E. Dunnam, and Dr. H. A. Van Rinsvelt. The author wishos to

think three for reading the manuscript and making suggestions which

resulted in improvements upon it.

Dr. J. E. Merrill offered advice, ras only lie could, on tho

Russell-Merrill model. Of particular value were discussions con-

cerning the application of the model to W UMa systcmzs.

The author would like to thank Mr. W. W. Richardson for doing

many of the final inked drawings. He also did. many drawings which

do not appear here which were used in various presentations and

papers. Mr. H. Schrader also i-orked in these efforts, competently

handling the photographic problems of making slides and pictures.

The cooperation from several persons at the Air Force Weapons

Laboratory, Kirtland Air Force Base, where the entire text ias

written, was essential in the completion of this work. The author

wishes to thank Lt. Col. M. Bacon, Major R. Pratc-r and Dr. A. Cal-

lender for their understanding xnd support during the final stages.

An extraordinary exercise in patience was accomplished by the

author's wife during the five years of gradueto study. She typed

all the drafts and the final copy of the manuscript,

The work is dedicated to the author's father vwho taught him

the moat important lessons of all.


iii















TABLE OF CONTENTS


Page


/.C, i;OW IEDGE TS . . . . . . . . . . .

LIST OF TA LES . . . . . . . . . . . .


LIST OF FIGUES . . . . . . . . . . .


ABSTRACT . . . . . . ... .


CHAPTER


I INTRODUCTION . . . . . . . .

11 INThU::EIT/.TON AND OBSERVATIONS . . .


. .


Dr. Reneis Sternwarte . . . .
Rosemary Hill Observatory .
Differential Photoelectric Yhotometry
General Observational Procedures .


III REDUCTIONS OF THE DATA . . . . .


Preliminary Steps ..
Differential Magnitudes
Light Time Correction .
The Period . . . .
Light and Color Curves .


IV AA CETI . . . . . . . . .


. . 4 q


History . . . . . . . .
Photoalectri c Observations . . . .
Light Elemnt . . . . . .
Color Curves . . . . . .
Light Curves . . . . . .
The Model . . . . . . . .
The Rectification of the Light Curves of
Solutions of the Light Curves of AA Ceti


O I l
. . .





AA Ceti
* . .


Suimary of Results and Suggestions for Fuxther
Work . . . . . .. . . . .


xii


6 .


I. . . .


4 0 . 4


II



IIIIIII








TABLE OF CONTENTS (coIntinuied)


CHAPTER Page

V UZ PUPPIS . . . . . . . . 92

History . ..... .. . . . . 92
Photoelectric Observation.: .. . . . .. 98
Light Elements ........ . 103
Color Curves ....... .. . . . 107
Light Curves ... . . . . . .. 107
Rectification of the LigUt Curv . . . 118
Solutions of the light CurcoS of UZ Puppis . . 120
Summary of RKsults and Sri;cti!''ns for Further
Work . . . . . . 148

APPENDIX . . . . . . . . . . . . . 149

LIST OF REFERE CES . . . . . . . . . . 170

BIOGRAPHICAL SKETCH . . ....... . . . ... 172












LIST OF TABLES


Table Page

). The equatorial coordinates, sp>ctral types, and cata-
logue designations for AA Coti, its companion star,
an- the cornparicon and check: stars used in this in-
ves tigation . . . . . . . . . 27

2 Sone early observatiorns of the double star ADS 1581 . 28

3 First oleor extinction coefficients and their prob-
able errors used in the product ion of data pertaining
to !.A Ceti . . . . . . . . . . . 32

4 The differential r.nnitudes between the comparison
arnd check stars used during the photometric investi-
gation of AA Ceti . . . . . . . 34

5 The differential rlnnitudes between the companion
and comparison stars and the weights used for each
measure:ent to calculate the average and probable
error . . . . .. . . . 35

6 TiMes of ninimun light of AA Ceti and the weights
usei to calculate the light elements . . . . . 38

7 Coefficients used in the rectification of the light
curves of AA Ceti . . . . . 64

8 Elements of the system AA Ceti . . . . .. 66

9 Results of analysis of the yellow light curves of
AA Ceti and RR Centuri . . . . . . . . 81

10 Average differences of the observed data and the
theoretical shapes over the yellow light curve of
AA Ceti . . . . . . 84

11 Average differences of the observed data and the
theoretical shares over the blue light curve of AA
Ceti . . . . . . . . .. 86

12 Average differences of the observed date. and the
t)eor(-:tical sh-.pzs over the ultraviolet light curve
of AA Ceti . . . . . . . . . .. 88






13 Spectroscopic orbital elements of UZ Puppis from
Struve (1945) . . . . . . . . . . 95

14 The coordinates and Bonner Durchnusterung numbers
of stars related to the investigation of UZ Puppis . . 98

15 The differential magnitudes between the comparison
and check stars used during the photometric inves-
tigation of UZ Puppis . . . . . . . . 99

16 First order extinction ccefficients and their prob-
able errors used in the reduction of data pertaining
to UZ Puppis . . . . . . . . . 102

17 Times of minimum light of UZ Puppis and the weights
used to calculate the light elements . . . . . 104

18 Coefficients used in the rectification of the light
curves of UZ Puppis . . . . .. . . . . 119

19 Elements of the system UZ Puppis assuming L3 0 . . . 133

20 Elements of the system UZ Puppis assuming L3= 0 . . .. 134

21 Average differences of the observed data and the
theoretical shapes over the yellow (L 3 0) light
curve of UZ Puppis . . . . . . . . 136

22 Average differences of the observed data and the
theoretical shapes over the yellow (L3 0) light
curve of UZ Puppis . . . . . . . . 138

23 Average differences of the observed data and the
theoretical shapes over the blue (L3 0) light curve
of UZ Puppis . . . . . . . . . . 140

24 Average differences of the observed data and the
theoretical shapes over the blue (L3= 0) light curve
of UZ Puppis . . . . . . . . . . . 142

25 Average differences of the observed data and the
theoretical shapes over the ultraviolet (L/ 0)
light curve of UZ Puppis ........... .... 144

26 Average differences of the observed data and the
theoretical shapes over the ultraviolet (L = 0)
light curve of UZ Puppis . . . 146

27 AA Ceti observations in yellow . . .. . .. . 150

28 AA Ceti observations in blue . . . . . . . 153








AA Cctl observations in ultraviolet . . .

StatiEstcal study of the solutions to the light
variations of AA Cti . . . . . . .

UZ Pi-pis observations in yellow . . . .

UI PuI.:is obscrv'ations in blue . . . ..

UZ Fuppis observations in ultraviolet . .

Statistical stu3y of the solutions to the light
variations of UZ Puppis . . . . . .


156


. 159

. . 160

. 163

. 166


169


v il


* . .


*

.*


*


* .












LIST OF FIGURES


Figure

1 A blink microscope used for the routine search of
sky patrol plates at the Dr. Remeis Sternwarte,
Banberg, West Germany . . . . . . .

2 The dome of the thirty-inch reflecting telescope
at the Rosenary Hill Observatory . . . . .

3 The photometer base and amplifiers attached to the
Cassegrain focus of the thirty-inch telescope . .

4 A chart recorded tracing of photometric neasure-
ments on AA Ceti and its comparison star taken on
the night of 10/29-30/72 . . . . . . .

5 A finding chart for AA Ceti, or BV 1481, and the
comparison and check stars used in this investi-
gation . . . . . . . . . . . .

6 The observed minus the calculated times of minimum
light of AA Ceti used in the calculation of the
light elements . . . . . . . . . .

7 The observed minus the calculated times of minimum
light of AA Ceti derived from the photoelectric
data only . . . . . . . .

8 The b-v curve for AA Ceti after subtraction of the
third light of the companion star . . .

9 The u-b curve for AA Ceti after subtraction of the
third light of the companion star . . . . .


Page


* .









* .



* .





* 9


10 The v light curve of AA Ceti after subtraction of
the third light of the companion star and adjust-
ment of the magnitude scale to 0.0 magnitude at
maximum light . . . . . . . 51

11 The b light curve of AA Ceti after subtraction of
the third light of the companion star and adjust-
ment of the magnitude scale to 0.0 magnitude at
maximum light . . . . . . . . . 53






12 The u light curve of AA Ceti after subtraction of
the third light of the companion star and iadjust-
ment of the magnitude scale to C,0 magnitude at
maxirrum light . . . . . . . . .

13 The theoretical solution of the yellow. light curve
plotted with the rectified data of the primary
eclipse of AA Ceti . . .. . . . .

14 The theoretical solution of the yellow light curve
plotted with the rectified data of the secondary
eclipse of AA Ceti . . . . . . . . .

15 The theoretical solution of the blue light curve
plotted with the rectified data of the primary
eclipse of AA Ceti . . . . . . . . .

16 The theoretical solution of the blue light curve
plotted with the rectified data of the secondary
eclipse of AA Ceti . . . . . . . .

17 The theoretical solution of the ultraviolet light
curve plotted with the rectified data of the pri-
mary eclipse of AA Ceti . . .

18 The theoretical solution of the ultrav~o3et light
curve plotted with the rectified data of the soc-
ondary eclipse of AA Ceti . . . .

19 An early light curve of UZ Puppis by Florja (1937)

20 Radial velocity curves of UZ Pappis by Struve (1945)


. 55


21 A finding chart for UZ Puppis and the comparison
star used in this investigation . . . . . 101

22 The observed minus the calculated tin:es of nininum
light of UZ Puppis which were found in this inves-
tigation . . . . . . . . . .. 1 106

23 The b-v color curve for UZ Pappis . . . . .. .109

24 The u-b color curve for UZ Puppis . . . . ... 111

25 The yellow light curve of UZ Puppis adjusted to 0.0
magnitude at maximum light . . . . . . . 113


26 The blue light curve of UZ Puppis adjusted to 0.0
magnitude at maxim un light . . . . . .

27 The ultraviolet light curve of UZ Puppis adjusted
to 0.0 magnitude at maximuitu light . . . . .


. . 115


. . 117






28 The theoretical solution of the yellow light curve
plotted with the rectified data of the primary
eclipse of UZ Puppis . . . . . . . .


. 122


29 The theoretical solution of the yellow light curve
plotted with the rectified data of the secondary
eclipse of UZ Puppis . . . . . . . . 124

30 The theoretical solution of the blue light curve
plotted with the rectified data of the primary
eclipse of UZ Puppis . . . . . . . . 126

31 The theoretical solution of the blue light curve
plotted with the rectified data of the secondary
eclipse of UZ Puppis . . . .. . . 128

32 The theoretical solution of the ultraviolet light
curve plotted with the rectified data of the pri-
nary eclipse of UZ Puppis. . . . . . ... 130

33 The theoretical solution of the ultraviolet light
curve plotted with the rectified data of the sec-
ondary eclipse of UZ Puppis . . . . . . . 132










Abstract of Dissertation Presented to the
Graduate Council of the University of Florida in Partial
Fulfillnent of the Requirements for the Degree of Doctor of Philosophy


PHOTOELECTRIC INVESTIGATIONS OF AA CETI AND UZ PUPPIES


By

Rayr.ond H. Bloomer, Jr.

June, 1973


Chairman: Dr. Kwan-Yu Chen
Major Dep"Lrtmenti Physics ar.d Astronomy


Tw:o clipsing binary stars were observed photoelectrically using

the thirty-inch reflecting telescope of the University of Florida at

the Rosemary Hill Observatory. The entire light variation of each star

was measured in three colors, and a total of over 1,400 useful obser-

vations wore made. A period study was undertaken for each star, and

solutions of the light curves were found based upon the Russell-Merrill

model.

The star, AA Ceti, was found to be a variable by the author in a

routine search of sky patrol plates at the Dr. Remois Stornwarte, Bam-

berg, West Germany. It is the brighter component of the visual binary

Aitken 1581. Photometric measurements during the winters of 1971 amd

1972, combined with times of faint light from the Bamberg plates,

showed the orbital period. of the system to be 0d53617353 0.00000050

p.e. IThe light curves which were constructed displayed W UMa-type var-

iations. The total secondary eclipse was flat at yellow and blue wave-


xii





lerlths, but the system became significantly fainter in ultraviolet

light as the total phase progressed. The light curves were rectified

to spherical star systems so that the Russoll-":orrill uodel could be

applied. The resulting solutions indicated that the secondary eclipse

was an occultation and that the stars were unusually different in size

for this type of binary system. Although the solutions were not de-

finitive, partly due to the small depth of the rectified eclipses, a

generlJ. description of the system has been determined and is suimmarized

by the following elements of the prolate ellipsoid model averaged over

tho three wavelength bands: a 0.53, a =. 0.18, c = 0.16, j 76??,
g s
L = 0.920, and Ls 0.080. The maxima after primary eclipse v&. dis-

placed from a phase of o025 which could possibly indicate that the

orbits are elliptical.

Although UZ Puppis had been studied spectroscopically and photo-

metrically by photographic methods, no reports of photoelectric light

curves were found in the Florida Card Catalogue of Eclipsing Binary

Stars. The star was observed photoelectrically on thi-teen nights.

A period study did not reveal a significant change in the period from;i

previously published values. The period derived from this photoelec-

tric data and. son old times of minir.um light gavs a refined period

of 0.79'85183 0.00000004. The light curves were sinilar in shape

to those of Lyrae and indicated that the eclipses were partial. The

center of the secondary eclipse was displaced from a phase of 0.5, and

the star showed a significant "periastron effect."- Solutions result-

ing front this investigation indicated that the secondary eclipse was

an occult.atIon of the smaller st and that there was a large amount

of third light present in the system. Solutions both ilth and without


xiii






the: r, .: ;.::. 1A, of t'irz-1 I ght owere found; the solutions allowing third

liGht irel:.~ in the following e3eonnts brscd upon the prolate ellip-

soid Icdel -- a -. v_..-iG over the three :avelcrgth bands a = 0.37,
g
as -: 0.32, c 0.14 85 7, L = 0.505, La 0.222, and L 3 0.273.

The I.nc3~i. .ti'i oLr the orbit froa this solution, combined with 0.

Strv,'o ;t.~ 3 v5..cicl-;y data, gave a nmss for each star of about 1.10

an-l c.! orbit.'L .o.l-.aj. euxis of approximately 1.6 x 106 }m.


xiv










CHAPTER I


INTRODUCTION


In 1783 Goodricke published the first correct interpretation for

the light variations of the eclipsing binary star, Algol. He explained

that the periodic decrease in brightness of 1.2 magnitudes followed by

a return to normal light within about ten hours could be caused by the

eclipse of one star by mother body. His insight (at the young age cf

nineteen) began a new field of study which has yielded a tremendous a-

rount of information about the stars. However, 190 years later one

finds that a great nany old questions remain unanswerdc, e.g., the

strange light variations of the bright system p Lyrae. In addition,

neiR problems have arisen largely due to the rapid increase in the nun-

ber of binary systems discovered and the inrproveent in the state--of-

th3-art of photometric and spectroscopic instrumentation.

A binary star system is defined here as two stars which are close

enough in space to appreciably affect one another's motion by their

gravitational fields. If the inclination of the plane of the orbit

with respect to the plane of the sky is close enough to ninety degrees,

each star will alternately eclipse the other with the subsequent loss

of light. Just how small the inclination can be and still cause e-

clipses depends upon the sizes of the two ctars relative to their

separation.

There aro several definitions of a "close" binary star system in





2



use today. In the broadest sense, a binary star system is considered

close if the two stars are close enough to effect one another's evo-

lution at some point in their lifetiners. This definition, proposed

by Plavoc (1970), allows stars sepa.-at.-d by nany zitronomical units

to be considered closo. In photonictric investigations of eclipsing

binary stars the tern "closo" usually dcspr1-cs a system which shows

large light variations due to the changiiG aspect of stars which ara

tidally distorted by one another. In this investigation the latter

moro-restricted definition will be assumed unless otherwise stated.

In recent years the number of known eclipsing binary stars has

increased vory rapidly due, in part, to photographic sky patrols such

rs the one at the Dr. Reoeis Sternwarte, Banlb'erg, Uest Germeany, which

is doscribc-d in the noet chapter. The dramatic increase in the es-

tizates of the percentcgo of all stars which are actually multiple

ctor systems has stinulatcd interest in the field. Other relatively

recent developments have led to a greater emphasis on the study of

close binary systems ani, in particular, eclipsing systems: (1) the

development of the photomultiplier tube, which has made possible

photometrlc me-suremonts routinely more precise than 0.01 magnitude

in typical csses, thus na:inj photonotrically determined physical

paranoters for the system of higher precision arid revealing hereto-

fore hidden photometric complications; (2) photometric and spectro-

scopic evidences of gaseous strea~is, rings, disks, and other peculiar-

ities in the system; (3) the suggestions that several types of

cataclysmic variable stars may, in fact, all be close binary cySter,,

e-cording to Plavec's definition, such as the novae (Starfield, 1971)

and th3 U GeCninorun star (Srk:, 1969); ar.- (4) the recent discovery








that some binary stars may be radio (e.g., Algol), and x-ray sources.

With this increased activity in the more esoteric phenomena re-

lated to binary stars and the astrophysical insight which can be ob-

tained from them it is of major importance that eclipsing binary stars

provide the only direct method available for determining the dimen-

sions of the stars (with the exception of the sun and inferences from

interferometry of a small number of supergiant stars). Few research

efforts in astronomy can be considered more important or fundamental

than finding the dimensions of the stars in various categories and

stages of evolution.

The dimensions extracted from the light variations are, however,

only as meaningful as the model assumed in the analysis. Some models

for eclipsing binary systems contain simplifying assumptions which

demand that they be applied, in general, to well-separated binaries

which have no strange peculiarities in their light variations. The

model of Russell and Merrill (1952) used in this work applies to this

situation. Since the majority of eclipsing binary stars do not fit

in this category, schemes have been developed to "rectify" or change

the system's light variations to ones which can be analyzed with a

simple model. In the Russell-Merrill model, however, a third light

(light which is emitted from a source other than one of the two stars

mutually eclipsing one another) can be assumed. This assumption adds

another degree of freedom to the model which could make up for improper

rectification procedures with the result being the extraction of in-

correct physical parameters or "elements" for the system. Many times

this third light is a large fraction of the total light of the system

and is not "visible" spectroscopically. Though today's procedures are








undoubtedly far- from being perfect, they iaust be applied and scruti-

nized until better methods can be developed to solve close binary star

light curves.

Now nodols have been developed based, for example, on the theore-

tical structure due to Lucy (1968); however, these models are imper-

fect also. Some problems in the analysis of close binary systems may

not yield to any models, such as intrinsic random variability of the

components and rapidly changing patterns of spots (bright or dark) on

their surfaces. At present these problems are generally ignored and

average to give some mean elements based upon several observing ses-

sions. For lack of a better understanding of close binary stars, this

seems to be the only recourse.

It was the intention of this work to investigate the light varia-

tions (including period studies) of two eclipsing binaries, one of

which vas discovered to be a variable by the author. The analysis of

the light curves was done by conventional methods using the Russell-

:c-rrill rodel. Certainly the observations of this epoch will con-

tinue to be of value even when eclipsing binaries are completely

understood.












CHAPTER II


INSTRUMENTATION AND OBSERVATIONS


Dr. Remeis Sternwarte


The eclipsing binary star having Bamberg variable number 1481,

now named AA Ceti (Kukarkin et al., 1972), was discovered to be vari-

able by "blinking" sky patrol plates at the Dr. Remeis Sternwarte,

Bamberg, West Germany. In this process two photographic plates of the

same area of the sky taken at different times were viewed alternately

in rapid succession through a microscope. If the plates were of simi-

lar quality and exposure, those objects which had not changed in

brightness between the times that the two plates were exposed displayed

no change in their image. Those objects which had varied in bright-

ness appeared to "blink" as a result of the change of plate density

and image diameter. The apparatus employed in the procedure described

above is shown in Figure 1,

The plates were taken with four-inch f/6 Tessar lenses mounted

either in six-camera units (at the Boyden Observatory, South Africa)

or in four-camera units (at Mount John Observatory, New Zealand).

Each camera covered a field of 13 x 130, and the optical axes of the

lenses were pointed thirteen degrees apart in declination. The typi-

cal one-hour exposure (one-half hour on each side of the meridian)

reached a photographic magnitude of about 14.0.

After a search was made of the star catalogues and it was





















I)
C)




H

H
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0

































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C))
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C,


C
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(



II


i l





7



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"\ I
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r. ^ / .* \ \ ? -
-; / *\ /*
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i.- .......~. .......~.~..~~.~. ...~ .~~..;.....*. .-. ,. )....~~._ i_ *








determined that the object had not been identified previously as a

variable, the author then assigned a Bamberg variable number to the

object, and a survey was then made of all available plates which

contained the new variable. Such a search, along with spectral types

available from, star catalogues, gave clues as to the nature of the

variation. For example, an object normally at maximum brightness

fading only occasionally was probably an Algol-type or p Lyrae-type

eclips-ing system. Those objects which spent as much time "bright"

as "faint" were probably either W UMa-type eclipsing stars or RR

Lyroe-type stars. Spectral types available for the brighter stars

usuallyy n 90m0) lent some key as to the nature of the variability.

From. the times of faint light available a period for the varia-

tion wa.s derived. This was done cssential.ly by trial and error

until a set of light elements was found i;hich satisfied all avail-

able data. Unfortunately, the poor timo precision of photographic

methods often led to many possible periods, particularly for short

period systems.


Rosemar y Hill Observatory


The photoelectric observations were made with the thirty-inch

Tinsley reflecting telescope at the Rosezary Hill Observatory of the

University of Florida. Located approximately twenty-five Iiles

southwest of Gainesvillo, Florida, the twenty-six-foot dome was con-

structetd on a natural oand ridge at an elevation above sea level of

about 130 feet, which is approximately fifty feet e.bove the surrourd-

ing terrain. The olevaticn offered some protection from early morn-

ing ground fog so common in this subtropical climate. This pollution-









free rural site provided a very dark sky often of photonetric quality.

The seeing disks were typically two seconds of arc in diameter. Many

of the characteristics of this location have recently been summarized

by Dr. A. G. Snith (1972). The dose is shown in Figure 2.

Tho photometer base, made by the Astro IMechanics Company, and am-

plifiers were attached to the telescope at the Cassegrain focus in the

configuration shown in Figure 3. After passing through a wide-field

viewing mechanism, the f/16 light beam was brought to a focus at the

diaphragm entrance to the photometer by moving the secondary mirror

parallel to the optical axis of the telescope. The diaphragm, used to

isolate the star being observed, was chosen to be either 25.0 or 32.5

second of arc in diameter depending upon how much sky brightness had

to be eliminated to achieve a reasonably high sky-plus-star to sky-

brightness ratio. Irumediately after the diaphragm in the optical path

was a small mirror which could be placed in the becm so that the dia-

phragm and star inage could be viewed through an, eyepiece to assure

proper focus and centering of the image. After being reflected by two

plane mirrors, the bean reached a six-position filter wheel in which

were placed three wide-band transmission filters (a Corning 9863, a

Cornizn 5030 and Schott GG 13, and a Corning 3384) which were the same

filters used by Johnson and h:1organ (1953) in the establishment of the

standard UBV photonetric system. The approximate effective wavelengths

of the Johnson-iMorgan system are 54803 44-'00, and 3650A; the filters

are characteristically 700 900A wide. Of course, ths response of the

system ras slightly different fron the stand.id system primarily due to

the use of a different photomultiplier tube.































Fig. 2. The dore of the thirty-inch reflecting telescope

at theo Rosoary Hill Observatory






























































































r ~ .w a


*1






f ,,
4


-S

"t


', I
ii


j


__ _
---lf"---. B-
--.~~


L'r!





















O



O
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0


0









19



















0
,o
















E4



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GI
aL




'C







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r1
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* C
11
.e
fl
t> +
E: .

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13


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II








i



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The filtered light hc2'i thc. js~ti through a Fabry lerT which

imaxed the light on the photoc thode of the photormultiplier tube, and

then through a flat glazs viijc..' at the entrance to the "cold box"

which housed the photomul'i.pjlier tul.. A! n FP:I 6256 B photomultiplier

tube- was used during the 1971 oL'-:!:rvtlic:r of UZ Puppis, and a similar

EMI 6256 S Has used for al'. the oth.r observations. These are two-

inch-dianeter thirteen-st:.-e tuis with S-13 photocathodes. Operated

at 1,000 volts (across the ca.th.' .: dynod chain, and anode), the dark

current vw;c alwa-ys rr.all co. pare' t tithl star or sky signal, so the

tubes wet.r oi ratd: at aticut c't tcre-raLture rather than coolirC with

dry ico. The signal fror. tLi phto-'ultiplier tube was then raplified

by a d/c aLuplifior. All but a foiu obsorve.tions were made with the

PA-1 amplifier designed by J. P. Oliver ajnd constructed at the Uni-

vcrrity of Florida for the pur-pos0c of stellar photonotry. The output

was then displayed on one channel of the Honeywell ElectroniK 16 dual

chann.-l strip chart recorder. The paper transport vas set at one

inch per minute and was checkIed several ti m3 during each observing

osssion with ITWV, the national tire ctandard, to insure that all tines

wero accurate to bettor than five seconds. A typical example of a

traclig is sho.nl in Figure 4. At least once each night the coarse

Gain s-ops of the PA-1 a:..plifier were calibr.ted using & built-in cur-

rent source. Fine gain st-',s i.-r.- u. < to be of constant calibra-

tion since very stable resistors (5 ppm/o drift) :wre used in the

construction. A calibration :.ade by D. H. ilartins under typical tem-

perature and humidity conditions was adopted. Drift of the resulting

calibration of any Gain setting wias founi to be never more than a few

















0





0
-r4






0
9)
e2









43















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P,




CO-4


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-ri
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to







sri
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+3


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Y
"_ o_ o_ a o o-oin a,-






Oo
0 o 09 o 0_ o o1 oc o0 01 o





















Q oo
E_





001 00 09 0^ 0i 09l 0 CC 07 01 0 0











E) -- --
--A


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cc 0: ____ Oj 0__


00o1 CM 08 o o 0 o


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001 4. -w- o0 0


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I








thousandths of a magnitude, comparable to the reading error of the

chart recording, so a mean calibration was used for all the obser-

vations.


Differential Photoelectric Photometry


The differential photoelectric photometry of variable stellar

sources is a well-developed procedure capable of reaching precisions

of the order of 0.01 magnitude or better with a minimum of system-

atic errors (Wood, 1963).

Due to small variations in the transparency of the atmosphere

over a time scale of minutes, a differential measurement between a

constant or "comparison" source (a near-by star) and the variable

star was used in this work. In this procedure the light of the com-

parison star was measured before ard after the variable star, and

its brightness was linearly interpolated to the variable star read-

ing so that a direct comparison could be made. This was done for

each color, independently. It was common practice also to make oc-

casional measurements of another star to "check" the constancy of

the comparison source. Both the comparison and check stars were

chosen for their similarity in brightness to the variable in order

to reduce the demands of changing amplifier gains too often as well

as for their similarity in color to simplify the reductions (see

Chapter III).

Of course, significant transparency variations, perhaps due to

a thin cloud, over a period of time shorter than the time between

comparison star measurements (typically, three to ten minutes) re-

sult in an increase in the scatter of the data. In addition to this








source of scatter, many components of close binary stars have been

known to be intrinsically variable over time scales of hours or less.

These facts contribute to the difficulty of a theoretical error anal-

ysis of photoelectric data. Further discussion of errors, except for

chart recorder reading errors discussed in the next section, will be

postponed until a discussion of the light curves.


General Observational Procedures


When the image of a star was centered and focused in the dia-

phragm of the photometer, readings were t&aen in yellow, blue, and

then ultraviolet light. Before each reading the centering of the

star was checked. This was necessary since the two stars of this in-

vestigation were always at a large zenith angle so that guiding cor-

rections had to be made due to variable refraction of the image

passing through the atmosphere. A thirty- to forty-five-socond trac-

ing was nornmally enough to establish the level of the noisy signal

to an accuracy of about 0.2 units (OT002 at full scale deflection)

when one fitted a line through the reading by eye. Then, in reverse

order, the came amplifier settings were used to measure the bright-

ness of the sky (plus dark current and zero-point of the recorder)

near the star. This procedure was followed for every star alternat-

ing between comparison and variable stars with an occasional set of

deflections for the check stars and "standard" stars. On some occa-

sions when the atmospheric transparency was reasonably constant ajA

the comparison star readings were not changing significantly over a

period of about five minutes, several sets of deflections ivero taken

on the variable star between each comparison star set. In all cases








an attempt was made to keep the deflections near full scale on the

chart recorder since the inevitable error in reading the chart (ap-

proximately 0.2 units) was a smaller fraction of the measurement

at full scale than with a smaller signal. Although it has been

choun that the time constant of the noisy signal has little effect

upon the reading errors (Young, 1968), a time constant of 0.5 sec-

onds was generally used as a comfortable compromise between large

"grass" in the signal and too much smoothing, which could mask the

effects of an unphotometric sky.

At the beginning of an observing session the Universal Time

from radio station WWV was narked on the chart. The transport rate

of the paper (one inch per minute) was chocked during the session

several times. Then each deflection was assigned a time by esti-

mating the center of the measurement. All times are accurate to

better than five seconds.













CHAPTER III


REDUCTIONS OF THE DATA


Preliminary Steps


The tracings on the strip chart recording were measured using a

strip of clear plastic scribed with a thin dark line. The line was

visually fitted to the signal, and the deflection was estimated to

the nearest 0.1 unit on a full scale of one hundred units. In most

cases this procedure was reproducible to within 0.2 units so that

the reading error for a deflection near full scale corresponded to

less than 0.01 magnitude in most cases. This same measuring tech-

nique was applied to the sky readings, and they were linearly inter-

polated to the star readings. Since the same gain was used for the

sky and star deflections, a direct subtraction could be made. This

difference and the Universal Time at the mid-point of each reading

were recorded and punched on computer cards for further reductions

which were done primarily on the IBM 360/65 computer. The computer

codes were initially developed at the University of Pennsylvania.

Some of the calculations were done with the Hewlett-Packard 9820A

calculator at the Air Force Weapons Laboratory, Kirtland Air Force

Base, using programs written by the author.








Differential Magnitudes


The first step in the computer reductions was to calculate the

first order extinction coefficients for each color on each night.

The extinction coefficients were derived from the comparison star

data. Those measurements which seemed obviously not to represent

the general linear extinction were removed, and a linear least-

squares fit to the remaining data (as a function of airmass) yielded

the extinction coefficients. Since the extinction was found to vary

by as much as a factor of two from one night to the next, it was

felt that mean coefficients would not be adequate. This policy re-

stricted useful nights to those in which enough comparison star data

was taken to achieve some precision in the determination of the ex-

tinction coefficients.

The difference in magnitude between the comparison and variable

star (or check star) was calculated allowing for differential extinc-

tion in light. If one defines the observed magnitude as m and the

magnitude of the star outside the earth's atmosphere as o then, to

a good approximation,

a nm + kX

where k is a positive number in units of magnitudes/airmass and X is

the airmass. The observed magnitude is, of course, dependent upon

the gain of the amplifier (G), which is calibrated in magnitudes,

and the deflection (S), such that

m G 2.5 log(S/100)

For these calculations the expression for the airmass from Hardie

(1962) was used:







X = sec Z 0.0018167(sec Z 1) 0.002875(sec Z 1)2

0.0008083(sec Z )3

where Z is the zenith angle of the object. The zenith angle is re-

lated to the latitude ( ) of the observer and the declination (6) and

hour angle (h) by

sec Z = [(sin 0 sin 6) + (cos 0 cos 6 cos h)]-1

The differential magnitude between the comparison star and variable

star, corrected for first order extinction, is then

m m = n a + k(X X )
var comp varo comp kvar comp

Of course, k would be different in wide-band filter photometry for

two stars of widely different color. In this work the comparison

stars and variable stars were chosen to be similar in color so that

second order extinction coefficients would be small and could be

neglected. In this way the differential magnitudes between the com-

parison star and the variable and check stars were computed.


Light Time Correction


As the earth moves in its orbit, the distance from the object

being observed changes. Since the velocity of light is finite, the

position of the earth in its orbit would then affect the timing of

an event such as an eclipse. To avoid this confusion, all times

were reduced to the arrival time of the signal at the sun, i.e., to

heliocentric time.

Following Binnendijl (1960), the correction added to the re-

corded geocentric time was At, where

At -0.005770 XE(cos 5 cos ') + Ye(tan e sin 6 + cos 6 sin ')








where X6 and YO are the Cartesian coordinates of the sun, a and 6 the

right ascension and declination of the variable star processedd to the

same equinox as X and Y ), and e is the obliquity of the ecliptic.


The Period


Times of minimum light were found in two ways. On nights when

data was available on both branches of an eclipse, the Hertzsprung

(1928) method of finding the geometrical center of the eclipse was

used. On nights when data was available on only one branch, the

tracing paper method was used to visually fit the partial data to an

eclipse whose mid-point was "known." Since this method is less pre-

cise than the Hertzsprung method, it was given lower weight in the

calculations.

If a preliminary period was available from previous work, as it

was for UZ Puppis, the relative cycle (or epoch) of each time of min-

imum could be calculated. For AA Ceti, however, the period had to be

estimated such that phases predicted by the elements agreed with all

the observations. A least-squares calculation was performed to find

the period (P) and the initial epoch (To) (and their probable errors)

so that times of future minima (T) could be predicted by the equation

T = T + P E

where E is obviously the number of cycles since the initial epoch.


Light and Color Curves


Once the light elements were found, the phase of each point was

calculated and all the data assembled into light curves. No "night

corrections" or adjustments were made between different sets of data.




24



If the comparison stars and variable stars had exactly the same

color, then the light curves could be subtracted from one another

directly to form the b-v and u-b vs. phase plots, i.e., the color

curves. Since this Is rarely the case, a correction should be made

for each night's data based upon standard stars. However, these

color-dependent constants were considered small, and a direct sub-

traction was made.













CHAPTER IV


AA CETI


History


The variable star BV 1481 was discovered to be a variable star

at the Dr. Remeis Sternwarte in a routine search of sky patrol plates

(Bloomer, 1971a). The object showed an amplitude of up to 0.5 magni-

tude, and on many plates (about 25 percent of them) the brightness

was below maximum light by a significant amount indicating that the

star was probably either a RR Lyrae-type star or a short period

eclipsing binary star. The late spectral type (F2) tended to support

the latter choice.

A set of light elements were calculated using times of faint

light from the plates (Bloomer, 1971b):

Min I = JD 2440566 + 0o733282 E

Subsequent photometry showed these elements to be incorrect.

K. Locher (1972) published a light curve from visual estimates

using these elements and pointed out that the eclipsing component is,

in fact, a member of a visual binary system (SAO 167450/167451). The

components do not differ greatly in brightness (Amag a 0.5) and are

separated by less than nine seconds of arc. He further identified

the variable as the southern-most component of the visual pair.

Recently BV 1481 was named AA Ceti by Kukarkin et al. (1972).









The catalogue n-r.s of the individual stars and the visual pair

as well as the coordinates of the stars are given in Table 1.

The visual binary systeiu was first studied, according to the

literature, in 1822 by J. Y. W. Herschel and James South (Burnham,

1880). Since then numerous publications concerning this pair have

revealed no significant change in the separation (slightly less than

nine seconds of arc according to most references) or position angle,

which remains near 306 degrees. Sone of those references and their

results are given in Table 2.* In this table d is the separation of

the stars, and e is the position aigle of the fainter star with re-

spect to the brighter o:ie and is measured counterclockwise from the

northern direction. The proper notions in both right ascension and

declination are so similar that the two stars are undoubtedly a

physical pair with a very long period. The proper motion of the

variable and companion stars according to the Smithsonian Astro-

physical Observatory Star Catalog are

va E r +00073/yr. :'010
var, C'
comp = +0oo057/yr. '.'0l
COmp, ci
and

ar = +0"'041 .'007
var, 6

comp, 6 +0:'0.58 1 '007

Since no photometric studies other than those discussed above

were found in the brief history of this variable star, the present

investigation was begun to find the photometric elements for the

system.

*It is interesting, indeed, that the variability of the brighter
component was not mentioned in the major double star catalogues.
















N
rx4f


p \ O%-
03
S 01 >


0 *


H
0
E-4










H-I


oa

0 C









.M .


C



S-4

Ul





'0

o8













0

o


C-
C1-










CO





'0












O
o



0
0*











0
CO


CoN
0
co






H
H 0


C->





CO
ri



0
0

I
0*









.0
0




cO
c.
C1-
Cp














0


*



N +'
N -0
(i. C)
S-
Cl-


0\
NO


I








\0
f-l


0.0
CO



U
a





I I
Co C)
M U 0


4)




o



0
'























5 I4
0 .





g o










a


I I



CO



co
ip
qp


0

C-








H
\0











0



0*



10
o
















0
0



Eo
O



















SOME EARLY


d

9"08



8'.'50
68 30


8"50


8'712


8:49


8'.'44


8747


TABLE 2

OBSERVATIONS OF THE DOUBLE STAR


8 Magnitude

306?5 8' 0-90O


305?0


3044


306?2 7To-7 25


306.4 7T0-7?25


305.0 7 T-7 4


306?9 7 2-7T4


ADS 1581


Reference

Burnham (1880)


Burnham (1880)


Burnham (1880)


Scott (1899)


Scott (1899)


Scott (1904)


Franks (1918)


Epoch

1822.69


1877. 86


1895.86


1897.893


1897.904


1902.895


1916.84








Photoelectric Observations


Useful photometric observations of AA Ceti were made on thirteen

nights during 1971 and 1792. The comparison star used was BD-210352.

As a check on its constancy, occasional measurements were made of the

check star BD-210339 during the 1972 observing season. The catalogue

names and the position of these stars are given in Table 1. Figure 5

is a finding chart for the stars.

As mentioned previously, the variable star is the brighter com-

ponent of the visual binary Aitken 1581. Since the companion star

of AA Ceti was so close (about nine seconds of arc), it was imprac-

tical to attempt to separate the stars in the diaphragm of the pho-

tometer, at least on a routine basis, so the great majority of the

measurements were made with both stars in the diaphragm. On nights

of good seeing one or more measurements were made with just the com-

panion star in the diaphragm. This was done to have an empirical

value of the third light of the system in each color and to act as a

check on possible variability of the companion star. In addition to

these special measurements, occasional measurements of the check

star were made to check the constancy of the comparison star.

The first step of the reductions, as discussed in the previous

chapters, was to find the first order extinction coefficients for

each color on each night. Observations which were apparently not

the result of smooth extinction effects were removed before the least-

squares fit was performed. The results of these calculations (and

the probable errors) are given in Table 3 in units of magnitudes per

unit airmass. On the night of 12-31-71/1-1-72 an estimate of the































Fig. 5. A finding chart for AA Ceti, or BV 1481, ard the

comparison and check stars used in this investigation
















i \ir-si -rl i~ I a1 -
0


Check Star,
\B


v Cell-


C)0
'Comporison Star


1481


0*


I I I I g I I I I I


I I I I I I I t I I I I I


2h00m h"50rn
RIGHT ASCENSION (1950.0)


-20r


*


-22C

0

C;)
o>
C
-23
z
0


-24
I-

0
-2
-25


0*


S*


Scl-(








TABLE 3


FIRST ORDER EXTI:CTTON; COEFFICIENTS AND THEIR PROBABLE ERRORS
USED IN THE REDUCTION OF DATA PERTAINING TO AA CETI


Date


11/ 5- 6/71


11/ 6- 7/71


11/ 8- 9/71


11/12-13/71


11/25-26/71


12/29-30/71


12/31/71-
1/1/72

9/ 2- 3/72


9/ 4- 5/72


10/28-29/72


10/29-30/72


1/ 8- 9/72
East Side


West Side


*12/ 8- 9/72


0.356
.018

0.343
.015

0.409
.025

0.243
.011

0.275
.020

0.274
.009


(est.) 0.35


0.451
.031

0.260
.035

0.253
.026

0.230
.019


0.244
.021

0.157
.028

0.313
1.169


kb

0.517
.021

0.515
.027

0.609
.028

0.394
.013

0.436
.019

0.437
.014


(est) 0.60


0.693
.035

0.422
.023

0.406
.043

0.391
.022


0.368
.011

0.286
.021

0.475
1.480


k
U
0.923
.023

0.902
.032

1.016
.029

0.777
.012

0.822
.027

0.812
.011


(est.) 1.00


1.099
.010

0.780
.067

0.805
.059

0.767
.030


0.721
.070

0.670
.015


No usable
data


*Very poor quality data









extinction was made due to the lack of data necessary to perform a

meaningful least-squares fit. With these extinction coefficients the

differential magnitudes between the comparison star and the combined

variable and companion, companion, and check stars were calculated as

described in Chapter III.

Table 4 lists the differential magnitudes (Amag) between the com-

parison star and check star. It is defined as the magnitude of the

check star ninus the comparison star. The probable errors listed at

the bottom of each column are the maximum likely intrinsic variations

of the comparison star or the check star. They are sufficiently small

that it can be said, at least on nights of observation, that the com-

parison star was suitably constant.

In an attempt to find an empirical value for the third light of

the system (that of the companion star), occasional measurements were

made on nights of good seeing. Unfortunately, nights of good seeing

were often of poor photometric quality. For example, a thin layer of

clouds may cause reduction of turbulence, thus improving the seeing,

but these clouds also cause a varying transparency of the atmosphere

making the night of poor photometric quality. Also the quality of

the tracing (due to imperfect centering of the companion star while

attempting to prevent light of the variable star from "leaking in")

was critical. For these two reasons, each measurement was weighted

with a number between one and five giving some indication of the

quality of the observation due to the photometric quality of the

night and the quality of the tracing. The product of these two num-

bers (divided by ten) was the adopted weight for the measurement in







TABLE 4

THE DIFFERENTIAL MAGNITUDES BETWEEN THE COMPARISON AM) CHECK STARS
USED DURING THE PHOTOMETRIC INVESTIGATION OF AA CETI

Date mag v nagb Amagu

9/ 2- 3/72 0.692 0.679 0.694
0.721 0.644 0.678
0.668


9/ 4- 5/72 0.698 0.671 0.685

*10/14-15/72 0.662 0.645 0.664

*10/16-17/72 0.733 0.701 0.692

10/28-29/72 0.712 0.676 0.662


10/29-30/72 0.703 0.664 0.690
0.703 0.676

11/ 8- 9/72 0.718 0.681 0.699
0.700 0.665 0.677
0.699 0.667 0.681

*12/ 8- 9/72 0.675 0.689 **

*12/ 9-10./72 0.704 0.649 0.641

Average 0.704 0.670 0.680

Probable Error .012 .010 .010


*Vory poor night; one-half weight
**No extinction information








TABLE 5


THE DIFFERENTIAL MAGNITUDES BETWEEN THE COMPANION AND COMPARISON STARS
AND THE WEIGHTS USED FOR EACH MEASUREMENT
TO CALCULATE THE AVERAGE AND PROBABLE ERROR


Date

11/12-13/71


11/25-26/71



12/29-30/71



12/31/71-
1/1/72


9/ 2- 3/72


9/ 4- 5/72


10/14-15/72



10/16-17/72


10/29-30/72



11/ 8- 9/72


Amag

0.218
0.302
0.307
0.359
0.367


0.385
0.352


0.252
0.231



0.254


0.287


0.191


0.185
0.162


0.206


0.242
0.269


0.233


Average 0.264


Weight


0.314
0.295



0.298


0.318


0.251


0.191
0.207


0.243


0.335
0.355


0.321


0.300


0.424
0.412



0.394


0.438


0.428


0.289
0.316


0.342


0.429
0.448


0.437


1.0
1.5
2.0
1.0
1.0


0.5
0.5


2.0
2.5



2.0


0.3


0.9


0.4
0.4


0.8


1.2
1.2


2.5


Probable Error


.036 t.026









each color, and that number is given in Table 5 along with the indi-

vidual observations, their weighted averages and probable errors.

The differential magnitudes are defined in the sense of companion

star minus comparison star.

It is apparent from the probable errors shown in Table 5 that

there is a great deal of scatter in the data possibly due to the dif-

ficulty of the measurement. However, the uncertainty in the third

light which was subtracted in the yellow results in a probable error

in the depth of the primary eclipse of only 0.017 magnitude for the

final light curve. Likewise for the blue and ultraviolet the uncer-

tainty in the final primary eclipse depths are only 0.011 and 0.010

magnitudes respectively, A careful study of the correlation of the

apparent "fluctuation" of the companion star's light from these meas-

-urements with respect to scatter in the light curve of both stars of

the visual binary seems to indicate that the variations are not "real,"

i.e., not attributable to intrinsic variations of the companion, but

rather caused by the difficulty of the measurement. The apparent

fading trend of the companion star on the night of 11/12-13/71, how-

ever, is difficult to explain in terms of observational difficulties

which would cause random scatter in the measurements. It would appear

then that extensive measurements should be made of the companion star

if improvement in the light curve of the visual binary system is at-

tempted. Since these measurements do not demonstrate that the "fluc-

tuations" of the light of the companion are systematic, it will be as-

sumed here that the companion star represented a constant source of

light. The best values of the third light given in Table 5 were then

subtracted to give the light of the variable component alone.









Light Elements


As was pointed out in a previous section, preliminary light

elements were found based upon times of faint light on the Bamberg

plates. Subsequent photoelectric observations showed these ele-

ments to be incorrect.

Individual times of mid-eclipse were determined from the

photoelectric data by the Hertzsprung (1928) method if sufficient

data was available on both branches of the eclipse. Otherwise,

an estimate was made using the tracing paper method of fitting the

partial data to an eclipse whose mid-point was "known." The ap-

proximate period was found by assuming for each available pair of

times that exactly n cycles (for two primary minima or two secon-

dary minima) or n plus one-half cycles (for a primary and a se-

condary) had taken place. Solving for several values of n (for

each pair) revealed one common period in the vicinity of 0.5 to

1.25 days. With this approximation the epoch of each observed e-

clipse was determined, and a linear least-squares calculation was

done to refine the elements. The final set of elements were cal-

culated using the best times available from the Bamberg plates and

all the photoelectric times of minimum light weighted as shown in

Table 6 (Bloomer, 1972a). The final light elements (and probable

errors) derived for this system were

Kin I = JD(hel.) 2441268.6869 + Od53617353 E

.0007 .00000050

The observed minus the calculated times of minimum light given in

Table 6 are plotted in Figures 6 and 7.








TABLE 6


TIMES OF MINIfUM LIGHT OF AA CETI
AND THE WEIGHTS USED TO CALCULATE THE LIGHT ELEMENTS


Julian Date (hel.)

2438728.319

38995.594

39006.549

39060.417

39383.507

39404.394

39444.359

39761.491

39768.483

39771.463

40530.392

40566.285

41261.7176
.7176
.7155

41264.6665
.6687
.6664

41268.6866
.6869
.6860

41281.5571
.5566
.5561

41315.6031
.6022
.6012


Epoch

-4738.0

-4239.5

-4219.0

-4118.5

-3516.0

-3477.0

-3402.5

-2811.0

-2798.0

-2792.5

-1377.0

-1310.0

- 13.0
- 13.0
- 13.0

- 7.5
- 7.5
- 7.5

0.0
0.0
0.0

+ 24.0
+ 24.0
+ 24.0

+ 87.5
+ 87.5
+ 87.5


Weight

2

1

1

1

1

2

1

2

1

1

1

2

5
5
5

4
4


4
5
5

5
4
4
4

4
4
4


O-C(days)*

+0.022

+0.015

-0.022

-0.039

+0.006

-0.018

+0.003

-0.012

+0.010

+0.041

+0.016

-0.015

+0.0010
+0.0010
-0.0011

+0.0009
40.0030
+0.0008

-0.0003
0.0000
-0.0009

+0.0020
+0.0015
+0.0010

+0.0010
+0.0001
-0.0009








TABLE 6 CONTINUED


Julian Date (hel.)

2441317.481

41563.853

41607.836

41620.664

41630.603


Epoch

+ 91.0

+ 550.5

+ 632.5

+ 656.5

+ 675.0


Weight

2

2

1

2

3


O-C(days)*

+0.002

+0.002

+0.019

-0.021

-0.002


* Observed minus calculated time af minimum predicted from
Min I JD(hel.) 2441268.6869 + 0.53617353 E





















a,



-4



V
U



0
c.,





40


%-q
F3









0


ES


wi
r4



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C
19













0 b
C

















.r- -H




t%)

%-q
rr4
0
-rl

4-


a,














r10
Q












S--
0


I I sl


-~-- I- ----


C













0


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4-
CO
O0
a.
00



0E
.o o



.L -
4- o
4-




C V>

o o
o m


0





aC


0 0 0 0 0
o o o o o

0 0 0 0
o d ()

+ (sAgp) c-o


I-I I I


0
0


0
0
oo
--CO












-0




0
0

0
O
0

^-





0



0
0


O)










00
00
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O

r0


0

O-
SCO
O



































+31
V















13
-4
a



43
C,




1,
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-4









IE
+3










0




C13
0
0
a0
+3



"-4
.5
C)






+3






P- 0

o 'd


o o

.4,

* V

a,



a,


+3









I I I









Q


SQ 0


0'


(sADp) 0-o


0


0~


-J
_I

0
GO
(\


-I
0*
0
0
0J

Nu


---- aaa~---~------ ---n ^-~m-ar-3err~ourtrruurrcr*.~ nnuus


-- --------- I








Color Curves


In order to display the average change in color of the star

system during a cycle, b-v and u-b curves were constructed after

the third light subtraction had been made. First, the phase of

each observation was calculated using the light elements discussed

previously. Then, since the b and v measurements were not made

simultaneously, the blue magnitudes were linearly interpolated to

the yellow measurements so that the subtraction could be made. The

same procedure was followed for the u-b curve. No night correc-

tions or seasonal adjustments were made in plotting all the data

together.

The b-v curve shown in Figure 8 exhibits a distinct reddening

of the system by about 0.06 magnitude during the primary eclipse.

Since the primary eclipse is, in most cases, the eclipse of the

hotter star, this is an expected result. There does not seem to

be a significant change during the secondary eclipse.

The u-b curve (Figure 9) shows the same general characteris-

tics although the reddening during the eclipse is only one-half as

large, and the scatter of the data seems greater.

The fact that the system does not become more blue during the

secondary eclipse (the eclipse of the cooler star) tends to indi-

cate that the relative contribution of the cooler star to the total

light of the system is small at all the wavelengths investigated.


















0
0



0
0
0

4,





0



(D
-c


43
4,






'4-1
0

0
9-



48
0
0


4)
9-4





0




C,
'.
0
'4-


0








-c


H














Id














4 4
++



4



+
4+
.4 4
-t





+4 + 4


4

4+
44*
4 4+4


* +1 +


444
+,t
4 4

+ +
+4++t*


m

a



3-



+l






J
in





+T
rJ1


I-
1

+
E3





B





ffl


+


1 -I----------- I I
I'0- '0 I 0+




















0






*-f
40
,:


r-




r:



o
o
0)








-r
*p


0
U


(I)




C,







!




48




4



+ +







+ 4 -- +
+ +
+


++: U]


+ +


+ t +
+ 4.
+


+



+ [



$+
4 + +


4+ + 4 4
4 +,1 4
4 + + +



+ +







+ 4 4
*
+
4..

44.
+4








Light Curves


The data points were plotted into light curves. No night cor-

rections or seasonal adjustments were made. The resulting curves

are shown in Figures 10 through 12. All the curves showed a con-

tinuous variation between the eclipses and a period of constant

light which lasted approximately fifty minutes during the secondary

eclipse with the exception of the ultraviolet light, which faded

linearly with phase as the eclipse progressed. There was a larger

apparent scatter during the secondary eclipse than in the rest of

the curves. However, an analysis of this region showed that there

were night-to-night changes in the level and shape of the light

variations at the bottom of the eclipse. Although there were in-

dications, then, of intrinsic variability of the stars, an average

of all the data was used for the solution as if the scatter had

been random.

There were no significant indications of a "periastron effect"

or of a displacement of the secondary minimum from a linear phase

of 0.5. It is apparent, however, that maximum light after primary

eclipse came later than a phase of 0.25.

The light curves can then be classified as essentially W UMa-

type curves with a total secondary eclipse (and annular primary)

with some photometric complications.





















0










0
S4









.c

+ J



















4-f .P
0

























o
O'
* 3




















X;
;s ~0




0
g )




o '0


00



S C)


o*


E0
0l 0
s ^


-s
H g
> +






o
-r +













*


+ '+ef4





+ t

+
+,+4
4+


+


4.
4
+v+


44 4

+4
+
S44


4+4


4\.+


4 +4 P


4++


.44+ .


4++


t I i I I


E_ 0-


h'0+


3thHJI3<3


0*0


























0
+)

























4-4
4-)
4C





























.r: 4-)

4) )




r4
01
























0
C,
C. t



'=~~






,0 a





q- 4

rx 0
0
'-




0
U




53






1+ 4
Sru



+

+ +







S+, +
++
++



4+







I





+++H
+ ++ +.
+










h,
--- I t















<+ ,+
+





'0- O'0 '0+ bh'E+ 9'0+ .

























a,
4A


ril
0
43
4-' .s:
bo
-~H





4>

4-3
0







ri
90


4 0
4-)
0



V
to
0










4.)

4' 0
4 4


Q 14
4-)
H



r 0
El























0
C.)

















+ +


4 +

4+



+3

4+


*+ +
+ 4.4
+ +4


+






+t-



++ 4 4
'i.


.4
4+ 4 4



+ A +


+ +4.
f *


+
+
4.


hI I Z
h" + S'0+


e"I+


5idiF- o


---t------i--------- r-- -I I L


v W I


F*0-


EO'0








The Model


In order to deduce the photometric elements or parameters des-

cribing a physical binary system, some model must be employed. The

model of Russell and Merrill (1952) adopted here assumes two spher-

ical stars which are darkened at the limb by a linear cosine relation

and which are moving in circular orbits. The darkening equation re-

lates the observed surface brightness (J) at some point on the disk

described by the angle y (the angle measured at the center of the

star between the direction of the observer and the point on the sur-

face) and the surface brightness at the center of the observed disk

(Jc); specifically

J J (l x + x cosy)

where x is the limb darkening coefficient, which may have a value

between zero and one.

Merrill (1950) has calculated tables of functions which describe

the shapes of eclipses in terms of the tabular values of the ratio

of the radii of the two stars, a quantity which describes the "depth"

or geometrical degree of eclipse taking place, and values of the

darkening coefficient 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0. Finding a

solution, then, is a multiparameter fitting problem. In order to

discuss the solution, the parameters involved must be enumerated.*

For the case of spherical stars moving in circular orbits and

darkened at the limb by the aforementioned linear cosine relation,

the following parameters must be determined

P the orbital period of the two stars,

*For a detailed explanation of the model the original papers of
Russell and Merrill are referenced here.








T the initial epoch necessary to predict the relative posi-
o
tions of the stars at some time; the initial epoch and

period are usually referred to as the "light elements,"

i the inclination of the orbit of the spherical star system

with respect to the plane of the sky,

L the fractional luminosity contribution of the smaller

star to the total light of the system,

L the fractional luminosity contribution of the "greater"
g
or larger star to the total light of the system,

L the light of a supposed third body if the solution does

not satisfy L + L = 1.0,
s g
r the radius of the larger star in units of the separation
g
of their centers,

r the radius of the smaller star in units of the separa-
s
tion of their centers,

or k = r/r ,
sg
x the darkening coefficient of the smaller star,
s
and

x the darkening coefficient of the larger star.
g
The darkening coefficients are usually assumed a priori based

upon theoretical considerations, although they are sometimes changed

to achieve a better fit to the data. If one of the eclipses is total,

i.e., a smaller star is completely eclipsed by a larger one, then L
S
is determined, and L = 1 L unless there is a third light known or
g s
suspected. Since the period and initial epoch are determined before-

hand, a star with a total eclipse has only three parameters to be de-

termined from the shape of the eclipse: i, r and r or k. For a
gs









system in which partial eclipses take place, four parameters must be

found with the addition of L or L (assuming L = 0).
s 0 3
The extent of eclipse may be described by the quantity

6 r
p r
r
s

where 6 is the apparent or projected separation of the centers of

the two stars, and po is defined so that when the eclipsed area is

maximum. (and light is minimum), p = o Then

p > 1 for no eclipse,

1 > Po > -1 for a partial eclipse,

and

p < -1 for a complete eclipse.

Another quantity which describes the degree of eclipse is a It is

expressed in terns of the light lost at internal tangency (1- 1i).

If the light lost in mid-eclipse is (1 L ), then

1- i
o
Si.


During a partial eclipse Lo never reaches Li, and therefore o < 1.
0 1 o
For an eclipse which is total, i.e., a total occultation of a smaller
oc
star, ao = 2. An eclipse which is annular, i.e., the transit of
o
tr
the smaller star across the face of the larger, will have a > 1.
0
The problem reduces, then, essentially to choosing the proper
tr oc
values of k, a and o (or po in the case of total eclipses since

oC then is fixed at unity) which describe the data of both eclipses.

A great deal of ti.e and insight is gained by utilizing the nomographs

of Merrill (1953) which graphically display the "shape functions" on








a grid of possible solutions (k's and po's). These functions,

Xx(k,co,n) and XY(k,c), were tabulated by Kerrill (1950) as men-

tioned at the beginning of this chapter.

Once the best values of k and po are found, then the physical

elements of the system may be calculated by solving the equations

below.

1 aoc
L = o (oc =1 for total eclipses)
s oc o
0

L = 1 L if L3 = 0


r2 cosec21 = sin2 e/ k(l po)(2 + k + ko)]


r2 cosec i(l + kp ) = cot i


The angle e is the point of external contact or the phase angle
e
at which the eclipse begins.

On the nomographs a possible solution is described by coor-

dinates V and H, and then

1- tr 1- oc
o o
L = and L 0
g V s oc

If a solution is chosen which satisfies the so-called "depth equa-

tion" that relates the maximum light lost in the eclipses, then V

will have a value such that L + L = 1. However, if the shape
g s
functions require a solution at a different value of V, then L +

Lg < 1, and the solution requires that L > 0. A solution which

causes L + L > 1 is clearly spurious.
s g
Although the linear cosine law of darkening and the assump-

tion of circular orbits are usually adequate for close binaries,








the extreme proximity of the stars and the resulting huge tide dis-

tortions make the assumption of spherical stars very poor indeed.

This is evidenced by the fact that the major light changes are some-

times due not to eclipse effects but to changing aspects of the

highly distorted stars. In order to remove non-eclipse effects, a

"rectification" or transformation is performed.

It has been demonstrated by Russell and Merrill (1952) that the

distortion of the component stars can be approximated by two similar

tri-axial elllpcoids. In fact, the stars will assume the shape of

their Roche limiting surfaces (the equipotential contours around the

stars); however, similar ellipsoids have been assumed in the in-

terest of tractability.

Russell and Merrill have also shown that the ellipsoids cause

distortions of the light curve (generally referred to as the "ellip-

ticity effect") which can be described by a truncated Fourier series:

L = a0 + al cos e + a2 cos 29e...

Actually the effects of limb darkening and gravity darkening (bright-

ening of the star's surface at the poles due to centripital flatten-

ing which results in a higher acceleration of gravity) complicate

the transformation somewhat, and a rectification of the phase as well

as of the light is necessary.

The so-called "reflection effect," a heating up of the facing

surfaces of the stars, also contributes cos ne terms to the light

variations. The coefficients can be estimated from theory (Russell

and Merrill, 1952) and are generally designated by Cn

None of these effects cause Bn sin n8 variations in the light

although they are sometimes observed to be present. Conventionally,








these "complications" in the light curve are simply subtracted in

the rectification process. This assumes that the complications ex-

tend through the entire cycle.

The rectification process, then, consists of, first, fitting

by the method of least-squares a truncated Fourier series to the

light variations outside of eclipse and finding the coefficients of

the series:

4 4
A = A0 + A cos nO + Y B sin nO
n=l n=l

These coefficients include the algebraic sun of the effects of el-

lipticity and reflection. The reflection coefficients are found

from the theory as described by Russell and Merrill, and then the

rectified light (L') and the rectified phase (6') of each observed

L and 0 are found by the following equations:

L' = (L + C + C1 cos + C2 cos 2 A3 cos 39

A4 cos 48 B1 sin 0 B2 sin 2 B3 sin 30

B sin 46) / (A + CO + (A2 + C2) cos 20)

and
2 sin2 -4(A-C )
sin 8' = sin where Z 2= 2+2
l-Zcos2 (A-CO-A 2 2)N

The quantity N is estimated to be 2.0, 2.2, 2.6, and 3.2 for the

values of x of 0.2, 0.4, 0.6, and 0.8 respectively. In these equa-

tions the value of -A1 is substituted for C1 so that the entire

cos 8 variation is removed. The resulting light is presumably the

light of two spherical stars whose radii have a ratio (k) which

corresponds to two ellipsoids whose similar axes have the ratio of








k, also. Likewise, the other spherical star elements of the system

have an interpretation in terms of the "real" ellipsoidal system

initially assumed.


The Rectification of the Light Curves of AA Ceti


The Fourier coefficients were found for all three light curves

by estimating the point of external contact to be 0P12 = 43?2 and

by performing a least-squares fit to the data outside of eclipse.

With the values of A2 resulting from this calculation, the cos 28

terms were removed from the unrectified light curves revealing a

change in the slope at a phase of slightly greater than 0Plll = 40.

The sine and cosine coefficients were found through n = 4 since the

exclusion of the higher order terms (n = 3 and 4) could result in

spurious values of the lower order terms as pointed out by Merrill

(1970).

From an initial rectification of each light curve the relative

depths of the primary and secondary eclipses were found. The ratio

of the depths approximates the surface brightness ratio of the two

components, and from this the theoretical reflection coefficients

were found.

A rectification was performed with the equations discussed in

the previous section assuming limb darkenings of 0.4, 0.6, and 0.6

for the v, b, and u curves respectively. The preliminary solutions

indicated that the darkening coefficients should have been chosen

to be smaller in order to achieve a good fit to the data, and so

the final rectification assumed darkenings of 0.2, 0.4, and 0.4.

The coefficients used for the final rectification are given in Table








7. Since the sine terms were all small, they were set equal to zero

in the final rectification.

In the process of finding the reflection coefficients using the

graphical method of Russell and Merrill (1952), an attempt was made

to find the spectral type of the fainter star; however, this proce-

dure depended very critically upon the rectification done, particu-

larly since the eclipses were extremely shallow. The primary component

was assumed to be F2, and the secondary component was found to have a

spectral type of F2, F6, and F9 based upon the v, b, and u curves re-

spectively.

This same problem of achieving a consistent and reliable result

for all the light curves occurred for another essentially W UM.a-type

star, RR Centauri. This star was found by several investigators

(Knipe, 1965, and Bookmeyer, 1968) to be very difficult to solve,

partly because the shallow eclipses did not allow definitive values

of the ratio of surface brightnesses to be determined. Merrill (1970)

found the spectral type of the smaller star in RR Centauri to be F3

if the brighter component were F2. However, Koch et al. (1970) stated

that the secondary spectrum was F9. Both were considering only yellow

observations.

The disparity of results for both AA Ceti and RR Centauri tends

to make the determination of the spectrum of the fainter component

of little value. In AA Ceti the secondary component is probably sev-

eral sub-classes later than the primary component. The many other

similarities between RR Centauri and AA Ceti will be discussed in the

next section.








TABLE 7


COEFFICIENTS USED IN THE RECTIFICATION OF THE LIGHT CURVES OF AA CETI


+0.87319
0.00607

-0.05371
0.00611

-0.13546
0.00943

-0.01735
0.00390

-0.01774
0.00498

-0.00245*
0.00134

+0.00452*
0.00203

+0.00672*
0.00206

+0.00233*
0.00244

+0.3083
(x0. 2)

+0.03100

+0.05371

+0.01030


b

+0.88441
0.00697

-0.04703
0.00697

-0.13551
0.01087

-0.01132
0.00447

-0.01494
0.00576

-0.00085*
0.00154

+0.00302*
0.00228

+0.00413*
0.00234

+0.00521*
0.00274

+0.2620
(x-0.4)

+0.03200

+0.04703

+0.01070


u

+0.89095
0.00700

-0.04197
0.00718

-0.11958
+0.01094

-0.00840
0.00457

-0.01111
0.00575

-0.00423*
0.00156

+0.00840*
0.00233

+0. 00295*
0.00236

+0.00221*
0.00282

+0.2504
(x-0o.4)

+0.03270

+0.04197

+0.01090


*Set equal to zero in rectification








Solutions of the Light Curves of AA Ceti


The rectified light curves were solved using the X functions and

nomographs. It was apparent from the values of X that the primary

eclipse was the transit and the secondary eclipse the occultation of

the somewhat cooler and smaller star. The nomographic solutions re-

vealed that the ratio of the radii was a small number and that the

eclipses were total, which agreed with the appearance of the unrec-

tified light curves. Independent solutions for each color were found

using the X functions, and a summary of the results is shown in Table

8. The values of 9e, which seemed to become larger at shorter wave-

lengths, are an average of the e from the primary and secondary e-

clipses. Figures 13 through 18 show the fit of the theoretical curves

generated by the elements in Table 7 and the rectified data.

The inclination of the orbit for the ellipsoidal system (j) was

found from the equation

cos j = cos2i(l Z)

It was assumed that the semi-major axis of the ellipsoid of revolu-

tion (a) was the value of r resulting from the model. Then, the

other axis of the equatorial cross section of the star (b) was found

from
tan i
tan j
It is possible to find the third dimension (c) of tri-axial stars;

however, with the uncertainties in these solutions based upon some-

what uncertain rectification procedures, the calculation would be of

little value,

It is customary to solve the light curves of totally eclipsing








TABLE 8


ELEMENTS OF THE SYSTEM AA CETI


x = x
s g
k

Po
tr
0
oc
0


r
g
r



j



06 (T solutions)

a
g
b
g
8

b
S
s
a-b
a


-Ltr
1-^ t
0
1-Lo

L
g
L
s

7 f-


v

0.2

0.26

-1.50

1.030

*1.000


0.55

0.14

70.3

737

40?7

14?7

0.55

0.45

0.14

0.12

0.184


0.065

0.079

0.921

0.079

0.786


b

0.4

0.34

-1.54

1.069

1.000


0.53

0.18

754

775

43?4

14?4

0.53

0.45

0.18

0.15

0.150


0.115

0.073

0.927

0.073

1.475


u

0.4

0.41

-1.37

1.050

1.000


0.52

0.21

76?9

78?7

45?6

122

0.52

0.45

0.21

0.18

0.141


0.165

0.089

0.912

0.089

1.727
























+3






4.)
0







H
r4








0
r-











H
0
+3














x
+3
0




-H
p4



:3

r-I

0
r
H
















0





0
-I

+3









0












0 H
-o C
0)

H
--

f
o p
--

:* i





~4)




68



-I + I+
S +
-+. +

-1
-I.
-+ +

+I :+


-. -I .
-II




+ .
+ + +
F-


+ I+



4 +t



-I.
4.





+
-t.
li- I--lI- I- I I I ii 'III I -- ----
50A 1 SN 1 i
3KD*l
)J -iJII

























-ri






o
-4
0
0









'tD
.C

4.)

0








.-
ob
r-

0
r-
1-p












0
0
r-




.C
c C




l0








0
^-0
rE-

















0

-I
a
o














S-p
^ ^

s
0 W















,

4-1


tn
















-Il.1



CI

-I.






+-I -I

-1*L-t




~rI. HI;N1 n






















0







14
0
0










P,





W
14,
H
0


0









0








,0 i
0
















4JP





0
0
-a
o '

















+) ~
0 r



0



4,









0
sr:







o




Id
4,




72




+ I- ++





4. +
+


+ +
+ + +
+
+ -i.


++ .


+ I+
t



+-

\ + +
-i .
\ +



-i N
+





I DIN Imnm





















4-4




--I
0


H





























0
.o










0
,.






0
-4-)


00
-o




















HO
0



a,






,0


.c
:i

co


0



0 0



i- 0








-I- )



c'-1

aS
4, ^









++
74





+ +




++ +
++

+ + m+


+ IEl


++


++ +
++
-t Yt J-t-
+ +








+ +
+ ++
S ++




+ + +-
+
++
+



++ +
- I I I I I I -+-----I----I--I----- ---
55'0
_115Dll !i lI I















Id
-H
CH


41

0











r--
4P
-H




















o
0
4,

0




0















o
C)
r4-1

























o

H
0



















O
O--


,"-
o


0 0







Sc
4, p

0 -l











H <
-1
-H
i-p









'ca
--)







+ +


+

S+ .1
+



+ +

\+

+ +
+ +



++ +
\ [ 1




+ I+
+ +

+

+\


++
+


- I 1 -I 1 I I I I I I I I I- t- I 1 I I J- I', I I I I-- I- -4-I- H I -
SB'5 6'i0 'N I-
,AII SDN I Ini














0
T1
4a

U
0)

CH
0













p4-

0)




P-4-



4-)
.C.
0
tt

















H
0


1-f
4)
H

0
















C4-1
0
z















0
-H
s:











4-)
r(' 0

U









.4-4 +)
0)
1- -4










H0
~o



r:4









43
06 l.










H 0










.9.)




78

+ T I.
it
+

+ +

+
++
+ +


+-\ +
+
+



+ +










+
+ +




+ + + +
+ + +








+
+
+ I + i

L+ I1N + + n-








stars using the Y functions of Russell and Herrill (1952). For

light curves of high precision this is advisable since the Y tables

can be linearly interpolated with higher precision than the X func-

tions. It was felt that the X functions were sufficient for the

eclipses being discussed here. As a check, however, the final

solutions (k's, c rs, and p 's) were used to generate the Y func-
o 0
tions, and they were plotted against the rectified data. These

theoretical fits were almost identical with those generated by the

X functions. Also, the elements resulting from the T procedure

were almost exactly the same as those resulting from the initial

X solutions.


The approximate

eclipses in terms of


functions which describe the shapes of the

the X(k, ) functions were:


v Primary


v Secondary


b Primary


b Secondary


u Primary


u Secondary


sin2(0 tr)


sin2e(a0c)


sin 2e(r)


sin2e(coc)


sin2e(otr)


sin2e(coc)


- 0.0670 0.2Y(0.26, tr) + 0.1500


= 0.0960


= 0.0690


= 0.0990


= 0.0830


= 0.1007


0.2T(0.26, aoo) + 0.2180


0.4(0.34, ar)


.4Y (.34, a c)


0.4(0.41, tr)


0.4Y(0.41, coc)


+ 0.1500


+ 0.2220


+ 0.1550


+ 0.1981


Since T functions are more

the values of 9i, the points of

based upon these functions.


easily interpolated at large y,

internal tangency, in Table 8 are


Due to the uncertainties in subtraction of third light, the








shallow eclipses, and the difficulty in the rectification procedures,

those solutions must be considered only preliminary.

It is interesting to note the similarities between the well-

known eclipsing binary RR Centauri and AA Ceti. Extensive photo-

electric yellow observations of RR Centauri by Knipe (1965) were

analyzed by Bookmeyer (1968) utilizing the Russell model and are

repeated here in Table 9 along with the results of the present anal-

ysis of the yellow light curve of AA Ceti. The problems encountered

by Knipe and later by Bookmeyer in rectification and solution of RR

Centauri are summarized by Koch et al. (1970, p. 60):


Her work did not lead to a unique solution, and the choice of
rectification procedures affects the ratio J /J In the first
solution given in our catalogue the Fourier series was trun-
cated after cos 20 which resulted in a positive but rather small
A term. If the third harmonic is included, one obtains A
-6.0110 and A1 < 0. Neither solution appears quite satisfactory
although a general picture of the system is fairly well-defined
for such a closely-interacting binary.


It would seem that some of the problems which afflict the solutions

of RR Centauri are present in AA Ceti. Of particular note is the

large size of A3 in both stars. Although there seems to be no shift

of the secondary minimum from a phase of 0?5, the maximum after pri-

mary is definitely shifted in AA Ceti to a later phase than the ex-

pected 0P25. As Bookmeyer points out in regard to RR Centauri,

this could be indicative of an eccentric orbit. If the orbit were

eccentric and viewed along the semi-major axis, then: (1) the e-

clipses would not be of equal duration; (2) the secondary minimum

would not be displaced from a phase of 05; and (3) the maximum

light would occur at a true anomaly of 0.25 in phase, not at the








TABLE 9


RESULTS OF ANALYSIS OF THE YELLOW LIGHT CURVES
OF AA CETI AND RR CENTAURI


AA Ceti RB Centauri
(Present Work) (Bookmeyer, 1968)
Solution B

L 0.921 0.889


L 0.079 0.111

a 0.55 0.58
g

b 0.45 0.52


a 0.14 0.21


b 0.12 0.19


x 0.2 0.6


k 0.26 0.36


1 70?3 72?4


e, 407 49?2


\ 14?7 98

tr
tr 1.03 1.043

0 1.00 1.00
0








mean anomaly of OF25 as predicted by the linear light elements.

These three conditions are evident in the light curves of both stars.

In the case of RR Centauri there is some evidence to deny that the

apparent eccentricity is real; this is not yet the case of AA Ceti,

and so the possibility remains. From the duration of the primary

(tl) and the secondary (t2) one can extract the eccentricity (e)

using the following equation:


1l+e tl
1- e t2

if the angle from the ascending node to the periastron point is

ninety degrees.

From the yellow curve e = 0.12, from the blue e = 0.10, and

from the ultraviolet e = 0.06, all of which are extremely large for

this type of binary. If the eccentricity is real, the star, of

course, would be valuable for apsidal notion studies; however, it

is felt in this investigation that the problems of rectification

and the third light subtraction are sufficiently severe to make a

determination of the eccentricity dubious, at best.

There appeared to be an inconsistency in the values of 7g/s

for the components of AA Ceti, which were calculated from
2
J L r
S L 2
J s r
s g
The value for the v curve was less than 1.0, which implied that

the larger star was the cooler one, and thus that the secondary

eclipse was a transit. This implication was just the opposite of

what had been assumed from the beginning of this investigation.








However, inspection of the rectified light curves revealed that the

rectified primary was indeed the shallower eclipse, and so the solu-

tions were internally consistent. It was just such an inconsistency

which caused Bookmeyer to re-analyze the light curves of RR Centauri

in the first place. Whether the roles of the two stars did reverse

between the v and b bands (which is a physical possibility) or wheth-

er the problems involved in rectification were asserting themselves

was not clear.

In order to evaluate the fit of the theoretical shapes to the

data, a program was written for the Hewlett-Packard 9820A calcula-

tor to find the average difference between the individual rectified

data points and the value of the theory at that phase. These aver-

age (0 C)'s and the average of their phases were found over ten-

degree intervals. They are shown in Tables 10 through 12, including

the number of data points (n) in each interval. Then the program

"de-rectified" the theoretical values using the final set of coef-

ficients from Table 7, and the average differences were calculated

for the unrectified light over the entire period.

It is apparent that the residuals are generally small and

change sign often, indicating that the fit is good over the entire

light variation. As was stated previously, ignoring the sine terms

in the rectification has little effect on the ultimate results as

is indicated by the small (0 C)'s.

A statistical study of the fit is shonm in Table 30 in the

Appendix where the root-mean-square deviations of the data from the

theoretical shape are given for the rectified primary eclipse, first

maximum, secondary eclipse, and second maximum in each color.








TABLE 10


AVERAGE DIFFERE1NCF3S OF THE OBSERVED DATA AND THE THEORETICAL SHAPES
OVER THE YELLOW LIGHT CURVE OF AA CETI


Rectified Light

Average 8' Average(0-C)

5?43 -0.002

14?58 -0.006

24?18 -0.005

34.19 -0.001

45?43 +0.013

55?38 +0.003
65?02 +0.005

75?78 -0.008

85?28 -0.015

95?49 -0.002

105?28 -0.008

115?36 -0.006

123?97 -0.006

135?97 +0.002
46? 10 +0.004

155?53 +0.012

164?95 -0.008

175.99 +0.006
186?19 -0.005

195.41 +0.001


Un-Rectified Light


n Average 8


4?52

12?21

20?49

29?49

40.19

50?33
60?77

73?08

84 34

96?58

108 17

119?66

128 98

141?17

150778

159?26

167?38

176?67

185?16

192?92


Average (O-C)

-0.001

-0.004

-0.004

-0.001

+0.011

+0.003

+0.005

-0.008

-0.015

-0.002

-0.008

-0.006

-0.005

+0.008

+0.003

+0.010

-0.007

+0.004

-0.004

+0.001


-0.013 7 200?59


2049.29


-0.010








TABLE 10 CONTINUED


Rectified Light

Average e' Average(O-C)

21469 -0.007

224?45 -0.014

236?80 +0.014

245?88 -0.001

270?70 +0.008

295?45 +0.005

304026 -0.021

314?10 -0.002

324?36 +0.009

335?90 +0.010

345?15 -0.002

356?19 +0.004


Un-Rectified Light


n Average 8


209?96

219?23

231?82

241?73

270?83

299?76

309?30

319?33

329?19

339?57

347 55

351?83


Average(O-C)

-0.006

-0.013

+0.013

-0.001

+0.008

+0.005

-0.019

-0.002

+0.007

+0.008

-0.001

+0.003








TABLE 11

AVERAGE DIFFERENCES OF THE OBSERVED DATA AN;D THE THEORETICAL SHAPES
OVER THE BLUE LIGHT CURVE OF AA CETI


Rectified Light


Un-Rectified Light


Average 0'

5?14

15?37

25?25

34?35

45?03

55?17

64084

75?71

85?55

95?11

104?98

115774

124?58

142 28

154.67

163?45

173?94

184?87

194?63

204?89

23 5.00


Average(0-C)

+0.003

-0.004

-0.001

-0.004

+0.009

-0.004

-0.001

-0.013

-0.017

-0.006

+0.008

-0.006

-0.004

+0.008

+0.019

-0.009

-0.007

+0.007

-0.008

-0.003

-0.004


n Average 0

8 4?42

8 13?29

.9 22?07

LI 30?26

L2 40?71

8 50?91

9 61?35

5 73?50

5 84?83

4 95?93

4 107?29

5 119?29

7 128?71

9 146?34

7 157?86

6 165?67

6 174?79

6 184l19

7 192?65

8 201?74

6 211?05


Average(O-C)

+0.003

-0.003

-0.001

-0.002

+0.008

-0.003

-0.001

-0.013

-0.018

-0.006

+0.008

-0.006

-0.004

+0.007

+0.016

-0.007

-0.006

,+0.005

-0.006

-0.003

-0.004




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