The Light Curves of W Ursae Majoris Systems
by
Ian Stuart Rudnick
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA IN PARTIAL
FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1972
~
'To my wife, Andrea
ACKNOWLEDGMENTS
The author sincerely expresses his appreciation to
his committee chairman and advisor, Dr. Frank Bradshaw.Wood,
for his comments and suggestions, which greatly aided the
completion of this work. The author wishes to thank Dr.
R. E. Wilson of the University of South Florida for provid
ing one of the synthetic light curves and for serving on
the author's committee. Thanks are also due to Drs. KY
Chen and R. C. Isler for serving on the author's committee.
The author expresses his gratitude to Dr. S. M.
Rucinski for providing the other synthetic light curve. The
author wishes to thank Drs. J. E. Merrill and J. K. Gleim
for their many helpful discussions. Thanks are also due to
R. M. Williamon and T. F. Collins for their help in ob
taining some of the data and for many enlightening conver
sations. W. W. Richardson deserves highest commendation
for his untiring work on the drawings.
The author extends his thanks to the Department of
Physics and Astronomy for providing financial support in
the form of graduate assistantships, and to the Graduate
School for support in the form of a Graduate School Fellow
ship;
The author is indeed grateful to his parents and
to his wife's parents for their encouragement.
The author's wife deserves more than appreciation
for her patience, encouragement, and hard work during five
years of school life. Her devotion and understanding helped
as nothing else could.
TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS. . . . . . . . . . . iii
LIST OF TABLES . . . . . . . .. . . vii
LIST OF FIGURES . . . . . . . . . . x
ABSTRACT . . . . . . . . . . . . ii
CHAPTER I INTRODUCTION. . .. . . . . 1
The Russell Model. . . . . . . .. 2
CHAPTER II FIRST SYNTHETIC LIGHT CURVE . . . 6
Rectification . . . . . . . . . 6
Solutions from Graphical Rectification . . . 22
Solutions from Least Squares Rectification . 44
Orbital Elements . . . . . . ... 55
Figures of the Components . . . . ... 58
Comparison of Solution with Input Parameters . 59
CHAPTER III SECOND SYNTHETIC LIGHT CURVE. .. . 64
Rectification. . . . . . . . . . 69
Solutions ...... .. .... . . . 75
Orbital Elements and Figures of the Components . 78
Comparison of Solutions with Input Parameters. . 78
CHAPTER IV THE SYSTEM OF 44i BOOTIS. . . . ... 87
History. . . . . . . . . . . 87
Visual Binary . . . . . . . . 89
Spectroscopic Binary . ..... .. . .. . 91
Eclipsing Variable . . . . . . .. 92
Instrumentation. . . . . . . 93
Observations. . . . . . .. 96
Reduction.:of Data. . . . . ... 98
Times of Minimum Light and the Period. 126
Variation in the period caused by
motion in a visual binary system . 126
Page
A recent period change . . ... .132
Light Variations. . . . . . ... 133
Rectification . . . . . ... 144
Orbital Elements. . . . . . ... 146
CHAPTER V SUMMARY AND CONCLUSIONS . . . ... .153
LIST OF REFERENCES .............. ..... 156
BIOGRAPHICAL SKETCH. .................. 161
LIST OF TABLES
Page
I Observations of First Synthetic Light Curve. ... 7
II Graphical Rectification Coefficients for First
Synthetic Light Curve. . . . . . . ... 13
III Least Squares Rectification Coefficients for
First Synthetic Light Curve. . . . . . ... 15
IV Rectified First Synthetic Light Curve Using Graphi
cal Rectification Coefficients . . . . .. 16
V Rectified First Synthetic Light Curve Using Least
Squares Rectification Coefficients . . . ... 17
VI Solutions with Different Values of Limb Darkening. 25
VII Solution for x = 0.8, 1 kpr= 0.18, 1 sec= 0.16,
o o
k = 0.45, po = 0.94, aoc = 0.9943, atr 0.9698. . 26
OO O0
VIII Solution for x = 1.0, 1 pr= 0.18, 1 sec= 0.16,
O O
k = 0.45, po = 0.90, ac= 0.9905, atr= 0.9375 . 27
IX Solutions for x = 0.6, 1 tr /oc 0.2142, (1 tr)+
o k 0
(1 oc) = 0.34. . . . . . . . .... .. 33
0
X Solution for x = 0.8, 1 _pr= 0.18, 1 zSec= 0.16,
O 0
k = 0.50, p = 0.6608, ac= 0.89, atr= 0.7915 . 34
XI Solution for x = 0.8, 1 jpr= 0.18, 1 sec= 0.16,
o o
=oc tr
k = 0.55, po =04798, a c= 0.78, a0 = 0.6659. . 35
XII Solution for x = 1.0, 1 Zpr= 0.18, 1 Asec= 0.16,
0 0
oc tr
k = 0.60, po = 0.3225, a oc= 0.68, atr= 0.5332 . 36
0 0
Page
XIII Shape Curves for 0e = 500, x0c = 3.955, Xtr
4.605, x = 0.8 and 0 = 45?9, X = 3.476,
xtr = 4.047, x = 0.8. . . . . . . ... 37
XIV Depth Curve for x = 0.8, 1 koc= 0.16,
1 tr= 0.18 . . . . . . . . . 4
o
XV Solution for x = 0.8, 1 _pr= 0.16, 1 _sec=
o o
oc
0.145, proc, k = 0.65, po = 0.1625, ao = 0.55,
atr= 0.4548 . . . . . . . . . 47
o
XVI Solution for x = 0.8, 1 pr= 0.16, 1 esec
o o
0.145, prtr, k = 0.65, po = 0.1271, 0c= 0.525,
tr= 0.4314 . . . . . . . . . 48
XVII Solution for x = 0.8, 1 pr= 0.16, 1 sec=
o o
0.145, k = 1.0, po = 0.1540, ao = 0.305 . .. 49
XVIII OC's from Solution for k = 1, po = 0.1540,
a = 0.305. . . . . . . . . . 51
o
XIX Solution for x = 0.8, 1 _pr= 0.165, 1 Rsec
o o
tr
0.145, k = 0.70, po = 1.429, asc= 1.0, ao =
0 0
1.067, T= 0.558590, L3 = 0.588 . . . .. 54
XX Orbital Elements for Solutions of First Synthetic
Light Curve . . . . . . . . 57
XXI Figures of the Components . . . . . .. 60
XXII Observations of Second Synthetic Light Curve. . 67
XXIII Rectification Coefficients for Second Synthetic
Light Curve . . . . . . . . .. 70
XXIV Rectified Second Synthetic Light Curve ...... 71
oc tr
XXV Solution for x = 0.4, k = 0.545, a = 1.0, a =
1.016, po=1.10,T = 0.3126, 1 egr= 0.265, 1 zsec=
0.233 . . . . . .. ... . . . . 76
viii
Page
oc
XXVI Solution for x = 0.4, k = 0.65, a = 1.0,
atr= 1.039, p = 1.538,T = 0.447537, 1 'pr=
0.265, 1 ksec= 0.233, L = 0.234. . . .. 79
L 3=
XXVII Orbital Elements for Solutions of Second Synthetic
Light Curve. . . . . . . . . ... 80
XXVIII Figures of the Components. . . . . ... 81
XXIX Input Parameters for Second Synthetic Light Curve 83
XXX Comparison Stars . . . . . . ... 97
XXXI Observations of 44i Bootis . . . . .. 99
XXXII Standard Stars . . . . . . . ... .112
XXXIII Recent Times of Minimum Light. . . . ... 127
XXXIV Times of Minimum Light with Corrections for Motion
in a Visual Binary Orbit .... . .......... 131
XXXV Solutions and Orbital Elements with x = 0.6,
1 pr= 0.23, 1 sec= 0.13, 0 = 41?0, and
o o e
m2/ml = 0.50 . . . . . . . . . 150
LIST OF FIGURES
Page
1. Observations of First Synthetic Light Curve. . . 8
2. Graphical Rectification Plots for First Synthetic
Light Curve. . . . . . . . .... . 12
3. Rectified First Synthetic Light Curve Using
Graphical Rectification Coefficients . . ... 19
4. Rectified First Synthetic Light Curve Using Least
Squares Rectification Coefficients . . . ... 21
5. Solution for x = 0.8, 1 kpr= 0.18, 1 isec= 0.16,
o o
oc tr
k = 0.45, po =0.94, o = 0.9943, t 0.9698. . 29
6. Solution for x =1.0, 1 . pr= 0.18, 1 ksec= 0.16,
O 0
oc tr
k = 0.45, po =0.90, a0 = 0.9905, a= 0.9375. . 31
7. Depth Curve and Shape Curves . . . . ... 39
8. Superpositions of Primary and Secondary Eclipses 43
9. OC's from Solution for k = 1, p = 0.1540,
S = 0.305 . . . . .. . . . . . 52
10. Observations of Second Synthetic Light Curve . .. 66
11. Rectified Second Synthetic Light Curve . . ... 74
12. Visual Binary Orbit of 44i Bootis. . . . ... 90
13. Comparison Star Extinction for July 1, 1970. ... .110
14. Secondorder Extinction for Albireo. . . . ... 114
15. Firstorder Yellow Extinction for Albireo. ... .117
16. Firstorder Blue Extinction for Albireo. . . . 119
17. Firstorder UltraViolet Extinction for Albireo .
18. UBV Transformation Coefficients . .
19. Variation in the Period of 44i Bootis B
Motion in a Visual Binary System. . .
20. 0C's from Pohl's Light Elements. . .
21. OC's from New Light Elements . . .
22. Yellow Light Curve of 44i Bootis. . .
23. Blue Light Curve of 44i Bootis. . .
24. The Light Curves of Different Authors
25. The Light Curves of Different Authors
26. Deformities of the Light Curve of 44i B(
Two Nights. . . . . . . .
Caused by
. o .
. . . 135
. . . 136
. . . 137
. . . 139
. . . 140
otis on
. 143
Page
121
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
The Light Curves of W Ursae Majoris Systems
by
Ian Stuart Rudnick
June, 1972
Chairman: Frank Bradshaw Wood
Major Department: Astronomy
Two synthetic light curves computed from theoretical
astrophysical models of W Ursae Majoris systems are discussed.
Solutions of these light curves, based on the geometrical
Russell model and the RussellMerrill method of solution of
the light curves of eclipsing binaries, are presented. The
relatively shallow minima caused by the partial eclipses
of the first synthetic light curve lead to a problem of
indeterminacy in the solution. The "observed" points in the
shoulders of both synthetic light curves fall below the
theoretical light curves predicted by the solutions. The
addition of third light in the solution of the two synthetic
light curves improves the fit of the solutions to the light
curves; however, there is no sound basis for adding this
third light. The orbital elements predicted by the Russell
Merrill solutions of the two synthetic light curves are not
at all close to the orbital elements used to generate these
light curves from the theoretical astrophysical models. In
particular the RussellMerrill solution underestimates the
sizes of the components. It is concluded that the Russell
model is not compatible with the theoretical astrophysical
models used to generate the synthetic light curves.
Observations of the system of 44i Bootis, an example
of a W Ursae Majoris system, are also discussed. The times
of minimum light indicate that an increase in the period
occurred in 1967.
xiii
CHAPTER I
INTRODUCTION
The W Ursae Majoris systems are eclipsing variable
stars whose light curves have maxima which are strongly
curved and minima which are nearly equal in depth. These
systems have periods which average approximately onehalf
day. Orbital solutions in the literature indicate that
W Ursae Majoris systems are close binaries whose separations
are less than the dimensions of the components. Many
complexities which are caused by the proximity of the com
ponents appear in the light curves (1). Most of the solu
tions of the light curves of W Ursae Majoris systems in
the literature are based upon a geometrical model first
proposed by Russell (2). Lucy has proposed that some or
most of the W Ursae Majoris systems may be true contact sys
tems, whose common boundary follows a single equipotential
surface (3). It is the purpose of this thesis to apply the
RussellMerrill method of solution of the light curves of
eclipsing binaries (2,4,5) to synthetic light curves from
astrophysical models similar to Lucy's model in order to
determine whether the geometrical Russell model is compati
ble with the astrophysical models. Lucy (3), Rucinski (6),
and Mochnacki and Doughty (7) have written computer programs
for computing theoretical light curves of W Ursae Majoris
system; Mochnacki and Doughty have published a trial and
error fit to the system AW Ursae Majoris using their program.
Wilson and Devinney (8) have published a general procedure
for computing light curves of close binaries which includes
the W Ursae Majoris systems as a special case. This pro
cedure is now being applied to selected contact systems and
the results will be published soon. Two synthetic light
curves have been computed for this study, one by Rucinski (6)
and one by Wilson (9). In addition, observations of the
system of 44i Bootis, an example of a W Ursae Majoris system,
are discussed.
The Russell Model
The Russell model and the RussellMerrill method of
solution of the light curves of eclipsing binaries are
discussed in detail by Russell and Merrill (2). A brief
discussion is given here in order to define the notation
used. The Merrill tables and nomographs for solution of
light curves of eclipsing binaries are based upon a spherical
model (4,5). It is assumed that the orbit is circular,
that the stars are spherical, and that they appear darkened
at the limb by a linear cosine darkening law. This limb
darkening (x) may differ for the two components. The light
of each star is constant for the spherical model; therefore
the light of the system outside of eclipse is also constant.
It is convenient to take the light of the system outside
of eclipse as the unit of light and the radius of the spheri
cal orbit as the unit of component dimensions. Then the
components are defined by the following parameters:
Radius of the larger (greater) star r
g
Radius of the smaller star r
s
Inclination of the orbit i
Light of the larger (greater)star L
g
Light of the smaller star L
s
with L + L = 1. If 6 is the longitude in orbit (from
g s
conjunction), then the apparent distance between the centers
of the disks is given by
62 = cos2i + sin2i sin2 6
Setting p = (6 r )/r the eclipse will be absent, partial,
or complete, for p > 1, 1 > p > 1, or p < 1 respectively.
Setting k = r / rg then 6 = r (1 + kp). The quantities k
and p are dimensionless, and their values completely define
the geometrical circumstances of a given phase.
If f and fs represent the fractions of the light of
the two stars which are obscured at any phase of the eclipse
of either, and k is the normalized value of the light re
ceived from the whole system:
a = L (l f ) + L (l f ) = 1 L f L f
For tabular purposesRussell and Merrill express these in
terms of two other functions a and T, where a is the ratio
of light lost at any phase of an eclipse to the loss at
internal tangency, and T is that of the latter to the whole
light of the star. Then for the light at any phase during
an occultation (the larger star in front)
1 oc = focL = Lsoc (xsk,p)
and during a transit (the smaller star in front)
1 tr = ftrLg = L T(x ,k)a t(x ,k,p)
It is convenient to use the Xfunctions of Russell
and Merrill to determine the solution of the light curve.
Defining n = a/a where the zero subscript refers to the
value of the parameter at mideclipse, the xfunctions are
given as
(x,k,a ,n) sin28(n) [l+kp(x,k,nao) 2
X(xk,a ,n)  2 T
sin28(n=0.5) [1+kp(x,k,0.5a )]
[l+kp(x,k,ao)]2
[l+kp(x,k,ao) 2
These xfunctions have been tabulated by Merrill (4).
Russell and Merrill have shown that a system consisting
of two similar triaxial ellipsoids with semiaxes a b ,
c and a = ka b = kb c = kc can be rectified to
the spherical model making certain approximations which
involve the gravity and reflection effects. The aaxis of
each ellipsoid is along the line joining the centers of the
components, the caxis is parallel to the axis rotation of
the system, and the baxis is in the third mutually perpen
dicular direction. A mean radius r = (a + b + c)/3 may
also be defined for these ellipsoids. The fundamental geo
metrical equations for this model (i.e. the Russell model)
may be written as
2 2 2 2 2
cos i + sin i sin 0 = a (1 + kp)
r r g
where ir and 9 are the rectified inclination and phase angle
(orbital longitude) respectively. This equation is identical
in form with the equation for spherical stars. Thus the
observed intensity and phase angle can be rectified in such
a way as to produce a rectified light curve which will be
nearly that produced by the eclipse of a pair of spherical
stars of radii a as = ka and inclination ir.
CHAPTER II
FIRST SYNTHETIC LIGHT CURVE
A synthetic light curve for a W Ursae Majoris type
eclipsing binary was generated by Dr. S. M. Rucinski from
Lucy's model (6). The "observational" data for this light
curve consisted of 37 values of the normalized light (or
flux) as a function of phase angle, with the phases given
in 50 intervals from 0 to 1800. The other half of the
light curve (1800 to 3600) was assumed to be symmetrical.
An additional ten points were generated later to define
better the centers of the eclipse regions of the light
curve. This "observational" data is listed in Table I
and plotted in Figure I. This synthetic light curve was
to be treated as observational data and solved by the
standard RussellMerrill method of solution of light curves
of eclipsing binaries. No additional information about the
nature of the derivation of this light curve was to be used
in the solution.
Rectification
The first step in the process of getting a solution
TABLE I
Observationsof First Synthetic Light Curve
PHASE INTENSITY PHASE INTENSITY
0.00 0.45242 95.00 0.99343
2.50 0.44826 100.00 0.98470
5.00 0.45216 105.00 0.96954
7.50 0.46429 110.00 0.94766
10.00 0.48257 115.00 0.92548
12.50 0.49728 120.00 0.90144
15.00 0.51875 125.00 0.86813
17.50 0.54518 130.00 0.83688
20.00 0.56516 135.00 0.80528
22.50 0.58812 140.00 0.77124
25.00 0.61087 145.00 0.73817
30.00 0.65572 150.00 0.70216
35.00 0.70708 155.00 0.66023
40.00 0.74969 157.50 0.64105
45.00 0.79267 160.00 0.62372
50.00 0.83087 162.50 0.60589
55.00 0.86916 165.00 0.58505
60.00 0.90446 167.50 0.56515
65.00 0.92940 170.00 0.55371
70.00 0.95640 172.50 0.54237
75.00 0.97754 175.00 0.53530
80.00 0.99117 177.50 0.53398
85.00 0.99632 180.00 0.53866
90.00 0.10000
8
0 4
e
o
a
.0 
0
4)
O
00
S i
o 
o 4 
rl
*Q
I
0
0 .,
0 0
0 H
**
0
*
0
0
0o
II W IA
 0 00 0
for the light curve was an analysis of the light outside
the eclipses in order to arrive at a rectification of the
light curve to the spherical model. Two methods were used
for this analysis: Merrill's graphical method (10)and
a least squares Fourier analysis of the material outside of
eclipse.
Let the light outside eclipse be represented by a
truncated Fourier series of the form:
I = AO + A0 cos + A2 cos 20 + A3 cos 30 + A4 cos 4,
(Since the light curve is symmetrical about 0 = 180, it
is not necessary to include sine terms in the above Fourier
series,) Following Merrill's graphical method, a and b
represent readings for 0 and 1800 0 on the light curve.
(The given "observational" points were used rather than
reading from a freehand curve since the scatter of the
points was small.) It immediately follows that
(a + b) = A + A2 cos 20 + A4 cos 4
1
(a b) = A1 cos9 + A3 cos 36
1 1
Letting C (a b) and C2 = (a + b), by simple trigo
nometric substitution,
C1 = (A1 3A3) cos 6 + 4A3 cos 36
C2 = (Ao A4) + A2 cos 26 + 2A4 cos2 268
Therefore a plot of C1 versus cos8 would have the form of
a cubic and a plot of C2 versus cos 26 would have the form
of a parabola. Such plots are given in Figure 2. The plot
of C1 versus cos 6 shows the presence of a considerable A3
(cos 36) term and indicates the possible presence of higher
order odd cosine terms. The plot of C2 versus cos 26 is
essentially linear, indicating that the A4 term is negli
gible. The plotted points seem to fall below this straight
line somewhere around 400, thus locating 9e (the value of
o at external tangency) to a first approximation. Values
of the Fourier cosine coefficients were then derived from
the plots and are listed in Table II.
Because of the similarity of the "colors" of the two
"stars" and the small difference in the depths of the two
minima, rectification coefficients for the reflection effect
were obtained in the following manner (2):
C1 = A1
C = 0.090 sin2 6
2 e
C2 = 0.030 sin2 e
The eclipses were assumed to be partial and values were
derived for 9e = 390; these values are listed in Table II.
4
O*
rl
44
0
4c
U O
H U
a),
tp
ri
.rl
P4
TABLE II
Graphical Rectification Coefficients for
First Synthetic Light Curve
00
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
A =
o
a
.4534
.4522
.4826
.5188
.5652
.6109
.6557
.7051
.7497
.7927
.8309
.8692
.9045
.9294
.9564
.9775
.9912
.9963
1
0.7990
A1 = 0.0232
A2 = 0.2010
A = 0.0147
A4 = 0
b
.5387
.5353
.5357
.5851
.6237
.6602
.7022
.7382
.7712
.8053
.8369
.8681
.9014
.9355
.9477
.9695
.9847
.9934
1
C = 0.0354
0
C1 = A1
C = 0.0118
0.3353
z = 0.2724
0.2180
C1
.0432
.0416
.0356
.0332
.0293
.0247
.0233
.0166
.0108
.0063
.0030
.0006
.0016
.0020
.0044
.0040
.0033
.0015
0
.4956
.4938
.5182
.5520
.5945
.6356
.6790
.7217
.7605
.7990
.8339
.8687
.9030
.9275
.9521
.9735
.9880
.9949
1
0.6
for x = 0.8
1.0
Also listed in Table II is the value of z, given by
4(A2 C2)
Z=
N(A C A + C2)
where N depends on the assumed limb darkening (x); values
were adopted such that N = 2.6, 3.2, or 4.0 when x is
assumed to be 0.6, 0.8 or 1.0 respectively.
Fourier coefficients were also computed by a least
squares Fourier analysis of the light outside of eclipses
(450 to 1350). The results of this analysis are listed in
Table III.
Both sets of rectification coefficients were then
used to compute a rectified light curve. The rectified
intensity is given by
I+C +C1 cos 9+C2 cos 20A3 cos 38A4 cos 40
I" e
A +Co+ (A2+C2) cos 20
and the rectified phase by
sin2 = sin2 8
1z cos2
The rectified points are listed in Tables IV and V
and plotted in Figures 3 and 4.
Several things are apparent from an examination of
TABLE III
Least Squares Rectification Coefficients for
First Synthetic Light Curve
A = 0.79532 + 0.00123
O0
A = 0.02208 + 0.00114
A = 0.20699 + 0.00195
A = 0.01304 + 0.00066
A = 0.00350 + 0.00090
OBSERVED THEORETICAL OC 6
INTENSITY INTENSITY
0.7927 0.7924 0.0003 45.0000
0.8309 0.8317 0.0008 50.0000
0.8692 0.8687 0.0005 55.0000
0.9045 0.9026 0.0019 60.0000
0.9294 0.9322 0.0028 65.0000
0.9564 0.9570 0.0006 70.0000
0.9775 0.9763 0.0012 75.0000
0.9912 0.9898 0.0014 80.0000
0.9963 0.9973 0.0010 85.0000
1.0000 0.9988 0.0012 90.0000
0.9934 0.9944 0.0010 95.0000
0.9847 0.9845 0.0002 100.0000
0.9695 0.9693 0.0002 105.0000
0.9477 0.9495 0.0018 110.0000
0.9255 0.9257 0.0002 115.0000
0.9014 0.8986 0.0028 120.0000
0.8681 0.8689 0.0008 125.0000
0.8369 0.8375 0.0006 130.0000
0.8053 0.8052 0.0001 135.0000
TABLE IV
Rectified First Synthetic Light Curve
Using Graphical Rectification Coefficients
O(x=.6) O(x=.8) O(x=l) I"( E) I"(180+0):,
0.000 0.8331 0.8493
3.066 2.931 2.826 0.8254 0.8413
6.125 5.857 5.651 0.8278 0.8410
9.173 8.775 8.468 0.8404 0.8480
12.204 11.680 11.277 0.8599 0.8599
15.212 14.570 14.074 0.8714 0.8703
18.194 17.439 16.857 0.8906 0.8916
21.143 20.286 19.624 0.9145 0.9124
24.058 23.108 22.371 0.9264 0.9268
26.933 25.901 25.099 0.9405 0.9388
29.768 28.665 27.803 0.9522 0.9517
35.405 34.092 33.141 0.9693 0.9778
40.658 39.382 38.373 0.9870 0.9905
45.825 44.540 43.498 0.9950 0.9959
50.810 49.536 48.513 0.9996 1.0003
55.573 54.406 53.423 0.9990 1.0009
60.279 59.152 58.234 1.0006 1.0014
64.793 63.781 62.953 1.0020 1.0054
65.000 0.9967 1.0018
70.000 0.9986 0.9995
75.000 1.0001 1.0009
80.000 1.0000 1.0001
85.000 0.9976 0.9983
90.000 1.0000 1.0000
TABLE V
Rectified First Synthetic Light Curve Using Least
Squares Rectification Coefficients
I I" 0
0.4524 0.8469 0.000
0.4483 0.8391 2.945
0.4522 0.8412 5.885
0.4643 0.8535 8.816
0.4826 0.8726 11.735
0.4973 0.8834 14.637
0.5188 0.9019 17.519
0.5452 0.9250 20.377
0.5652 0.9359 23.208
0.5881 0.9488 26.011
0.6109 0.9595 28.781
0.6557 0.9745 34.221
0.7051 0.9904 39.518
0.7497 0.9968 44.668
0.7927 1.0004 49.673
0.8309 0.9992 54.538
0.8692 1.0006 59.273
0.9045 1.0021 63.891
0.9294 0.9971 68.405
0.9564 0.9994 72.831
0.9775 1.0012 77.186
0.9912 1.0014 81.488
0.9963 0.9990 85.753
1.0000 1.0012 90.000
0.9934 0.9990 94.248
0.9847 1.0003 98.513
0.9695 1.0002 102.815
0.9477 0.9982 107.170
0.9255 0.9998 111.596
0.9014 1.0031 116.110
0.8681 0.9992 120.727
0.8369 0.9994 125.463
0.8053 1.0002 130.328
18
Table V continued.
II" 9
0.7712 0.9977 135.332
0.7382 0.9952 140.483
0.7022 0.9857 145.780
0.6602 0.9631 151.220
0.6411 0.9521 153.990
0.6237 0.9417 156.792
0.6059 0.9290 159.624
0.5851 0.9097 162.482
0.5652 0.8897 165.364
0.5537 0.8804 168.266
0.5424 0.8694 171.185
0.5353 0.8630 174.116
0.5340 0.8639 177.056
0.5387 0.8722 180.000
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1"
either of the rectified light curves. The first is that
the amount of scatter of the data points in eclipse is much
larger than was originally expected. This scatter will be
treated as "observational" error for the present and no at
tempt will be made to explain it either in terms of the
model or the method of computation of the data. Another
obvious feature of the rectified minima is the "brightening"
at the centers of the eclipses. Again no attempt will be
made at present to explain this effect; however, its pres
ence' creates a serious uncertainty in both the depths of
the eclipses and the credibility of the points near the
center in terms of the Russell model.
Solutions from Graphical Rectification
The rectified light curve produced by applying the
rectification coefficients derived by the graphical method
was used for the first attempts at a solution. The phase
was rectified with a z based on N = 2.6 corresponding to a
darkening x = 0.6, and the first solutions were tried on the
x = 0.6 nomograph. The depths of the eclipses were chosen
as 1 r pr = 0.18 and 1 k sec = 0.16 and the following
o o
values were read off the plot of the rectified light curve:
2 pr 2 sec
n 0 sin20 6 pX 0 sin 2 xe
0.2 33?0 .29663 2.268 32?6 .29027 1.901
0.5 21.2 .13077 1 23.0 .15267 1
0.8 12.2 .044658 .342 15.0 .066987 .439
By choosing values of O(n = .5) and O(n = .8)
at o + 0?5, a permissible range of .301 < pr(n = .8) <
.387 and .394 < se(n = .8) < .488 was found. Since
sec(n = .8) > (n = .8), it appears that the primary
eclipse is a transit and the secondary eclipse is an occul
tation. Then the values needed for the depth line on the
nomograph are
(1 oc) + (1 tr) = 0.34
o 0
1  tr oc = 0.2142
o o
1 tr oc + (1/50) oc oc 02180
1 /o + (1/50)1 o /o = 0.2180
Taking these values and the X(n = 0.8) values given above
to the x = 0.6 nomograph, an intersection of the depth line
and the permissible values of both X contours was found
oc tr
with k = 0.45, po =.98, = .9987, r = .9928. When
these elements are taken to the X tables, they produce the
following points on the light curve:
n c sin2 0 Xtr sin2O
0.0 2.823 .44122 4106 3.374 .44122 41?6
0.2 1.725 .26961 31.3 1.913 .25016 30.0
0.5 1 .15629 23.3 1 .13077 21.2
0.8 .448 .070020 15.3 .376 .049170 12.8
Comparing these values with those taken from the
"observed" light curve, several things are apparent. First,
the fit from the halfway point down is moderately satis
factory for this preliminary stage of solution. Second,
the fit at the shoulders is very bad; the computed curve
is much narrower than the "observed" curve. The x(n < .5)
values for both eclipses need to be increased a significant
amount while making only small changes in the X(n > .5)
values, in order to fit the "observed"light curve with the
chosen depths of the eclipses.
Since the nomographic solution on darkening
x = 0.6 as described above was not satisfactory, other
possibilities were explored. The first approach was to
try nomographic solutions with other values of darkening.
Table VI summarizes the results of these attempts. The
main conclusion from this exploration of solutions with
different values of darkening is that the fit will improve
with increasing darkening. There are two reasons for this
improvement. First the X(n < .5) values for the transit
eclipse tend to increase with increasing darkening, and
second, the value of z is smaller for larger darkening.
With a smaller z, the rectification of the phase tends to
make the shoulders narrower relative to the halfwidth
than with a larger z. Thus, it appeared that a darkening
of x = .8 or x = 1 should be used for further trial
solutions. The results of these solutions with k = .45
(see Tables VII and VIII and Figures 5 and 6) were not at
all satisfactory. It was still not possible to fit the
TABLE VI
Solutions with Different Values of Limb Darkening
x 0.2 0.4 0.6 0.8 1.0
k 0.4625 0.4625 0.45 0.45 0.45
Po 0.93 0.90 0.98 0.94 0.90
acoc 0.9868 0.9799 0.9987 0.9943 0.9905
a tr 0.9809 0.9627 0.9928 0.9698 0.9375
o
X9C(n=.8) 0.401 0.405 0.448 0.452 0.452
tr(n=.8) 0.383 0.370 0.376 0.361 0.351
Xo(n=.2) 1.812 1.799 1.725 1.712 1.703
tr(n=.2) 1.867 1.908 1.913 1.957 1.991
TABLE VII
Solution for x = 0.8, 1
k = 0.45, po = 0.94,
oc
n X
2.827
2.047
1.712
1.203
1
.817
.452
.276
sin2
.47732
.34562
.28905
.20312
.16884
.13794
.076316
.044600
 pr
0
0 =
0
= 0.18,
0.9943,
1 sec= 0.16,
tr
S= 0.9698
o
0 tr sin2
4307
36.0
32.5
26.8
24.3
21.8
16.0
12.2
3.650
2.464
1.957
1.265
1
.766
.361
.181
.47732
.32222
.25592
.16542
.13077
.100170
.047208
.023669
43?7
34.6
30.4
24.0
21.2
18.5
12.6
8.9
0.95 .1734 .029277 9.9 .0925 .012096 6.3
TABLE VIII
Solution for x =
k = 0.45, p =
o
1.0, 1 ~r = 0.18, 1 sec = 0.16
0 0
0.90, ac = 0.9905, tr = 0.9375
0 o
n oc xoc sin2
0.0 3.801
0.1 2.257
0.2 1.625
0.4 .7274
0.5 .3590
0.6 .0191
0.8 .6275
0.9 .9684
0.95 1.1648
2.911
2.054
1.703
1.205
1
.811
.452
.263
.47732
.33675
.27922
.19750
.16396
.13302
.074151
.043116
43?7
35.5
31.9
26.4
23.9
21.4
15.8
12.0
5.412
3.082
2.186
.9965
.5439
.1465
.5306
.8290
3.939
2.532
1.991
1.273
1
.760
.351
.171
.47732
.30687
.24132
.15430
.12119
.092118
.042584
.20755
0
43.7
33.6
29.4
23.1
20.4
17.6
11.9
8.3
.9717 .0851 .010315 5.8
90 tr Xtr sin2
.1539 .025236 9.1
 a o01
00 r H O
1 1 +
H C ) H 4)1
04 00
H C 0'0 0
a 0o
CII NC
II N .. 0 o
u co a) u
, p I
4) o o4 r4 *d
co * 4.
II o )u
1 (D 
O aa O
0 II O 0
0
r'4
F c
o o n D
IH '' 1r
I
0
0
E
0.
I.
e
+
Q
1 _I I + + 
+
+
0
 I s ?~P 51 
UI)
O 0
o H 0
II 0 0)
o 0)
0 E 0 0
a o a
M in n(u
o o( rI .
0
II 0 rj
H 4J
o s w
o L
o L 0 M 0
m nu mW 
II o J I u
U) m a) r
mo o a) aCo
0 0 A 0 C
iI Za u ifd +
1h I *+ _0 +
o r
 = 0 0
'4
OC
: O O
_I _
shoulders of the eclipses, especially the primary eclipse,
while,at the same time, fitting the rest of the eclipse
curves. In particular, with k = .45, the solutions for
a circular orbit were too wide at the halfway point of
the secondary eclipse and too narrow on the shoulders of
the primary eclipse, with lesser problems elsewhere.
Because of the problems described above, it was
necessary to abandon the chosen value of k = 0.45 and to
explore other possibilities of k on the x = 0.6 nomograph,
staying on the depth line given above. Table IX summarizes
the results of these explorations. From an inspection of
this table, it was decided to attempt trial solutions with
k in the range .50 to .60 for darkenings x = 0.8 and total
darkening. Some of these solutions are given in Tables
X, XI, and XII. While these solutions are an improvement
over previous ones, the major problem of the fit of the
shoulders, especially in the primary eclipse, has not been
alleviated.
Because of the recurrent problem with the fit of
the shoulders of the eclipses, it was felt that a different
approach might prove helpful. This approach was to choose
a O(n = .5) for the primary eclipse and a 0 From these
values, X(n = 0)'s were derived. These X's were taken to
the x = 0.8 tables and values of k and oc were obtained
0
and these shape curves were plotted (see Table XIII and
Figure 7). The depth curve derived from
oc (1 oc) + (1 tr)/
06 0 0 O
TABLE IX
Solutions for x = 0.6, 1 
1.00
.0956
.3400
.3400
.346
.346
1.969
1.969
tr /oc =0.2142
0 0
(1 _tr) + (1 oc) =0.34
o o
.85
.0000
.4136
.3788
.346
.344
1.963
1.976
.75
.1000
.4900
.4363
.347
.343
1.956
1.982
.60
.3538
.6800
.6018
.354
.343
1.927
1.986
.55
.4820
.7700
.6897
.361
.344
1.902
1.983
.50
.6789
.8900
.8224
.379
.348
1.853
1.972
k
PO
po
0 tr
Xc (n=.8)
tr
x (n=.8)
oc(n=.2)
tr (n=.2)
tr
x (n=.2)
.45
.9800
.9987
.9928
.448
.376
1.725
1.913
TABLE X
Solution for x = 0.8, 1 _pr
o
k = 0.50, po = 0.6608, ao
n Xc sin 2
3.217
2.261
1.838
1.241
1
.781
.384
.195
.47732
.33547
.27271
.18413
.14837
.115880
.056976
.028933
4307
35.4
31.5
25.4
22.7
19.9
13.8
9.8
= 0.18,
= 0.89,
tr
0 X
3.746
2.524
1.995
1.277
1
.758
.345
.166
1 sec = 0.16,
o
tr
a = 0.7915
sin 2
.47732
.32161
.25421
.16272
.12742
.096585
.043960
.021152
43"7
34.6
30.3
23.8
20.9
18.1
12.1
8.4
0.95 .0992 .014719 7.0
0816 .010398 5.9
TABLE XI
Solution for x = 0.8, 1 _pr
0
k = 0.55, po = 0.4798, aoc
O~O
n,: X sin2
3.382
2.347
1.892
1.254
1
.771
.366
.180
.47732
.33124
.26703
.17698
.141135
.108815
.051655
.025404
43?7
35.1
31.1
24.9
22.1
19.3
13.1
9.2
= 0.18,
= 0.78,
1 sec= 0.16,
o
tr
atr = 0.6659
O
S Xtr sin 2
3.764
2.535
2.002
1.278
1
.756
.343
.164
.47732
.32147
.25388
.15207
.126812
.095870
.043497
.020797
0
437
34.5
30.3
23.7
20.9
18.0
12.0
8.3
0.95 .0893 .012603 6.4
.0803 .010183 5.8
TABLE XII
Solution for x =
k = 0.60, p =
O
0oc
4.495
2.910
2.276
1.4016
1.0562
.7471
.2032
.0424
oc
X00
x
3.582
2.392
1.916
1.259
1
.768
.360
.175
1.0, 1 
0.3225,
sin2
.47732
.31875
.25532
.16777
.133255
.102340
.047972
.023320
pr =
0
soc=
43?7
34.4
30.3
24.2
21.4
18.7
12.7
8.8
0.18, 1 Zsec = 0.16,
tr
0.68, 0tr = 0.5332
o
,tr
6.349
4.186
3.349
2.233
1.8073
1.4350
.0843
.5327
tr
x
3.982
2.562
2.012
1.280
1
.756
.342
.163
sin20
.47732
.30710
.24117
.15343
.119869
.090621
.040995
.019539
4061 .0800 .009590 5.6
0
43"7
33.7
29.4
23.1
20.3
17.5
11.7
8.0
0868 .011567 6.2
0.95 .1602
TABLE XIII
Shape Curve
Xtr
a (pr)
.9232
.8543
.8140
Shape Curve
tr
oc (pr)
.9293
.7810
.6862
for e = 500, Xoc = 3.955,
= 4.605, x = 0.8
a C(pr) a (sec)
o o
.8680 .8009
.8108 .6289
.8140 .5301
for 0e = 45?9, x0c = 3.476,
= 4.047, x = 0.8
aoc(pr)
.8541
.6877
.6092
.5747
.5828
(oc(sec)
.6786
.4568
.3700
.2767
.2245
.1913
k
0.6
0.65
0.7
0.8
0.9
1.0
.8313
.9789
P4 5 O
(1) a) $A 4 4J I
E 0 W 40J rd 0 t4
E1 r) :J a P4 n 0
0 04 E E 0
0 04 r 0 0 O 0
a)M iOH fa
(D n..Icli u u o ]
> 0 (n E u a a .
:J 4 (d >O o
an r) p14 i U UiH o
0D 1 10) 0)
4J a "i 4 r a 1 14)
0 Oi O MC .O
03O* 3OQH o
4000 .4O 030
.0 O O MM a) U u0
r. r a 43J rd DC (D
fo 3 a)a)) 044 )a r
Q0)4 3 C 444, )4. 44 4) 0
a)40 4) 0 40 4+
>0 U  F
> o u *' *' e 4
P H4 a0) '(o) 0 r
0 r) 00 04* 040 < 4
S*H 0 ,C 0 ) ard Lo. ao
4 .4J C tLo a. n
.) 104) to a) rO 04
4 H (d F II II 11 0) *
Pa lHO 0 03 a )4 H
0 0 03 4 0 4. 0 400 0 40 0
03
tP
4
F*4
oc
with k(a q ) taken from the x = .8 tables was also
derived and plotted (Table XIV and Figure 7). An inspection
of this plot shows that for 0e = 50 and e = 45?9, the
intersection of the shape curves for the two eclipses in
both cases lies above the depth curve. This implies the
possibility of a solution from the shapes alone, abandoning
the depth curve and therefore the assumption that L1 + L2= 1.
This socalled third light solution is indeed one way to
produce a theoretical light curve that will fit the observed
curve. There is, however, no real justification for assuming
the presence of this third light in the present case and
therefore, this possibility was rejected.
It had been noticed from the first plotting of this
rectified light curve that a number of the points in the
primary and secondary eclipses were similar (i.e. for a given
value of 0, I" was nearly the same for both eclipses). In
attempting solutions for k = 1, this similarity became
even more apparent and it was decided to superimpose the
plots of the two eclipses (see Figure 8). The result was
remarkable, from the shoulders down to a depth of approxi
mately n = 0.8, the two eclipses were virtually identical;
deeper than this point, there was a sharp divergence. Since
this type of behavior is not possible in a system described
by the Russell model, either the points near mideclipse had
to be completely abandoned, or the rectification itself
might be at fault, and a new rectification could be tried.
It was decided to try to find a solution on the other
TABLE XIV
Depth Curve for x = 0.8, 1 o = 0.16,
0
1 = 0.18
Saoc k
0o o
0.195 1.00 .425
0.2 .98 .433
0.25 .82 .510
0.3 .713 .565
0.4 .58 .650
0.5 .50 .724
0.6 .447 .788
0.7 .408 .848
0.8 .38 .901
0.9 .357 .952
1.0 .34 1.000
to
*H
a) 0a
*dl >
OH
rdi a) 0
C 0  Qt
) H .4 H
S4 )1
0 Ol
UP (U P 0
O t o>
4a () rd
0 P o o
1i 0 0
* 0 *HU
0 a 0
4 a> Q4
p)4 HO 9
04l to 0)
0 0C Q .0
E 12 0 4 0
C
1 1
r
o'
o+
0+
o+
o +
o +
 1 o
_I~ I_ ~P~
rectification mentioned above, i.e. the one whose coefficients
were derived by the least squares method of Fourier analysis.
Solutions from Least Squares Rectification
There are several differences apparent in looking
at the two curves produced by the two different sets of
rectification coefficients. Outside the eclipses, the
residuals are more or less evenly distributed throughout
the whole curve in the least squares rectification, with
the sum of the squares of the residuals smaller than that
from the graphical rectification, as expected. Inside the
eclipses the two curves are no longer virtually identical
from the shoulders down to an approximate level of n = 0.8;
in general, for any given phase the primary is deeper than
the secondary. The greatest change in the new rectification,
however, is in the depths of the two eclipses. Both
eclipses are significantly shallower, the depth of the
primary eclipse going from about 0.18 to 0.16 and the depth
of the secondary eclipse going from about 0.16 to 0.145.
A nomographic solution of this second rectified
light curve was then attempted. The light curve was rec
tified in phase with a value of z based on N = 3.2 corre
sponding to a darkening x = 0.8 and the solution was tried
on the x = 0.8 nomograph. The value of x = 0.8 was chosen
for convenience since there is neither a nomograph nor a
set of X tables for complete darkening and the previous
explorations of the light curve seemed to indicate a large
value for the darkening. The depths of the eclipses
were chosen as 1 pr = 0.16 and 1 Asec = 0.14 and the
o o
following values were read off the plot of the rectified
light curve (Figure 4):
n Opr sin2Epr pr sec sin2 sec sec
0.2 30?0 .25000 2.038 31?5 .27300 2.032
0.5 20.5 .12265 1 21.5 ;13432 1
0.8 12.0 .043227 .352 13.5 .054497 .406
Since xsec(n = 0.8) > XPr(n = 0.8), it seems that the
primary eclipse is a transit and the secondary eclipse is
an occultation as before. The values needed for the depth
line on the nomograph are
1 tr
o = 0.1860
koc
0
tr oc
1 r 1 a
S+ 0.1893
koc 50 oc
O O
o o
Taking these values and the x(n = 0.8) values given above
to the x = 0.8 nomograph, an intersection of the depth line
and both X contours was found with k = 0.45, p = 0.75,
oc tr
o = 09395, and a = 0.8522. Using the X tables, the
following points on the light curve were derived from these
parameters.
oc 2 tr 2
n X sin 9 X sin29 0
0.0 3.037 .44706 42?0 3.645 .44706 42?0
0.2 1.780 .26202 30.8 1.967 .24125 29.4
0.5 1 .14720 22.6 1 .12265 20.5
0.8 .407 .059912 14.2 .352 .043173 12.0
The situation here is quite similar to that found with
the previous rectification. The computed curve is defi
nitely narrower at the shoulders than the observed curve.
The fit of primary eclipse from the halfway point down is
fairly satisfactory. The lower half of the computed curve
for the secondary eclipse is considerably wider than the
observed curve.
These difficulties are basically the same as those
encountered in the solution based on the first rectification.
Furthermore, the situation here is one common to solutions
of many systems with relatively shallow eclipses, i.e.
solutions with a wide range of values of k differ very
little in the light curves which they produce. For example,
in this particular case, if the depths of the two eclipses
and the value of the external tangency point, e are fixed,
then solutions for the cases: 1) k = .65 primaryoccultation,
2) k = .65 primarytransit, 3) k = 1 (see Tables XV, XVI,
and XVII and Figure 4), differ in the whole course of the
light curve by no more than approximately 0?6. This means
that distinguishing among these possible solutions is quite
TABLE XV
Solution for x = 0.8, 1 pr = 0.16, 1 sec = 0.145,
0 0
proc, k = 0.65, po= 0.1625,
n Xoc sin2
.48255
.331
.26564
.17353
.13728
.104884
.048461
.023475
0 X
44?0
35.2
31.0
24.6
21.7
18.9
12.7
8.8
aO= 0.55,tr= .45s48
tr sin2
sin O S
3.696
2.507
1.986
1.275
1
.758
.344
.165
.48255
.32731
.25929
.16646
.13056
.098064
.944913
.021542
44?0
34.9
30.6
24.1
21.2
18.3
12.2
8.4
0.95 .0840 .011532 6.2
3.515
2.418
1.935
1.264
1
.764
.353
.171
.0807 .010536 5.9
TABLE XVI
Solution for x = 0.8,
prtr, k = 0.65, p =
n ocx sn
n X sin2
3.505
2.415
1.934
1.264
1
765
.354
.171
.48255
.33249
.26626
.17402
.13768
.105321
.048737
.023542
1 pr = 0.16, 1 sec = 0.145,
o o
0.1271, 0oc = 0.525, tr = 0.4314
0 tr sin2
44?0
35.2
31.1
24.7
21.8
18.9
12.8
8.8
3.671
2.497
1.981
1.274
1
.758
.345
.165
.48255
.32823
.26040
.16747
.13145
.099638
.045350
.021689
0
44?0
35.0
30.7
24.1
21.3
18.4
12.3
8.5
0.95 .0841 .011578 6.2 .0811 .010661 5.9
TABLE XVII
Solution for x = 0.8, 1 pr = 0.16,
k = 1.0, p = 0.1540, a
k = 1.0, po = 0.1540, ao =
n
0.0
0.1
0.2
0.4
0.5
0.6
0.8
0.9
0.95
x
3.626
2.475
1.969
1.271
1
.760
.347
.167
.0818
1 Asec = 0.145,
0.305
sin 0
.48255
.32938
.26204
.16915
.133081
.101142
.046179
.022225
.010886
0
44?0
35.0
30.8
24.3
21.4
18.5
12.4
8.6
6.0
_ _
i
difficult.
Because of the above difficulties, a provisional
solution with the parameters 1 pr = 0.16, 1 ksec = 0.145,
O O
k = 1.0, ao = .305, and po = .1540 was chosen. The computed
curve was plotted (Figure 4) and values of observed minus
computed (0 C) for each observed point were derived graph
ically. These (0 C) values are listed in Table XVIII and
plotted as a function of phase in Figure 9. From an inspec
tion of this plot, it appears that a higher order cosine
term (e.g. cos 9 0) is present, especially in the primary
eclipse.
The possibility of using Kitamura's method for
the solution of eclipsing binary light curves (11) was also
explored. Kitamura's method has the advantage of using the
whole light curve to find a provisional solution rather
than using a few selected points as is done in the nomo
graphic method. Since both methods of solution are based
upon the same geometrical model for the eclipses, their
final results should agree. Even though the two methods
might produce different provisional solutions, a careful
and thorough analysis based upon these different preliminary
solutions should finally produce the same final solution.
Any difference in the results has to be caused by improper
application of one of the methods. Since solutions using
Kitamura's method should not differ from those derived
using the RussellMerrill method, Kitamura's method was
TABLE XVIII
OC's from Solution for k = 1, p = 0.1540, a = 0.305.
0 OC(O) 0C(180 +0)
0.000 .007 .017
2.945 .003 .007
5.885 .007 .000
8.816 .003 .002
11.735 .003 .002
14.637 .000 .004
17.519 .003 .002
20.377 .011 .007
23.208 .006 .005
26.011 .004 .002
28.781 .001 .001
34.221 .007 .003
39.518 .006 .001
 I
S
S
0
So
n 
o o
0 0 0
do o
I 0o
o0
0
!
LA
In
O
o o
0
'4
O o
I
rH
0
0
41
o 0
CD
o
0
,I
0
n
ur
S _
I
not pursued any further.
Before going on to find the geometrical elements
corresponding to these possible solutions, a further
exploration of the possibility that the eclipses are com
plete might be in order. By making this assumption, it
is possible to find a solution from the shapes of the
eclipses alone (i.e. by not assuming that L1 + L2 = 1).
While this'?introduction of third light seems somewhat
unwarranted, it is presented here for the sake of com
pleteness. The shapes of the eclipses were used to derive
the solution given in Table XIX and plotted in Figure 4.
This solution assumes that the eclipses are complete and,
in fact, takes the limiting situation of central eclipses
(i.e. i = 90). These assumptions require that more than
half of the total light of the system come from some
unknown third body (L3 = 0.588). This solution seems to
fit the light curve about as well as the previously men
tioned solutions.
In rectifying the phase, the ellipticity z is
a function of a parameter N which goes as
N = (15 + x)(1 + y)
15 5x
where x is the limb darkening and y is the gravity darkening
of the star being eclipsed. In the rectification of the
phase described above, a value of N = 2.6 corresponding to
TABLE XIX'
Solution for x = 0.8, 1 Opr =
o
k = 0.70, p = 1.429, ao 1.0, a
S= 0.558590, L = 0.588.
0.165, 1 
tr
o
ksec = 0.145,
1.067
= 1.067,
Xoc sin20
sin
3.339
2.281
1.834
1.231
1
.798
.452
.297
.217
.48255
.32965
.26505
.17790
.14452
.11533
.065323
.042922
.031361
tr
0
.1040 .015030 7.0
.1396 .015033 7.0
0.937=n.
1
0.0
0.1
0.2
0.4
0.5
0.6
0.8
0.9
0.95
44?0
35.0
31.0
24.9
22.3
19.9
14.8
12.0
10.2
sin2
.48255
.30508
.23293
.14075
.107688
.080228
.037691
.021107
.012298
4.481
2.833
2.163
1.307
1
.745
.350
.196
.1142
0
44?0
33.5
28.9
22.0
19.2
16.5
11.2
8.4
6.4
a limb darkening of x = 0.6 implies that a gravity darkening
of y = 1.0 was chosen. Dr. S. Rucinski (6) has suggested
that a value of gravity darkening y = 0.32 might be more
appropriate for this system. This possibility was explored
and it was found to have no significant effect on the
problems encountered in finding a solution for this system.
Therefore, it was decided to proceed with the solutions
already derived.
Orbital Elements
The next step was to derive orbital elements for
the eclipsing system from the geometrical eclipse parameters.
These elements can most easily be derived from the formulas
given by Merrill (5):
2 sin2 e
g 2 2 2
g (1 + k) (1 + kP) cos e
a= ka
s g
cos ir = (1 + kp )a
L =1 oc oc
s o o
L =1 L
g s
The derived orbital elements for the various solutions are
listed in Table XX.
Mean errors (P = JZ(0 C)2/(n 1)) were derived
for the "observed" points from each of the four possible
solutions. These mean errors are listed in Table XX. The
mean error has almost the same value for all three of the
partial eclipse solutions (with no third light added). The
value of the mean error for the complete eclipse solution
(with third light) is slightly smaller than the values of
the mean error for the other three solutions. Since these
four solutions with very different values of the geometrical
elements produce computed light curves which fit the ob
served light curve more or less equally well, it would not
be meaningful to apply the method of differential corrections
to find a least squares solution of the light curve. In
the absence of a formal least squares solution, it is not
possible to determine mean errors for each of the geome
trical elements. From the results presented in Table XX,
however, it is possible to estimate a reasonable range of
each element. For the three partial eclipse solutions
(without third light) the approximate ranges of these
elements are: .38 < a < .46; .29 < a < .38; 640 < ir < 660;
.53 < L < .72; and .27 < L < .48. With the addition
g s
of third light the inclination can increase up to the
limiting value of 900 and solutions are possible anywhere
within this range of inclinations. In the solutions
TABLE XX
Orbital Elements for Solutions of First Synthetic Light Curve
proc prtr
k .65 1.0: .65 .70
a0c .5500 .3050 .5250 1.0
0
atr .4548 .3050 .4314 1.067
0
po .1625 +.1540 .1271 1.429
a .457 .382 .459 .409
g
a .297 .382 .299 .286
s
cos i .40890 .44083 .42139 0
i 65?9 63?9 65?1 90
r
L .709 .525 .724 .277
g
L .291 .475 .276 .135
s
L 0 0 0 .588
p .0045 .0044 .0044 .0037
presented here, the value of 8 was chosen as 440 in order
to best fit the whole light curve. The observations on
the shoulders of the eclipses alone indicate that this
value might be increased by several degrees. The effect
of increasing 0 is to increase the values of a and
e g
a (i.e. to increase the sizes of the stars relative to
their separation) and to decrease the inclination of the
orbit.
Figures of the Components
The figures of the components can now be computed
from the formulas given by Merrill (10):
r + (0.17 + 1.19 m/mg)r4 = a
bg = a 1.53(m /m )r4
c = 3r a b
g g g g
These formulas require that the mass ratio of the compo
nents be known. Since this mass ratio was not known, a
more indirect method was used to set limits on both the
figures of the components and their mass ratios. This
method assumes that no dimension of the star may exceed the
size of the Roche lobe it is contained within. The dimen
sions of the Roche lobes for various mass ratios are
tabulated in the literature (e.g., Kopal (12)). The
figures of the components were computed for a number of
different mass ratios for each of the solutions. (The
two solutions for k = .65 were combined for this purpose
since the values of a and a in each were nearly equal.)
These figures were compared with the size of the Roche lobe
in order to determine the limiting mass ratios which would
satisfy the criterion that the sizes of the stars must not
exceed the sizes of their Roche lobes. These limiting
mass ratios and the figures of the components derived
using them for each of the solutions are listed in Table
XXI.
Comparison of Solutions with Input Parameters
It is now possible to compare the solutions
derived above with the parameters used to generate the
light curve. The light curve was produced using Lucy's
model for the light curves of W Ursae Majoris stars. In
this model the surfaces of the two stars are Roche equipo
tential surfaces and for the Rucinski light curve the
stars share a common envelope which fills the outer
contact surface. The constant which defined this surface
is C = 3.5591 with the mass ratio q = m /m = 0.4. The
other parameters used in deriving the light curve are the
inclination of the orbit i = 820, the effective temperature
of the surface Te = 57000, the wavelength of the light
TABLE XXI
Figures of the Components
.65 .65
.28 .57
.458 .458
.439 .429
.442 .428
.417 .401
.298 .298
.273 .283
.267 .281
.254 .270
1.0
.87
.382
.361
.359
.342
.382
.357
.354
.335
1.0 .70
1.15 .25
.382 .409
.357 .397
.354 .400
.335 .382
.382 .286
.361 .263
.359 .257
.342 .246
k
ms/mg
a
g
r
g
b
g
c
g
a
s
r
s
b
s
s
.70
.96
.409
.381
.378
.356
.286
.278
.276
.272
A = 5500 A, the limb darkening u = 0.6, and the gravity
darkening y = 0.32. The size of the equipotential surface
(and therefore the stars) was also given. These values
are b = 0.531, c = 0.482, bs = 0.363, and cs = 0.337 .(61).
These parameters can be compared with the results
of the solutions as listed in Table XX and Table XXI. The
inclination of the orbit suggests that a solution using
the RussellMerrill method which has this inclination will
include the presence of some third light. The figures of
the components (with q = 0.4) for such a solution can be
of roughly the same shape in the yzplane (the plane con
taining the b and c axes of the components) as the shapes
of the lobes of the outer contact surface in the yzplane.
The RussellMerrill dimensions of the components in this
plane relative to the separation of the components are
considerably smaller than the dimensions of the lobe of
the outer contact surface in the yzplane. Although it is
possible, as previously mentioned, to increase slightly
the dimensions of the components by increasing the values
of 0 in the solution, it is not possible to increase the
e
dimensions of the components to anywhere near the size of
the outer contact surface and still have the value of 0
compatible with the light curve. Using the Russell model
of similar prolate ellipsoids, it is possible to propose
a system consisting of two components whose dimensions in
the yzplane are nearly as large as the dimensions of the
lobes of the outer contact surface in the yzplane. These
prolate ellipsoids would be in contact (i.e., a + a = 1).
g s
There area number of objections to this proposed system.
First of all, it is not possible for real stars in contact
to have ellipsoidal shapes. Even if such a system were
physically possible, it would produce eclipses with 0 = 90,
in contrast with the values (440< e < 50) given by the
light curve. Even the physically realistic system with
the two components filling their Roche lobes (inner contact
surfaces) produces eclipses that are wider (for i = 82)
than those in the light curve (12). In this case the sizes
of the components in the yzplane are also considerably smaller
than the sizes of the lobes of the outer contact surface in
the yzplane.
As the above discussion indicates, the solutions
produced by applying the RussellMerrill method to the light
curve give very different values for the orbital elements
than the parameters used to generate the light curve. This
is especially true if third light is not included in the
solution. Then, not only are the stars much smaller than
the outer contact surface, but the inclination of the orbit
is also quite different from the inclination used in de
riving the light curve. Leaving aside the question of
third light, since the amount of third light cannot be
determined from the light curve alone, it is apparent that
the RussellMerrill solution of the light curve is com
pletely unsuccessful in predicting the elements used to
derive the light curve from Lucy's model. Thus these two
models for the light curves of W Ursae Majoris stars seem
to be incompatible. While the models can produce similar
light curves, both of which resemble the light curves
observed for W Ursae Majoris type stars, the orbital elements
which they use to produce these similar light curves are
quite different. Since most of the solutions of the light
curves of W Ursae Majoris type stars which are in the
literature are based upon the geometrical model lying
behind the RussellMerrill method of solution (or, equiv
alently, other methods based upon the same model), much
caution is in order in looking at these solutions. If
Lucy's model represents the actual physical situation for
a W Ursae Majoris type star, then most of the solutions in
the literature probably do not realistically represent the
actual physical situation present in the W Ursae Majoris
type stars.
CHAPTER III
SECOND SYNTHETIC LIGHT CURVE
A second synthetic light curve for a W Ursae Majoris
type eclipsing binary was generated from a theoretical model
by Dr. R. E. Wilson (9). The "observational" data for this
light curve consisted of 51 values of the differential
stellar magnitude as a function of phase, with the phases
given in intervals of onehundredth of the period from
phase 0.00 to phase 0.50. The remaining half of the light
curve from phase 0.50 to phase 1.00 was assumed to be
symmetrical. This "observational" data is plotted in Figure
10 and listed in Table XXII. (A normalization constant
equal to 0.2049 was subtracted from each of the magnitudes.)
Also listed in this table are the corresponding light values
and phase angles in degrees for each data point. The syn
thetic light curve was to be treated as observational data
and a solution was to be obtained by using the standard
RussellMerrill method of solution for light curves of
eclipsing binaries. No additional information about the
nature of the derivation of this light curve was to be used
in the solution.
0
O
e
09
0
id d o
I I I o
kq rI: C 'CC
o o d
0o o o o
~ ~ ~ ~
_ __ *
TABLE XXII
Observations of Second Synthetic Light Curve
PHASE Am0.2049 I 6
0.00 0.6900 0.5297 0.0
0.01 0.6893 0.5300 3.6
0.02 0.6857 0.5318 7.2
0.03 0.6398 0.5547 10.8
0.04 0.5725 0.5902 14.4
0.05 0.5012 0.6303 18.0
0.06 0.4333 0.6709 21.6
0.07 0.3718 0.7100 25.2
0.08 0.3176 0.7464 28.8
0.09 0.2707 0.7793 32.4
0.10 0.2308 0.8085 36.0
0.11 0.1972 0.8339 39.6
0.12 0.1691 0.8558 43.2
0.13 0.1457 0.8744 46.8
0.14 0.1252 0.8911 50.4
0.15 0.1060 0.9070 54.0
0.16 0.0881 0.9221 57.6
0.17 0.0712 0.9365 61.2
0.18 0.0557 0.9500 64.8
0.19 0.0419 0.9621 68.4
0.20 0.0298 0.9729 72.0
0.21 0.0197 0.9820 75.6
0.22 0.0115 0.9895 79.2
0.23 0.0057 0.9948 82.8
0.24 0.0016 0.9985 86.4
0.25 0.0000 1.0000 90.0
0.26 0.0006 0.9994 93.6
0.27 0.0037 0.9966 97.2
0.28 0.0087 0.9920 100.8
0.29 0.0160 0.9854 104.4
0.30 0.0256 0.9767 108.0
0.31 0.0374 0.9661 111.6
0.32 0.0510 0.9541 115.2
Table XXII continued.
PHASE Am0.2049 I 6
0.33 0.0667 0.9404 118.8
0.34 0.0841 0.9255 122.4
0.35 0.1028 0.9097 126.0
0.36 0.1231 0.8928 129.6
0.37 0.1459 0.8743 133.2
0.38 0.1715 0.8539 136.8
0.39 0.1998 0.8319 140.4
0.40 0.2322 0.8075 144.0
0.41 0.0694 0.7803 147.6
0.42 0.3115 0.7506 151.2
0.43 0.3621 0.7164 154.8
0.44 0.4172 0.6810 158.4
0.45 0.4779 0.6439 162.0
0.46 0.5426 0.6067 165.6
0.47 0.6066 0.5720 169.2
0.48 0.6574 0.5458 172.8
0.49 0.6672 0.5409 176.4
0.50 0.6693 0.5399 180.0
Rectification
As in the previous synthetic light curve solution,
the first step is an analysis of the light outside the
eclipses in order to rectify the light curve to the spher
ical model. This analysis was done by a leastsquares
Fourier analysis of the light outside of eclipse (48 to
1320). The results of this analysis are listed in Table
XXIII. The light values were then rectified by the
formula:
I+Co+C1 cos 8+C2 cos 28A3 cos 36A4 cos 40
A +Co+ (A2+C2) cos 20
and the phase angles were rectified by the formula
sin2 = sin 2/(1 z cos2 )
where z = 4(A2 C2)/N(Ao C A2 + C2) with N = 2.2,
corresponding to a limb darkening x = 0.4. The rectification
coefficients for reflection were determined from the statis
tical formula given by Russell and Merrill (2): Co = 0.072
sin2 e = 0.0398, C1 = A1, and C2 = 0.024 sin2e = 0.0133,
with e = 48. Following Merrill (10), the A3 and A4
terms were removed by subtraction. The rectified points are
listed in Table XXIV and plotted in Figure 11.
TABLE XXIII
Rectification Coefficients for Second
Synthetic Light Curve
A' = 0.86563
o
A = 0.00390
A2 = 0.13528
A3 = 0.00382
A = 0.00098
THEORETICAL
INTENSITY
0.8910
0.9069
0.9222
0.9366
0.9500
0.9622
0.9729
0.9820
0.9894
0.9949
0.9984
0.9999
0.9994
0.9967
0.9920
0.9853
0.9767
0.9662
0.9541
0.9405
0.9256
0.9096
0.8928
+ 0.00012
+ 0.00009
+ 0.00018
+ 0.00005
+ 0.00007
0.0001
0.0001
0.0001
0.0001
0.0000
0.0001
0.0000
0.0000
0.0001
0.0001
0.0001
0.0001
0.0000
0.0001
0.000
0.0001
0.0000
0.0001
0.0000
0.0001
0.0001
0.0001
0.0000
50.4000
*54.0000
57.6000
61.2000
64.8000
68.4000
72.0000
75.6000
79.2000
82.8000
86.4000
90.0000
93.6000
97.2000
100.8000
104.4000
108.0000
111.6000
115.2000
118.8000
122.4000
126.0000
129.6000
OBSERVED
INTENSITY
0.8911
0.9070
0.9221
0.9365
0.9500
0.9621
0.9729
0.9820
0.9895
0.9948
0.9985
1.0000
0.9994
0.9966
0.9920
0.9854
0.9767
0.9661
0.9541
0.9404
0.9255
0.9097
0.8928
TABLE XXIV
Rectified Second Synthetic Light Curve
I I" a
0.5297 0.7353 0.000
0.5300 0.7347 4.233
0.5318 0.7341 8.453
0.5547 0.7583 12.647
0.5902 0.7960 16.805
0.6303 0.8370 20.917
0.6709 0.8760 24.972
0.7100 0.9106 28.965
0.7464 0.9396 32.889
0.7793 0.9624 36.741
0.8085 0.9792 40.518
0.8339 0.9905 44.219
0.8558 0.9970 47.845
0.8744 0.9996 51.398
0.8911 1.0001 54.881
0.9070 1.0002 58.298
0.9221 1.0000 61.653
0.9365 0.9999 64.950
0.9500 1.0000 68.196
0.9621 1.0000 71.397
0.9729 1.0001 74.559
0.9820 1.0000 77.687
0.9895 1.0002 80.789
0.9948 1.0000 83.870
0.9985 1.0001 86.939
1.0000 1.0001 90.000
0.9994 1.0001 93.062
0.9966 0.9999 96.130
0.9920 1.0000 99.212
0.9854 1.0002 102.314
0.9767 1.0001 105.442
0.9661 0.9999 108.604
0.9541 1.0001 111.804
Table XXIV continued.
I I" 0
0.9404 1.0000 115.051
0.9255 1.0000 118.348
0.9097 1.0002 121.703
0.8928 1.0001 125.119
0.9743 0.9989 128.602
0.8539 0.9958 132.156
0.8319 0.9908 135.782
0.8075 0.9826 139.483
0.7803 0.9701 143.260
0.7506 0.9533 147.112
0.7164 0.9291 151.036
0.6810 0.9012 155.028
0.6439 0.8686 159.084
0.6067 0.8331 163.195
0.5720 0.7979 167.354
0.5458 0.7708 171.548
0.5409 0.7681 175.768
0.5399 0.7681 180.000
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Solutions
An examination of the rectified light curve reveals
several things. The most obvious is that the eclipses are
apparently complete. Furthermore it appears that the
primary eclipse is the occultation (total) eclipse and the
secondary eclipse is the transit (annular) eclipse. Closer
examination reveals a very slight brightening in the center
of the primary eclipse. The flatness of both eclipses
indicates that the limb darkening is probably small. The
depths of the two eclipses were chosen as 1 pr = 0.265
and 1 Asec = 0.233 for the purpose of obtaining a pre
0
liminary nomographic solution. Then the following values
were read off the plot of the rectified light curve:
n 0pr sin20 Xpr asec sin2 sec
0.2 33.5 <.30463 1.841 33.1 .29823 1.986
0.5 24.0 .16543 1 22.8 .15017 1
0.8 15.8 .074137 .448 14.4 .061847 .412
XPr(n = 0.8) > Xsec(n = 0.8), confirming that the primary
eclipse is the occultation eclipse and the secondary
eclipse is the transit eclipse. These X values along with
a depth line 1 itr/Zsec = 0.3176 were taken to the total
0 0
eclipse portion of the x = 0.4 nomographs and a satisfactory
intersection of the three curves was found. This intersection
oc
implied a solution with the values k = 0.545, oc = 1.0,
tr
ao = 1.016, p = 1.10, and T= 0.3126. The results of this
solution are listed in Table XXV and plotted in Figure 11.
TABLE XXV
Solution for x = 0.4, k = 0.545, aoc = 1.0, a tr = 1.016,
S 0
pr = sec=
p = 1.10,T= 0.3126, 1 pr 0.265, 1 se 0.233
0 0
oc
n X
0.0
0.1
0.2
0.4
0.5
0.6
0.8
0.9
0.95
2.977
2.180
1.785
1.224
1
.800
.447
.279
.190
sin 20
.50000
.36613
.29979
.20557
.16795
.13436
.075074
.046858
.031911
0 tr
0 X
45?0
37.2
33.2
27.0
24.2
21.5
15.9
12.5
10.3
3.364
2.395
1.921
1.256
1
777
.398
.231
.1426
sin 9
.50000
.35597
.28552
.18668
.14863
.11549
.059155
.034334
.021195
0
45?0
36.6
32.3
25.6
22.7
19.9
14.1
10.7
8.4
1.0 .0623 .010463 5.9
0.984=ni
1
.069 .010255 5.8
A brief inspection of this plot shows that the fit
is reasonably satisfactory except at the shoulders of the
eclipses. At the shoulders, the computed curve is sig
nificantly narrower than the "observed" points. This, in
essence, is the same situation that was found in the solu
tion of the first synthetic light curve. Further refine
ments of the solution, consistent with the assumption of
complete eclipses and the retention of the depth line
1 .tr
X = 0.3176, will not significantly improve this
oc
situation. It seems that there is probably some incom
patibility between the Russell model of prolate spheroids as
rectified to the spherical model and the theoretical model
which was used to generate the light curves. If the assump
tion that the theoretical model more closely approximates
physical reality for W Ursae Majoris stars is warranted, then
the Russell model has serious faults when applied to these
systems. It is well known, of course, that the Russell model
is only a rough approximation to the very close binary systems.
The real question, which this study will attempt to answer
partially, is whether a solution of such a system derived
from the Russell model will provide an adequate representation
of the true orbital elements of the binary system.
One way to somewhat improve the fit of a solution
to the'"observed" points is to abandon the depth line and
I
do a solution only from the shapes of the eclipses. This
may mean the introduction of the light of a third body to
the system. In the present case, the introduction of third
light (L3 = 0.234) yields the central eclipse solution listed
in Table XXVI and plotted in Figure 11. This solution im
proves the fit somewhat, though the shoulders of the com
puted curve are still too narrow. There seems to be no sound
observational reason for adding this third light. In about
20% of the solutions of eclipsing binary light curves in the
literature (13), some third light is present. This third
light solution is presented here in order to demonstrate
how the addition of third light affects the solution. Unless
there is some real physical source of this third light, how
ever, the necessity for its introduction to improve the so
lution only serves to point out the inadequacies of the
model on which the solution is based.
Orbital Elements and Figures of the Components
These two solutions, from the formulas given above,
were then used to derive orbital parameters and limiting
mass ratios for the assumption that neither component exceeds
the size of its Roche lobe. The results of these derivations
are listed in Tables XXVII and XXVIII.
Comparison of Solutions with Input Parameters
It is now possible to compare the solutions derived
TABLE XXVI
Solution for x = 0.4, k = 0.65, ac = 1.0, atr = 1.039,
po = 1.538,T = 0.447537, 1 Ypr = 0.265
1 sec = 0.233, L = 0.234
O 3
oc 2 tr 2
n xc sin2O 0 Xr sin2O
0.0 3.131 .52965 46?7 3.600 .52965 46?7
0.1 2.250 .38061 38.1 2.502 .36812 37.4
0.2 1.822 .30821 33.7 1.976 .29073 32.4
0.4 1.229 .20790 27.1 1.265 .18612 25.6
0.5 1 .16916 24.3 1 .14713 22.6
0.6 .800 .13533 21.6 .775 .11403 19.8
0.8 .462 .078152 16.2 .413 .060765 14.3
0.9 .314 .053116 13.3 .263 .038695 11.3
0.95 .240 .040598 11.6 .1867 .027469 9.5
1.0 .141 .023852 8.9
0.962=n. .162 .023835 8.9
1
TABLE XXVII
Orbital Elements for Solutions of Second
Synthetic Light Curve
k 0.545 0.65
aoc 1.0 1.0
0
tr 1.016 1.039
0
PO 1.10 1.538
a 0.466' 0.441
g
a 0.254 0.287
s
cos i 0.18663 0
r
i 792 900
r
L 0.735 0.501
9
L 0.265 0.265
s
L3 0 0.234
TABLE XXVIII
Figures of the Components
ms/mg .52 .17 .66 .25
a .466 .466 .441 .441
g
r .437 .451 .413 .426
g
b .437 .455 .412 .429
g
c .408 .432 .386 .408
g
a .254 .254 .287 .287
s
r .245 .233 .276 .264
s
b .244 .228 .274 .258
s
c .237 .217 .267 .247
s
above with the parameters used to generate the light curve.
The theoretical light curve is a trial and error match to
Broglia's observations of RZ Comae (14). This light curve
was generated by Dr. R. E. Wilson from the parameters
listed in Table XXIX. In this model, as in the model
used to generate the first synthetic light curve, the sizes
of both the components exceed the sizes of their Roche lobes.
In this case the boundary of the components lies along a
common envelope which is. smaller than the outer contact
surface. Broglia's observations of RZ Comae were solved
by Binnendijk (15). The solution of the synthetic light
curve (without third light) as listed in Table XXVII and
Table XXVIII is fairly close to Binnendijk's solution of the
real. observations of RZ Comae. This would seem to
indicate that this solution is the one which follows from
the Russell model and that it is not greatly sensitive to
effects of judgment and details of procedure. However
the true parameters used to derive the synthetic light
curve are not at all close to the solution given above or
to Binnendijk's. In fact, it is possible to derive a
theoretical light curve from the true parameters using the
Russell model. Such a light curve would look very different
from the synthetic light curve derived from the same para
meters using Wilson's model. The most obvious difference
is that the Russell model theoretical light curve would
have significantly wider and deeper eclipses than those
TABLE XXIX
Input Parameters for Second Synthetic Light Curve
i 86?00
L1 0.3149
L2 0.6851
x1 0.40
x2 0.40
r (pole) 0.2992
ri(side) .3132
rl (back) .3505
Xeff 5500 A
gl 1.00
g2 1.00
T1 5500K (polar)
T2 5563K(polar)
m2/m1 2.200
r2 (pole) 0.4287
r2(side) .'4579
k 0.69
r2 (back) .4880
01 5.449
02 5.449
A1 1.00
A2 1.00
3 0.000
present in the given synthetic light curve. The difference
in depth is especially obvious since the light lost in the
total eclipse is equal to the light of the smaller star. In
addition the annular eclipse in the theoretical Russell
Model light curve derived from the true parameters would be
deeper than the total eclipse, the reverse of the situation
in the synthetic light curve.
A comparison of the solutions derived above using
the RussellMerrill method (Table XXVII and Table XXVIII)
with the true parameters used to derive the synthetic light
curve (Table XXIX)' reveals that the RussellMerrill
solution of the light curve is completely unsuccessful in
predicting the elements used to derive the light curve
from Wilson's model. The situation here is similar to that
found in the solution of the first synthetic light curve.
Wilson's model, like Lucy's model, seems to be incompatible
with the Russell model as applied to the light curves of
W Ursae Majoris stars. Again, although Wilson's model and
the Russell model will produce similar light curves, both
of which resemble the light curves observed for W Ursae
Majoris type stars, the orbital elements which they use to
produce these similar light curves are quite different.
Unfortunately, there is no way to determine from the two
synthetic light curves whether Lucy's model and Wilson's
model are compatible. The evidence of the solutions of
the two synthetic light curves makes it clear that some
caution is in order in looking at published solutions for
the light curves of W Ursae Majoris type stars. Most of
these solutions are based on the geometrical model which
lies behind the RussellMerrill method of solution (or,
equivalently, other methods based upon the same model).
The solutions from this geometrical model (i.e. the Russell
model) are not compatible with the parameters used to
derive similar theoretical light curves from astrophysical
models such as Lucy's or Wilson's. While it is not possible
to determine in this study which of the three models best
represent the real situation in W Ursae Majoris type stars,
there is little doubt that there are serious problems in
applying the Russell model to these stars.
One of the problems encountered in the solutions of
both synthetic light curves was the fit of the solutions at
the shoulders of the eclipses. In both cases the "obser
vations" at the shoulders lie below the theoretical curves,
and the eclipses seem to be of longer duration observa
tionally than predicted by the Russell model. According to
Bookmyer "this particular discrepancy between the Russell
model and observations occurs in many W Ursae Majoris
systems" (16). Binnendijk also reports that "it has been
determined that in many light curves the observations
defining the shoulders of the eclipse curves are fainter
than expected from the Russell model" (1). Both authors
interpret this effect as evidence for a permanent distortion
86
in the shapes of the components; the facing hemispheres
of the components are more elongated than the Russell model
ellipsoids. Since the synthetic light curves show this
feature which is common to most W Ursae Majoris systems,
perhaps this is further evidence that the Russell model
does not apply to these systems.

PAGE 1
The Light Curves of W Ursae Majoris Systems by Ian Stuart Rudnick A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1972
PAGE 2
UNIVERSITY OF FLORIDA 3 1262 08666 451 2
PAGE 3
To my wife, Andrea
PAGE 4
ACKNOWLEDGMENTS The author sincerely expresses his appreciation to his conmiittee chairman and advisor, Dr. Frank BradshawJ Wood , for his comments and suggestions, which greatly aided the completion of this work. The author wishes to thank Dr. R. E. Wilson of the University of South Florida for provide ing one of the synthetic light curves and for serving on the author's committee. Thanks are also due to Drs. KY Chen and R. C. Isler for serving on the author's committee. The author expresses his gratitude to Dr. S. M. Rucinski for providing the other synthetic light curve. The author wishes to thank Drs. J. E. Merrill and J. K. Gleim for their many helpful discussions. Thanks are also due to R. M. Williamon and T. F. Collins for their help in obtaining some of the data and for many enlightening conversations. W. W. Richardson deserves highest commendation for his untiring work on the drawings. The author extends his thanks to the Department of Physics and Astronomy for providing financial support in the form of graduate assistantships, and to the Graduate School for support in the form of a Graduate School Fellowship*
PAGE 5
The author is indeed grateful to his parents and to his wife's parents for their encouragement. The author's wife deserves more than appreciation for her patience, encouragement, and hard work during five years of school life. Her devotion and understanding helped as nothing else could.
PAGE 6
TABLE OF CONTENTS Page ACKNOWLEDGMENTS . Hi LIST OF TABLES . vii LIST OF FIGURES . x ABSTRACT xii CHAPTER I INTRODUCTION 1 The Russell Model 2 CHAPTER II FIRST SYNTHETIC LIGHT CURVE 6 Rectification. 6 Solutions from Graphical Rectification 22 Solutions from Least Squares Rectification .... 44 Orbital Elements 55 Figures of the Components 58 Comparison of Solution with Input Parameters ... 59 CHAPTER III SECOND SYNTHETIC LIGHT CURVE. . 64 Rectification , 69 Solutions .......Â„ 75 Orbital Elements and Figures of the Components . . 78 Comparison of Solutions with Input Parameters. . . 78 CHAPTER IV THE SYSTEM OF 44i BOOTIS 87 History 87 Visual Binary 89 Spectroscopic Binary 91 Eclipsing Variable 92 Instrumentation. . 93 Observations 96 Rieductioncdf Data 98 Times of Minimxam Light and the Period. . 126 Variation in the period caused by motion in a visual binary system . . 126
PAGE 7
Page A recent period change 132 Light Variations. 133 Rectification . 144 Orbital Elements 146 CHAPTER V SUMMARY AND CONCLUSIONS 153 LIST OF REFERENCES 156 BIOGRAPHICAL SKETCH 161
PAGE 8
LIST OF TABLES Page I Observations of First Synthetic Light Curve,; .... 7 II Graphical Rectification Coefficients for First Synthetic Light Curve 13 III Least Squares Rectification Coefficients for First Synthetic Light Curve 15 IV Rectified First Synthetic Light Curve Using Graphical Rectification Coefficients 16 V Rectified First Synthetic Light Curve Using Least Squares Rectification Coefficients .... 17 VI Solutions with Different Values of Limb Darkening. . 25 VII Solution for x = 0.8, 1 Â£P^= 0.18, 1 Z^^^= 0.16, o o ' k = 0.45, p^ = 0.94, aÂ°^ = 0.9943, a^^= 0.9698. . . 26 VIII Solution for x = 1.0, 1 Â£P^= 0.18, 1 il^Â®^= 0.16, o o ' k =0.45, p^ = 0.90, aÂ°^= 0.9905, a^^= 0.9375 ... 27 IX Solutions for x = 0.6, 1 Z^^/9P^= 0.2142, (1 l^^) + o o o (1 1Â°^) =0.34 33 X Solution for x = 0.8, 1 l^^= 0.18, 1 1^^^= 0.16, o o ' k = 0.50, p = 0.6608, aÂ°^= 0.89, a^^= 0.7915 ... 34 XI Solution for x = 0.8, 1 Z^^= 0.18, 1 1^^^= 0.16, o o ' k = 0.55, p^ =0.4798, aÂ°^= 0.78, a^^= 0.6659. ... 35 XII Solution for x = 1.0, 1 Â£P^= 0.18, 1 Â£^Â®^= 0.16, o o k = 0.60, p = 0.3225, aÂ°^= 0.68, a^^= 0.5332 ... 36 ^ o o o VI 1
PAGE 9
Page OP "tT" XIII Shape Curves for 0^ = 50Â°, X = 3.955, x = oc 4.605;X = 0.8 and = 45?9, x = 3.476, ^^ = 4.047, X = 0.8 37 XIV Depth Curve for x = 0.8, 1 Â£^ = 0.16, 1 1^^= 0.18 41 XV Solution for x = 0.8, 1 i^^= 0.16, 1 !i^^^= oc 0.145, proc, k = 0.65, p = 0.1625, a^ = 0.55, a^^= 0.4548 47 o jpr_ Â„ ,, , Â„sec XVI Solution for x = 0.8, 1 5,^ = 0.16, 1 H^ 0.145, prtr, k = 0.65, p = 0.1271, aÂ°^= 0.525, a^''= 0.4314 ..48 o XVII Solution for X = 0.8, 1 2p^= 0.16, 1 Z^^^= 0.145, k = 1.0, p = 0.1540, a = 0.305 49 ^o o XVIII 0C's from Solution for k = 1, p = 0.1540, a = 0.305 51 o XIX Solution for x = 0.8, 1 1^^= 0.165, 1 ^q^*^= 0.145, k = 0.70, p = 1.429, aÂ°^= 1.0, aj^ = ^o o o 1.067, T= 0.558590, L^ = 0.588 ^^ XX Orbital Elements for Solutions of First Synthetic Light Curve 57 XXI Figures of the Components 60 XXII Observations of Second Synthetic Light Curve Â• Â• Â• 67 XXIII Rectification Coefficients for Second Synthetic Light Curve 70 XXIV Rectified Second Synthetic Light Curve j^. OC tr XXV Solution for x = 0.4, k = 0.545, a =1.0, aÂ„ = ' o o 1.016, p =1.10,T = 0.3126, 1 2.g^= 0.265, 1 i^^^= 0.233 76 Vlll
PAGE 10
Page oc XXVI Solution for x = 0.4, k = 0.65, a^ = 1.0, a^^= 1.039, p = 1.538,T = 0.447537, 1 il?^= o o ' o 0.265, 1 1^^^= 0.233, L,= 0.234 79 XXVII Orbital Elements for Solutions of Second Synthetic Light Curve 80 XXVIII Figures of the Components 81 XXIX Input Parameters for Second Synthetic Light Curve 83 XXX Comparison Stars . . 97 XXXI Observations of 44i Bootis 99 XXXII Standard Stars . 112 XXXIII Recent Times of Minimum Light 127 XXXIV Times of Minimiam Light with Corrections for Motion in a Visual Binary Orbit ,...,i.,.,. 131 XXXV Solutions and Orbital Elements Mth x = .6 , 1 _ jiP^= 0.23, 1 Â£^Â®Â°= 0.13, = 41?0, and o o ' e m /m^ = 0.50 150
PAGE 11
LIST OF FIGURES Page 1. Observations of First Synthetic Light Curve 8 2. Graphical Rectification Plots for First Synthetic Light Curve 12 3. Rectified First Synthetic Light Curve Using Graphical Rectification Coefficients . 19 4. Rectified First Synthetic Light Curve Using Least Squares Rectification Coefficients 21 5. Solution for X = 0.8, 1 Sp^= 0.18, 1 l^^^= 0.16, o o k = 0.45, p^ =0.94, aÂ°^= 0.9943, a^^= 0.9698. ... 29 6. Solution for x =1.0, 1 Â£P^= 0.18, 1 il^Â®Â°= 0.16, ' o o ' k = 0.45, p =0.90, 0.Â°^= 0.9905, a^^= 0.9375. ... 31 7. Depth Curve and Shape Curves 39 8. Superpositions of Primary and Secondary Eclipses . . 43 9. 0G's from Solution for k = 1, p = 0.1540, a = 0.305 Â°. . 52 o 10. Observations of Second Synthetic Light Curve .... 66 11. Rectified Second Synthetic Light Curve 74 12. Visual Binary Orbit of 44i Bootis 90 13. Comparison Star Extinction for July 1, 1970 110 14. Secondorder Extinction for Albireo 114 15. Firstorder Yellow Extinction for Albireo 117 16. Firstorder Blue Extinction for Albireo 119
PAGE 12
Page 17. Firstorder UltraViolet Extinction for Albireo . . 121 18. UBV Transformation Coefficients 123 19. Variation in the Period of 44i Bootis B Caused by Motion in a Visual Binary System 130 20. 0C's from Pbhl's Light Elements 134 21. 0C's from New Light Elements 135 22. Yellow Light Curve of 44i Bootis 136 23. Blue Light Curve of 44i Bootis 137 24. The Light Curves of Different Authors 139 25. The Light Curves of Different Authors 140 26. Deformities of the Light Curve of 44i Bootis on Two Nights 143
PAGE 13
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 The Light Curves of W Ursae Majoris Systems by Ian Stuart Rudnick June, 1972 Chairman: Frank Bradshaw Wood Major Department: Astronomy Two synthetic light curves computed from theoretical astrophysical models of W Ursae Majoris systems are discussed. Solutions of these light curves, based on the geometrical Russell model and the RussellMerrill method of solution of the light curves of eclipsing binaries, are presented. The relatively shallow minima caused by the partial eclipses of the first synthetic light curve lead to a problem of indeterminacy in the solution. The "observed" points in the shoulders of both synthetic light curves fall below the theoretical light curves predicted by the solutions. The addition of third light in the solution of the two synthetic
PAGE 14
light curves improves the fit of the solutions to the light curves; however, there is no sound basis for adding this third light. The orbital elements predicted by the RussellMerrill solutions of the two synthetic light curves are not at all close to the orbital elements used to generate these light curves from the theoretical astrophysical models. In particular the RussellMerrill solution underestimates the sizes of the components. It is concluded that the Russell model is not compatible with the theoretical astrophysical models used to generate the synthetic light curves. Observations of the system of 44i Bootis , an example of a W Ursae Majoris system, are also discussed. The times of minimum light indicate that an increase in the period occurred in 1967.
PAGE 15
CHAPTER I INTRODUCTION The W Ursae Majoris systems are eclipsing variable stars whose light curves have maxima which are strongly curved and minima which are nearly equal in depth. These systems have periods which average approximately onehalf day. Orbital solutions in the literature indicate that W Ursae Majoris systems are close binaries whose separations are less than the dimensions of the components. Many complexities which are caused by the proximity of the components appear in the light curves (1) . Most of the solutions of the light curves of W Ursae Majoris systems in the literature are based upon a geometrical model first proposed by Russell (2) . Lucy has proposed that some or most of the W Ursae Majoris systems may be true contact systems, whose common boundary follows a single equipotential surface (3). It is the purpose of this thesis to apply the RussellMerrill method of solution of the light curves of eclipsing binaries (2,4,5) to synthetic light curves from astrophysical models similar to Lucy's model in order to determine whether the geometrical Russell model is compati
PAGE 16
ble with the astrophysical models. Lucy (3), Rucinski (6), and Mochnacki and Doughty (7) have written computer programs for computing theoretical light curves of W Ursae Majoris system; Mochnacki and Doughty have published a trial and error fit to the system AW Ursae Majoris using their program. Wilson and Devinney (8) have published a general procedure for computing light curves of close binaries which includes the W Ursae Majoris systems as a special case. This procedure is now being applied to selected contact systems and the results will be published soon. Two synthetic light curves have been computed for this study, one by Rucinski (6) and one by Wilson (9) . In addition, observations of the system of 44i Bootis, an example of a W Ursae Majoris system, are discussed. The Russell Model The Russell model and the RussellMerrill method of solution of the light curves of eclipsing binaries are discussed in detail by Russell and Merrill (2) . A brief discussion is given here in order to define the notation used. The Merrill tables and nomographs for solution of light curves of eclipsing binaries are based upon a spherical model (4,5). It is assumed that the orbit is circular, that the stars are spherical, and that they appear darkened at the limb by a linear cosine darkening law. This limb
PAGE 17
darkening (x) may differ for the two components. The light of each star is constant for the spherical model; therefore the light of the system outside of eclipse is also constant. It is convenient to take the light of the system outside of eclipse as the unit of light and the radius of the spherical orbit as the unit lof component dimensions. Then the components are defined by the following parameters: Radius of the larger (greater) star r y Radius of the smaller star r Inclination of the orbit i Light of the larger (greater) star L y Light of the smaller star L with L + L =1. If 9 is the longitude in orbit (from g s conjunction) , then the apparent distance between the centers of the disks is given by 2 2 2 2 6 = cos i + sin i sin 6 Setting p = (6 r )/r , the eclipse will be absent, partial, g s or complete, for p>l, l>p>l, orp< 1 respectively. Setting k = r / r then 6 = r (1 + kp) . The quantities k ^ s g g and p are dimensionless, and their values completely define the geometrical circumstances of a given phase. If f and f represent the fractions of the light of g s the two stars which are obscured at any phase of the eclipse of either, and i is the normalized value of the light re
PAGE 18
ceived from the whole system: Â« = L^d f^) + L^d f^) = 1 Lgf^L^f^ For tabular purposes Russell and Merrill express these in terms of two other functions a and x, where a is the ratio of light lost at any phase of an eclipse to the loss at internal tangency, and x is that of the latter to the whole light of the star. Then for the light at any phase during an occultation (the larger star in front) 1 Â£ = f Lg = Lgtt (Xg,k,p) and during a transit (the smaller star in front) 1 _ 5,tr ^ ftr^^ ^ Lg x(Xg,k)a ^^(Xg,k,p) It is convenient to use the xfunctions of Russell and Merrill to determine the solution of the light curve. Defining n = a/a^ where the zero subscript refers to the value of the parameter at mideclipse, the X'f unctions are given as X(x,k,a^;n)= ^^^^^in) ^ [l+kp(x,k,nao)]^ sin2e(n=0.5) [1+kp (x,k,0 .5a^) ] ^ _ [l+kp(x,k/ao)]^ [l+kp(x,k,a^)]2
PAGE 19
These x~functions have been tabulated by Merrill (4). Russell and Merrill have shown that a system consisting of two similar triaxial ellipsoids with semiaxes a , b , g g c and a = ka , b = kb , c = kc can be rectified to g s g' s g s g the spherical model making, certain approximations which involve the gravity and reflection effects. The aaxis of each ellipsoid is along the line joining the centers of the components, the caxis is parallel to the axis rotation of the system, and the baxis is in the third mutually perpendicular direction. A mean radius r = (a + b + c)/3 may also be defined for these ellipsoids. The fundamental geometrical equations for this model (i.e. the Russell model) may be written as 2 2 2 2 2 cos i + sin x sin 9 = a (1 + kp) r r g where i and are the rectified inclination and phase angle (orbital longitude) respectively. This equation is identical in form with the equation for spherical stars. Thus the observed intensity and phase angle can be rectified in such a way as to produce a rectified light curve which will be nearly that produced by the eclipse of a pair of spherical stars of radii a , a = ka , and inclination iÂ„. g s g r
PAGE 20
CHAPTER II FIRST SYNTHETIC LIGHT CURVE A synthetic light curve for a W Ursae Majoris type eclipsing binary was generated by Dr. S. M. Rucinski from Lucy's model (6) . The "observational" data for this light curve consisted of 37 values of the normalized light (or flux) as a function of phase angle, with the phases given in 5Â° intervals from 0Â° to 180Â°. The other half of the light curve (180Â° to 360Â°) was assvimed to be symmetrical. An additional ten points were generated later to define better the centers of the eclipse regions of the light curve. This "observational" data is listed in Table I and plotted in Figure I. This synthetic light curve was to be treated as observational data and solved by the standard RussellMerrill method of solution of light curves of eclipsing binaries . No additional information about the nature of the derivation of this light curve was to be used in the solution. Rectification The first step in the process of getting a solution
PAGE 21
TABLE I Observations of First Synthetic Light Curve PHASE INTENSITY PHASE INTENSITY 0.00
PAGE 23
for the light curve was an analysis of the light outside the eclipses in order to arrive at a rectification of the light curve to the spherical model. Two methods were used for this analysis: Merrill's graphical method (10) and a least squares Fourier analysis of the material outside of eclipse. Let the light outside eclipse be represented by a truncated Fourier series of the form: I = A + A, cos + A^ cos 29 + A, cos 36 + A. cos 4( o 1 2 3 4 (Since the light curve is symmetrical about 9 = 180Â°, it is not necessary to include sine terms in the above Fourier series i') Following Merrill's graphical method, a and b represent readings for 9 and 180" 6 on the light curve. (The given "observational" points were used rather than reading from a freehand curve since the scatter of the points was small.) It immediately follows that 2(a + b) = A^ + A^ cos 29 + A^ cos 4( p(a b) = Acos9 + A^ cos 3( Letting C, = ^(a b) and C^ = yC^ + b) , by simple trigonometric substitution, C, = (A, 3A^) cos + 4A2 cos 36
PAGE 24
10 C2 = (Aq A^) + A2 cos 26 + 2A^ cos^ 29 Therefore a plot of C^^ versus cos 6 would have the form of a cubic and a plot of Q.^ versus cos 20 would have the form of a parabola. Such plots are given in Figure 2. The plot of C^ versus cos 9 shows the presence of a considerable A(cos 30) term and indicates the possible presence of higher order odd cosine terms. The plot of Q^ versus cos 20 is essentially linear, indicating that the A^ term is negligible. The plotted points seem to fall below this straight line somewhere around 40Â°, thus locating (the value of 9 at external tangency) to a first approximation. Values of the Fourier cosine coefficients were then derived from the plots and are listed in Table II. Because of the similarity of the "colors" of the two "stars" and the small difference in the depths of the two minima, rectification coefficients for the reflection effect were obtained in the following manner (2): ^1 = ^1 C = 0.090 sin^ 9 o e C, = 0.030 sin^ 2 e The eclipses were assumed to be partial and values were derived for 0^ = 39Â°; these values are listed in Table II,
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p
PAGE 26
12
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13 TABLE II Graphical Rectification Coefficients for First Synthetic Light Curve e
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14 Also listed in Table II is the value of z, given by 4(A2 C2) N(^o ^o ^2 ^ ^2^ where N depends on the assiomed limb darkening (x) ', values were adopted such that N = 2.6, 3.2, or 4.0 when x is assxuned to be 0.6, 0.8 or 1.0 respectively. Fourier coefficients were also computed by a least squares Fourier analysis of the light outside of eclipses (45Â° to 135Â°) . The results of this analysis are listed in Table III. Both sets of rectification coefficients were then used to compute a rectified light curve. The rectified intensity is given by I+C +C, cos e+C^ cos 29 A^ cos 36 A^ cos 4( I" = ^^ ^ ,^.^^ ^^^ . .V.2 Â— ^. Â«3 v,^^ ... Â«4 A +C + (A^+Co) cos 2( 00 2 2 and the rectified phase by sin^e = ^ I 2. 1z cos f The rectified points are listed in Tables IV and V and plotted in Figures 3 and 4. Several things are apparent from an examination of
PAGE 29
15 TABLE III Least Squares Rectification Coefficients for First Synthetic Light Curve
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16 TABLE IV Rectified First Synthetic Light Curve Using Graphical Rectification Coefficients 0(x=.6) 0(x=.8) 0(x=l) I" ( e ) I" (180+9 P; 0.000
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17 TABLE V Rectified First Synthetic Light Curve Using Least Squares Rectification Coefficients 0.4524 0.4483 0.4522 0.4643 0.4826 0.4973 0.5188 0.5452 0.5652 0.5881 0.6109 0.6557 0.7051 0.7497 0.7927 0.8309 0.8692 0.9045 0.9294 0.9564 0.9775 0.9912 0.9963 1.0000 0.9934 0.9847 0.9695 0.9477 0.9255 0.9014 0.8681 0.8369 0.8053 I" 0.8469
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18 Table V continued. 0.7712 0.7382 0.7022 0.6602 0.6411 0.6237 0.6059 0.5851 0.5652 0.5537 0.5424 0.5353 0.5340 0.5387 I" 0.9977
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19
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1
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21
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22 either of the rectified light curves. The first is that the amount of scatter of the data points in eclipse is much larger than was originally expected. This scatter will be treated as "observational" error for the present and no attempt will be made to explain it either in terms of the model or the method of computation of the data. Another obvious feature of the rectified minima is the "brightening" at the centers of the eclipses. Again no attempt will be made at present to explain this effect; however, its presence^ creates a serious uncertainty in both the depths of the eclipses and the credibility of the points near the center in terms of the Russell model. Solutions from Graphical Rectification The rectified light curve produced by applying the rectification coefficients derived by the graphical method was used for the first attempts at a solution. The phase was rectified with a z based on N = 2.6 corresponding to a darkening x = 0.6, and the first solutions were tried on the X = 0.6 nomograph. The depths of the eclipses were chosen as 1 ij^^ =0.18 and 1 l^^^^ =0.16 and the following values were read off the plot of the rectified light curve: n sin^e x^"" e sin^e x^Â®"" 0.2 33?0 .29663 2.268 32?6 .29027 1.901 0.5 21.2 .13077 1 23.0 .15267 1 0.8 12.2 .044658 .342 15.0 .066987 .439
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23 By choosing values of 0(n = .5) and 9{n = .8) PIT at 9 + 0?5, a permissible range of .301 <_ % (n = .8) <_ .387 and .394 < ^^^^(n = .8) < .488 was found. Since ^secjj^ = .8) > x^^^^ ~ '^^ ' ^^ appears that the primary eclipse is a transit and the secondary eclipse is an occultation. Then the values needed for the depth line on the nomograph are (1 JlÂ°Â°) + (1 ll"^) =0.34 1 il^/iÂ°^ = 0.2142 1 Â£oVilÂ°'' + (1/50) 1 Â£Â°''/iiÂ°Â°= 0.2180 Taking these values and the x(n = 0.8) values given above to the X = 0.6 nomograph, an intersection of the depth line and the permissible values of both x contours was found with k = 0.45, p =.98, a^^ = .9987, a^^ = .9928. When these elements are taken to the x tables, they produce the following points on the light curve: XÂ°^ 0.0 2.823 0.2 1.725 0.5 1 0.8 .448 Comparing these values with those taken from the "observed" light curve, several things are apparent. First, Â• 2Â„ sin
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24 the fit from the halfway point down is moderately satisfactory for this preliminary stage of solution. Second, the fit at the shoulders is very bad; the computed curve is much narrower than the "observed" curve. The x (n < .5) values for both eclipses need to be increased a significant amount while making only small changes in the x(n > .5) values, in order to fit the "observed" light curve with the chosen depths of the eclipses. Since the nomographic solution on darkening X = 0.6 as described above was not satisfactory, other possibilities were explored. The first approach was to try nomographic solutions with other values of darkening. Table VI summarizes the results of these attempts. The main conclusion from this exploration of solutions with different values of darkening is that the fit will improve with increasing darkening. There are two reasons for this improvement. First the x (n < .5) values for the transit eclipse tend to increase with increasing darkening, and second, the value of z is smaller for larger darkening. With a smaller z, the rectification of the phase tends to make the shoulders narrower relative to the halfwidth than with a larger z. Thus, it appeared that a darkening of X = .8 or X = 1 should be used for further trial solutions. The results of these solutions with k = .45 (see Tables VII and VIII and Figures 5 and 6) were not at all satisfactory. It was still not possible to fit the
PAGE 39
25 TABLE VI Solutions with Different Values of Liitib Darkening X
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26 TABLE VII Solution for x = 0.8, 1 il^^ = 0.18, 1 Jl^Â®^= 0.16, k = 0.45, p = 0.94, aÂ°^ = 0.9943, a^^ = 0.9698 n
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27 TABLE VIII Solution for x = 1.0, 1 l^^ = 0.18, 1 Â£^Â®^ =0.16 ' o o k = 0.45, p = 0.90, aÂ°^ = 0.9905, a^^ = 0.9375 '^O o o 0.0 3.801 2.911 .47732 43?7 5.412 3.939 .47732 43.7 0.1 2.257 2.054 .33675 35.5 3.082 2.532 .30687 33.6 0.2 1.625 1.703 .27922 31.9 2.186 1.991 .24132 29.4 0.4 .7274 1.205 .19750 26.4 .9965 1.273 .15430 23.1 0.5 .3590 1 .16396 23.9 .5439 1 .12119 20.4 0.6 .0191 .811 .13302 21.4 .1465 .760 .092118 17.6 0.8 .6275 .452 .074151 15.8 .5306 .351 .042584 11.9 0.9 .9684 .263 .043116 12.0 .8290 .171 .20755 8.3 0.95 1.1648 .1539 .025236 9.1 .9717 .0851 .010315 5.8
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29 M O
PAGE 45
31
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32 shoulders of the eclipses, especially the primary eclipse, while, at the same time, fitting the rest of the eclipse curves. In particular, with k = .45, the solutions for a circular orbit were too wide at the halfway point of the secondary eclipse and too narrow on the shoulders of the primary eclipse, with lesser problems elsewhere. Because of the problems described above, it was necessary to abandon the chosen value of k = 0.45 and to explore other possibilities of k on the x = 0.6 nomograph, staying on the depth line given above. Table IX summarizes the results of these explorations. From an inspection of this table, it was decided to attempt trial solutions with k in the range .50 to .60 for darkenings x = 0.8 and total darkening. Some of these solutions are given in Tables X, XI, and XII. While these solutions are an improvement over previous ones, the major problem of the fit of the shoulders, especially in the primary eclipse, has not been alleviated. Because of the recurrent problem with the fit of the shoulders of the eclipses, it was felt that a different approach might prove helpful. This approach was to choose a (n = .5) for the primary eclipse and a 9 . From these values, X (ii 0)'s were derived. These X's were taken to oc the X = . 8 tables and values of k and a were obtained o and these shape curves were plotted (see Table XIII and Figure 7) . The depth curve derived from aÂ° = (1 1Â°"") + (1 lT)/q^
PAGE 47
33 TABLE IX Solutions for x = 0.6, 1 l^^/n!^^ =0.2142 o o (1 9.^) + (1 iiÂ°^) =0.34 k 1.00 .85 .75 .60 .55 .50 .45 Pq .0956 .0000 .1000 .3538 .4820 .6789 .9800 a^oc .3400 .4136 .4900 .6800 .7700 .8900 .9987 a^tr .3400 .3788 .4363 .6018 .6897 .8224 .9928 XÂ°^(n=.8) .346 .346 .347 .354 .361 .379 .448 x''^^(n=.8) .346 .344 .343 .343 .344 .348 .376 oc X (n=.2) 1.969 1.963 1.956 1.927 1.902 1.853 1.725 X^^(n=.2) 1.969 1.976 1.982 1.986 1.983 1.972 1.913
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34 TABLE X Solution for x = 0.8, l;iP^ = 0.18, 1l^^^ = 0.16, o o k = 0.50, p = 0.6608, aÂ°^ = 0.89, a^^ = 0.7915 n
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35 TABLE XI Solution for x = 0.8, 1 l^^ = 0.18, 1 l^^^= 0.16, o o ' k = 0.55, p^ = 0.4798, aÂ°^ = 0.78, a^^ = 0.6659 '^o ' o o n:
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36 TABLE XII Solution for X = 1.0, 1 Ji^^ = 0.18, 1 H^^^ = 0.16, k = 0.60, p = 0.3225, aÂ°^ = 0.68, a'^^ = 0.5332 ' '^O o o , oc oc . 2n r\ . tr tr . 2^ Â„ n t\) X sin ij; X sin 0.0 4.495 3.582 .47732 43?7 6.349 3.982 .47732 43?7 0.1 2.910 2.392 .31875 34.4 4.186 2.562 .30710 33.7 0.2 2.276 1.916 .25532 30.3 3.349 2.012 .24117 29.4 0.4 1.4016 1.259 .16777 24.2 2.233 1.280 .15343 23.1 0.5 1.0562 1 .133255 21.4 1.8073 1 .119869 20.3 0.6 .7471 .768 .102340 18.7 1.4350 .756 .090621 17.5 0.8 .2032 .360 .047972 12.7 .0843 .342 .040995 11.7 0.9 .0424 .175 .023320 8.8 .5327 .163 .019539 8.0 0.95 ;16b2 .0868 .011567 6.2 .4061 .0800 .009590 ::5;6
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37 TABLE XIII Shape Curve for = 50Â°, xÂ°^ = 3.955, X^^ = 4.605, X = 0.8 k
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39
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40 oc with k(a , q ) taken from the x = .8 tables was also derived and plotted (Table XIV and Figure 7) . An inspection of this plot shows that for =50Â° and 0^ = 45?9, the ^ e e intersection of the shape curves for the two eclipses in both cases lies above the depth curve. This implies the possibility of a solution from the shapes alone, abandoning the depth curve and therefore the assumption that L, + L2= 1. This socalled third light solution is indeed one way to produce a theoretical light curve that will fit the observed curve. There is, however, no real justification for assuming the presence of this third light in the present case and therefore, this possibility was rejected. It had been noticed from the first plotting of this rectified light curve that a number of the points in the primary and secondary eclipses were similar (i.e. for a given value of 0, I" was nearly the same for both eclipses) . In attempting solutions for k = 1, this similarity became even more apparent and it was decided to superimpose the plots of the two eclipses (see Figure 8) . The result was remarkable, from the shoulders down to a depth of approximately n = 0.8, the two eclipses were virtually identical; deeper than this point, there was a sharp divergence. Since this type of behavior is not possible in a system described by the Russell model, either the points near mideclipse had to be completely abandoned, or the rectification itself might be at fault, and a new rectification could be tried. It was decided to try to find a solution on the other
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41 TABLE XIV oc Depth Curve for x = 0.8, 1ii =0.16,
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43
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44 rectification mentioned above, i.e. the one whose coefficients were derived by the least squares method of Fourier analysis. Solutions from Least Squares Rectification There are several differences apparent in looking at the two curves produced by the two different sets of rectification coefficients. Outside the eclipses, the residuals are more or less evenly distributed throughout the whole curve in the least squares rectification, with the sum of the squares of the residuals smaller than that from the graphical rectification, as expected. Inside the eclipses the two curves are no longer virtually identical from the shoulders down to an approximate level of n = 0.8; in general, for any given phase the primary is deeper than the secondary. The greatest change in the new rectification, however, is in the depths of the two eclipses. Both eclipses are significantly shallower, the depth of the primary eclipse going from about 0.18 to 0.16 and the depth of the secondary eclipse going from about 0.16 to 0.145. A nomographic solution of this second rectified light curve was then attempted. The light curve was rectified in phase with a value of z based on N = 3.2 corresponding to a darkening x = 0.8 and the solution was tried on the X = 0.8 nomograph. The value of x = 0.8 was chosen for convenience since there is neither a nomograph nor a
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45 set of X tables for complete darkening and the previous explorations of the light curve seemed to indicate a large value for the darkening. The depths of the eclipses were chosen as 1 l^^ = 0.16 and 1 iljÂ®^ = 0.14 and the o o following values were read off the plot of the rectified light
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46 parameters.
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47 TABLE XV Solution for x = 0.8, 1 i^^ = 0.16, 1 5^^^ = 0.145, proc, k = 0.65, p^= 0.1625, aÂ°^= 0.55,a^^=^v4548 n
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48 TABLE XVI Solution for X = 0.8, 1 l^^ = 0.16, 1 l^^^ = 0.145, ' o ' o ' prtr, k = 0.65, p = 0.1271, aÂ°^ = 0.525, a^^ = 0.4314 ^ '^O o o n
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49 TABLE XVII Solution for X = 0.8, 1 ii^^ = 0.16, 1 i^^^ = 0.145, k = 1.0, p^ = 0.1540, a = 0.305 n X 0.0 3.626 0.1 2.475 0.2 1.969 0.4 1.271 0.5 1 0.6 .760 0.8 .347 0.9 .167 0.95 .0818 2 sin
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50 difficult. Because of the above difficulties, a provisional' solution with the parameters 1 i^^ = 0.16, 1 Â£^Â®*^ = 0.145, k = 1.0, a^ = .305, and p^ = .1540 was chosen. The computed curve was plotted (Figure 4) and values of observed minus computed (0 c) for each observed point were derived graphically. These (0 C) values are listed in Table XVIII and plotted as a function of phase in Figure 9 . From an inspection of this plot, it appears that a higher order cosine term (e.g. cos 9 0) is present, especially in the primary eclipse. The possibility of using Kitamura's method for the solution of eclipsing binary light curves (IT) was also explored. Kitamura's method has the advantage of using the whole light curve to find a provisional solution rather than using a few selected points as is done in the nomographic method. Since both methods of solution are based upon the same geometrical model for the eclipses, their final results should agree. Even though the two methods might produce different provisional solutions, a careful and thorough analysis based upon these different preliminary solutions should finally produce the same final solution. Any difference in the results has to be caused by improper application of one of the methods. Since solutions using Kitamura's method should not differ from those derived using the RussellMerrill method, Kitamura's method was
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51 TABLE XVIII 0C's from Solution for k = 1, p. = 0.1540, a = 0.305. OCO) OC{180 +0) 0.000 .007 .017 2.945 .003 .007 5.885 .007 .000 8.816 .003 .002 11.735 .003 .002 14.637 .000 .004 17.519 .003 .002 20.377 .011 .007 23.208 .006 .005 26.011 .004 .002 28.781 .001 .001 34.221 .007 .003 39.518 .006 .001
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52
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53 not pursued any further. Before going on to find the geometrical elements corresponding to these possible solutions, a further exploration of the possibility that the eclipses are complete might be in order. By making this assumption, it is possible to find a solution from the shapes of the eclipses alone (i.e. , by not assuming that L, + L^ = 1) . While this introduction of third light seems somewhat unwarranted, it is presented here for the sake of completeness. The shapes of the eclipses were used to derive the solution given in Table X IX and plotted in Figure 4 . This solution assumes that the eclipses are complete and, in fact, takes the limiting situation of central eclipses (i.e., i = 90Â°). These assumptions require that more than half of the total light of the system come; from some unknown third body (L_ = 0.588). This solution seems to fit the light curve about as well as the previously mentioned solutions. In rectifying the phase, the ellipticity z is a function of a parameter N which goes as N = (15 + X) (1 + y) 15 5x where x is the limb darkening and y is the gravity darkening of the star being eclipsed. In the rectification of the phase described above, a value of N = 2.6 corresponding to
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54 TABLE Xix: Solution for X = 0.8, 1 Ji^^ = 0.165, 1 ii^Â®^ = 0.145, k = 0.70, p^ = 1.429, aÂ°^ = 1.0, a^^ = 1.067,
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55 a limb darkening of x = 0.6 implies that a gravity darkening of y = 1.0 was chosen. Dr. S. Rucinski (6) has suggested that a value of gravity darkening y = 0.32 might be more appropriate for this system. This possibility was explored and it was found to have no significant effect on the problems encountered in finding a solution for this system. Therefore, it was decided to proceed with the solutions already derived. Orbital Elements The next step was to derive orbital elements for the eclipsing system from the geometrical eclipse parameters. These elements can most easily be derived from the formulas given by Merrill (5) : 2 o sm a^ = ^ (1 + k)^ (1 + kp )^cos^0 o e a = ka s g cos i^ = (1 + kp^)ag T , Â„OC / oc L =11 /a s o ' o L^ = 1 L g s
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56 The derived orbital elements for the various solutions are listed in Table XX. Mean errors (y = \/z(0 C)^/(n 1)) were derived for the "observed" points from each of the four possible solutions. These mean errors are listed in Table XX. The mean error has almost the same value for all three of the partial eclipse solutions (with no third light added) . The value of the mean error for the complete eclipse solution (with third light) is slightly smaller than the values of the mean error for the other three solutions. Since these four solutions with very different values of the geometrical elements produce computed light curves which fit the observed light curve more or less equally well, it would not be meaningful to apply the method of differential corrections to find a least squares solution of the light curve. In the absence of a formal least squares solution, it is not possible to determine mean errors for each of the geometrical elements. From the results presented in Table XX, however, it is possible to estimate a reasonable range of each element. For the three partial eclipse solutions (without third light) the approximate ranges of these elements are: .38
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57 TABLE XX Orbital Elements for Solutions of First Synthetic Light Curve
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58 presented here, the value of was chosen as 44Â° in order e to best fit the whole light curve. The observations on the shoulders of the eclipses alone indicate that this value might be increased by several degrees. The effect of increasing is to increase the values of a and e g a (i.e., to increase the sizes of the stars relative to their separation) and to decrease the inclination of the orbit. Figures of the Components The figures of the components can now be computed from the formulas given by Merrill (10) : r + (0.17 + 1.19 m /m )r^ = a g s^ g' g g bg= ag 1.53(m^/mg)r^ c^ = 3r a b g g g g These formulas require that the mass ratio of the components be known. Since this mass ratio was not known, a more indirect method was used to set limits on both the figures of the components and their mass ratios. This method assumes that no dimension of the star may exceed the size of the Roche lobe it is contained within. The dimensions of the Roche lobes for various mass ratios are
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59 tabulated in the literature (e.g., Kopal (12)). The figures of the components were computed for a number of different mass ratios for each of the solutions. (The two solutions for k = .65 were combined for this purpose since the values of a and a in each were nearly equals ) These figures were compared with the size of the Roche lobe in order to determine the limiting mass ratios which would satisfy the criterion that the sizes of the stars must not exceed the sizes of their Roche lobes. These limiting mass ratios and the figures of the components derived using them for each of the solutions are listed in Table XXI. Comparison of Solutions with Input Parameters It is: now possible to compare the solutions derived above with the parameters used to generate the light curve. The light curve was produced using Lucy's model for the light curves of W Ursae Majoris stars. In this model the surfaces of the two stars are Roche equipotential surfaces and for the Rucinski light curve the stars share a common envelope which fills the outer contact surface. The constant which defined this surface is C = 3.5591 with the mass ratio q = m /m = 0.4. The 5 y Other parameters used in deriving the light curve are the inclination of the orbit i = 82Â°, the effective temperature of the surface T = 5700Â°, the wavelength of the light
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60 TABLE XXI Figures of the Components k
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61 X = 5500 A, the limb darkening u = 0.6, and the gravity darkening y = 0.32. The size of the equipotential surface (and therefore the stars) was also given. These values are b = 0.531, c = 0.482, b =0.363, and c =0.337 :(60 g g s s These parameters can be compared with the results of the solutions as listed in Table XX and Table XXI. The inclination of the orbit suggests that a solution using the RussellMerrill method which has this inclination will include the presence of some third light. The figures of the components (with q = 0.4) for such a solution can be of roughly the same shape in the yzplane (the plane containing the b and c axes of the components) as the shapes of the lobes of the outer contact surface in the yzplane. The RussellMerrill dimensions of the components in this plane relative to the separation of the components are considerably smaller than the dimensions of the lobe of the outer contact surface in the yzplane. Although it is possible, as previously mentioned, to increase slightly the dimensions of the components by increasing the values of in the solution, it is not possible to increase the dimensions of the components to anywhere near the size of the outer contact surface and still have the value of compatible with the light curve. Using the Russell model of similar prolate ellipsoids, it is possible to propose a system consisting of two components whose dimensions in the yzplane are nearly as large as the dimensions of the
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62 lobes of the outer contact surface in the yzplane. These prolate ellipsoids would be in contact (i.e., a + a^ = 1) . There are a number of objections to this proposed system. First of all, it is not possible for real stars in contact to have ellipsoidal shapes. Even if such a system were physically possible, it would produce eclipses with e^ = 90Â°, in contrast with the values (44Â°< Q^ < 50Â°) given by the light curve. Even the physically realistic system with the two components filling their Roche lobes (inner contact surfaces) produces eclipses that are wider (for i = 82Â°) than those in the light curve (12). In this case the sizes of the components in the yzplane are also considerably smaller than the sizes of the lobes of the outer contact surface in the yzplane. As the above discussion indicates, the solutions produced by applying the RussellMerrill method to the light curve give very different values for the orbital elements than the parameters used to generate the light curve. This is especially true if third light is not included in the solution. Then, not only are the stars much smaller than the outer contact surface, but the inclination of the orbit is also quite different from the inclination used in deriving the light curve. Leaving aside the question of third light, since the amount of third light cannot be determined from the light curve alone, it is apparent that the RussellMerrill solution of the light curve is completely unsuccessful in predicting the elements used to derive the light curve from Lucy's model. Thus these two
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63 models for the light curves of W Ursae Ma j oris stars seem to be incompatible. While the models can produce similar light curves, both of which resemble the light curves observed for W Ursae Majoris type stars, the orbital elements which they use to produce these similar light curves are quite different. Since most of the solutions of the light curves of W Ursae Majoris type stars which are in the literature are based upon the geometrical model lying behind the RussellMerrill method of solution (or, equivalently, other methods based upon the same model) , much caution is in order in looking at these solutions. If Lucy's model represents the actual physical situation for a W Ursae Majoris type star, then most of the solutions in the literature probably do not realistically represent the actual physical situation present in the W Ursae Majoris type stars.
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CHAPTER III SECOND SYNTHETIC LIGHT CURVE A second synthetic light curve for a W Ursae Ma j oris type eclipsing binary was generated from a theoretical model by Dr. R. E. Wilson (9) . The "observational" data for this light curve consisted of 51 values of the differential stellar magnitude as a function of phase, with the phases given in intervals of onehundredth of the period from phase 0.00 to phase 0.50. The remaining half of the light curve from phase 0.50 to phase 1.00 was assumed to be symmetrical. This "observational" data is plotted in Figure 10 and listed in Table XXII. (A normalization constant equal to 0.2049 was subtracted from each of the magnitudes;) Also listed in this table are the corresponding light values and phase angles in degrees for each data point. The synthetic light curve was to be treated as observational data and a solution was to be obtained by using the standard RussellMerrill method of solution for light curves of eclipsing binaries. No additional information about the nature of the derivation of this light curve was to be used in the solution. 64
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en u Â•H p p CO o o 0) w m o CQ o H P (d > M Q) m Xi O t en H
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66
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67 TABLE XXII Observations of Second Synthetic Light Curve PHASE
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Table XXII continued. 68 PHASE
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69 Rectification As in the previous synthetic light curve solution, the first step is an analysis of the light outside the eclipses in order to rectify the light curve to the spherical model. This analysis was done by a leastsquares Fourier analysis of the light outside of eclipse (48Â° to 132Â°) . The results of this analysis are listed in Table XXIII. The light values were then rectified by the formula : I+C +Ct cos e+C^ cos 2eAcos 3eA, cos 49 jii _ o 1 2 3 4 A +C + (A^+C^) cos 26 O O 2 2 and the phase angles were rectified by the formula 2 2 2 sin = sin 6/(1 z cos 6) where z = 4 (A2 C2)/N(Aq C^ A2 + C2) with N = 2.2, corresponding to a limb darkening x = 0.4. The rectification coefficients for reflection were determined from the statistical formula given by Russell and Merrill (2) : C = 0.072 sin^e = 0.0398, C, = A,, and C= 0.024 sin 6 = 0.0133, e 11 2 e with 6 =48Â°. Following Merrill (10), the A^ and A^ terms were removed by subtraction. The rectified points are listed in Table XXIV and plotted in Figure 11.
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70 TABLE ; XXI 1 1 Rectification Coefficients for Second Synthetic Light Curve
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71 TABLE XXIV Rectified Second Synthetic Light Curve I I" 0.5297
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72 Table XXIV continued, I" 0.9404
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74
PAGE 89
75 Solutions An examination of the rectified light curve reveals several things. The most obvious is that the eclipses are apparently complete. Furthermore it appears that the primary eclipse is the occultation (total) eclipse and the secondary eclipse is the transit (annular) eclipse. Closer examination reveals a very slight brightening in the center of the primary eclipse. The flatness of both eclipses indicates that the limb darkening is probably small. The depths of the two eclipses were chosen as 1 5,^ = 0.265 Sep and 1 a = 0.233 for the purpose of obtaining a preliminary nomographic solution. Then the following values were read off the plot of the rectified light curve: 0P^ sin^e xP"" e^^^ sin^Q f^"" 0.2 33.5 Â•=^..30463 1.841 33.1 .29823 1.986 0.5 24.0 .16543 1 22.8 .15017 1 0.8 15.8 .074137 .448 14.4 .061847 .412 X^^(n = 0.8) > x^^*^ (n = 0.8), confiinning that the primary eclipse is the occultation eclipse and the secondary eclipse is the transit eclipse. These x values along with a depth line 1 i^^/SL^^^ = 0.3176 were taken to the total ^ o o eclipse portion of the x = 0.4 nomographs and a satisfactory intersection of the three curves was found. This intersection oc implied a solution with the values k = 0.545, a =1.0, a^^= 1.016, p = 1.10, and t= 0.3126. The results of this o ^o solution are listed in Table XXV and plotted in Figure 11.
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76 TABLE XXV Solution for x = 0.4, k = 0.545, a Â°^ = 1.0, a ^^ = 1.016, = 1.10, T= 0.3126, 1 ll"^ = 0.265, 1 " i? = = 0.233 n
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la A brief inspection of this plot shows that the fit is reasohably satisfactory except at the shoulders of the eclipses. At the shoulders, the computed curve is significantly narrower than the "observed" points. This , in essence, is the same situation that was found in the solution of the first synthetic light curve. Further refinements of the solution, consistent with the assumption of complete eclipses and the retention of the depth line 2Â— = 0.3176, will not significantly improve this *o situation. It seems that there is probably some incompatibility between the Russell model of prolate spheroids as rectified to the spherical model and the theoretical model which was used to generate the light curves. If the assumption that the theoretical model more closely approximates p physical reality for W Ursae Majoris stars is warranted, then the Russell model has serious faults when applied to these systems. It is well known, of course, that the Russell model is only a rough approximation to the very close binary systems. The real question, which this study will attempt to answer partially, is whether a solution of such a system derived from the Russell model will provide an adequate representation of the true orbital elements of the binary system. One way to somewhat improve the fit of a solution to the' "observed" points is to abandon the depth line and
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78 do a solution only from the shapes of the eclipses. This may mean the introduction of the light of a third body to the system. In the present case, the introduction of third light (L. = 0.234) yields the central eclipse solution listed in Table XXVI and plotted in Figure 11. This solution improves the fit somewhat, though the shoulders of the computed curve are still too narrow. There seems to be no sound observational reason for adding this third light. In about 20% of the solutions of eclipsing binary light curves in the literature (13) , some third light is present. This third light solution is presented here in order to demonstrate how the addition of third light affects the solution. Unless there is some real physical source of this third light, however, the necessity for its introduction to improve the solution only serves to point out the inadequacies of the model on which the solution is based. Orbital Elements and Figures of the Components These two solutions, from the formulas given above, were then used to derive orbital parameters and limiting mass ratios for the assumption that neither component exceeds the size of its Roche lobe. The results of these derivations are listed in Tables XXVII and XXVIII. Comparison of Solutions with Input Parameters It is now possible to compare the solutions derived
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79 TABLE XXVI Solution for X = 0.4, k = 0.65, aÂ°^ = 1.0, a^^ = 1.039, P = 1.538,T = 0.447537, 1 i^^ = 0.265> sec 1 i^^^ = 0.233, L3 = 0.234 n
PAGE 94
80 TABLE XXVII Orbital Elements for Solutions of Second Synthetic Light Curve k
PAGE 95
TABLE XXVIII Figures of the Components m^/nig
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82 above with the parameters used to generate the light curve. The theoretical light curve is a trial and error match to Broglia's observations of RZ Comae (14). This light curve was generated by Dr. R. E. Wilson from the parameters listed in Table XXIX. In this model, as in the model used to generate the first synthetic light curve, the sizes of both the components exceed the sizes of their Roche lobes. In this case the boundary of the components lies along a common envelope which is. smaller than the outer contact surface. Broglia's observations of RZ Comae were solved by Binnendijk (15) . The solution of the synthetic light curve (without third light) as listed in Table XXVII and Table XXVIII is fairly close to Binnendijk 's solution of the real : observations of RZ Comae. This would seem to indicate that this solution is the one which follows from the Russell model and that it is not greatly sensitive to effects of judgment and details of procedure. However the true parameters used to derive the synthetic light curve are not at all close to the solution given above or to Binnendijk' s. In fact, it is possible to derive a theoretical light curve from the true parameters using the Russell model. Such a light curve would look very different from the synthetic light curve derived from the same parameters using Wilson's model. The most obvious difference is that the Russell model theoretical light curve would have significantly wider and deeper eclipses than those
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83 TABLE XXIX Input Parameters for Second Synthetic Light Curve i
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84 present in the given synthetic light curve. The difference in depth is especially obvious since the light lost in the total eclipse is equal to the light of the smaller star. In addition the annular eclipse in the theoretical Russell Model light curve derived from the true parameters would be deeper than the total eclipse, the reverse of the situation in the synthetic light curve. A comparison of the solutions derived above using the RussellMerrill method (Table XXVII and Table XXVIII) with the true parameters used to derive the synthetic light curve (Table XX IX)) reveals that the RussellMerrill solution of the light curve is completely unsuccessful in predicting the elements used to derive the light curve from Wilson's model. The situation here is similar to that found in the solution of the first synthetic light curve. Wilson's model, like Lucy's model, seems to be incompatible with the Russell model as applied to the light curves of W Ursae Majoris stars. Again, although Wilson's model and the Russell model will produce similar light curves, both of which resemble the light curves observed for W Ursae Majoris type stars, the orbital elements which they use to produce these similar light curves are quite different. Unfortunately, there is no way to determine from the two synthetic light curves whether Lucy's model and Wilson's model are compatible. The evidence of the solutions of the two synthetic light curves makes it clear that some
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85 caution is in order in looking at piiblished solutions for the light curves of W Ursae Majoris type stars. Most of these solutions are based on the geometrical model which lies behind the RussellMerrill method of solution (or, equivalently, other methods based upon the same model) . The solutions from this geometrical model (i.e. the Russell model) are not compatible with the parameters used to derive similar theoretical light curves from astrophysical models such as Lucy's or Wilson's. While it is not possible to determine in this study which of the three models best represent the real situation in W Ursae Majoris type stars, there is little doubt that there are serious problems in applying the Russell model to these stars. One of the problems encountered in the solutions of both synthetic light curves was the fit of the solutions at the shoulders of the eclipses. In both cases the "observations" at the shoulders lie below the theoretical curves, and the eclipses seem to be of longer duration observationally than predicted by the Russell model. According to Bookmyer "this particular discrepancy between the Russell model and observations occurs in many W Ursae Majoris systems" (16) . Binnendijk also reports that "it has been determined that in many light curves the observations defining the shoulders of the eclipse curves are fainter than expected from the Russell Model" (1) . Both authors interpret this effect as evidence for a permanent distortion
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86 in the shapes of the components; the facing hemispheres of the components are more elongated than the Russell model ellipsoids. Since the synthetic light curves show this feature which is common to most W Ursae Majoris systems, perhaps this is further evidence that the Russell model does not apply to these systems.
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CHAPTER IV THE SYSTEM OF 44i BOOTIS History The system of 44i Bootis was discovered as a visual binary by William Herschel in 1781. On the discovery night he entered in his observing book: Double. Considerably unequal. Both white. With 227 they seem almost to touch, or at most 1/4 diameter of S. asunder; with 460, 1/2 or 3/4 diameter of S. This is a fine object to try a telescope, and a miniature of Alpha Geminorum. Position 29Â° 54' n. following (17) . The variation of one or both components was suspected by several early observers of the visual pair. These early observations are summarized by Miss Agnes Clerke as follows: On June 16, 1819, Struve noted a difference of two magnitudes between the components; of one invariably 182233, but of only half a magnitude 183338. Argelander found them exactly equal, June 6, 1830. To Dawes in April 1841, the attendant star seemed a shade brighter than its primary, which was rated as of fifth magnitude. Duner's observations at Lund, 186875, confirm their relative variability, causing the disparity between them to range from 0.4 to 87
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88 1.3 magnitude; and he points out that they appeared to Herschel considerably unequal in 1781, but perfectly matched in 1787. Both stars were yellow in 1875, but the tint of the smaller was at times less deep than at others. Admiral Smyth marked it as "lucid gray" in 1842; Webb and Secchi respectively found it blue in 1850 and 1859; Webb and Engelmann reddish in 1856 and 1865. The principal star has often been considered as pure white. The spectriom belongs to Class I. The photographic magnitudes of the pair, as determined at Paris in 1886, are 5.3 and 6. Engelmann concluded the smaller component to vary from magnitude 5 to 7 , the larger from 5 to 6. They revolve in a period of 261 years, the plane of their orbit passing nearly through the sun. The periastron passage was in 1783. They possess at least four times the solar luminous intensity (18) . The fainter component of the visual pair was found to have a spectrum resembling that of W Ursae Majoris by members of the Mount Wilson staff in 1921. In 1926 J. Schilt observed the star photographically and found that this component is an eclipsing variable, as suspected, with a period a little greater than a quarter of a day. In 1929 G. P. Kuiper confirmed the variability found by Schilt from older plates of the visual pair taken by Hertzsprung in 1915 and Munch in 1922 and 1926. There have been many investigations of the light variation of the fainter component of the visual double since that time (19) .
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89 Visual Binary 44i Bootis is a well known visual binary,? 19 09 ADS 9494. The most recent discussion of the orbit was by W. D. Heintz in 1963. His final elements as reported by J. Meeus (20) are p = 246.20 years a = 4'.'100 T = 2042.00 i = +84?50 e = 0.360 w = 237S90 TT = OVOSO ^ = 228?50 d where P is the period of revolution, a is the semi major axis of the orbit, T is the year of periastron passage, i is the inclination of the orbit, e is the eccentricity of the orbit, w is the longitude of periastron, tt^ is the dynamical parallax, and ^Â±s the position angle of the ascending node. Because of the high inclination the apparent ellipse is very elongated and the angular distance between the components is less than onehalf second of arc at this time (see Figure 12) . The dynamical parallax agrees well with the trigonometric and spectroscopic parallaxes quoted by Binnendijk (19) . TT = 0'.'076 + 0'.'005 tr 7T = 0'.'073 + 0'.'005 sp We will call the brighter component star A. Its spectral
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90 Figure 12. Visual Binary Orbit of 44i Bootis, (20) .
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91 type is F9 (21) and, according to Binnendijk, there are indications that its light shows a slight variability. The fainter component of the visual double star has a spectral type G4 (21) with rotationally broadened absorption lines like those found in W Ursae Majoris stars. We will call the fainter component (the eclipsing variable) star B, and its brighter and fainter components star Bl and star B2 (or sometimes star 1 and star 2) respectively. From the visual orbit and a mean parallax of 0'.'076 we find for the sum of the masses: ^A + ^Bl + ^B2 = 2.59 . Spectroscopic Binary The eclipsing system is also a spectroscopic binary. D. M. Popper (22) observed the radial velocity curves and found the following results : Kg^ = 115.4 +1.6 km/sec K^^ = 231.1 + 1.9 km/sec e = ^2/^Bl ^ Â°^Â° '^^ a^^sin i = 0.43 x 10^ km ^32^^^ i = 0.85 x 10^ km Mg^ sin i = 0.77 Â© ^2^^" ^ ^ 0.39Â© Where K is the: amplitude of the radial velocity variation, e is the eccentricity of the orbit, M is the mass of the
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92 star, a is the semi major axis of the orbit, and i is the inclination of the orbit. A more recent solution of the radial velocity curves was made by L. Binnendijk; however, it is based on older observations made by W. E. Harper. Following A. H. Batten (23), Popper's solution was chosen because it is based on a greater number of observations. From the line intensities Popper extimated that the Iximinosity ratio of the components is 0.6 and most certainly lies between the values 0.5 and 0.85. Therefore in our notation L^ = 0.62 and L2 = 0.38. On the other hand, Petrie gives .06 + .05 for the magnitude difference between the components of the spectroscopic binary or 0.95 +0.04 for the luminosity ratio; thus L^ = 0.51 and L2 = 0.49. Eclipsing yariable : A large number of observations of the light variation of 44i Bootis B have been made since this variation was discovered. These observations have shown the presence of strange deviations from the mean light curve. There have also been several apparently abrupt changes in the period of the system. There are no published solutions made from observations in more than one color. For these reasons, it was decided that further observations were in k order.
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93 Instriamentation The observations were taken at the Rosemary Hill Observatory of the University of Florida. The telescope used was a 30inch reflector manufactured by the Tinsley Company. The telescope has a prime focal length of 120 inches. It has interchangeable secondary mirrors which allow observing at either the Newtonian focus or the Cassegrain focus. A sixinch refractor is mounted alongside the reflector for use as a finderscope. The photometer was mounted at the Cassegrain focus with an effective focal ratio of f/16. The optical and mechanical parts of the photometer were built by Astromechanics. As the light rays pass from the telescope into the photometer, they travel first through a small aperture, then through a filter, and finally through a Fabry lens before impinging on the phototube. The diaphragm which provides aperture of 0.33, 0.51, 0.93, 1.52, 1.98, 2.54, 2.89, 3.90, 4.89, and 5.85 millimeters, which, with a focal length of 480 inches, correspond to angular aperture of 5.4, 8.3, 15.2, 25, 32.5, 41.5, 47, 64, 80, and 96 seconds of arc. The 32.5second: aperture was used in these observations. A diagonal mirror may be placed in the light path following the aperture. When the diagonal mirror is in the light path, the light rays are deflected to a viewing eyepiece. The outline of
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94 the particular aperture in use can be seen when the eyepiece is in use, as the aperture is illuminated by a small bulb. This facilitates centering the star in the aperture. When the diagonal mirror is withdrawn from the light path, the light path is unobstructed and a signal can be generated. When the mirror is withdrawn to take a photoelectric reading, the bulb is switched off by a microswitch. The starlight then passes through a filter. The filters are mounted on a wheel so that the observer can select one of the ultraviolet (Corning 9863) , blue (Corning 5030and Schott GG 13) , and yellow (Corning 3389) filters, which correspond to the JohnsonMorgan UBV system (24) . The Fabry lens serves to keep the same portion of the photocathode illuminated despite the effect of seeing. The filtered light rays impinge oh the cathode of the EMI 6256B multiplier phototube at the end of the light path. A high voltage power supply was used to provide a potential difference of 1500 volts between the cathode and the anode of the photomultiplier. (Unfortunately, the photometer was not wired properly by Astromechanics and there was no potential applied to the cathode. Thus there was no potential difference between the cathode and the first dynode; the whole 1500 volts' potential difference was applied between the first dynode and the anode. This fault was not discovered until after the observations had been completed; however, the quality of the observations does
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95 not seem to have been affected by the lack of potential on the cathode of the phototube ..) The output of the photomultiplier tube is fed into a DC amplifier. The DC amplifier is used to amplify the small current from the phototube to a level detectable by the recorder. The amplifier was of standard design with an electric circuit similar to the one described by Whitford (25) . The size of the resistors is determined so as to change the amplification in coarse steps of 2.5 magnitudes, from to 12.5 and in fine steps of 0.25 magnitudes from to 2.50, where magnitude here refers to stellar magnitude. Different combinations of these resistors are used to produce the suitable signal on the recorder chart. The combination of resistors desired is selected with two rotary switches mounted on the amplifier. The time, constant is controlled by a third rotary switch which gives a choice of four time constants. The output of the DC amplifier is used to drive the pen of the dualchannel recorder built by Honeywell. Only one channel of the recorder was used; it has a pen which marks a continuous stripchart to correspond with the signal current from the amplifier. The chart is set to run at a rate of one inch per minute. The observatory is equipped with a high frequency receiver for receiving WWV time signals; the time was recorded on the chart several times during the night. Readings of a barometer, a thermometer, and a hygrometer were also made several times a
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96 night and recorded on the chart. Observations The light of the triple system was measured by centering the system in a diaphragm approximately thirty seconds of arc in diameter. Observations were obtained on five nights in the spring and summer of 1969 and 1970. Additional observations were made in the spring of 1971 for times of minim;am light. The star HR 5581 was used as a comparison star throughout. Its constancy had been thoroughly checked by J. Stebbins and C. M. Huffer, and later by 0. J. Eggen, by comparing it with star HR 5635 (26) (see Table XXX) . Binnendijk (19) later reconfirmed the constancy of the comparison star; thus, only one check star reading was taken each night. The observations were taken in a regular sequence: comparison star, sky deflection, variable star, sky, variable, comparison, etc. Each object was observed through both blue and yellow filters, with individual readings lasting approximately thirty seconds. Ultraviolet observations were made on only one night. It was decided to observe only in blue and yellow light, since the short period of the system requires observations closely spaced in time. The star deflections were read from the chart to an accuracy of three places (e.g. 54.7) with the sky plus dark current deflections subtracted. The time for each observation
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97 CN
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98 was read from the chart to the nearest 5 seconds. A series of programs for the IBM 360 computer was used to convert the raw data read from the charts into the final light curve. The difference in magnitude, Am, between the variable star and the comparison star at the time of a variable star reading was determined from the following relation: where i?,(comp.) is the apparent light of the comparison star and Jl(var.) that of the variable star; and this ratio is, of course, equal to the ratio of chartreadings of the two stars with sky readings subtracted. Linear interpolation was used wherever values between readings were needed. The magnitude differences together with the times of observation in heliocentric Julian days are given in Table XXXI. These magnitude differences were corrected for firstand secondorder extinction and transformed to the JohnsonMorgan UBV standard system in the manner described below. Reduction of Data After observations have been made of the variable and comparison stars, it is necessary to remove the effects of the earth's atmosphere from these observations. It is also most useful to transform the data to a standard
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99 TABLE XXXI Observations of 44i Bootis Yellow Observations Hel JD
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100 Table XXXI continued. Hel JD PHASE Am 2440000+ .71806 0.39315 .71985 0.39983 .75596 0.53467 .75868 0.54482 .77622 0.61031 .77801 0.61700 .77980 0.62368 392.57818 0.62973 .57997 0.63641 .59051 0.67577 .59219 0.68204 .60029 0.71229 .60220 0.71942 .63269 0.83326 .63455 0.84021 .64676 0.88580 .64872 0.89312 .65781 0.92706 .65943 0.93311 .67019 0.97329 .67210 0.98042 715.59120 0.99047 .59462 0.00324 .60816 0.05380 .61076 0.06351 .62262 0.10779 .62546 0.11840 .63727 0.16249 .63975 0.17175 .65081 0.21305 .65376 0.22407 .66481 0.26533 .66765 0.27593 .67928 0.31935 .68188 0.32906 .69421 0.37510 .69664 0.38417 .70735 0.42416 .71007 0.43432 .72147 0.47689 > .72506 0.49029 769.59105 0.35379 ; 59325 0.36200 .59829 0.38082 .60066 0.38967 0.
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Table XXXI continued, 101 Hel JD
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102 Table XXXI continued. Hel JD
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Table XXXI continued. 103 Hel JD
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Table XXXI continued. 104 Hel JD 2440000+ PHASE Am 715.59016 .59566 .60723 .61163 .62181 .62650 .63634 .64051 .65011 .65474 .66400 .66846 .67847 .68275 .69334 .69651 .70659 .71105 .72072 .72592 769.59152 .59291 .59875 .60020 .60656 .60795 .61357 .61467 .61953 .62080 .62850 .62983 .63573 .63677 .64215 .64331 .64921 .65060 .65541 .65645 .66177 .66258 .66686 0.98659
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Table XXXI continued. 105 Hel JD 2440000+ PHASE Am,' .66820 .67236 .67352 .67821 .67954 .68504 .68614 .68967 .69071 .69580 .69638 .70182 .70292 .70685 .70778 .71200 .71322 .71721 .71831 .72352 .72502 .72902 .73006 .73480 .73561 .74013 .74105 .74673 .74754 .75321 .75408 .76004 .77263 .78284 1102.58535 .58686 .59444 .59559 .60080 .60231 .60873 .61006 0.64186 0.65739 0.66172 0.67923 0.68420 0.70473 0.70884 0.72202 0.72591 0.74491 0.74708 0.76739 0.77150 0.78617 0.78964 0.80540 0.80996 0.82485 0.82898 0.84841 0.85402 0.86895 0.87283 0.89053 0.89356 0.91043 0.91387 0.93508 0.93810 0.95927 0.96252 0.98478 0.03179 0.06991 0.777 0.802 0.772 0.779 0.797 0.813 0.819 0.799 0.797 0 0 0 0 0.777 0.770 0.794 0.781 0.796 0.789 0.788 0.790 0.770 0.762 791 0.763 0.775 776 0.723 0.694 0.725 0.715 0.689 0.651 0.717 0.694 0.689 0.692 0.721 0.763 0.751 0.776 772 0.776 0.770 0.764 758
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106 Table XXXI continued. Hel JD 2440000+ PHASE Am .61515 .61660 .62187 .62314 .62766 .62870 .63385 .63506 .64073 .64183 .65694 .65850 .66625 .66770 .67384 .67505 .68101 .68263 .68761 .68900 .69432 .69519 .70051 .70184 0 0 0.739 0.745 0.732 0.733 0.723 0.725 0.695 0.702 0.668 0.654 0.611 616 0.654 0.665 681 0.691 0.714 0.716 0.718 0.727 0.737 0.735 0.749 739 0 Ultraviolet Observations 715.58808 .59658 .60619 .61250 .62089 .62749 .63547 .64137 .64936 .65590 .66325 .66944 .67760 .68362 0.97882 0.01056 0.04644 0.07001 0.10133 0.12598 0.15577 0.17780 0.20764 0.23206 0.25950 0.28261 0.31308 0.33556 0.600 0.610 0.677 0.710 0.718 0.684 0.696 0.690 0.703 0.729 0.733 0.716 0.715 0.713
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107 Table XXXI continued. Hel JD PHASE Am 2440000+ .69247 0.36860 0.828 .69832 0.39045 0.689 .70567 0.41789 0.648 .71203 0.44164 0.644 .71996 0.47125 0.642 .72702 0.49761 0.634
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108 observational system. This transformation makes the results independent of the choice of telescopephotometer system, thereby making direct comparison of data taken with different instrumental systems feasible. The methods used for these reductions will closely correspond to those described by Hardie (28) . Passage through the earth's atmosphere has two effects on the light from a star. First, it is diminished and second, it is reddened. The magnitude of these effects is dependent on the amount of atmosphere the light passes through. These extinction effects can be expressed as m = m k'X k"CX (1) o where m is the magnitude of the star outside the atmosphere, m is the magnitude observed after the light has passed through a path length X of the atmosphere, with X expressed in units of air mass at the zenith of the observer, C is the color index of the star, and k' and k" we shall call the first and second order extinction coefficients. The relative air mass, X, in units of the air mass at the zenith is a function of the secant of the zenith distance, z. The value of sec z may be determined through the relation . sec z = (sincf) sin6 + cosp coscj) cosh) (2)
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109 in which (j) is the observer's latitude, while 5 and h are the declination and hour angle of the star. After sec z is found as above, then X may be determined by the formula (28) X = sec z 0.0018167 (sec z 1) 0.002875 (sec z 1) 0.0008083 (sec z 1) ^ , (3) To determine the firstorder (or principal) extinction one may measure a nonvariable star over a wide range of air mass. Then the extinction coefficient may be derived from a plot of the measured magnitude of the star, m, versus the air mass X. In differential photometry of variable stars, the comparison star may serve as the constant star used to determine the firstorder extinction coefficients , provided that the observations cover a sufficiently large range of air mass. Only on one of the five nights of observation discussed above (July 1, 1970) were the observations taken through a sufficiently large range of air mass for extinction to be determined accurately. The values so determined (by a least squares fit of the data) were k' , = 1.15 and k' = 1.53. Plots of the y b comparison star magnitude versus air mass for that night are shown in Figure.13. For the other four nights, it was necessary to adopt the following mean extinction coefficients: k'^ = 0.4, k'j^ = 0.6, k' = 1.0. The secondorder extinction coefficients are a 2
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110 7.0 6.5 Figure 13. Comparison Star Extinction for July 1, 1970.
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Ill measure of the wavelength dependence of the extinction. They may be determined by observing a pair of stars of widely differing color index over a large range of air mass. Then if the two stars have essentially the same position, the differential measures of their magnitude and color will follow these relations: Am = Am k" ACX (4) o AC^ = AC k"^ ACX (5) Therefore, plots of Am versus (ACX) and of AC versus (ACX) will give lines whose slopes are k" and k" . These secondorder extinction coefficients are substantially constant over short periods of time (e.g. one observing season) . The two stars of the visual binary Albireo (Beta Cygni AB) were used to determine the secondorder coefficients. This system was chosen since both stars are relatively bright, one star is blue and the other is red, and both stars are JohnsonMorgan standard stars (see Table XXXII) . The two stars were observed on two nights (July 19, 1970 and August 16, 1970) through a total range of about four air masses. When the data was reduced and plotted (Figure 14) , it showed the presence of a nonnegligible secondorder extinction in the V (visible or yellow) wavelength band. This extinction is not expected; however, it might be attributed to the high water vapor
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112 TABLE XXXII Standard Stars STAR Sp o Tau G8III RA(1970) 3^23"'lif7 Dec (1970) V BV UB e Tau KOIII 4 26 51.7 TT"^Ori F6V 4 48 12.6 C Oph 09.5V 16 35 30.2 a Oph A5III 17 33 32.4 3 Oph K2III 17 41 59.3 Y Lyr B9III 18 57 49.2 3 Cyg A (KO) 19 29 31.8 3 Cyg B B8V 19 29 31.8 K Aql B0.5III 19 35 16.1 3 Aql C8IV 19 53 50.3 a Del B9V e Aqr AlV li Cap M2IA 7 4 Aqr (B9) a Peg B9V 1 Psc F7V 20 38 14.6 20 46 03.1 21 42 35.2 22 51 51.8 23 03 15.8 23 38 24.3 8Â°55.'5 3.59 0.89 0.62 19 06.9 3.54 1.02 0.88 6 54.6 3.19 0.45 0.01 10 30.5 2.56 0.02 0.86 12 34.8 2.08 0.15 0.10 4 34.7 2.77 1.16 1.24 32 38.8 3.25 0.05 0.09 27 53.8 3.07 1.12 0.62 27 53.8 5.11 0.10 0i=32 7 05.3 4.96 0.01 0.87 6 19.8 3.71 0.86 0.48 15 48.3 3.77 0.06 0.22 9 36.4 3.77 0.01 0.04 58 38.5 3.99 2.41 2.40 11 46.4 5.81 0.08 0.32 15 02.6 2.49 0.05 0.06 5 27.8 4.13 0.51 0.00
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o U ro i C! rH Â•rH 13 VD p in U Q) Q) iH O O M M 0H I U 0H U rH 0) OpH W CO (U M Â•H
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114 1
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115 content of the atmosphere at the time the observations were taken (relative hiamidity usually greater than 95%) . Since this effect appeared to be an atmospheric effect, the k" term was retained and the working equations used were modified to include it. The secondorder extinction coefficients derived from the data by a least squaresifit were k"^ = 0.028, k"^^ = 0.034, and k"^^ =0.022. Principal extinction coefficients were also derived from the observations of Albireo in order to better determine mean values for these coefficients. This data is plotted in Figures 1517. Once the extinction coefficients have been determined, they may be applied to the observed magnitudes in the three UBV color bands (u, b, v) to obtain the extraatmosphere values of the stellar magnitudes denoted by the zero subscripts: v^ = V k'^X k"^(b v)X (6a) b^ = b k'j^X k"^{h v)X (6b) u = u k'^X k" (u b)X (6c) where k"^ = k^^^ + k"^ and k"^ = k"^^ + k"^. In order to be able to compare observations taken with different equipment, it is necessary to transform these quantities to a standard system (i.e. the JohnsonMorgan UBV system) . This transformation can be represented
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O a\ 0) H S^ T! Â• HO HO XJ r^ *^ rH r^ rH cTi iH o en < H tn rH 4J M ^ CO Â•' * O o^ P o vu in rH tjlt^ rH OH to rl 3 ^ ^ 3 +) 13 < CT. Cn Â•>H <; tPrH " Â•H >i P CQ C U Id &i H >in5 C U +> CQ tP rt ffl H u 0) P
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117
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H O Â•> TS Â• i< rH 4J iH m C 3 13 O Id Â•H H !3 P ^H < CHrH Â•> P fl U Id X tn rH W >i (Â« ^ fl O P ffl tn 0) CU >i 3 (Â« CQ H U rH P I d H rH pq Q)
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119
PAGE 134
U >i
PAGE 135
121
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122 by the following equations: V = Vq + eCB V) + c^ (7a) B V = y(b v)^ + ^^ {7b) U B = ip(u b)^ + ^j^ (7c) where U, B, and V are the magnitudes on the standard system; e, u, and iJj are the transformation coefficients; and the C's represent zeropoint differences between the natural system and the standard system. One way to determine these transformation coefficients is to observe a niunber of stars with known standard magnitudes and color indices. These stars should be chosen to cover as wide a range of color indices as possible. The plots of V v^ versus B V, B V versus (b v) , and U B versus (u b)^ for each star will have slopes of e, y, and ^ respectively. Seventeen stars chosen from the list of Johnson (29) were observed and used to determine the transformation coefficients. These stars with their standard magnitudes and colors are listed in Table XXXII. The observations were corrected for extinction and plots were made of V v versus B V, (B V) (b v) versus B V, and (U B) (u b)^ versus U B (see Figure 18). Then e, y , and ij; were determined by a least squares method. The zeropoint coefficients were not determined. The values of the transformation coefficients were e = 0.134, y= 1.072 and ijj = 1.''020.
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123 0.5 VVo 1.0Â•1.5 Â« 1.0 0.0 1.0 2.0 BV 2.0 (BV) lbv)o 1.5 1.0
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124 Once the extinction and trans forroat ion coefficients have been determined, it is useful to have a set of working equations which combine all the reductions. In the study of variable stars, it is customary to select a comparison star close to the variable in the sky and similar in color and magnitude. By doing this the differential measures are less dependent on extinction, thereby the extinction corrections are usually small. Considering equations (6) as applied to both variable and comparison star and expressing them in differential magnitudes and color indices: Av = Av k' AX k" A(b v)X (8a) o V v Ab = Ab k', AX k" A (b v)X (8b) o b b Au = Au k' AX k" A(u b)X (8c) o u u where X is the mean air mass and thirdorder terms are ignored. Considering equations (7) as applied to both stars, and rewriting them in terms of standard magnitudes: AV = Av + eijA(b v)i (9a) o o AB = Av + (y + ey) A (b v)^ (9b) AU = Av + (u + ey) A(b v)^ + ipA (u b)^ (9c) Equations (8) and (9) may be combined to give the final
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125 working equations used in the reductions: AV = Av + (ey(l k"^X) k"^X)A(b v) k'^AX (10a) AB = Av + (y + eyd ^\^^) " k"^X)A(b v) k'^^AX (10b) AU = Av + (y + eyd k"j^^X) k"^X)A(b v) + iP(l k"^j^X)A(u b) k'^AX (10c) The time is the most accurately observed quantity. The time in hours, minutes, and seconds was converted into decimals of a day with the day itself denoted by its Julian Day number. These times in geocentric Julian day; must be converted into heliocentric Julian days because of the varying position of the earth in its orbit around the sun. The light from a star may reach the earth as much as eight minutes earlier and eight minutes later than it reaches the sun, due to the finite velocity of light. This correction must be made in order not to introduce a period change in the observed variable star. This correction is: t = O'?005775((cos6 cosa)X + (tane sin6 ; + cos6 sina)Y) where a and 6 are the right ascension and declination of the star, e is the obliquity of the ecliptic, X and Y are the rectangular coordinates of the sun for the given date, and 0cl005775 is the time in which light travels one astronomical unit (see Binnendijk (30) for derivation).
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126 Times of Minimum Light and the Period The times of minimum light were obtained from the curves of magnitude difference as a function of time by the tracing paper method (31) for the five nights of observation. An additional time of minimvim light was obtained from the observations made on May, 30, 1971 by the method of Kwee and Van Woerden (32). The times of minimum light are averages of the values obtained with the blue and yellow filters. They are listed in Table XXXIII along with other recent times of minimiim light. Tabulations of earlier times of minimum light are given by Purgathofer and Prochazka (33) and by Schneller (34) . Variation in the period caused by motion in a visual binary system Since 44i Bootis B is one component of a visual binary system, the observed times of minimum light will vary sinusoidally about the true time of minimiam light measured with respect to the center of gravity of the binary ".system. That is, the observed time of minimum light will differ from the true time with respect to the center of gravity of the system by an amount equal to the stellar light time corresponding to the distance of the variable from the center of gravity projected on the line of sight. Figure 19 shows the effect of this correction on the selected times of minimum light listed in Table XXXIV. The correc
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127 TABLE XXXIII Recent Times of Minimum Light Hel JD
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128 Table XXXIII continued. Hel JD
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Table XXXIII continued. 129 Hel JD 2400000+ 0C 0C REFERENCES 392. 420. 661. 675. 675. 678. 679. 679. 681. 681. 682. 686. 697. 697. 698. 698. 699. 699. 700. 700. 700. 700. 701. 701. 714. 714. 753. 769. 769. 780. 1055. 1055. 1102. 1138. 1139. 1141. 6820!? 7971 4273 4875 6200 5661 3684 5037 3774 5119 4502 4668 7184 8526 7890 9213 7284 8617 3943 6648 8009 9324 7347 8709 5891Â° 7299 c 4231 6286Â° 7628Â° 471 3820 5165 6556 406 345 4886 7017 7122 8020.5 8073 8073.5 8084.5 8087.5 8088 8095 8095.5 8099 8114 8156 8156.5 8160 8160.5 8163.5 8164 8166 8167 8167.5 8168 8171 8171.5 8219 ' 8219.5 8364 8424.5 8425 8465 9491.5 9492 9668 9801.5 9805 9813 +0.0125 +0.0071 +0.0061 +0.0071 +0.0046 +0.0048 +0.0036 +0.0050 +0.0040 +0.0046 +0.0056 +0.0050 +0.0084 +0.0087 +0.0077 +0.0061 +0.0098 +0.0092 +0.0061 +0.0089 +0.0110 +0.0086 +0.0075 +0.0097 +0.0068 +0.0137 +0.0077 +0.0104 +0.0107 +0.0064 +0.0060 +0.0066 +0.0103 +0.0075 +0.0092 +0.0103 199 304 1202.5 1255 1255.5 1266.5 1269.5 1270 1777 1277.5 1281 1296 1338 1338.5 1342 1342.5 1345.5 1346 1348 1349 1349.5 1350 1353 1353.5 1401 1401.5 1546 1606.5 1607 1647 2673.5 2674 2850 2983.5 2987 2995 +0.0047 0.0011 0.0047 0.0049 0.0063 0.0062 0.0074 0.0060 0.0070 0.0064 0.0055 0.0061 0.0029 0.0026 0.0035 0.0051 0.0015 0.0021 0.0051 0.0025 0.0003 0.0027 0.0038 0.0015 0.0046 +0.0023 0.0043 0.0016 0.0013 0.0058 0.0093 0.0087 0.0054 0.0086 0.0070 0.0059 d 48 49 46 46 46 46 46 46 46 46 46 43 43 43 43 43 43 50 43 43 43 43 43 d d 49 d d 49 46 46 d 51 51 51 a Computed from min = JD 2438513.4166 + 0'?2678143E b Computed from min = JD 2440339.3817 + 0^26781731E c Used to compute new light elements d Present work
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130 JD 2420000 2430000 2440000 0C 0^20h 0.15 0.10 0.050.20 Â• UNCORRECTED MINIMA MINIMA CORRECTED, L = + 84Â°5 o MINIMA CORRECTED, L = 84!5 1910 Â•f Â• Â•f Â• o o o Â• o 0.00 Â• t 0.05 0.10 0.15 + + * * Â• Â• Â« Â• Â• + + O O o o o o J. J_ 1920 1930 1940 1950 I960 1970 1980 Figure 19. Variation in the Period of 44i Bootis B Caused by Motion in a Visual Binary System.
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131 en ronooinronLnnnc>j(NrorHinmnnforon^ voo^o^cNo^^>g^t^lnoo^^nroorrlncMVDO^Ln^^^^^^^ aSooO'*inoooo^vorH"*>i>r^<^CT Â•H +J (U c A:: u c 0) 0) U en o u O Ti voinr~vD^'^I'oon oo rr r^ oo o irt CM vo 00 O cjOOOOOOOOOOOOOOOOilrHrHHrHrHrH I I I I I I I I I I + % EH M O Â« HO incMvooir^rHrHoocMOOooooo{rÂ»t^rM^cr>o>acno^oo ^00CM'*lnt^C0000^0^^^00r^Ot^C0CMOCMVÂ£>a^CM^V0 OCDOiliHrHHrHiHrHHHrHHrHOOOOOOOHHrH oooooooooooooooooooooooo * I I I I I I r^'*inin'^cM!^'^'=i"ooin'^oooovÂ£(roro*in*ooooooin'* vocJ^^^Â«^ooc^cM'Â•<*Â«>ocn.l^oMÂ•^lnt^Or^o^'*Â«^ooo^o^l rHrHCMOsJCMOOOOOOOOOO'*^ininininVDW3VD>i>VX)VOVJ3r>rCJ^c^lC^>o^o^CJ^ala^CT^o^CTlO^o^c^c^^0^o^o^c^lo>inHt^in^ooooo"5i"t^ooinooinoot^inHt~~o nj Ta<^oovDr^ocr>iHincMrHr^cMoocMOrHt^cMoovD(NrHrO'*oinovDCMr~oovo oor~Â»oovoooiH'!i'vx>ooorHooiHooincT>rHCMLnr~OrHooÂ«* rH^,_icMCMCMoooo'Â«3''*ininininvovDVDVDr~r^r^r~ Q
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132 tions correspond to the visual orbital elements given above with the reasonable mass ratio M /(M, + IVL, + Mg2) =0.4 with both positive and negative inclinations of the visual binary system being plotted. The computed times of minimum light are derived from an early set of light elements given by Eggen (26) : Min = JD2421113.2588 + 0
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133 by Pohl (41) : Primary minimum = JD2438573.4166 + 0'?26781430 E These (0C) values are listed in Table XXXIII along with the number of elapsed periods since the given epoch. Figure 20 is a plot of these (0C) values as a function of the number of elapsed periods. From this plot it seems apparent that a sudden increase in the period occurred sometime in 1967. Twenty times of minimum light observed after this time were used to derive new light elements: Primary minimum = JD2440339 . 3817 + 0^26781731 E +3 +24 The (0C) values and the number of elapsed periods from these light elements are listed in Table XXXIII and plotted in Figure 21. This new period represents an increase of about 0.26 seconds from Pohl's period. The phases of the observations were now computed from these light elements and are listed in Table XXXI. Light Variations When the results of the individual nights were plotted as a function of phase and combined (see Figures 22 and 23), several things became apparent. The first was the expected general shape of the light curve; i.e. typical, of a W Ursae Majoris type variable. The second was an apparent shift of the zeropoint of the magnitudes from
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134 Ol Â• Â« TÂ™ o o m si) 609 CJ g go Â«*Â» Â«Â» ^'^ Â•Â» o Â» o ro 5; 00 (O o + ro in 00 ro CM II c E in O d + ^15 o5 Â«(?\i* Â• 9 Â• Â• in O O d + in o O d I o o o to o o o o o o CM o o o CVI I o o o I o o 5' d I p A en Â•H A o e o 0) u Â•H
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135 ro co 1^ (0 CJ o + 00 ro a> ro ro O ^ CM o 3 O o m ^ ^ o d + n" e^ eo ..^* Â•Â• '8 Â« ^ Â• O I o in O O d a O O d I o 8 Id o CJ o o o o o o CM I o o Oro 5' o I p Â•H g o u m u o
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136 Â•H
PAGE 151
137 in < X a.
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138 night to night; that is, the whole light curve goes slightly up and down. The third feature which appears in the light curve is a slight depression shortly after maximum light in the observations taken on the night of July 1, 1970. The main feature in the mean light curve is the difference in the height of the two maxima. The secondary maximiim (i.e. the one following the shallower eclipse) is obviously higher than the primary maxim\am. This has not been true in all previous observations of the system. In fact, the light curve is subject to change as is shown by Figures 24 and 25 which give the mean light curves of the different authors who have observed the system. The earlier light curves up to and including Binnendijk's are reproduced from Binnendijk's paper (19). All phases are made zero at primary minimum. The zero points of the magnitudes are adjusted to give the best agreement with Binnendijk's mean light curve, but the magnitude scale of the authors was not altered. Schilt observed the eclipsing star with respect to the bright visual component photographically. To produce the light curve shown in Figure 24 Binnendijk made normal points from these long runs which gave a good light curve. Following Kuiper he omitted three short runs of plates of poor quality. The light of star 1 was added in the computation to arrive at the combined light of the system using Schilt 's value of 0.76 mag. for the difference
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139 19J6 pg Schilt I9283D pg Ocsterhoff 1930 pe Siebbiris end Huffer . J 1035 pe Shapley ond Colder 1337 pv PIcj) 1933 pe rjikonov 1947 pe Eqgen i9535Â« ?e Binnendijk Figure 24. The Light Curves of Different Authors, (19)
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140 Am 19546 SCHMIDT AND Pe SCHRICK ^' Â• Â• _ 1961 WEHLAU AND LEUNG pe 1962 CATALANO AND P^ SAITTA 1965 CHEN AND
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141 between the visual components. The same was done with the observations of the minima by Oosterhoff (52) . All later authors had to measure this combined light because the separation of the stars became too small. The light curve by Stebbins and Buffer, observed in 1930, was computed by Eggen (26) . The normal points of the light curve of Shapley and Calder were only computed for the observations taken in 1935. In Plaut's light curve (53) Binnendijk recomputed the normals around the maxima. Nikonov gives light curves in the blue and infrared; Binnendijk reproduces the former. Normal points were made from the published graphs of Eggen' s observations. This was possible except for the gap in the figure, where the deviations from a mean curve are as large as 0.04 mag. Binnendijk also omitted the deep and shifted secondary minimum which Eggen observed on March 6, 1947 from this mean curve. Binnendijk computed the normal points of his own observations after the zero points of each nightVs observations had been adjusted to force agreement of the maxima. Schmidt and Schrick (54) observed the system in three colors and gave their observations in two groups: Group I included only good observations; and Group II included the remaining poorer observations. The normal points given here were computed from the Group I observations taken at an effective wavelength of 450 my. Wehlau and Leung (35) give light curves in yellow and blue light; the former is reproduced here. Catalano and Saitta (36)
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142 observed the system in Ha light; their normal points are reproduced here. Chen and Rekenthaler (55) , Brown and Pinnington (42) , and the author observed the system in yellow and blue light. In all cases normal points were computed from the yellow observations. An inspection of these mean light curves shows that both the relative heights of the maxima and the shape of the light curve change. It seems apparent from reading the literature that the changes in the shape of the light curve occur on a very short time scale. Chen and Rekenthaler (56) reported a variation in the depth of the secondary minimum of approximately 0.033 mag. from one cycle to the next, i.e. in a period of less than six and onehalf hours. Other authors have also reported night to night changes in the light curve. Eggen, for example, observed a deepened (0.04 mag) and shifted secondary minimum on the night of March 16, 1947. Both Brown and Pinnington and the author have observed similar slight depressions shortly after secondary maximum on the nights of March 6, 1968 and July 1, 1970 respectively (see Figure 26) . These variations in the light curve and especially the deformity of the light curve at maximum light seem to be present in most of the observations of the system. Variations of light curves of eclipsing binaries of the W Ursae Majoris type are not uncommon. In fact, the many irregular period changes of the system suggest that the system is unstable, and there
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143 .85 PHASE .00 AM, 0.95 0.90 0.85 .65 .70 .75 .80 MARCH 67, 1968 .85 PHASE Figure 26. Deformities of the Light Curve of 44i Bootis on Two Nights.
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144 fore that variability of the light curve is not unexpected. In this case, however, the picture is further complicated by several indications in earlier observations that the brighter noneclipsing component of the visual pair is slightly variable. Untangling these two effects on the combined light will probably not be possible until the two components of the visual binary can be observed separately. Re c t if i cat ion In order to proceed conveniently towards a solution of the light curve, it was necessary to convert the data from stellar magnitudes to light values and to free the observations from the light of star A. There are three independent measures of the difference in magnitude Am^g between component A and the eclipsing system B at maximum light (21) : X ^^AB Author B 0.76 Schilt V 0.63 Kuiper V 0.70 Wallenquist A difference of 0.76 mag. in blue light and 0.66 mag. in yellow light was used to remove the light of component A from the light curves. These values are also consistent with the spectral types of the stars as reported by Kurpinska and Van't Veer (21).
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145 The yellow light curve was chosen for the first attempts at a solution. The light in the maxima of the eclipsing system was now written in the form: I = A + A^cosQ + A_cos26 + ACOS38 + A.cosf O 1 2 J 4 + B,sine + B2sin2e + B^sinSB + B^sin4e. The coefficients were then derived by using Merrill's graphical method as follows: A = 0.7985 o A^ =0.0038 B^ =0.0253 A2 =0.0976 B2 = 0.0050 A^ = 0.0055 ^3 = Â° A. = B. = 4 4 The intensities were then rectified by the formula: I A, cose A,cos3e B, sine B^sin2e rll _ Â± f Â± ^ A + ACOs2e O 2 and the phase angles were rectified by the formula 2 2 2 sin = sin e/(l z cos e) where z = 4Ao/N{A A^) with N = 2.6 corresponding to a Z O z limb darkening x = 0.6.
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146 Orbital Elements The rectified light curve shows two unequal minima with light losses of 1 ^^^ =0.23 and 1 ^ =0.13. After ^ o o a coefficient of limb darkening x = 0.6 was assiimed for each component, solutions were made with several values of k, the ratio of radii. As the Merrill nomographs reveal, for shallow minima of partial eclipses a large change in k introduces only a slight change in the shape of the eclipses. This means that for a great part of the partial eclipse portion of the nomographs, the shape (x) curves run nearly parallel to or coincident with the depth line. If the differences in the depths of the two eclipses are not great, then the depth lines for the two possible cases (the primary eclipse either an occultation or a transit with the secondary eclipse the other) will not differ greatly, becoming coincident when the two eclipses are equally deep. This indeterminacy, which is illustrated by the nomographs, means that solutions must be tried for a large range of values of k. Solutions were made for k = 1.0 and for the two cases of k =0.50 (primary an occultation and primary a transit) . These solutions almost cover the whole range of k possible with the assumed eclipse depths and the restriction to partial eclipses. Later investigation showed that for other possible values outside this range (including complete eclipses) , the size of one of the components
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147 would exceed the size of its Roche lobe. Therefore these values of k cover the whole range of permitted solutions. Theoretical light curves were computed for the three values of k and compared with the rectified observations. The sums of the squares of the mean deviations for all three cases were nearly the same. Therefore the solution for these observations is indeterminate over the whole range of values of k. Previous solutions in the literature also mention this problem of indeterminacy. Eggen found a range of indeterminacy of k between the two values of 0.9, although all his solutions have one component that exceeds or nearly exceeds the size of its Roche lobe. Binnendijk found a "satisfactory" solution with k = 0;70 when he used 0.76 for the difference in magnitude between the eclipsing system and the brighter component of the visual binary. However, when he used 0.63 for this difference in magnitude, he found so large a range of possible k's that the solution was indeterminate. Since the light of the brighter component of the visual binary cannot be measured more accurately at this time, and, in fact, this light might even be slightly variable, it does not seem likely that further observations at this time will produce a better r determined solution. There are three solutions of photoelectric light curves of 44i Bootis B in the literature. Eggen' s solution was made from the unfiltered photoelectric light curve
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148 observed by Stebbins and Huffer in 1930. Binnendijk's light curve was observed with a yellow filter that does not correspond to a standard system. Catalano and Saittra observed the system in Ha light. These three light curves can be compared with the author's yellow light curve. The light curves, as illustrated in Figure 24 and Figure 25, for the four sets of observations show somewhat different shapes. It is difficult to make any direct comparisons of the light curves since some of the authors have included zeropoint adjustments in their data. It is apparent, however, that differences in shape do occur in the various light curves. These zeropoint adjustments, and the differing amounts of third light make direct comparisons of the various solutions somewhat meaningless. The best that can be said is that all the solutions suffer from the lack of precise knowledge of the two components of the visual binary system and from the poor determinacy of solutions when the stars are highly distorted and the geometrical depth is low. The rectification coefficients and the light losses in the rectified minima differ in the various solutions. This is not surprising since both the rectification coefficients and the light losses in the rectified minima depend upon the shape of the light curve outside of eclipse and the amount of third light removed from the light curve. In the case of the author's yellow light curve, the changes in
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149 the shapes of the light curve on the five nights the observations were made add greatly to the uncertainty of all the parameters. The light losses in the rectified minima are not well determined; however, this uncertainty in itself had littles effect on the determinacy of the solution. The limb darkening coefficient for both components was chosen as x = 0.6 following the value determined for the sun which is of similar spectral type. AH three previous solutions used a value of limb darkening x = 0.8. Changing the liirdD darkening to this value does not significantly affect the results of the solution. There are considerable differences in some of the orbital elements for the three solutions listed in Table XXXV. While the general problem of indeterminacy is typical of t shallow minima or partial eclipses, the very large range of possible solutions may be attributed to the large amount of scatter in the observations. This large amount of scatter makes it virtually impossible to improve the determinacy of the solution. Orbital elements were determined ^for these three possible solutions of the yellow light curve. The figures of the components were determined using the mass ratio m^/m, = 0.50 taken from Popper's spectroscopic elements. These orbital elements are listed in Table XXXV. The sizes of the Roche lobes for the mass ratio ^2/"!^^ = 0,50 are y^ = 0.441, z^ = 0.414, ^2 = 0.313, and z^ =0.300. (The
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150 TABLE XXXV Solutions and Orbital Elements with x = 0.6, 1 It. = 0.23, 1 i^^^ = 0.13, = 41?0, and 111^/111^ =0.50 o e z J.
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151 band caxes of the stars are in the yand zdirections respectively;) Thus in the solution where k = 0.50 and the primary eclipse is an occultation, the size of the brighter star just exceeds the size of its Roche lobe. For the other case where k = 0.50, neither component exceeds the size of its Roche lobe. In the case where k = 1.0, the fainter component is larger in size than its Roche lobe. The Roche lobe is, in effect, the limiting surface for a dynamically stable star in a binary system where the two stars have distinct surfaces. If both components fill their lobes, which may often be the case for W Ursae Majoris systems, the components will share a common surface. The eclipsing system of 44i Bootis B is very likely near this limit of stability. Certainly the changes in the period and the light curve indicate that the system is not stable. Since the solution of the yellow light curve was virtually indeterminate, no solution was made from the slightly poorer quality blue light curve. Further observations of the system of 44i Bootis for the purpose of determining orbital elements will probably not be productive until the two components of the visual binary can be observed separately. When this is possible, the problems caused by the uncertainty of the difference of magnitude of the two visual components and the possible variability of the brighter component should be resolvable. Then, it is possible that very accurate photoelectric observations might provide a better determined solution for the orbital elements. Even under the best circumstances, however, the
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152 basic problem of indeterminacy in shallow minima of partial eclipses will remain. Until the time when the visual components can be observed separately, the combined system should be observed in order to determine the times of minimxom light, and to detect from these times of minimum, any further period changes.
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CHAPTER V SUMMARY AND CONCLUSIONS The light curves of three W Ursae Ma j oris sy terns, two generated from? theoretical astrophysical models and one which was observed, have been studied using the RussellMerrill method of solution of light curves of eclipsing binaries. The observed system, 44i Bootis B, is, in many ways, typical of W Ursae Majoris systems. The light curve of the system changes with time in an apparently irregular manner. The period of the system also shows irregular variations. These variations, which are common in W Ursae Majoris systems may indicate that these systems are not dynamically stable. In addition 44i Bootis B is a member of a visual binary system. Three major problems were encountered in the solution of the light curves. The light curves of 44i Bootis B and the first synthetic system had relatively shallow minima caused by partial eclipses. This led to a problem of indeterminacy in the solutions. The "observed" points in the shoulders of both synthetic light curves fell below the theoretical light curves predicted by the solutions. In all three light curves there was a problem with third light. In 153
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154 the case of 44i Bootis B, there was a known visual companion whose light was included in the observations; however, the amount of this third light was not precisely known. The addition of third light in the solution of the two synthetic light curves somewhat improved the fit of the solutions to the light curves. There was, however, no sound basis for adding this third light. In addition to these problems, the orbital elements predicted by the solutions of the two synthetic light curves using the RussellMerrill method were not at all close to the orbital elements that were used to generate these light curves from the theoretical astrophysical models. In both cases the RussellMerrill solutions predicted that the stars were smaller and more detached than the actual stars used to generate the light curves. The results summarized above led ' to the conclusion that the Russell model is not compatible with the theoretical astrophysical models used to generate the synthetic light curves. It is not possible to determine from this study which of the models more closely represents the actual conditions present in W Ursae Majoris systems. The solutions of the light curves of W Ursae Majoris systems which are in the literature should be regarded as crude approximations. Better solutions may be possible when the nature of the variations in the periods and light curves of these systems are determined. In this light, it is suggested that precise multicolor photoelectric observations of a few selected
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155 systems , preferably those showing complete eclipses , carried out as continuously as possible and extending over a period of several years , would be more meaningful than observations of many systems solely for the purpose of deriving orbital elements. These observations, together with simultaneous spectroscopic observations, could add greatly to the knowledge of the nature of the W Ursae Majoris systems.
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LIST OF REFERENCES 1. Binnendijk, L. , "The Orbital Elements of W Ursae Majoris Systems," in Vistas in Astronomy 12 ed. by Arthur Beer (Pergamon Press, Oxford, 1970). 2. Russell, Henry Norris , and Merrill, John Ellsworth, "The Determination of the Elements of Elcipsing Binaries," Contributions from the Princeton University Observatory No. 26 (1952). 3. Lucy, L. B. , "The Light Curves of W Ursae Majoris Stars," Astrophysical Journal 153, 877 (1968) . 4. Merrill, John Ellsworth, "Tables for Solution of Light Curves of Eclipsing Binaries," Contributions from The Princeton University Observatory No. 23 (1953) . 5. Merrill, John Ellsworth, "Nomographs for Solution of Light Curves of Eclipsing Binaries," Contributions from The Princeton University Observatory No. 24 (1953). 6. Rucinski, S. M. , Private Communication. 7. Mochnacki, S. W. , and Doughty, N. A., "A Model for the Totally Eclipsing W Ursae Majoris System AW UMa," Monthly Notices of the Royal Astronomical Society 156 , 51 (1972) . 8. Wilson, Robert E., and Devinney, Edward J., "Realization of Accurate CloseBinary Light Curves : Application to MR Cygni," Astrophysical Journal 166, 605 (1971). 9. Wilson, Robert E., Private Communcation . 10. Merrill, John E. , "Rectification of Light Curves of W Ursae MajorisType Systems on the Russell Model," in Vistas in Astronomy 12^ ed. by Arthur Beer (Pergamon Press, Oxford, 1970). 11. Kitamura, Masatoshi, Tables of the Characteristic Functions of the Eclipse" and the Related DeltaFunctions for Solution of Light Curves of Eclipsing Binary Systems (University of Tokyo, Tokyo, 1967). 156
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157 12. Kopal, Zdenek, Close Binary Systems (John Wiley and Sons Inc., New York, 1959). 13. Gyldenkerne, Kjeld, and West, Richard M. , ed. , Mass Loss and Evolution in Close Binaries (Proceedings of the International Astronomical Union Colloquixim No. 6, Copenhagen University Publications Fund, Copenhagen, 1970) . 14. Wilson, Robert E., Private Communication. 15. Binnendijk, L. , "The Orbital Elements of RZ Comae," Astronomical Journal Â£9 , 154 (1964) . 16. Bookmyer, Beverly B., "A Study of Photoelectric Observations of SW Lacertae," Astronomical Journal 70 , 415 (1965). 17. Herschel, William, "Catalogue of Double Stars," Philosophical Transactions of the Royal Society of London, 72, 118 (1782) . 18. Clerke, Agnes, "A Historical and Descriptive List of Some Double Stars Suspected to Vary in Light," Nature 39, 55 (1888) . 19. Binnendijk, L. , "The Light Variation and Orbital Elements of 44i Bootis," Astronomical Journal 60^, 355 (1955) . 20. Meeus, Jean, "Some Bright Visual Binary StarsII," Sky and Telescope 41, 88 (1971) . 21. Kurpinska, M. , and Van't Veer, F. , "Etude Photometrique en Huit Couleurs de la Binaire ^ Eclipse 44i Bootis," Astronomy and Astrophysics Â£, 253 (1970) . 22. Popper, Daniel M. , "Five Spectroscopic Binaries," Astrophysical Journal 97^, 394 (1943) . 23. Batten, Alan H. , "Sixth Catalogue of the Orbital Elements of Spectroscopic Binary Systems," Publications of the Dominion Astrophysical Observatory 13, 119 (1967) . 24. Johnson, H. L., and Morgan, W. W. , "Fundamental Stellar Photometry for Standards of Spectral Type on the Revised System of the Yerkes Spectral Atlas," Astrophysical Journal 117' 313 (1953) . 25. Whitford, A. E. , "Photoelectric Techniques," in Encyclo pedia of Physics 54 ed.by S. Fliigge (SpringerVerlag , Berlin, 1962) .
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158 26. Eggen, Olin J., "The System of 44i Bootis," Astrophysical Journal 108 , 15 (1948) . 27. Cannon, Annie J. and Pickering Edward C. , "The Henry Draper Catalogue," Annals of the Astronomical Observatory of Harvard College 956 (19201) . 28. Hardie, Robert H. , "Photoelectric Reductions," in Astronomical Techniques ed. by W. A. Hiltner (University of Chicago Press, Chicago, 1962) . 29. Johnson, Harold L. , "A photometric System," Annales D' As trophy sique 18, 292 (1955). 30. Binnendijk, Leendert, Properties of Double Stars (University of Pennsylvania Press, Philadelphia, 1960) . 31. Wood, Frank Bradshaw, "Observation of Eclipsing Variables," in Photoelectric Astronomy for Amateurs ed. by F. W. Wood (The Macmillan Company, New York, 1963) . 32. Kwee, K. K. , and Van Woerden, H. , "A Method for Computing Accurately the Epoch of Minimum of Eclipsing Variable," Bulletin of the Astronomical Institutes of the Netherlands 12, 327 (1956) . 33. Purgathofer, A., and Prochazka, F., "Periodenschwankungen Bei Kurzperiodischen Bedeckungsverander lichen," Mitteilungen der UniversitatsSternwarte Wien 13^, 151 (1967) . 34. Schneller, H. , "Die Periode von i Bootis," Astronomische Nachrichten 288 , 183 (1965) . 35. Wehlau, William, and Leung, KamChing, "Photoelectric Observations of i Bootis , " Journal of the Royal Astronomical Society of Canada 5Â£, 105 (1962) . 36. Catalano, S., and Saitta, T. , "Observazioni ed Elementi Orbitali del Sistema 44i Bootis B," Memorie Delia Society Astronomica Italiana 3^, 3 (1964) . 37. Pohl, E., and Kizilirmak, A., "Beobachtungsergebnisse an Verander lichen Sternen," Astronomische Nachrichten 288, 69 (1964). 38. Schneller, H. , "i Bootis," Information Bulletin on Variable Stars 57 (1964) . 39. Popovici, C. , "Photoelectric Minima of Eclipsing Variables," Information Bulletin on Variable Stars 148 _ (1966) .
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159 40. Schneller, H. , "44i Bootis," Information Bulletin on Variable Stars 144 (1966) . 41. Kizilirmak, A., and Pohl, E. , "Minima and New LightElements for Eclipsing Binaries," Astronomische Nachrichten 2S1, 111 (1969) . 42. Brown, B. M. K. , and Pinnington, E. H., "Photoelectric Observations of Three W Ursae Majoris Systems," Astronomical Journal 74, 538 (1969) . 43. Johns ton, Kenneth, Private Communication. 44. Popovici, C. , "Photoelectric Minima of Eclipsing Variables , " ' Information Bulletin on Variable Stars 322 (1968) . 45. Pohl, E., and Kizilirmak, A., "Photoelectric Minima of Eclipsing Binaries," Information Bulletin on Variable Stars 45_6 (1970) . 46. Bergeat, J., Lunel, M. , Sibille, F., and Van't Veer, F>> "Sudden Changes in the Period of the Eclipsing Contact Binary 44i Bootis," Astronomy and Astrophysics 17, 215 (1972)>. 47. Popovici, C. , "Photoelectric Minima of Eclipsing Variables," Information Bulletin on Variable Stars 41_9 (1970) . 48. Scarf e, C. D. , and Brimacombe, J., "Photoelectric Observations of Two W UMa Systems , " Astronomical Journal 76 , 50 (1971) . 49. Kizilirmak, A., and Pohl, E. , "Photoelectric Minima of Eclipsing Binaries," Information Bulletin on Variable Stars ^2D (1971) . 50. Popovici, C. ," Photoelectric Minima of Eclipsing Variables," Information Bulletin on Variable Stars 508 (1971) . 51. Pohl, E., and Kizilirmak, A., "Photoelectric Minima of Eclipsing Binaries," Information Bulletin on Variable Stars 647 (1972) . 52. Oosterhoff, P. Th. , "Photographic Observations of Six Minima of 44i Bootis B," Bulletin of the Astronomical Institutes of the Netherlands 9, 11 (1939).
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160 53. Plaut, L. , "Photometric Observations of 44i Bootis," Bulletin of the Astronomical Institutes of the Netherlands 9, 1 (1939) . 54. Schmidt, H., and Schrick, K. W. , "Untersuchungen an W Ursae MajorisSternen, III. Lichtelektrische Beobachtungen von i Bootis," Zeitschrift Fur Astrophysik, 43, 165 (1957) . 55. Rekenthaler, Douglas A., Photoelectric Observations of the Eclipsing Variables W Orionis, 44i Bootis, and Delta Librae (M. S. Thesis, University of Florida, 1965) , 56. Chen, KY., and Rekenthaler, D. A., "Photoelectric Photometry of 44i Bootis," Quarterly Journal of the Florida Academy of Sciences 29, 1 (1966).
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BIOGRAPHICAL SKETCH Ian Stuart Rudnick was born October 30, 1942, in Chicago, Illinois. He attended primary schools in Chicago and St. Petersburg, Florida before moving to Coral Gables, Florida. He graduated from Coral Gables Senior High School in June, 1960. He received his first three years of college at Massachusetts Institute of Technology after which he transfered to the University of Miami, from which he received the degree of Bachelor of Science cum laude with a major in physics in June, 1965. He then entered the graduate school of the same institution and received the degree of Master of Science with a major in physics in June, 1967. In September, 1968, he entered the graduate school of the University of Florida and began a study and research program leading to the degree of Doctor of Philosophy with a major in astronomy. During this period he held both graduate assistanships and a Graduate School Fellowship. Ian Stuart Rudnick is married to the former Andrea Warshaw, and has one son. He is a member of Sigma Pi Sigma. 161
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, :Un scope and quality, as a dissertation for the degree of Doctor of Philosophy, Frank Bradshaw Wood, Chairman Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. >Vm' AA^ KY. q^en Associate Professor of Astronomy I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation; and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. Ralph C. Isle? Associate Professor of Physics
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I certify that I have read this study and that in my opinion it conforms to acceptable standards of scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor of Philosophy. ,,/^1.Â„^o.^^ r , L'^.'vL Robert E. Wilson Professor of Astronomy University of South Florida This dissertation was submitted to the Department of Physics and Astronomy in the College of Arts and Sciences and to the Graduate Council, and was accepted as partial fulfillment of the requirements for the degree of Doctor of Philosophy. Dean, Graduate School June, 1972
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