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. K-Y
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
O-O 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, pr-oc, 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, pr-tr, 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 O-C'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. O-C'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. Second-order Extinction for Albireo. . . . ... 114
15. First-order Yellow Extinction for Albireo. ... .117
16. First-order Blue Extinction for Albireo. . . . 119
17. First-order Ultra-Violet Extinction for Albireo .
18. UBV Transformation Coefficients . .
19. Variation in the Period of 44i Bootis B
Motion in a Visual Binary System. . .
20. 0-C's from Pohl's Light Elements. . .
21. O-C'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 Russell-Merrill 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 Russell-Merrill 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 one-half
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
Russell-Merrill 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 Russell-Merrill 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 X-functions 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 mid-eclipse, the x-functions 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 x-functions have been tabulated by Merrill (4).
Russell and Merrill have shown that a system consisting
of two similar triaxial ellipsoids with semi-axes 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 a-axis of
each ellipsoid is along the line joining the centers of the
components, the c-axis is parallel to the axis rotation of
the system, and the b-axis 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 Russell-Merrill 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
I-I 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 20-A3 cos 38-A4 cos 40
I" e
A +Co+ (A2+C2) cos 20
and the rectified phase by
sin2 = sin2 8
1-z 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 O-C 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
~
1HO
r-I
-H
a
*d
4
-*
4-U
U,
0
*H
S4
IOO
Cl
*d
*l *i
*u
U -I
U
>1
h-d
4-1r
cca 18
*H *r
>H I
*H *
+1+
U U
0 tt
o
3nm
m-,W ,'L O .0
$) 0 0o 0 1i o I -r
U> -* p aa a O
1 0 a4 -H 11O a)0
0 -rHa r:. u Zo
0-4 ~0 O 0[ n
4-1U U .4 W 0 Ir-4
, -0 4 0 0 O
tnu u 04 n r r 0i-
.i wH E 0 0
O C Or-l Q t m
0 -O 0 a-C n
0 l 0 V WO 4P L)
-HO a) o H Io D
-HI 4J H U ) )- 4 $ w
4 M 4+) -H. 4J U0 4
H 4- 0H
c *d a r- r- II 6 *
4J-1 r, OO 000
C -H -W r-I mW o Wo
>1 r-i4 r1 L-
) U. 11 00 -
-m C 00 4o
-H V W U) V F $4 Wr 0
m o 0 O : -W II
0 0) H 0M4 0 $ 40 M
5ri V W O 0 -I
(d n 0 fa o
S-H- 4-) C0
>1 Mrl 0 ) r
4-1 0 0 +1 0 H
4J W) M 3 a4 U
u mi a)0 r- i I (mII
) a),c-rI o o k !-
(X 1-4 Ei in U W 0 0 1 4
rlU
3OH OA
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 half-way 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 half-width
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 i-n n(u
o o( r-I .
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
i-I 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
E-1 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 O-i O -MC .O
03O* 3OQH o
-4000 .-4-O 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 44-4, )4. 44 4-) 0
a)40 4) 0 4-0 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 so-called 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 mid-eclipse 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
rd-i 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
-1-i 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 half-way 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 primary-occultation,
2) k = .65 primary-transit, 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
pr-oc, 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,
pr-tr, 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 Russell-Merrill method, Kitamura's method was
TABLE XVIII
O-C's from Solution for k = 1, p = 0.1540, a = 0.305.
0 O-C(O) 0-C(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
4-1
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
pr-oc pr-tr
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 Russell-Merrill 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 yz-plane (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 yz-plane.
The Russell-Merrill 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 yz-plane. 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 yz-plane are nearly as large as the dimensions of the
lobes of the outer contact surface in the yz-plane. 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 yz-plane are also considerably smaller
than the sizes of the lobes of the outer contact surface in
the yz-plane.
As the above discussion indicates, the solutions
produced by applying the Russell-Merrill 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 Russell-Merrill 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 Russell-Merrill 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 one-hundredth 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
Russell-Merrill 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 Am-0.2049 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 Am-0.2049 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 least-squares
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 28-A3 cos 36-A4 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
'o)
0 4J-
-H 4-3 U0
4> I 4- 0
- .-rd 4 4-) Q) 5-c
04- ( 0 O -i
-4 o O U)
U- 0 01 a H 1
-H C- -H, 0 H-
4i-)4 ( n 04-)
,CO r- O
mO -l Ol.
.o 0 0-1 0 0-
9 r-I 4 -) -/
0 ) () G) 4-)
U $4 -- 4 0C
0 -H w 4-) -1r-1
OO 0 U)
o o 0 9/
a -a1 m o o
S*u d t o
I o 0 > '4-
-i m 4 o0 -P
0 0 m 4-1 7
S.0 ,Q C *-HI$4
,t O--. O l-
Z E-1 0 04H -4-A
r4
0'
-r
(U
+o
*
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 Russell-Merrill method (Table XXVII and Table XXVIII)
with the true parameters used to derive the synthetic light
curve (Table XXIX)' reveals that the Russell-Merrill
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 Russell-Merrill 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.