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

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4

-*

4-U

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IOO

Cl

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cca 18

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0 tt

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m-,W ,'L O .0__

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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-

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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

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m o 0 O : -W II

0 0) H 0M4 0 $ 40 M

5ri V W O 0 -I

(d n 0 fa o

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>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.