<%BANNER%>
Near infrared and radio observations of selected close binary stars
CITATION THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/AA00011851/00001
 Material Information
Title: Near infrared and radio observations of selected close binary stars
Physical Description: xii, 162 leaves : ill. ; 28 cm.
Language: English
Creator: Florkowski, David Robert, 1949-
Publication Date: 1980
 Subjects
Subjects / Keywords: Double stars   ( lcsh )
Astronomy thesis Ph. D   ( lcsh )
Dissertations, Academic -- Astronomy -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 158-161.
Statement of Responsibility: by David Robert Florkowski.
General Note: Typescript.
General Note: Vita.
 Record Information
Source Institution: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 000098776
oclc - 06841149
notis - AAL4224
System ID: AA00011851:00001

Full Text










NEAR INFRARED AND RADIO OBSERVATIONS OF
SELECTED CLOSE BINARY STARS






BY

DAVID ROBERT FLORK6WSKI




















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

1980

































Dedicated to my parents,

Walter and Erna Florkowski














ACKNOWLEDGEMENTS

I would like to express my gratitude to the many people who helped

make this dissertation possible.

First, I would like to thank my parents, Walter and Erna Florkowski,

for nurturing my childhood interest in astronomy. I am indebted to the

chairmen of my committee, Drs. Kwan-Yu Chen and Stephen T. Gottesman.

Dr. Chen instructed me on photometric observing and helped make some

of the observations of TW Cassiopeiae. He also provided guidance and

encouragement during the analysis of the light curves. Dr. Gottesman

suggested to me the topic of radio stars and helped me learn about radio

interferometry. Without his encouragement the research on radio stars

would not have been possible. I would also like to thank Drs. John E.

Merrill and F. Bradshaw Wood for their advice on many occasions.

Special thanks go to Whit Ludington for his help with the computer

program WINK. He modified this program in several ways, also improving

its performance. The use of his display program WAVE is also greatly

appreciated. I acknowledge from my officemates, Michael Desch and

David Gordon, their consolation and consultation during my years as a

graduate student.

Travel to national observatories was made possible by grants from

the National Science Foundation (AST 7807169 and 7801278) and from the








Southern Regional Education Board. Finally, I would like to thank the

directors of Cerro Tololo Inter-American Observatory* and the National

Radio Astronomy Observatory** for the observing time at these facilities.














































*Operated by the Association of Universities for Research in Astronomy,
Inc., under contract with the National Science Foundation.

**Operated by Associated Universities, Inc., under contract with the
National Science Foundation.














TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS . . . . . . . . . . . .. .iii

LIST OF TABLES . . . . . . . . .. . . ... .vii

LIST OF FIGURES ................... ...... ix

ABSTRACT . . . . . . . . .. . . . .. xi

CHAPTER ONE INTRODUCTION . . . . . . . . . 1

CHAPTER TWO ANALYSIS OF ECLIPSING BINARY LIGHT CURVES . . 5

Advantages of V, R, I Observations . . . . . . 5
Solution by Differential Corrections .. . . . .. 9
Nomographic Solutions . . . . . . . .. ... 12
Rectification of Light Curves . . . . . . ... 14
The WINK Differential Corrections Program . . . . .. 17

CHAPTER THREE OBSERVATION AND ANALYSIS OF AS VELORUM . . .. 20

History .. .... ...... .... ... ........ 20
Cerro Tololo Observations . . . . . . . .... 20
Light Elements for AS Velorum . . . . . . .. 25
Light Curves and a Nomographic Solution. . . . .. 29
Solutions Determined with WINK . . . . . .. . 30
Second Order Effects . . . . . . . .. ... 55
Conclusions. . . . . . . . . . . . 58

CHAPTER FOUR OBSERVATION AND ANALYSIS OF TW CASSOPEIAE . . 60

Recent History ....................... 60
Rosemary Hill Observations . . . . . . . ... 61
Rectification of the i Light Curve . . . . . ... 68
Nomographic Solutions ................... 70
WINK Solution . . . . . . . . ... . . . 71

CHAPTER FIVE RADIO OBSERVATIONS AND DISSUCSSION OF HD 193793 106

Introduction . . . . . . . . . . . 106
Radio Interferometry and Aperture Synthesis . . . . 107
Radio Observations of HD 193793. . . . . . . .116
Infrared Observations of HD 193793 . . . . . .. 125
Discussion . . . . . . . . .. . . . 127








Page

APPENDIX A SOUTHERN STANDARD STARS . . . . . . 131

APPENDIX B VISUAL OBSERVATIONS OF AS VELORUM . . ... 132

APPENDIX C RED OBSERVATIONS OF AS VELORUM . . . ... 137

APPENDIX D INFRARED OBSERVATIONS OF AS VELORUM . . ... 142

APPENDIX E VISUAL OBSERVATIONS OF TW CASSIOPEIAE ...... .147

APPENDIX F RED OBSERVATIONS OF TW CASSIOPEIAE . . ... 151

APPENDIX G INFRARED OBSERVATIONS OF TW CASSIOPEIAE .... 154

APPENDIX H FLUX DENSITIES OF THE CALIBRATORS . . ... 157

BIBLIOGRAPHY ................... ....... 158

BIOGRAPHICAL SKETCH ....................... 162














LIST OF TABLES


Table Page

1. AS Velorum and Its Comparison and Check Stars . . . . 23

2. Times of Minima for AS Velorum . . . . . .... 25

3. Nomographic Solution for the i Light Curve . . . ... 30

4. Independent WINK Solutions for Each Color . . . ... 32

5. Adopted Three Color Solution . . . . . . .... 33

6. Visual Observations of AS Velorum . . . . . .... 35

7. Red Observations of AS Velorum. . . . . . .. .39

8. Infrered Observations of AS Velorum . . . . .... 43

9. Parameters for the Distribution of the Residuals . . .. .53

10. Solutions of TW Cassiopeiae . . . . . . .... 61

11. TW Cassiopeiae and Its Comparison and Check Stars . . .. .62

12. Fourier Coefficients for the i Light Curve . . . ... 70

13. Nomographic Solutions . . . . . . . .... .71

14. Preliminary Solution of TW Cassiopeiae . . . . .. 74

15. Visual Observations of TW Cassiopeiae . . . . .... 86

16. Red Observations of TW Cassiopeiae . . . . .... 90

17. Infrared Observations of TW Cassiopeiae . . . .... 94

18. Blue Observations of TW Cassiopeiae from McCook (1971) . 98

19. Visual Observations of TW Cassiopeiae from McCook (1971) 100









Table Page

20. HD 193793 and Its Calibrators . . . . . . ... 118

21. Journal of Green Bank Observations . . . . . ... 118

22. Flux Densities for HD 193793 . . . . . . ... .121

23. Journal of VLA Observations . . . . . . . ... .123

24. 4.9 GHz Flux Densities for HD 193793 . . . . ... 124


viii














LIST OF FIGURES

Figure Page

1. Illustration of the geometric quantities 6 and Po . . 7

2. The wavelength dependence of the ratio of surface brightness 8

3. The mean photographic light curve of AS Velorum . . .. .21

4. Visual observations of AS Velorum made at Cerro Tololo . 26

5. Red observations of AS Velorum made at Cerro Tololo . . 27

6. Infrared observations of AS Velorum made at Cerro Tololo .28

7. Comparison between the visual normal points and the
theoretical points .... . . . . . . . . 47

8. Comparison between the red normal points and the
theoretical points . . .... .. . . * * 48

9. Comparison between the infrared normal points and the
theoretical points ............ ....**** ..... 49

10. Residuals for visual . . . .... . . . . 50

11. Residuals for red ..................... .... . 51

12. Residuals for infrared ............ ....... 52

13. Histograms of the residuals . . . .... . . 54

14. Spectral response curves ...... .. . . * * 63

15. Visual observations of TW Cassiopeiae made at Rosemary
Hill Observatory ...................... 65

16. Red Observations of TW Cassiopeiae made at Rosemary
Hill Observatory ............ ........... . 66

17. Infrared observations of TW Cassiopeiae amde at
Rosemary Hill Observatory . . . . . . . .. .67

18. Residuals for the light elements of TW Cassiopeiae .... .69

19. Blue observations of TW Cassiopeiae from McCook (1971) . 72

20. Visual observations of TW Cassiopeiae from McCook (1971) . 73








Figure Page

21. Comparison between the visual normal points and the
theoretical points ................... .. 75

22. Comparison between the red normal points and the
theoretical points ................... .. 76

23. Comparison between the infrared normal points and the
theoretical points ................... .. 77

24. Comparison between the infrared normal points and the
theoretical points ................... .. 78

25. Comparison between the visual normal points from McCook
(1971) and the theoretical points . . . . .... 79

26. Residuals for visual . . . . . . . .... 81

27. Residuals for red . . . . . . . . ... .. .82

28. Residuals for infrared ................... 83

29. Residuals for blue (McCook 1971) . . . . . ... 84

30. Residuals for visual (McCook 1971) . . . . .... 85

31. A schematic diagram of an interferometer . . . ... 109

32. Orientation of i and to . . . . . . . .... 111














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


NEAR INFRARED AND RADIO OBSERVATIONS OF
SELECTED CLOSE BINARY STARS

By


DAVID ROBERT FLORKOWSKI

June 1980

Chairman: Kwan-Yu Chen
Co-Chairman: Stephen T. Gottesman
Major Department: Astronomy

Photoelectric observations in V, R, I were made of two eclipsing

stars, AS Velorum and TW Cassiopeiae. The light curves were analyzed

with the Merrill nomographs and a modified version of Wood's (A. J.,

76, 701) differential corrections program WINK. For AS Velorum the

following elements were determined: i = 88?6, rh = .20, rc = .11,

T = 9000K, and Tc = 3600K. In the analysis of TW Cassiopeiae, the

B, and V data of McCook (A. J., 76, 449) were also used. While several

solutions would fit an individual light curve, no solution could be

found to satisfy all the data. A preliminary solution that nearly

fits the data is i = 77, rh = .29, re = .27, Th = 11600K, and Tc =

5600K.

The Wolf-Rayet binary star HD 193793 was detected at centimeter

wavelengths with the Green Bank radio interferometer. Subsequent

observations were made with this instrument and with the completed

portion of the Very Large Array. Variations in the flux density of








HD 193793 were observed. The radio data together with published infra-

red observations were used to describe the ejection of material via

a stellar wind from this binary system.














CHAPTER ONE

INTRODUCTION

Much of our knowledge about stars comes from the study of binary stars.

Values for stellar sizes and masses are obtained from observations of

eclipsing systems. Knowledge of stellar masses is particularly important

because this parameter largely determines the rate of stellar evolution.

In special cases, the interior density distribution for the components of a

binary system can be determined. This result can be used to test evolution-

ary models. It is now known that if the components of a binary system are

close together, they will not evolve independently, i.e., not in the same

way as single stars. The evolution of one star can affect its companion

by transferring mass to the other star, and/or by ejecting mass from the

binary system. The evolution of close, or interacting, binaries is a sub-

ject that has received a great deal of attention. It is now realized that

in extreme cases stars quite unlike single stars can be produced. Novae

and other cataclysmic variable stars, binary X-ray sources that have a

collapsed component (i.e., a neutron star, or a black hole), and Wolf-

Rayet stars are some samples.

The analysis of the light curves of eclipsing stars has never been

more important. Knowledge about the more "normal" interacting binaries

is needed so that the processes that give rise to the more peculiar systems

can be understood. In many cases binary stars consist of two stars that

have quite dissimilar temperatures. This type of binary presents special

problems in the solution of their light curves. Most observations of

binary stars are made in the wavelength region between 3600 and 5500 A.

1








In this interval often 99% of the light of the whole system is emitted by

the hotter star. Thus, one can learn very little about the cooler star

from these observations. This is unfortunate because the cooler star is

of astrophysical interest. The closeness of the hotter companion causes

strong proximity effects: irradiation of the facing hemispheres, and tidal

distortion of the stars. Rucinski (1969) studied the irradiation of the

cooler star, and has predicted that this heating effect will change the

structure of the convective envelope of the cooler star and reduce its

bolometric albedo to 0.5. Pustynik and Toomasson (1973) noted that strong

heating of the envelope of the cooler star causes large departures from

local thermodynamic equilibrium which results in abnormal limb darkening.

Observations made in longer wavelength regions, the red and infrared, will

increase the contribution of the cooler star to the luminosity of the whole

system. Thus, the proximity effects on the cooler star can more readily

be detected, and the above theoretical predictions can be checked. However,

proximity effects are second order effects, and a thorough knowledge of the

geometry of the binary system is necessary for their study. Observations

at long wavelengths, because of the added information about the cooler star,

yield better determined light curve solutions. Grygar and Horak (1974)

have discussed the advantages of multicolor light curves in the study of

binary stars. The geometry of the binary system should be wavelength

independent. For example, the orbital inclination and the sizes of the

stars should be the same for short or long wavelength observations. In the

analysis of multicolor light curves these parameters can be required to be

the same for all light curves. This constraint greatly aids in determining

values of wavelength dependent parameters, including those that describe

second order effects. Light curves that cover widely separated wavelength

regions are of great benefit to the understanding of binary stars.





3

Huang (1973) has reviewed the evidence for circumstellar material and

its location in close binary systems. Since then, infrared and radio

observations have been made of close binary systems. These observations

show that large clouds of ionized material can surround some binaries.

Theoretical work has indicated the importance of mass loss from the system.

Plavec (1973) noted that some binary systems, for example U Sagittae, cannot

be the result of evolutionary mass transfer within the system. In addition

some mass ejection from the system is needed in order to account for the

present configurations of these binaries.

One mechanism for ejecting material from a massive, close binary system

is a stellar wind. Some massive early-type stars are observed to have rates

of mass ejection that are a billion times larger than that of the Sun.

Early type binaries are of interest because stellar evolution with mass loss

via stellar winds may result in the formation of Wolf-Rayet stars (Chiosi,

Nasi, and Bertelli 1979; Vanbeveren et al. 1979). Also, early type stars

are similar to the optical components of binary X-ray sources. Stellar

wind models have been used (Conti 1978; Cester and Mezzetti 1976) to see if

winds are responsible for the observed X-ray luminosities. However, the

mathematical description of stellar winds is rather uncertain. Cassinelli,

Castor and Lamers (1978) have reviewed the status of current models. Also,

the mechanism that causes stellar winds in early type stars is not under-

stood. Both radiation pressure and the dissipation of mechanical energy

have been proposed.

Spectral line profiles, in the visible and far ultraviolet regions,

have been used to study the properties of stellar winds. Some assumptions

about the line forming region must be made in this type of analysis and

given the complexity of the problem it is not surprising that different

researchers get very different results. For example, in a study of the






4

stellar wind of P Cygni, Kuan and Kuhi (1975) found the stellar wind to

be decelerating. In a similar study of the same star van Blerkom (1978)

found an accelerating wind. It has been noted (Klein and Castor 1978;

van Blerkom 1978) that infrared and radio emission from the ejected

material could be used to determine the rate of mass ejection independ-

ently of temperature. These observations could also be used to understand

the stellar wind in the region where the observed spectral lines are formed.

Thus, for the stellar wind problem infrared and radio observations can be

used to complement optical observations.

In the first part of this dissertation (chatpers 2-4) the analysis

of two eclipsing binaries, TW Cassiopeiae and AS Velorum, is presented.

Visual, red, and near infrared observations were made of these binaries

and parameters that describe the binary stars were determined from these

data. In the second part (chapter 5) a study of the Wolf-Rayet binary

system, HD 193793, is presented. Centimeter radio observations and

published infrared observations of HD 193793 were used to study its rate

of mass ejection via stellar wind and the variations of the ejection rate

with time.















CHAPTER TWO

ANALYSIS OF ECLIPSING BINARY LIGHT CURVES

Advantages of V, R, I Observations

Any discussion of the analysis of light curves is made difficult by the

large number of methods available. Some methods of solution are of limited

utility and are of historical interest only, while other methods are widely

used today. The monograph edited by Tsesevich (1973) gives a comprehensive

review of all methods of solution, while Sahade and Wood (1978) primarily

review methods currently used. The discussion here is limited to one

approach to the analysis of light curves and is not meant to imply that

other, equally good, approaches do not exist.

As mentioned in the introduction, special problems are encountered in

the analysis of binary systems having stars of very dissimilar temperatures.

For this case the depth of secondary eclipse will be shallow when compared

to the depth of primary eclipse. The shallow depth causes the eclipse shape

and the value of the depth to be poorly known. Also, the position of

secondary eclipse and its width relative to primary eclipse will be uncertain.

The latter are especially important because they are indicators of a non-

circular orbit. A distorted radial velocity curve can also indicate an

eccentric orbit. However, the presence of circumstellar material can cause

a similar distortion. The information obtained from secondary eclipse is

the best evidence for an eccentric orbit.

The depth of an eclipse is a measure of how much light is being blocked

from view at the time of mideclipse. This loss depends on the amount of

surface area that is blocked from view, and also on the ratio of the brightness

5








per unit area for the hotter and cooler stars. The first factor is geo-

metrical in nature and is explained in the following way. For a binary

star with a circular orbit, it is convenient to take the distance between

the center of the stars as a unit of distance. The apparent distance

between the centers of the stellar disks, 6, is given by


2 = cos2i + sin2icos, (1)


where i is the orbital inclination, and ( is the orbital phase angle. A

parameter, p, used to measure the amount of overlap between the two stars

can be defined as


p = (6 r )/rs, (2)


where rg is the radius of the greater star, and rs is the radius of the

smaller star. In figure 1 these relationships are illustrated.

For a given binary system the value of the overlap parameter, po

(the subscript zero denoting the time of mideclipse), is fixed by the

geometry between the binary system and the observer. Thus, the depth of

the eclipse cannot be changed by altering po. However, the depth also

depends on the ratio of surface brightnesses which is wavelength dependent.

So, by changing the wavelength of observation the depth of the eclipse can

be changed. Assuming that the energy distribution of stars can be repre-

sented by a black body, figure 2 shows the wavelength dependence of the

ratio of surface brightnesses for a typical binary having stars of tempera-

tures 12000 K and 5500 K. The graph shows that for the I band the ratio is

much smaller than for the U, B, V, bands. Thus an I light curve would have

a deeper secondary eclipse than U, B, or V light curves. It is true that

primary eclipse would be shallower, but its depth would still be rather deep.


































og


Fig. 1. Illustration of the geometric quantities 6 and p .
Top: 6, the apparent distance between the centers of
the stars is indicated. Bottom: the distance of the
limb of the larger star to the center of the smaller
star at the time of mideclipse is shown. This dis-
tance expressed as a fraction of the rs is p .










50-

Th 12000 K

B T, 5500K


Jc
h


V


10-
-R




5-





0.5 1.0 1.5 2.0
WAVELENGTH
(microns)
Fig. 2. The wavelength dependence of the ratio of the surface
brightness of the hotter star, Jh, and of the cooler
star, Jc.








The benefits of a deeper secondary eclipse greatly outweigh the effects on

the depth of primary eclipse. A light curve at I would yield more certain

information about the cooler star than shorter wavelength curves. Observa-

tions at shorter wavelengths are useful when one wants to study the hotter

star alone (the light from the cooler star being negligible) outside eclipse.

Ideally U, B, V, R, I observations would give the advantages of both short

and long wavelengths, and the intermediate colors would allow one to smooth-

ly follow the variations with wavelength of certain parameters. In practice

it is usually too difficult to cover adequately the rapid variations during

eclipse in five colors; usually only three color observations can be made.

The colors V, R, I were selected because of (1) the above mentioned advant-

ages of I, (2) the wide wavelength interval, 3500 A, between V and I

(compared with 1900 A between U and V), and (3) R would be a good inter-

mediate color being nearly midway between V and I.


Solution by Differential Corrections

Computer programs for light curve synthesis and differential corrections

are now often used in the analysis of light curves. In general a model hav-

ing m parameters (x x2, . xm) is assumed, and the light intensity

that an observer would see at a given time can be expressed as I = I(xI,

x2, . Xm). An approximate solution Ia is chosen as a starting point,

and an equation of condition is formed:

a aI aI
AI = I ob- = Ax1 + Ax + . + 2 Ax (3)
obs a ax. 1 x 2 x m'

where the light residual, AI, is the difference between the observed and

the calculated light intensities. The correction for a given parameter,

Axi, is defined as


Axi xi (improved)- x (initial).








For all N observations the method of least squares is used to minimize

the sum of the squares of the AI's. The normal equations can be written

in the matrix form:


[A][C] = [R]. (5)


The matrix [A] has elements that are products of the partial derivative

coefficients

N aI aI
A.. = Z ax ax (6)
IJ k=l Xik jk


The column matrix [C] has elements that are the required corrections to the

parameter values of the approximate solution

N
C = Axik. (7)
k=l


The matrix [R] has elements that are the residuals of light intensity

weighted by the derivative coefficients

N a1
R.= E lak (8)
k=l k a

-l
Equation (5) can be multiplied by the inverse of matrix [A], [A]-,

yielding:


[C] = [A]-1[R]. (9)


Thus, the set of corrections can be evaluated and a new approximate solution

can be calculated. New light residuals between the observations and the new

theoretical intensities are calculated, and if the fit is unsatisfactory the

least squares procedure is repeated. In some cases the convergence of the

solution is very difficult. The elements of [A] differ by several orders of








magnitude, which makes inversion of the matrix difficult. Also, correla-

tions between the corrections can make [A] singular. In practice computational

difficulties do not permit simultaneous adjustment of all of the parameters.

Wilson and Biermann (1976) suggest that convergence can be improved by separat-

ing the correlated parameters into two groups. One group is adjusted while

the other is held fixed, then on the next iteration the first group is held

fixed and the second group is adjusted. Sobieski and White (1973) give

examples of correlation between parameters and have calculated correlation

coefficients. Sometimes certain parameters are always held fixed at what are

believed to be "good" values, and sometimes this can lead to spurious results.

For example, in their analysis of Algol, Wilson et al. (1972) used a mass

ratio value of 5.0. They rejected an indirectly determined value of 4.6

(Hill et al. 1971) since the former value gave negligible residuals. Recently,

Tomkin and Lambert (1978) determined directly the mass ratio of Algol. Their

value 4.60.1 is in good agreement with that of Hill et al. A solution may

fit the observations, but there may be other solutions that will also fit.

The equation of condition neglects second and higher order derivatives, so

the starting values must be close to the true values. A poor starting value

may mean the solution converges to a minima, but it is not the deepest

minimum. The parameters of a model, while they are convenient for computa-

tional purposes, may not be directly observable quantities. Thus it is

difficult to determine the starting values. Convergence problems make the

analysis of a light curve a matter of trial and error.

The use of least squares assumes that the chosen model is the correct

one. Unfortunately there is no general consensus as to which model is correct.

We do not have detailed knowledge of the physical processes that occur in a

binary system. Thus the choice of the model affects the values of the

parameters of the solution. Grygar and Horak (1974) have collected from the








literature solutions of various light curves of the binary star Algol.

Their table shows, for example, that published values for the inclination

range from 80.40 to 82.40. Similarly, Soderhjelm (1974) collected pub-

lished solutions by various authors of Hall and Hardie's light curve of

MR Cygni and noted that there were systematic differences in the solutions.

These differences in parameter values were often much larger than the quoted

errors.

In binary star models it is often assumed that information about single

stars is adequate (and relevant) for binaries even though proximity effects

are present. Muthsam (1978) has calculated model atmospheres for the com-

ponents of interacting binary systems and has concluded that the atmospheric

structure of a star is quite sensitive to irradiation by its companion.

Also, Pustynik and Toomasson (1973) have noted departures from normal limb

darkening, and Rucinski (1969) has predicted changes in the bolometric

albedo which are caused by irradiation. The use of least squares for

differential corrections may yield the "best" solution for a given model,

but it does not insure that the true solution for the actual binary is found.

Therefore one should be aware that some systematic discrepancies are possible.

Study of such discrepancies for a number of binary systems could lead to in-

sight in how to improve the assumed model.


Nomographic Solutions

In the discussion above it is clear that a good approximate solution

would be a great help in the convergence of the differential corrections.

A graphical method has been developed by Merrill (1953), based on the

Russell model, for the solution of light curves. This method is quite useful

in determining an initial approximate solution to be used in differential

corrections. Detailed examples of the use of this method have been published

(Merrill 1963, 1978) so only a brief summary will be given here. The








nomograph is divided into two regions, one for partial eclipses and one for

complete eclipses. Every point in these regions represents a possible light

curve solution. The coordinates are chosen so that the eclipse depths are

variables in a linear equation. For partial eclipses the equation is


1 Itr = VIoc H(l Ioc), (10)
o o o


and for complete eclipses the equation is


V = (1 Itr )/ (11)
o o


where H and V represent the horizontal and vertical coordinates, respective-

ly, and the superscripts "tr" and "oc" represent transit and occulation

eclipses, respectively. The shape of an eclipse can be specified by a

function X which is defined as


X = sin2(n=.5)/sin2 (n=.8), (12)


where 0(n=.5) is the phase angle at which the eclipse depth is equal to

0.5 times the mideclipse depth, and 0(n=.8) is the phase angle at which

the eclipse depth is equal to 0.8 times the mideclipse depth. Plotted on

the nomographs are contours for Xoc and Xt Using the observed eclipse

depths and the observed values of X for primary and secondary eclipse, one

can find a point of intersection where the observed X curves cross the

observed depth line. Of course one does not know before hand if primary

eclipse is an occulation or a transit or, in many cases, whether the eclipses

are partial or complete. Each combination must be tired. For most cases

only one combination will result in an intersection, i.e., only one solution

will be possible. In practice the values of Xpri and Xsec, as well as the

depths of the eclipses, will be somewhat uncertain. Instead of an inter-

section at a single point, a region of intersection will be possible, and








the light curve solution lies within the boundaries of this region. The

determinacy of the solution can be judged by the size of this region.


Rectification of Light Curves

The nomograph method is valid for spherical stars only, and the ob-

served binary system may have significant distortions. The presence of

distortions, also known as proximity effects, is indicated by curvature in

the light curve between the eclipses. A procedure called rectification has

been developed (e.g., Russell and Merrill 1952) to compensate for these

effects. Basically a harmonic analysis is performed on the out-of-eclipse

regions to determine the Fourier coefficients to be used in various equa-

tions. These relations are applied to the light curve and the modified

eclipse curves will be equivalent to an eclipse by spherical stars.

Physically the distortions are caused by tidal interactions and by the rapid

rotation of the stars (hereafter called "ellipticity"), as well as the heat-

ing caused by irradiation of the sides of the stars facing each other

(hereafter called "reflection"). The reflection effect is a rather complex

process in which the incident radiation is modified and re-radiated. This

process is dependent on the spectral energy distribution of the incident

light and on the atmospheric structure of the irradiated star. Present

theories cannot adequately explain this effect in detail.

In theory the light variations outside of an eclipse can be represented

by


I(W) = Ao + Alcoso + A2cos, (13)


where the A1 coefficient is predominately due to the reflection effect, and

the A2 coefficient is predominately due to ellipticity. For observations

of an actual binary this equation may not be adequate to represent accurately








the observed curvature outside of eclipse. Assymmetries may be present,

and they must be represented by sine terms, and if the stars are highly

distorted then higher harmonic cosine terms must also be included. A more

general expression for the out-of-eclipse variations is given below

N N
I($) = Ao + Z A cosm4 + E Bmsinmo. (14)
m=l m=l


In harmonic analysis by least squares the choice of the harmonic expansion

affects the values of the coefficients. The gaps in the phase coverage,

where the eclipses are, make the sizes of the coefficients interrelated.

For example, if only up to cos24 terms were included, then any cos30

variation that is present in the data will modify the value of the derived

coefficient of the cos# term. Merrill (1970) discusses this problem and

recommends the inclusion of terms to at least the fourth harmonic (N=4).

Three constants that are used in describing the irradiation effects

for partial eclipses are


C = 0.090sin2e. (15)


Cl = -Al' (16)



C2 = 0.030sin2 e' (17)


and for complete eclipses are


C = 0.072sin2 (18)


Cl = -A1' (19)


C2 = 0.024sin2 e'
6 e







where 1e is the phase angle at which external tangency occurs. To rectify

the observations, one first subtracts the sine terms

4
I'() = I() E B sinmo. (21)
m=l m


The curvature is now symmetric, and theory states that reflection and

ellipticity can be removed using


I'(0) A4cos4o A3cos3o + Co Alcos) + C2cos2(
Ir( ) = Ao + Co + (A2 + C2)cos20 (22)


for all of the observations. The rectification should be checked to see if

the rectified light intensity, Ir, is actually constant outside of eclipse.

The phases during each eclipse are rectified using Merrill's transformation


sin2 r = sin2 /(l zcos2e). (23)


The photometric ellipticity, Nz, is given by


Nz = -4(A2 C2)/(A0 C0 A2 + C2) (24)


According to Hosokawa (1957) N is related to the limb darkening coefficient,

x, and the gravity darkening coefficient, y, by


N = (15 + x)(l + y)/(15 5x). (25)


Further information on the rectification procedure and its theoretical basis

is given by Merrill (1970) and by Guinan (1976). The rectified intensities

and phases, Ir(Or), are now equivalent to an eclipse by spherical stars, and

use of the nomograph method is now valid. A solution having coordinates

(H,V) can be determined in the same manner as described above. Besides the








X contours, there are also plotted on the nomographs contours for the over-

lap parameter, po, and contours for the ratio of radii, k = rs/rg The

radii of the stars can be determined with


r = sin~re/(l + k)2 (1 + kpo) cos2re), (26


r = kr (27)


where Ore is the rectified phase angle where external tangency occurs. The

orbital inclination can be determined with the aid of


cosi = (1 + kPo)r (28)


The relative luminosities of the stars are given by


L = (1 = Ic)/oc, (29)
s 0 0


L = (1 Ior)/V, (30)
g o


where for partial eclipses a00 = V/(V+H), and for complete eclipses aoc = 1.
o 0
These parameter values can be used as a starting point for further refine-

ment by differential corrections.


The WINK Differential Corrections Program

After nomographic solutions were obtained, the light curves of AS

Velorum and TW Cassiopeiae were analyzed with the differential corrections

program WINK. Wood (1971, 1972) gives a detailed description of the com-

puter program and its use. Only a few interesting features and some

unpublished revisions to the program will be noted here. The two stars,

designated as "A" and "B," can be described by two different sets of

parameters, an astrophysicall" set and a "model" set. The latter is used








only as an aid in translating a solution based on the Russell model, e.g.

a nomographic solution, into a set of astrophysical parameters. The astro-

physical parameters are convenient for computational purposes, but most of

them cannot be directly related to observable quantities. In WINK only the

astrophysical parameters can be used as variables in the adjustment pro-

cedure.

The stars are assumed to be in synchronous rotation with their axes

perpendicular to the orbital plane, and it is assumed that the stars can be

represented by triaxial ellipsoids. Chandrasekhar (1933) showed that a

rotating, tidally distorted star can be represented by an ellipsoid whose

semimajor axes are functions of the polytropic index of the star and of the

mass ratio of the binary. Wood uses truncated versions, of order r of the

expansions derived by Chandrasekhar. Merrill (1970) citing Reuning (private

communication) states that the inclusion of terms of the order of r and r5

will only change the value of the semiaxes by a few part in a thousand. As

an example of the equations used by WINK, the semimajor axes for star A are

given by


aA = rA[l + 1/6(1 + 7q)A2A rA3], (31)


bA = rA[l + 1/6(1 2q)A2A rA3], (32)


cA = rA[l 1/6(2 + 5q)A2A rA], (33)


where q is the mass ratio, M/MA, rA is the unperturbed radius of star A,

and A2A is a slowly varying function of the polytropic index of star A.

The value of the polytropic index for star B can be different from the value

for star A; hence dissimilar ellipsoids can be calculated by this program.





19

The treatment of the reflection effect that is described by Wood (1973)

has since been modified. Clausen, Gyldenkerne, and Gronbech (1976) discuss

the new treatment of the reflection effect. The spectral energy distribu-

tion of the two stars can be represented by a black body law, or by a

spectral distribution obtained from a model atmosphere.

For the analysis of AS Velorum and TW Cassiopeiae theoretical limb

darkening coefficients tabulated by Al-Naimiy (1978) were used initially.

The gravity darkening exponent was fixed at 0.25 for stars with radiative

envelopes, and at 0.08 for stars with convective envelopes. For the reflec-

tion effect the bolometric albedo was fixed at 1.0 and 0.5 for stars having

radiative and convective envelopes, respectively. In general, the geometric

parameters of the binary system were adjusted first. Then, other parameters

were adjusted while the geometric parameters were held constant. For a

given trial solution, plots were made of observed and computed light curves

and their residuals for all three colors. Further adjustments were made

after careful examination of the plots for the three colors. The validity

of the above assumptions and the results of the analysis of the individual

systems are discussed in the next two chapters.















CHAPTER THREE

OBSERVATION AND ANALYSIS OF AS VELORUM

History

Life many other southern stars, very little is known about AS Velorum

(698.1935, CoD -38 4504, HD 71872). Its variability was discovered by

O'Leary and O'Connell (1935). On the basis of their photographic observa-

tions they classified AS Velorum as an eclipsing binary of the Algol type.

Gaposchkin (1953) published a mean light curve (Figure 3) and normal points

that were based on visual estimates of photographic plates. A period of

1155788874 was used, and no other information about the observations was

given. The secondary eclipse is displaced, mideclipse occurring at about

phase 0.60, and it is about 0.08 deep. The light curve outside of eclipse

is flat, and only a small amount of scatter is evident. The displaced

secondary eclipse and the relatively short period makes the study of this

binary attractive. Although AS Velorum is not very faint, V = 8.6, no

photoelectric observations have been made before this research project.


Cerro Tololo Observations

AS Velorum was observed at Cerro Tololo Inter-American Observatory

(hereafter CTIO) during February and March, 1977. Cerro Tololo is located

in Chile, 80 km southeast of the coastal city of La Serena, at an elevation

of 2210 m (7250 ft). The observations were made with the Lowell Observatory

0.6 m (24 in) reflector. A photomultiplier tube (ITT FW118) with a S-1

photocathode, cooled with dry ice, was used. This tube was selected so that

the instrumental system would be close to Johnson's standard photometric












































































006 o' :6 026 0; Oh:16 U16
3HdJUao010H.d :31nL IHNOU


" H
g




o o\
m rl




*0 d
a
.




















a
Cd


















C,
rf







d






a a
H



&
3
dl




a a








" dO
',-, *H




*2 IJ









S
0 0g







a.
-- C
dl'6








system. The CTIO filters that were available are

1) for V, a Corning 3384 and a 9780;

2) for R, a Schott KG-1/4, an OG-5/2, and a RG-6/1.5;

3) for I, a Schott RG-715/3, and a RG-780/1.

Instead of using the above V filter combination, a custom made Corion Corpora-

tion "infrared suppressing" filter was used. A much higher signal could be *

obtained with it. This filter was also used for the observations of TW

Cassiopeiae, and its transmission properties will be discussed in the next

chapter. The CTIO filters for R and I were used for the AS Velorum observa-

tions. The photometric measurements were made with a SSR photon counter.

Each measurement was converted into a magnitude using


mag = -2.51og [(N* Ns)/NC], (34)


where mag is magnitude, N* is the number of counts from the star plus sky,

N is the number of counts from the sky alone, and NC is a constant that was

selected to make the instrumental magnitudes close to the standard magnitudes.

The counting rates were low, and the correction for "dead time," i.e., a

multiple pulse count (Fernie 1976), was negligible.

Differential observations were made of AS Velorum using a nearby star

as a comparison star. In order to minimize systematic errors, the comparison

star was selected so that it is close to the position of AS Velorum, and is

about the same magnitude and color as AS Velorum. Using this criterion a

second star was selected to be a check star. Information about these stars

can be found in Table 1.








TABLE 1

AS Velorum and Its Comparison and Check Stars


Star SAO RA Dec Spec V V-R R-I
(1950)
AS Vel 199271 8h 26m 27.6s -380 48' 19'.'2 A3 8m63 m22 m09

Comparison 199189 8 25 5.2 -39 15 50.1 B9 8.18 .06 -.03

Check 199240 8 31 4.4 -38 36 21.8 AO 9.56 .07 -.02



The differential observations of AS Velorum consist of 20 sec integra-

tions made in the following sequence: (V, R, ; (I, R, V)s; (V, R, )v;

(I, R, ) s; (V, R, I) The subscripts c, v, and s represent comparison star,

variable star (AS Velorum), and sky respectively. In order to follow closely

the rapid changes in magnitude during an eclipse three sets (instead of one

as given above) of variable star measurements were made between the compari-

son star measurements. This use done only for primary eclipse. Unfortunately

for secondary eclipse this could not be done because the eclipse is so shallow

that one could not recognize, from the raw data, that an eclipse was occurring.

For a given night the principle extinction coefficient was determined from the

comparison star magnitudes. On one night the second order (color) extinction

coefficient was determined using a method described by Hardie (1962). A

visual double, having spectral types B9 and G5, was observed over a large

range in airmass. From these differential measurements the color extinction

coefficient was determined. Only the V-R color index had a large enough co-

efficient, -0.055, to be measureable. For the R-I color index the color ex-

tinction was negligible. As Hardie recommends, the second order extinction

was assumed to be the same for each night. Atmospheric extinction had very

little effect on the differential photometry of AS Velorum. At a large

airmass, for example X = 2, the difference between the airmass of the








variable star and that of the comparison star was only -0.008. The values

of the principle and second order extinction coefficients were small, and

when multiplied by the very small differential airmasses resulted in quite

negligible corrections. The scatter in the photometry is probably due to

changes in the atmospheric transparency from one comparison star measure-

ment to the next comparison star measurement.

Observations of standard stars were made so that the instrumental

photometric system could be transformed into Johnson's standard system.

The standard stars that were selected from Johnson et al. (1966) included

a wide range of colors, and were close to the position of AS Velorum.

Information about the standard stars that were used can be found in

Appendix A. A least squares procedure was used to determine the transforma-

tion equations. In order to distinguish between the instrumental and the

standard systems, lower case and upper case letters will be used respectively.

For the restricted method of least squares the independent variable is

assumed to be free from error. Thus the magnitudes and colors of the

standard stars, which define the standard system, were used as the independ-

ent variable, while the instrumental system was used as the dependent

variable. The following linear equations were found to fit the observations

(v r) = 0.97(V R) -0.04, (35)
.02 .01

(r i) = 0.76(R I) -0.04. (36)
.03 .01

For V v, a parabolic equation was required to fit the observations.


V v = 0.25(V R)2 0.39(V R) + 0.123. (37)
+.03 .03 .006

The equations of transformation are found by rearranging the above equations

giving,








v R = 1.03(v r) + 0.41, (38)

R I = 1.32(r i) + 0.53, (39)

V = v + 0.25(v r)2 0.19(v r) + 0.005. (40)


The standard stars in the southern sky have not been well observed. The

values of the magnitudes and color indices are averages of only 3 to 5

measurements. Johnson et al. (1966) gives the value for the error of a

single measurement. These errors imply that the average values are only

known to about 0.01. The use of the restricted method of least squares

assumes that the independent variable is free from error. In practice one

usually allows errors in the independent variable if they are an order of

magnitude smaller than the errors in the dependent variable. This condition

is not satisfied for the standards in the southern sky. At the present time,

it is believed that a reliable transformation is not possible, and so the

observations of AS Velorum were left in the instrumental photometric system.


Light Elements for AS Velorum

In the three weeks that AS Velorum was observed, insufficient time had

elapsed to improve on the value of the orbital period. The value given by

Gaposchkin (1953) was therefore adopted. Although several primary eclipses

were observed, only two of them had sufficient phase coverage to determine

times of minima. The bisection method was used to determine the times of

minima given below.

TABLE 2

Times of Minima for AS Velorum

HJD color
2443201.6975 v
.6970 r
.6970 i
2443212.6010 v
.6026 r
.6030 i


1--




















~_ 44.1*


. . ... . .

09..... 4 L.4











0 i' O'3 0o0o o0o 0O 00o 1
7iB' I A :30nnlNlUH y1130


CsC:~'
*Y
:







:;
:.


: i
" '


~~
:.

.~
i

':
r.
;s

.;j


:t:
: :






!~











: :
r:
i:
~ :i


0










r4
9 3



0



-a






rl




o
0















*ri


01 I1






27

















d0
.. . .; .***

o'. o.! .. . .
.:".' t5



S'






a a
...4


, s

o aC

Cu
** 0Cu







o.. o


o Q>
Cu
s $
0 :








.. , =o .. ,
-.



'. *






'g ^**. ..

,,.*:* *" ..v v
"""** ..d' '

0 P
|u





OE'O Oh'O t 5O n00 o OL!O 0os0o o'o'
03U : 3Gnu I .U i y17o30







28























c3
0

...0
14














o o44
'. \. o






P4




















ol'o 09, o OtO o 0
a: 0
















I "1








The WINK program was used to find the best value for the time of conjunc-

tion using all the observations. With this value and the times of minima

above the following light elements were adopted:


Min I = 2443212.6020 + 1.55788740 E. (41)


Light Curves and a Monographic Solution

Using the above light elements v, r, i light curves of AS Velorum were

plotted (figures 4 6). On these graphs each point represents a single

observation. The secondary eclipses in v and r are very shallow, so only

for the i light curve was a nomographic solution attempted. In order to

reduce the observational scatter, the individual observations were averaged

to form normal points. Since the observations during primary eclipse were

made in groups of three, the normal points were formed from averages of three

points. No evidence for an eccentric orbit could be found in the light curve,

so the orbit was assumed to be circular. Outside of the eclipses the light

curve is rather flat. However the shoulders of secondary eclipse are slightly

higher than the shoulder of primary eclipse. This indicates that some reflec-

tion effect is present. For a preliminary solution it was judged that recti-

fication of the i light curve was not needed.

A plot of the normal points, in intensity units, showed that the depth

of primary and secondary eclipse was 0.34 and 0.03, respectively. The values

of X were found to be very nearly equal, 0.50 for primary and 0.51 for second-

ary. While it is true that Xoc is greater than Xtr, the estimated uncertain-

ties in the X values are large enough so that either eclipse might be a

transit or an occulation eclipse. Therefore each combination was tried,

and it was found that the eclipses are complete and that primary is a transit

eclipse. The nomographic solution is given in the table below.








TABLE 3

Nomographic Solution for the i Light Curve


k .58 Po -1.67

r .22
g x 0.2
g
r .12 xs 0.2 (assumed)

1 89;5 0e 19?8

L .97
g J
sec .081
L .03 Jpri



This solution was used as a starting point for differential corrections

with the WINK computer program.


Solutions Determined with WINK

The nomographic solution for i was converted into the WINK "model"

parameters so that the differential corrections procedure could be used to

refine the preliminary solution. The conversion cannot be done exactly

because of differences in the way that the WINK parameters and the Russell

model parameters are defined. For example, the relative luminosities of

the Russell model are not included in the sets of "model" or astrophysicall"

input parameters that are used in WINK. In order to describe the luminosi-

ties in WINK the radii and the temperatures of the stars must be known. It

is not easy to relate the temperature used in WINK to an observable quantity.

One could specify the temperature of star A (the hotter star) and the ratio

of surface brightnesses of stars B and A. However, the temperature of star

A, as defined for WINK, is the effective temperature of star A at the sub-

earth point when viewed at phase 0.25. The use of an observable quantity





31

such as a color index or a spectral type gives a mean temperature for the

whole stellar disk. One must also note the difference between color

temperature and effective temperature, as well as considering the effects

of proximity effects on spectral type and color index. The surface bright-

ness in WINK also refers to the subearth point when viewed at phase 0.25.

When one uses the depths of the eclipses to determine the ratio of the

surface brightnesses, the surface brightness is an average brightness over

the area of the star that is eclipsed. One usually assumes that the approxi-

mations made in the translation of the Russell model parameters into the

WINK parameters do not introduce any significant errors. One hopes that

the effects of differences in definition are minor, and that the trans-

lated solution is a good starting point.

Initially one uses the starting solution to compute a theoretical

light curve and compares it with the observed light curve. Some insight

may be gained as to how to adjust the solution with the differential

corrections procedure. At first the bolometric albedo, the gravity

darkening coefficient, and the limb darkening coefficient for each star

are held fixed. Adjustment of these parameters results in only minor im-

provements of the fit. It was also assumed that the temperature of the

hotter star could be held fixed because it was reliably known, from its

spectral type, to have a temperature of 9000 K. The computational problems

mentioned in Chapter 2 makes convergence difficult. The three light curves

can only be adjusted separately by WINK, and one must be aware that the

wavelength independent parameters should have nearly the same values for

each color. As noted, Wilson and Biermann (1976) suggest that the convergence

can be aided by dividing the parameters into two groups, and adjusting each

group separately. For example, the inclination and the radii could be one

group, while the temperature of the cooler star and the mass ratio could








be the other group. The choice of the parameters that one picks for a

group is greatly influenced by what color light curve one is adjusting.

The i light curve is most sensitive for adjusting the temperature of

the cooler star, and the v light curve is most sensitive for adjusting

the mass ratio. However the refinement of the solution by differential

corrections is, to a large extent, a matter of trial and error. At inter-

mediate stages theoretical light curves for all three colors were computed

and compared with the observations. From these comparisons some physical

insight may be found, and the least squares adjustment can be "guided"

into a good fit of the observations. In Table 4 an independent solution

for each color is given. They are the best fitting solutions for a given

color. Star A, the hotter star,is designated by the subscript h, and star

B, the cooler star, is designated by the subscript c. In the table r* is

the unperturbed radius, and k is the ratio of the unperturbed radii.

Internal errors for the adjustable parameters are given.

TABLE 4

Independent WINK Solutions For Each Color

parameter v r i


i 89!5 o6 90?2 172 88?6 ?4
rh* .199 .001 .1995 .0001 .198 .0001
k .565 .002 .5675 .0009 .572 .002
ah .200 .200 .200
bh .199 .200 .199
ch .198 .199 .198
a .113 .114 .144
c
b .112 .113 .113
c
c .112 .113 .113
Th 9000++ 9000++ 9000++
Tb 3600++ 3600++ 3600 77


++assumed value that was held fixed.








TABLE 4 continued


parameter v i

xh .5++ .35++ .2++
x .5++ .35++ .2++
Lh .995 .989 .979
L .005 .011 .021
c
Mc/M .459 0.55 .374 .07 .440++


++assumed value that was held fixed.


With these solutions some further refinements were made and a "compromise"

set of parameters was adopted. The adopted solution (Table 5) was required

to have the same values for the wavelength independent parameters for each

color and still fit the three light curves.

TABLE 5

Adopted Three Color Solution


88.6
.199
.201
.200
.199

9000++


Mc/Mh
k
a
c
b
c
c
c
T
c


.40
.569
.114
.113
.113

3600


.35++
.35++
.989
.010


++fixed at this value.






34

Using the adopted solution, theoretical light curves were calculated

for each color. The computed points and the normal points for primary and

secondary eclipse are plotted in Figures 7 9. Also, for the entire phase

range the residuals, taken in the sense of observed minus computed, are

plotted in Figures 10 12, and are listed together with the observed normal

points in Tables 6 8. The graphs illustrating the residuals versus phase

show no obvious trends. This does not insure that the residuals are

actually random. In order to see if small systematic effects are present,

the residuals were examined statistically. It was not considered possible

to study meaningfully the phase dependence of the residuals. Systematic

deviations that are phase dependent may be present, but deviations from the

model used in WINK and systematic errors in the nightly observations cannot

be distinguished. So only the distribution of the size of the residuals was

considered. For each color a histogram of the residuals was formed. For

the three sets of residuals the mean, standard deviation, skewness, and

kurtosis were computed. Table 9 gives the results of these calculations.











TABLE 6


VISUAL OBSERVATIONS OF AS VELORUM
NORMAL POINTS AND RESIDUALS
FOR THE WINK SOLUTION


PHASE


0.000258
0.003066
0.003860
0.004904
0.006802

0.009094
0.010436
0.013444
0.015108
0.017831

0.019995
0.022081
0.024070
0.025250
0.026629

0.028393
0.029844
0.031191
0.033098
0.034993

0.035865
0.039328
0.040965
0.042193
0.043655

0.045465
0.046618
0.048918
0.050724
0.053116

0.054598
0.056668
0.058844
0.062127
0.064435

0.067403
0.072277
0.078031
0.079487
0.084102

0.089312
0.093568
0.099217
0.103676
0.108448
0.111686
0.114975
0.120305
0.128057
0.143833


NORMAL POINTS
(MAGNITUDE)

1.002
1.007
1.004
1.002
1.008

1.002
0.975
0.972
0.955
0.896

0.866
0.821
0.793
0.780
0.747

0.723
0.703
0.683
0.649
0.625

0.624
0.575
0.569
0.557
0.538

0.539
0.527
0.520
0.508
0.520

0.515
0.521
0.508
0.531
0.509

0.528
0.521
0.514
0.517
0.515

0.515
0.521
0.513
0.503
0.511

0.520
0.518
0.515
0.499
0.509


RESIDUALS
(MAGNITUDE)

-0.00618
0.00077
-0.00104
-0.00080
0.00971

0.01047
-0.01010
0.00431
0.01006
-0.00595

0.00153
-0.00640
0.00094
0.00956
-0.00021

0.00446
0.00683
0.00640
-0.00202
-0.00187

0.00732
-0.00380
0.00453
0.00252
-0.00560
0.00690
0.00099
0.00324
-0.00561
0.00630

0.00126
0.00805
-0.00515
0.01801
-0.00427
0.01478
0.00864
0.00137
0.00486
0.00333

0.00336
0.00943
0.00251
-0.00769
C.00074

0.00997
0.00790
C.00608
-0.00957
0.00115











TABLE 6 CONTINUED


VISUAL OBSERVATIONS OF AS VELORUM
NORMAL POINTS AND RESIDUALS
FOR THE WINK SOLUTION


PHASE

0.159894
0.173802
0.180579
0.184936
0.189729

0.194042
0.198216
0.202913
0.208836
0.219269

0.240915
0.249803
0.253949
0.258697
0.263273

0.268507
0.273655
0.278433
0.284959
0.295454

0.308415
0.315747
0.323643
0.330922
0.338043

0.356974
0.374889
0.388355
0.399911
0.406822

0.412305
0.417484
0.422520
0.428257
0.433999

0.442539
0.451477
0.458765
0.465641
0.472507

0.478523
0.486661
0.492241
0.499659
0.506321

0.514394
0.519595
0.524377
0.529946
0.534512


NORMAL POINTS
(MAGNITUDE)

0.506
0.512
0.506
0.495
0.499

0.507
0.506
0.506
0.500
0.502

0.517
0.509
0.493
0.503
0.491

0.497
0.495
0.500
0.504
0.502

0.507
0.506
0.514
0.499
0.504

0.505
0.506
0.513
0.503
0.505

0.498
0.516
0.506
0.500
0.498

0.512
0.515
0.509
0.514
0.515

0.519
0.519
0.523
0.513
0.520

0.522
0.523
0.516
0.511
0.517


RESIDUALS
(MAGNITUDE)

-0.00046
0.00671
0.00091
-0.00985
-0.00560

0.00232
0.00173
0.00154
-0.00410
-0.00170

0.01397
0.00661
-0.00945
0.00037
-0.01172

-0.00604
-0.00738
-0.00243
0.00106
-0.00138

0.00331
0.00253
0.01001
-0.00602
-0.00126

-0.00104
-0.00123
0.00521
-C.00534
-0.00332

-0.01121
0.00663
-0.00332
-0.00940
-0.01165

0.00189
0.00459
-0.00269
-C.00019
-0.00135

0.00077
-0.00025
0.00420
-0.00653
0.00059

0.00331
0.00485
-C.00137
-0.00428
0.00305











TABLE 6 CONTINUED


VISUAL OBSERVATIONS OF AS VELCRUM
NORMAL POINTS AND RESIDUALS
FOR THE WINK SOLUTION


PHASE

0.538020
0.541789
0.544945
0.549169
0.554490

0.561976
0.569478
0.576652
0.583689
0.591170

0.598700
0.605788
0.611935
0.615588
0.620663

0.625850
0.630211
0.635341
0.640462
0.646593

0.654095
0.660918
0.668254
0.674460
0.681257

0.687124
0.696364
0.705506
0.710909
0.715960

0.722045
0.729114
0.741572
0.760471
0.772256

0.781253
0.788886
0.796372
0.806617
0.820237

0.828528
0.841064
0.850993
0.859361
0.873337

0.885106
0.890903
0.898698
0.904853
0.909196


NORMAL POINTS
(MAGNITUDE)

0.510
0.511
0.506
0.504
0.508

0.504
0.503
0.495
0.517
0.505

0.509
0.505
0.507
0.509
0.504

0.506
0.501
0.505
0.494
0.510

0.502
0.510
0.508
0.509
0.501

0.505
0.505
0.508
0.495
0.504

0.505
0.506
0.507
0.506
0.502

0.500
0.504
0.505
0.508
0.514

0.513
0.502
0.509
0.512
0.513
0.517
0.512
0.512
0.517
0.512


RESIDUALS
(MAGNITUDE)

-0.00292
-0.00125
-0.00491
-0.00715
-0.00282

-C0.00623
-0.00690
-0.01477
0.00828
-0.00402

0.00108
-0.00272
-0.00057
0.00183
-0.00308

-0.00129
-0.00544
-0.00156
-0.01168
0.00385

-0.00346
0.00519
0.00304
0.00482
-0.00329

0.00164
0.00167
0.00455
-0.00772
0.00098

0.00198
0.00285
0.00457
0.00254
-0.00104

-0.00352
-0.00001
0.00037
0.00339
0.00836

0.00687
-0.00492
0.00201
0.00433
0.00403

0.00679
0.00199
0.00175
0.00604
0.00046











TABLE 6 CONTINUED


VISUAL OBSERVATIONS OF AS VELCRUM
NORMAL POINTS AND RESIDUALS
FOR THE WINK SOLUTION


PHASE

0.911588
0.915655
0.920868
0.922992
0.927628

0.930015
0.933166
0.936494
0.939259
0.941265

0.944346
0.946700
0.948617
0.951219
0.953403

0.955955
0.957775
0.960701
0.962452
0.964540

0.967150
0.968086
0.971565
0.973268
0.975142

0.977584
0.978847
0.980438
0.983001
0.984783

0.986409
0.988218
0.989807
0.991529
0.992867

0.994254
0.996118
0.997762
0.998732


NORMAL POINTS
(MAGNITUDE)

0.519
0.525
0.510
0.514
0.519

0.515
0.526
0.522
0.531
0.521

0.512
0.521
0.518
0.510
0.532

0.533
0.563
0.583
0.607
0.621

0.653
0.671
0.714
0.750
0.773

0.826
0.855
0.876
0.923
0.952

0.952
0.974
0.980
0.989
0.986

0.995
0.990
0.998
1.010


RES ICUALS
(MAGNITUDE)
0.00790
0.01275
-0.00186
0.00131
0.00655

0.00252
0.01335
0.00858
0.01745
0.00795

-0.00099
C.00740
0.00452
-0.00726
0.00639

-0.00723
0.00868
0.00333
0.00986
-0.00036

-0.00120
0.00416
-0.00427
0.00472
-0.00487
0.00487
0.01116
0.00330
0.00785
0.00830
-0.01395
-0.00433
-0.00611
-0.00397
-0.01214

-0.00619
-0.01470
-C.00900
0.00233










TABLE 7


RED 03SERVATIONS OF AS VELORUM
NORMAL POINTS AND RESIDJALS
FOR THE WINK SOLUTION


PHASE


0.002620
0.003762
0.004706
0.006813
0.009325

0.010650
0.313733
0.015346
0.018086
0.020222

0.022359
0.024316
0.025473
0.026885
0.029629

0.030097
0.031460
0.033350
0.0352?39
0.036094

0.033563
0.041183
0.342413
0.043907
0.045716

0.046830
0.049169
0.050998
0.053390
0.055509

0.057489
0.060110
0.363338
0.064683
0.0'7684

0.372519
0.078260
0.079752
0.084350
0.090418

0.095575
0.100856
0.105154
0.109113
0.113388

0.115688
0.123010
0.132997
0.148805
0.165374


NORMAL POINTS
(MAGNITUDE)

0.832
0.81
0.927
0.615
0. 830

0.805
0.806
0.734
0.740
0.710

0.652
0.633
0.614
0.606
0.563

0.529
0.514
0.503
0.472
0.456

0.432
0.403
0.398
0.399
0.385

0.383
0.372
0.366
0.368
0.361

0.364
0.365
0.344
0.353
0.356

0.367
0.359
0.376
0.357
0.3 36

0.369
0.354
0.353
0.361
0.350

0.370
0.354
0. 344
0.355
0.352


RESIDUALS
(MAGNITUDE)

0.0075a
-0.00480
0.00492
-0.00404
0.01612

-0.00445
0.00978
0.00802
0.00449
0.00815

-0.01305
0.00123
0.00138
0.0164A
0.00102

-0.01080
-0.00670
0.30744
-0.00034
-0. .0630

0.00647
-0.00757
-0.00290
0.00935
0.00712

0.01167
0.01002
0.00631
0.00850
0.00148

0.00446
0.00577
-0.01510
-0.00554
-0.30250

0.00683
0.00071
0.01819
-0.00096
-0.00182

0.01187
-0.03321
-0.30409
0.00450
-0.00653

0.01358
-0.00201
-0.01058
0.00084
-0.00056









TALE 7 CONTINUED


RED OBSERVATIONS OF AS VELORUM
NORMAL POINTS AND RESIOUALS
FOR THE WINK SOLUTION


PHASE


0.177678
0.182201
0.186617
0.191430
0.196001

0.203159
0.209063
0.219496
0.241159
0.250019

0.254199
0.258943
0.263537
0.268783
0.273899

0.279206
0.287977
0.300846
0.311119
0.318524

0.326497
0.333672
0.341535
0.365391
0.379746

0.393319
0.402241
0.408869
0.414411
0.419520

0.424745
3.430142
0.437509
0.449358
0.45695

0.462922
0.470981
0.476381
0.484612
0.490787

0.497030
0.504704
0.511655
0.513481
0.523L54

0.528486
0.532908
0.536176
0.539119
0.543099


NORMAL POINTS
(MAGNITUDE)

0.357
0.356
0.351
0.360
0.353

0.356
0.349
0.351
0.343
0.341

0.347
0.358
0.350
0.351
0.355

0.346
0.346
0.346
0. 39
0.364

0.339
0.349
0.348
0.352
0.355

0.356
0.350
0.359
0.361
0.356

0.346
0.355
0.354
0.365
0.364

0.349
0.355
0.365
0.359
0.359

0.364
0.359
0.381
0.369
0.353

0.353
0.355
0.349
0.362
0.359


RESIDUALS
(MAGNITUDE)

0.00433
0.00387
-0.00040
0.00844
0.00176

0.00519
-0.00154
0.00080
-0.00662
-0.00820

-0.00272
0.00885
.).00130
0.00164
0.00615

-0.00335
-0.00291
-0.00319
-0.00090
3.01422

-0.01049
-0.00159
-0..00204
0.00074
0.00311

0.00349
-0.00246
0.00569
0.00768
0.00310

-0.00749
0.00163
-0.00029
0.01038
0.00771

-0.01005
-0.00914
-0.00228
-0.01139
-0.01139

-0.00682
-0.01131
0.01043
-0.00002
-0.01404

-0.01077
-0.00606
-0.01075
0.00423
0.00303









T43LE 7 CONTINUED


RED OBSERVATIONS OF AS VELORUM
NORMAL POINTS AND RESIDUALS
FOR T.E WINK SOLUTION


PHASE


0.546680
0.550947
0.557134
0.564959
0.571795

0.579709
0.586S34
0.593577
0.601947
0.607704

0.614231
0.619538
0.624658
0.628991
0.633985

0.638598
0.644709
0.651841
0.658989
0.665399

0.672743
0.679356
0.683363
0.693339
0.702744

0.709451
0.714942
0.719784
0.727214
3.736723

0.753678
0.769657
0.773904
0.796366
0.794038

0.302243
0.817225
0.825904
0.837636
0.847781

0.856256
0.a69306
0.q82000
0.889290
0.895891

0.903701
0.901026
0.911302
0.915307
0.919533


NORMAL POINTS
(MAGNITUDE)

0.349
0.349
0.346
0.348
0.345

0.338
0.345
0.341
0.352
0.345

0.350
0.360
0.349
0.348
0.346

0.349
0.346
0.345
0.345
0.348

0.356
0.351
0.347
0.349
0.349

0.350
3.344
0.353
0.346
3.350

0.347
0.359
0.356
0.348
0.346

0.344
0.361
0.338
0.336
0.351

0.358
0.353
0.349
0.359
0.364

0.367
0.355
0.370
0.362
0.357


RESIDUALS
(MAGNITUDE)

-0.00608
-0.00541
-0.00795
-0.00602
-0.00810

-0.01513
-0.00817
-0.01202
-0.00069
-0.00757

-0.00162
0.00848
-0.00212
-0.00356
-0.00488

-0.00252
-0.00511
-0.00535
-0.00562
-0.00230

0.00609
0.00077
-0.00238
-0.00020
0.00017

0.00057
-0.00528
0.00414
-0.00325
0.00069

-0.00278
0.00937
0.00555
-0.00227
-0.00468

-0.00715
0.00950
-0.01456
-0.01724
-0.00225

0.00392
-0.00212
-0.00704
0.00230
0.00759

0.00942
-0.00164
0.01249
0.00376
-0.00157










TA3LE 7 CONTINUED


RED OBSERVATIONS OF AS VELORUM
NORMAL D3INTS AND RESIDUALS
FOR TIE WINK SOLUTION


PHASE


0.924941
0.928433
0.931626
0.933995
0.938191

0.940273
0.942291
0.945905
0.947856
0.951047

0.952647
0.955758
0.957317
0.960068
0.962036

0.963760
0.966886
0.967920
0.970636
0.973017

0.074615
0.975948
0.978651
0.980137
0.982379

0.984543
0.986158
0.987926
0.989477
0.991252

0.992722
0.993929
0.995618
0.997696
0.998531

0.999923


NORMAL POINTS
(MAGNITUDE)

0.353
0.360
0.357
0.369
0.365

0.367
0.365
0.361
0.371
0.359

0.380
0.380
0.398
0.431
0.439

0.468
0.507
0.514
0.549
0.587

0.620
0.668
0.687
0.710
0.764

0.785
0.793
0.803
0.813
0.310

0.805
0.821
0.813
0.810
0.822

0.820


RESIDUALS
(MAGNITUDE)

-0.00525
0.00118
-0.00197
0.00972
0.00594

0.00775
0.00577
0.00162
0.01175
-0.00373

0.01098
-0.00673
-0 00006
0.00919
-0.00263

0.00767
0.00779
0.00178
-0.00234
-0.00020

0.00663
0.01388
0.00462
0.00228
0.02081

0.01115
-0.00159
-0.00173
0.00336
-0.00482

-0.01308
0.00041
-0.00986
-0.01371
-0.00241

-0.00448






43

TABLE 8


INFRARED OBSERVATIONS OF AS VELCRUM
NORMAL POINTS AND RESICUALS
FOR THE lINK SOLUTION


PHASE


0.000126
0.002di9
0.003946
0.0CC49C3
0.007008

0.009524
0.010853
0.013941
0.015551
0.018317

0.02043 .
0.022566
0.024535
0.C025734
0.027104

0.028653
0.030302
0.031642
0.033551
0.035430

0.0363C9
0.039777
0. 04144
0.042618
0.0441 Cd

0.04591C
0.047040
0.049372
0.051217
0.053610

0.055704
0.057690
0.060313
0. C64261
0.065291

ZU.C6296
0.07541C
0.079043
0.081519
0.085929

0.C90996
0 .CC5820
0.101045
C.1363 7C
0.109369

0.1136 0
0.116396
L.12522J
0. 13779
0.155089


NORMAL POINTS
(MAGNITUDE)

0.706
0.708
0.709
0.707
0.712

0.712
0.702
0.690
0.678
0.524

0.607
0.558
0.529
0.512
0.5U7

C.47C
0.449
0.419
0.399
0.373

0.372
0.320
0.321
0.300
0.300

0.275
0.272
0.269
0.277
0.268
0.266
0.278
0.257
0.254
0.261

0.259
0.256
0.268
0.263
0.255

0.269
0.282
0.265
0. 266
0.276

C.270
0.268
0.278
0.267
0.266


RESICUALS
(MAGNITUDE)

0.00184
0.00515
0.00625
0.00470
0.01138

G.0145e
0.00626
0.00378
0.01117
-0.00643

C.C 082
-0.00672
-0.00277
-0.00126
0.01575

0.C052S
C.C0475
-0.00737
-C.C0195
-0.00515

0.00344
-0.01175
C. 0401
-C.00743
0.00416

-0.00775
-0.00571
0.00119
C.01092
0.00220

-C.00013
0.01225
-C.00816
-0.01101
-C.00426

-0.00600
-0.00893
0.00283
-0.00195
-C.0C923

0.00533
0.01834
0.00128
0.00298
0.01315

C.C0710
0.C0539
0.01564,
0.00581
C.00807









TALE 8 CONTINUED


INFRARED CBSEPVATIONS OF AS VELCRUM
NORMAL POINTS AND RESICUALS
FOR THE %INK SOLUTION


0.169944
0.179527
0.183928
0.1E81 69
0.193156

0.197326
0.201719
0.207541
0.215289
0.234354

0.247968
0.2530Q1
0.257721
0.2621a4
0.267500

0.272148
0.277173
0.283032
0.291344
0.305744

0.313261
0.32191 1
0.3290C4
0.335977
0.34d975

0.270939
0.384349
,.396989
0.4C5365
0.41C7C5

0.416334
0.421C14
0.427332
0.432736
0.441533

0.451973
0.459311
0.466193
0.473036
0.479214

0.467226
0.492727
0.500143
0.50b747
0.514929

C.S20090
0.524874
0.53C443
0.534141
0.537554


0.267
0.257
0.251
0.262
0.261

0.257
0.256
0.266
0.263
0.251

0.255
0.252
0.256
0.257
0.256

0.264
0.255
0.242
0.254
0.253

0.258
0.266
0.260
0.259
0.251

0.256
0.255
0.262
0.263
0.250

0.259
0.254
0.259
0.265
0.257

0.263
0.263
0.271
0.278
0.296

0.304
C.283
0.284
0.2 86
0.296

0.278
0.281
0.277
C.270
0.277


0.00751
-0.00249
-0.00756
0.00305
0.0024C

-0.00079
-0.00199
0.00800
0.00556
-0.C0555

-0.00146
-0.00431
0.O0071
0.io01e1
0.00071

C.CC827
-C.00057
-0.01323
-C.OC121
-0.00209

0.00284
0.01025
0.00476
C.00287
-0.00475

0.00005
-0.00213
0.00487
0.00546
-0.00687

C.C0190
-G.00331
C.C0114
C.00683
-C.00074

0.00413
-0.00006
C.C0187
0.00146
0.00320

0.01700
-C.00404
-0.00326
-0.00104
C.OC994

-C.00520
C.C0274
C.00450
0.CC060
0.01076





45


TABLE 8 CONTINUED


INFRARED OBSERVATIONS OF AS VELCRUM
NORMAL POINTS AND RESICUALS
FOR THE WINK SOLUTION


0.54C6)1l
0.544205
0.548336
0.552648
0.56C2oO

0.5o7528
0.574856
0.582276
0.589209
0.597295

0. 603703
0.6108C4
0.616012
0.621132
0.626349

C. e30t9g
0.635851
0.d40943
0.647C46
0.654633

0.661370
u.668719
0 .67496.3
0. c1722
0.687640

0.6 96867
0.705963
0.711354
0.716431
0.722591

0.729591
0.741901
0.76C943
0.772721
0. 7 1793

0.789412
C.7968ad
C.807225
0.60739
0. 29016

0.841623
C.851521
0.862835
0.88a2166
0.889499

0.896106
C.S 039C
0.908233
3.911478
0.915516


0.277
0.274
0.261
0.272
0.255

0.269
0.259
0.246
0.263
0.261

0.259
0.253
0.256
0.251
0.261

0.255
0.257
0.250
0.250
0.253

0.256
0.256
3.250
0.253
0.261

0.253
0.2'50
0.263
0.259
).261

0.26C
0.261
0.262

0.257
0.257
0.271
0.251
0.252

0.256
C.266
0.258
0.255
0.273

0.261
0.263
0.266
0.265
0.369


0.01394
0.01333
0.00284
0.01406
-C.00251

0.01153
0.00129
-0.01114
0.00573
C.00353

C.00209
-C.00355
-C.00053
-0.00582
C.00460

-0.00152
0.00078
-0.00604
-0.00642
-C. C0299

C.00032
-0.0C017
-C.00519
-C.00271
C.00515

-C.C0257
0.00098
C. C0799
C.00386
0.00543

-0.00505
0.00861
0.00622
0.00388
C.00450

-0.00066
-0.00132
0.0125i
-C.0077C
-0.00744
-0.00254
C.0056S
-C.C0379
-0.00715
C.01012

-C. 0028
-C. CC5C
0.001d7
0.OC110
C.C0429










TABLE 8 CONTINUED


INFRARED OBSERVATIONS OF AS VELCRUM
NORMAL POINTS AND RESIOUALS
FOR THE WINK SOLUTION


0.919730
0.922303
0.927333
0.930554
0.933684

0.936958
0.939686
0.941826
0.944772
0.947035

0.949045
0.951721
0.953870
0.956439
0.958194

0.961167
0.962830
0.965819
C.Sc8060
0.97;833

03.73246
0.974820
0.977005
0.978850;
0.980336

0.983163
C.9852C2
0.966801
0.968185
0.989684

0.991447
0.992936
C.994130
C.99S5804
0.997900

0.998736


0.257
0.264
0.259
0.263
0.255

0.262
0.261
0.252
0.260
0.270

0.276
0.277
0.301
0.308
C.306

0.350
0.361
0.411
0.417
0.467

0.502
0.520
0.556
0.596
0.608

0.657
0.687
0.691
0.692
0.708

C.703
0.711
0.712
0.702
0.706

0.707


-0. 0751
-0.00063
-C.00576
-0.00249
-0.01042

-0.00315
-0.00476
-0.01324
-0.00564
0.00378

0.01051
0.00547
C.C1794
0.C0817
-C.C0734

0.00840
0.C0135
C.01737
-0.00449
C.00676

0.00565
-C0.0127
-0.00156
0.00904
-0.C0273

C.CC614
C.01056
C.C0042
-C.0021t
0.01112

0.00393
C.01043
0.01009
-0.00072
0.00262

C.00384





























0



H t

10 02








HO




S. -
,* o


* t rI I0
.-0



















h'O O
S o 0



ss 101 02

(0'-



H

r~~~~h.O~~~ 04DO. O)( L. 80HtO U'
~uns~I 02nr ~ ~ at~3












































































0 E' OhVO UG0


OhO 0O0 0;) OL:O 0'0
o3a : 3niiu'IUIi 1ii17]l


0 m











S ui




00
d 0

























00
Oiva

.0

H
44,0


a ow
PB>*
5 0



r-4
ta-i 03






BS
o 0ow









* a
u u
0n 3 '3















o~c
a rl0'
5
0 Paa u


9900
U0

9..1 s































'o

.0




*t LoC
















w c
S0)P
1-lw



















'Sa
to H


















ova ,o G









PI
** . 01 4
*r) P






*F 140)








F.




























































































4-


I bj


Zo' a20o-


10'l- 00'0 (O 0 '0 0i
['Ui NI '3-D) STUnUI J3y IancIA


EO0'


'
'













,































*,






r.

.
,
.
~



,


































S

























14

0
4 4*










.: **.







*0"- 0'- 0'o, 0'. 8 "
NI 3





M o N.
:---- 4











f00- -OO- 00- affo ooa so-a
*'J ~ .1 '30 g~IUS3J C


















































'13

.44


I


0
p-i
C4

H
b3
I0
Dl
03

c'
H-

00
a4


EO'O- :0o0- ll'0- OO') 10'1) 0o'0 0 i0
f',SW I[ '3-01 S1itln S3;' 3J -'u'jili


s

0



O




.-..


..








TABLE 9

Parameters for the Distribution
of the Residuals


parameter v r i


mean O'0007 0'0002 O'002

standard deviation 0O006 0'007 0o007

skewness -0.004 0.115 0.215

kurtosis 2.92 2.70 2.69

number of residuals 189 186 186



For each color the mean of the residuals is essentially equal to zero.

At the 2% confidence level only the distribution of the i residuals is

significantly skewed. At the 1% confidence level none of the sets of

residuals shows any significant kurtosis. That is, the percentage of large

and small residuals is what would be expected for a random sample of a

population that has a Gaussian distribution. Figure 13 shows each histogram

and a Gaussian curve that has the same standard deviation.

Assuming that the sources of error in the variable and check star

observations are identical, one can determine the size distribution of the

errors in observation from the observations of the check star. From these

measurements one would expect the normal points of AS Velorum to have

standard deviations of 0.007, 0'008, and 0O007 for v, r, and i respectively.

A comparison of these values with those in the table above shows that the

differences between the observed and theoretical points are very nearly the

same size as observational scatter. On the basis of the distribution of

the size of the residuals the adopted WINK solution for AS Velorum agrees

quite well with the observations.














I I I I I

IIIII`



-- 1










/.t































A O n LO
N -

AON3nO38J


00
0 0



a,
cu
3
N -0
0)
0.
L a
'0
a

,0
2
4)

0 0

" Q I







0)





-0 ,44
a,





Sn
o sa





*H
- "I
P140






55

Second Order Effects

The success of the adopted solution in fitting the observations prompts

consideration of second order effects that were neglected in the above

analysis. As mentioned earlier, the physical processes involved in the

proximity effects are of theoretical interest. Unfortunately the observa-

tions of AS Velorum show that only very small proximity effects are present

in this binary system. The adopted solution shows that for each star its

a, b, and c semiaxes are very nearly equal, i.e., the stars are very nearly

spherical. Variations in the local effective temperature caused by varia-

tions in the local acceleration of gravity are negligible. Thus, AS

Velorum is not a suitable system for studying the gravity darkening effect.

Close inspection of the i lightcurve shows that the shoulders of secondary

eclipse are about 0T01 higher than the shoulders of primary eclipse.

Despite its inconspicuous size, a refined analysis of the reflection effect

was attempted. The large difference in the surface temperatures of the two

stars suggests that the value of the bolometric albedo of the cooler star

could be determined by differential corrections. For this determination

the bolometric albedo, the radius, and the temperature of the cooler star

were adjusted simultaneously with the WINK program. It was found that the

value determined for the bolometric albedo greatly depended on what start-

ing value was selected. Starting with the adopted solution, with the bolo-

metric albedo set at 0.5, the adjustment procedure yielded a value for the

albedo of 0.38 0.09. Using the adopted solution as a starting point, but

with the albedo set at 1.0, a value of 0.38 0.08 was found. These two

results are in good agreement. However, when a value for the albedo of 0.01

was used as a starting point a value of 0.03 .10 was found. This deter-

mination strongly disagrees with the other two determinations. Thus, a

reliable value of the bolometric albedo of the cooler star could not be






56

found from the i light curve of AS Velorum. The observed size of the reflec-

tion effect in this binary system is too small for an accurate determination

of the value of the bolometric albedo.

The absence of strong proximity effects is a great advantage in the

study of limb darkening. Grygar, Cooper, and Jurkevich (1972) have reviewed

the problems associated with the determination of coefficients for limb

darkening. They point out that this second order parameter is closely cor-

related with the radius of the eclipsed star. Various combinations of limb

darkening and the orbital elements can fit the observations. AS Velorum is

a binary system for which the determination of limb darkening coefficients

is practical because

1. the proximity effects are small or absent,

2. the eclipses are complete,

3. primary eclipse is a transit,

4. the orbit is circular,

5. and the scatter in the observations is small.

For the adopted solution the limb darkening coefficients for both stars

were held fixed, and were assumed to be the same for both stars. The values

that were used are listed by Al-Naimiy (1978) for T = 9000 K and were slightly

rounded. Since primary eclipse for AS Velorum is a transit, the determination

of the limb darkening of the hotter star is more favorable. The curvature at

the bottom of primary eclipse is an indication of the limb darkening of the

hotter star. Using the discussion by Grygar, Cooper, and Jurkevich as a

guide, the limb darkening coefficient for the hotter star, the orbital

inclination, and the radii of both stars were adjusted simultaneously with

the WINK program. The adjustment procedure yielded the following coeffici-

ents for v, r, and i respectively: 0.58 .01, 0.29 .08, and 0.34 .08.

It is important to keep in mind that the quoted errors are internal errors,

and do not indicate uncertainties caused by errors in the other adjusted








parameters (i, rh*, k) and possibly by systematic errors. Al-Naimiy lists

the following theoretical values for v, r, and i respectively: 0.45, 0.34,

and 0.29. There is, at best, modest agreement between the two sets of co-

efficients.

Some discrepancies between the observed and theoretical coefficients

should be expected. In the light curves of AS Velorum small assymmetries

are present near mideclipse. These small but systematic deviations, because

they occur from phases -.01 to .01, will have a large influence on the

values of the limb darkening coefficients. The WINK program was set to

represent stellar fluxes with black body fluxes which may not be adequate.

Strictly speaking, the theoretical coefficients are not physically the same

as the observed coefficients. The observed values are based on broad band

photometry which refers to the stellar flux over a wide wavelength interval.

The theoretical coefficients refer to monochromatic fluxes and may not have

the same value over a large wavelength interval. The theoretical coeffici-

ents from Al-Naimiy are based on a grid of model atmospheres by Carbon and

Gingerich (1969). For this particular temperature, 9000 K, line blanketing

was not considered in the calculation of the model atmosphere. The pre-

sence of absorption lines in a given wavelength interval affects the stellar

flux in two ways (Hack and Struve 1969). First, the line blocks some of the

continuum and reduces the emergent flux in this wavelength interval. Second,

the absorbing material re-radiates isotropically the absorbed energy. In-

stead of escaping from the star, the back scattered radiation heats the

photosphere. This back-warming effects raises the continuum level above the

level it would have in the absence of absorption lines. These processes

can have a significant effect on the stellar flux for some wavelength inter-

vals. Ardeberg and Virdefors (1975) have derived line blanketing coefficients

to the v filter, the line blanketing coefficient ranges from 005 to 0?01.






58

For the r and i filters the coefficients are insignificant. Al-Naimiy's

theoretical limb darkening coefficient for v may be affected by the neglect

of the effects of line blanketing in the model atmosphere calculation.

Recall that the v coefficient had the largest discrepancy between the ob-

served and theoretical values.

The above discussion does not consider the effects of rotational

broadening. If the components of a binary system are in synchronous rota-

tion, rotational broadening will be important for short period binary systems.

The use of model atmospheres based on nonrotating, single stars may not be

valid. In summary, it is not clear how to interpret the differences between

the observed and the theoretical limb darkening coefficients for AS Velorum

because both parameters are influenced by assumptions that may not be

adequate. More comparisons for other binary systems are needed before any

generalizations can be made.


Conclusions

The work presented here is the first photoelectric light curves and

photometric analysis made for AS Velorum. The depth of secondary eclipse is

very shallow, in i only 0O03 deep. The v observations show that the depth

of secondary is so shallow that it is indistinguishable from the observa-

tional scatter. If AS Velorum had been observed in the U, B, V spectral

region, as is often done, the presence of a shallow secondary eclipse would

not have been noticed. A very different picture of this binary system would

have been formed from the analysis of U, B, V observations. The binary would

have been thought to consist of two stars of equal temperature and to have an

orbital period of twice Gaposchkin's value. By observing in v, r, and i the

presence of the cooler star in the binary system was detected. AS Velorum

is unusual in that the temperature of the cooler component is quite low.

The adopted solution is consistent with the three color observations.





59

However, the depth and shape of secondary eclipse are poorly known (even in

i) and the adopted solution depends very heavily on primary eclipse. Our

knowledge about the cooler star (and the binary system) would be improved

with observations made at longer wavelengths, e.g., 1.6p and 2.2p. More

times of minima are needed to improve the value of the orbital period and

to see if the period value changes with time. Finally, radial velocity

measurements of this system are desirable so that the masses and absolute

dimensions of the components can be determined.














CHAPTER FOUR

OBSERVATION AND ANALYSIS OF TW CASSIOPEIAE

Recent History

The history of TW Cassiopeiae (BD + 650 289, HD 16907) is long and

rather confused. McCook (1971), and Cester and Pucillo (1972) summarize

the early work, explaining the confusion in the classification of the

light variations and in the value of the orbital period. Today, we now

know that TW Cassiopeiae is an eclipsing binary with an orbital period

of 11428 and consists of two stars of very dissimilar temperature.

Unfortunately that is all that we know for sure about this star. Des-

pite three recent photometric studies (McCook 1971, Cester and Pucillo

1972, and Kandpal 1975) it is not clear whether the hotter star is

larger or smaller than the cooler star. All three studies are based

on observations in the U, B, V regions. The light curves of McCook, and

Cester and Pucillo show significant curvature outside of eclipse. In

contrast, the light curves of Kandpal are flat outside of eclipse. What

is even more unusual is that for Kandpal's data the depth of secondary

eclipse in U is deeper than in B or V. Cester and Pucillo's data show

the depths of secondary eclipse in the expected order: v is deeper

than b, and b is deeper than u. McCook only observed TW Cassiopeiae in

b and v. One explanation for this discrepancy is that the U filter used

by Kandpal has a significant red leak. It is known that this can occur,

and failure to correct for the contaminating red light would result in

a spurious depth for secondary eclipse. The three studies give contra-

dictory pictures of TW Cassiopeiae. The analyses by McCook and by





61

Cester and Pucillo give partial eclipses and that primary eclipse is

an occulation. On the other hand, the analysis by Kandpal gives com-

plete eclipses with primary eclipse being a transit. Table 10

summarizes the solutions obtained in these studies.


TABLE 10

Solutions of TW Cassiopeiae


Cester and Pucillo McCook Kandpal

i 72;6 73?2 i 87?5

k .72 .772 k .338

a .328 .321 r .338
g g
b .322 .316
g
a .236 .248 r .224
s s
b .232 .244

S b v b U B V

L .112 .134 .172 .097 .158 .916 .942 .916

Ls .888 .866 .828 .903 .842 .084 .058 .084


The discrepancies in these results prompted us to observe TW Cassiopeiae

in v, r, and i.


Rosemary Hill Observations

TW Cassiopeiae was observed by Chen and Florkowski during 1975 to

1977. The observations were made at the University of Florida's

Rosemary Hill Observatory, which is located 6.5 km south of Bronson,

Florida at an elevation of 40 m (130 ft). A 0.76 m (30 in) reflector

was used with a dry ice cooled EMI 9684B photomultiplier tube. The

i and r filters were manufactured by Infrared Industries. The v

filter is a custom made "infrared suppressing" filter from Corion






62

Corporation. These filters used with the S-1 photocathode of the EMI

tube give a photometric response that is close to that of the standard

system. The spectral response curves for the S-1 photocathode and the

filter-tube combinations are shown in figure 14. For the observation

a DC amplifier was used. The output was simultaneously recorded on a

strip chart and also (after employing a voltage to frequency converter)

on to a magnetic tape in digital form. Each measurement was converted

into magnitudes using


mag= Gc + Gf 2.51og[N, Nc/NC] (42)


where mag is magnitude, Gc and Gf are coarse and fine gains of the DC

amplifier, N, is the number of counts from the digital system for the

star plus sky, N is the number of counts for the sky alone, and NC is

a constant selected to make the instrumental magnitudes close to the

standard magnitudes. The measurements of the stars were made on the

same coarse gain setting to eliminate the effect of drift in the value

of the coarse gain. The resistors for the fine gain steps were very

stable. Calibrations of them showed negligible changes with time. The

observations of TW Cassiopeiae were made differentially, using a nearby

star of nearly the same magnitude and color as a comparison star. A

second star was used as a check star. Information about these stars

can be found in table 11.

TABLE 11

TW Cassiopeiae and Its Comparison and Check Stars

Star BD RA Dec mv Spec
(1950)
TW Cas +65 289 2h 41m 44? +65 30'59'.' 8.7 B8
comparison +64 337 2 33 35. 64 7 24. 8.2 B8
check +65 291 2 42 58. 65 30 46. 8.5























2
w
o 0.4
IVI
a -

U 0.3
2


i. R








0.4 0.6 0.8 1.0 1.2

WAVELENGTH IN MICRONS



Fig. 14. Spectral response curves for the S-1 photocathode,
and the filter-tube combinations.








The differential observations were made in the following order:

(i, r, v)c; (, r, )s; (, r, ); (X, i); (i, r, v); where the

subscripts c, s and v stand for the comparison star, the sky, and the

variable star (TW Cassiopeiae) respectively. The principle extinction

coefficient star is far enough away from the position of TW Cassiopeiae

so that differential extinction is not negligible. For example, at X =

2 the difference between the air mass of the variable and that of the

comparison star is -0.04. The main source of scatter in the observa-

tions is variations, both short and long term, in the sky transparency.

On a number of nights observing had to stop because of the worsening

sky conditions. Attempts to determine the second order (color) extinc-

tion coefficients were hampered by variations in sky transparency. Rather

than use an unreliable value, no correction for color extinction was applied

to the data. Since the comparison star is of the same spectral class as TW

Cassiopeiae, the differential color extinction probably is small. For

similar reasons the observations were left in the instrumental photometric

system. The transformation equations that were determined were judged to

be too unreliable to use.

The observations of TW Cassiopeiae were made during 1975 to 1977.

These observations are shown in figures 15 17. The phase interval

0.10 to 0.40 was observed exclusively during 1976 1977. During this

season the sky conditions were often poor, and as a consequence there is

large scatter in the light curve for these phases. The v light curve

is affected worst, and the i light curve is affected least. To help

reduce the scatter, pairs of observations were averaged together to form

normal points. Examination of the light curves show that there is not

evidence for an eccentric orbit.







65























o
''"" 0 i







H





a-








(0

la




a







(0










.~~~ ~ *.*** """"s .


S" s'o : SE 30 Shn s saD0
It'nSIA :3JfnliNajb 81130


SL'o E' O;







66











o
S '* '



..?':. .





"a


'-a '"





""" i
.' ; 0










a ,
'a
*y: -s





m


'a
o

".': a







,! .' ".
I I '' I" Ir '
S0)
** U

a 0


H
. i ;


4 4
'V




,~ 'a '^
0
ra



a an*. ** *r
.^ '
H
a



aS~O awn sa a an na an


Os'0 Oh'O OS'O 0930 iL i s 0 U
03H 9;ij1'1 i Iri ij 11 m130


a~ OO'i







*67










.. .. ,






*...',. .
~. o



..* ": '
:.^ 0









*:. o -r
0

T"


.:." oo
':*^ -to


>'* C
S to
to
~ ~to
:': ** o <

."*" o "" .
1. ^ *

tO
o O
to
on 0
^: '
Sa



s i:
a


:**< " g
.4
o to






*. *
0






.^\" ..5
o* g
*E a H



.:"4




OS'o0 0 9 OL 'O 08'0 O6' 0O0'1 oI'Il o00'
03YUB MII :OnlIINf )H U1730








For most of the primary eclipses that were observed the phase

coverage was too incomplete for times of minima to be determined. From

one eclipse the following time of minima was found: 2443106.764.

Light elements were determined from photoelectric observations, from

1966 to 1979:


Min I = 2441441.3405 + 1.42832295 E. (43)


The residuals for these light elements are shown in figure 18. The

value of the period is quite close to the value determined by McCook,

1"4283224. The scatter in the residuals is somewhat large but there

is no indication of a change in the orbital period in this time interval.

The above light elements were used to calculate phases for the Rosemary

Hill data.


Rectification of the i Light Curve

The light curves of TW Cassiopeiae show noticeable curvature out-

side of eclipse. Therefore, the light curves must be rectified before

they can be solved using the Merrill nomographs. Only the i light

curve was rectified. With the large depth of secondary eclipse, this

color was considered to be more suitable for solution using the nomo-

graphs. The i normal points were converted into light intensity units,

and a harmonic analysis using least squares was made of the normals

outside of eclipse. In this analysis both sine and cosine terms up

to and including the fourth harmonic were considered. The table below

gives the results of the harmonic analysis.



















































































00o0 90Do hio'o o0o 00'0 'on IhO,- 90 D-
( I-OXIJSIbU HI slunaisjy


to
W
g
.g




44
0
ta
to



m
to


U

'4.
S41








to
*I a
.6





SIn
.o* .a







8
'- H







o_ a
0




C I
C, H

00

"-4







0 0

1 '
2





70

TABLE 12

Fourier Coefficients for the i Light Curve


AO 0.596(1)

A -0.008(1) B1 -0.0011(5)

A2 -0.011(2) B2 0.0002(6)

A3 0.001(1) B3 -0.0010(6)

A4 0.005(1) B4 0.0015(7)



For the purposes of rectification it was assumed that A3=B2=B3=0.

In addition the following constants were used (assuming partial

eclipses) in the rectification formulae:

0e = 2828, (44)
C0 = 0.02020, (45)

C2 = 0.00673, (46)

z = 0.05276. (47)

With the above constants the light curve was rectified by the method

described in chapter 2. The rectified intensities outside of eclipse

were checked, and the light curve was found to be suitably flat.

Nomographic Solutions

From the rectified light curves the depths of primary and secondary

eclipses were estimated to be 0.35 and 0.10, respectively. The eclipse

curves are poorly defined, and the values determined for X were very

uncertain. The nominal values were estimated to be Xri = .29, and
sec
X = .27, but these values are very unreliable. While it is true that

Xoc > Xtr the large uncertainties make it quite possible that each
eclipse could be either a transit or an occulation. Considering the

past photometric studies, this ambiguity is not surprising. It was






71

found that the depth line does not cross either of the nominal X contours.

However, the estimated uncertainties in X made a wide range of inter-

sections possible for partial eclipses. A very subjective choice of an

intersection point was made for both cases, primary being a transit and

primary being an occulation.


TABLE 13

Nomographic Solutions


Ore 306 x g-==0.2
Solution I: pri = tr Solution II: pri = oc

po -.30 P, -.12

k .84 k .86

r .30 r .30
g g
rs .25 rs .25

i 770 i 74"



Both solutions I and II were used as starting points for differential

corrections using the WINK program.

WINK Solution

To aid in the analysis of TW Cassiopeiae, the b and v observations

published by McCook (figures 19, 20) were included. Besides giving one

an additional color light curve, both light curves are significantly

better in quality than the Rosemary Hill observations. For solution II

the computed light curves showed a large reflection effect which is not

present in the r and i data. For this case the cooler star, being the

larger star, is strongly affected by irradiation from its companion.

No adjustments with WINK could resolve the discrepancy with the r and

i light curves. In the case of solution I, the reflection effect is not






















...













. "* a .
o H o ,






S .
i,4,
4,


c. .

$, HH








..a'' ..S
o a













I^ I
o (0
3



..0
0
o H
,'* s







a 0%
C H
90

H



0' 0 oa oa o.o or.]o ,- 00 09





Jn70 :J Hflj [ f 1.FJ q I".]
;< .s E


-r.. *

**'..... "




3m0~' :':***sY 3






73




















I-


.V











., t
r w









. *
C11



















a"
^. r I







o .0



o W


0 *
... ... C






74

a problem. The cooler star, being the smaller star, has a small amount

of surface area to be heated. The resultant reflection effect is much

smaller than for the case of solution II. Adjustments were found to

fit the r and i observations. However, no solution could be found to

fit satisfactorily the observations in all four colors. A compromise

solution is given below which comes close to fitting the observations.


TABLE 14

Preliminary Solution of


i

rh*

ah
bh

ch

Th

b

.50-

.78+

.95

.05

this value


77?2

.284

.29

.29

.28

11600++


v

1- .42++

- .65++

.92

.08


TW Cassiopeiae


Mc/Mh

k

a
c

b

cc

T
c

r

.34-+-

.52++-

.89

.11


.65

.92

.27

.26

.26

5500


i

.25+-+

.45+-

.85

.15


As in the case for AS Velorum, rh* is the unperturbed radius of the

hotter star, and k is the ratio of the unperturbed radii. Using the

above solution theoretical light curves were calculated. The computed

and the observed normal points for primary and secondary eclipse are

compared in figures 21 25. Also, for the entire phase range the

residuals, taken in the sense of observed minus computed, are plotted


Xh
x
c

Lh
L
c
4+fixed at






75























w
A
'ig




a
a
0








rH 4-
p.

















o ) "

w 0,
0










ba









P41

.. .. ai S
*tnSIA N*II U1'3-







76
















d)
Az












. .. G
I,'


A D









4)


S
0 L)



04O
o j 0




03d .
am
014*
0U"4















--(-- h-- ----- t --I -- I --- I --- -"-| -- | --- -- | --- -- *- -I --- h -- --- **
OS'0 Oh'0 O)'0 OS'0 Ol'O OS'0 09 0'O C8 0 04)U'I '
03Yd ':3r03110404u 0



























d a


P,44-

.. 0

)
wo









= 40




















C4
;



.. -
o 'a





















.0H




























:: . .










o .. 0
-rl


a
r4











.. 4.






o to


,a
S 4






o>2o o f










M 10 :30MQN,0914 LllJ
3110O 3OflI]:YN )!I1O





























,0








o at


0
41.
aWa









do
,0 a1
O 0-


0
,(0



*4 43 CO
m 0J w
..







44 0
o Co 0

Sr4 0




S s .0 s 0 a*







l -f1 0 4 0




~Bn~la N3nI4U 1








in figures 26 30. The residuals are also listed together with the

normal points in tables 15 19.

Inspection of the plots of the residuals gives some indication as

to why a satisfactory solution could not be found with WINK. In

McCook's data, because of the smaller scatter, one can see that the

residuals for phases 0.10 0.40 are not symmetric with those for

phases 0.60 0.90. This assymmetry may be present in the Rosemary

Hill data, but it is hidden in the large scatter. Struve (1950)

mentions that the hot component has an unusually slow velocity of rota-

tion, but he does not give a value. For a synchronously rotating

coordinate system the hotter component would appear to be in retro-

grade rotation. The model used in WINK assumes that the stars are in

synchronous rotation and that the longest semiaxes are in alignment.

This assumption may not be valid for TW Cassiopeiae. Further spectro-

scopic and photometric observations are needed. In particular near

infrared observations would be valuable because, as was found in this

study, the proximity effects are very useful constraints on the possible

light curve solutions. Deviations from a symmetric model, if confirmed

with more observations, could give one insight about binary stars in

general.






81






















V





s



*' *



a






'4 -
o* *







S. S*
,+
: . i **












50 0- 0'0- ItO'0- 20 0- 0o0 Z1 0 hn00 90
"')Uj N I '3-01 5'i ll '1 3al I "lU lSIA l [;































I



.5
:



































~
.











t



.Z
:


tI-UWi N '-I 1 Sl Ul'JIS3Y OZ0


'








4
'



~

t
I


'i



t


8 '-4
0
44







t-H
P 4

[a a

*rT4
CO
o 0]
N


90
io S



(-.


























































































gao0 hQ '- I'-
(CJahY HI


L.i





t










r
..



.
t











~
4




















I


j-0:)0 flA hT3:0 U
"3-o0 $7blynS3'y cmua-INI































t

:i
.r










~
t
r
:=
~.
t



'ii


90"0- h'd- O'o- 00'0 O:0 hD'O
c'B MII '3-Ut S Iyn (ll S3Y 3n13


,, ~
:






..












.I


tt:



:


t.


r-.

t-14
a'
H
a
a


r U
to
ci


: u










C4
z

1




*H



.0
a

Pa
a





H
41

'a
H







bb



























1










TABLE 15


VISUAL OBSERVATIONS CF T CASSIOPEIAE
NORMAL POINTS AND RESIDUALS
FOR THE WINK SOLUTION


PHASE


0.004236
0.012842
0.017118
0. (23081
0.033314

0.042380
0.050542
0.059103
0.064464
0.068272

3.072835
0.074194
0.080540
0.083944
O.C89398

C.093051
0.098884
0.103628
0.108084
0.116146

C.128748
0.141078
0.148324
0.158720
0.163723

0.171897
0.176058
0.182427
0.186628
0.193479

0. 176R6
0.205125
0.210o84
0.216621
0.222992

0.230196
0.235402
0.239692
0.244563
0.252368

0.2591 7
0.2640'18
0.260486
0.272783
0.277934

0.261960
0.287050
0.293266
0.300846
0.307429


NORMAL POINTS
(MAGNITUDE)

0.769
0.741
0.679
0.607
0.504

0.422
0.364
0.277
0.251
0.203

0.132
0.167
0.156
0.128
0.138

0.123
0.181
0.12o
0.144
0.151

0.146
0.135
0.170
0.118
0.149

0.129
0.134
0.117
0.122
0.129

0.151
0.118
0.131
0.126
0.123

0.137
0.110
0.095
0.116
0.104

C.102
0.123
0.083
0.112
0.111

0.114
0.112
0.097
0. 110
0.091


RES ICU ALS
(MAGNITUDE)

0.019
0.032
0.002
-0.015
-0.013

-0.002
0.017
0.002
0.013
-0.011

-0.007
-0.01e
-0.003
-0.022
-0.007

-0.021
3.038
-0.015
0.004
0.014

0.012
0.005
0.042
-0.008
0.025

0.007
0.013
-0.002
0.004
0.013

0.035
0.004
-0.012
0.014
0.012

0.027
C.001
-0.014
0.008
-0.003

-0.C06
0.016
-0.024
0.005
0.005

0.007
0.005
-0.010
0.303
-0.017










TABLE 15 CONTINUED


VISUAL 08SERVATIONS OF Tw CASSIOPEIAE
NORMAL POINTS AND RESIDUALS
FOR THE WINK SCLUTICN


PHASE


0.317096
0.224607
0.334784
0.341299
0.347775
0.353929
0.362194
0.367924
0.374243
0.380653

0.287017
0.393290
0.404652
0.423012
0.433913

0.444005
0.449039
0.454689
0.4599d5
0.465418

0.470442
0.475367
0.480552
0.485400
0.490343

3.4'6584
0.499987
0.504642
0.509960
0.516184
0.521386
0.525262
0.525328
0.532959
0.536848

0.541973
0.547700
0.550855
0.555951
0.559610

0.564136
0.567717
0.573595
0.577679
0.583441
0.586151
0.591578
0.599285
0.602996
C.606339


NORMAL POINTS
(MAGNITUDE)

0.120
0.098
0.115
0.119
0.119

0.122
0.127
0.146
0.136
0.139

0.116
0.135
0.125
0.119
0.143

0.152
0.152
0. 156
0.159
0.165

0.161
0.175
0.130
0. 196
0.180

0.176
0.176
0.177
0.196
0.185

0.174
0.191
0.172
0.171
0.167

0.171
0.155
0. 169
0.153
0.150

0.139
0.150
0.129
0.122
0.118

0.120
0.126
0.123
0.132
C.145


RESICUALS
(MAGNITUDE)

0.011
-0.012
0.004
0.007
0.006

0.008
0.012
0.030
0.020
0.321

-0.003
0.015
0.004
-3.008
0.008

0.006
0.001
-0.002
-0.005
-0.00O

-0.015
-0.006
-0.006
-0.003
-0.012

-0.018
-0.019
-0.017
-0.006
-0.003

-0.011
0.010
-0.005
-0.001
-0.001

0.00o
-0.000
0.01O
0.007
0.009

0.002
0.016
0.00C
-0.005
-0.006
-0.004
3.004
0.002
0.312
0.025






88


TABLE 15 CONTINUED


VISUAL O8SERVATICNS CF TW CASSIOPEIAE
NORMAL POINTS AND RESIDUALS
FOR THE WINK SOLUTION


PHASE

0.611191
0.617187
0.621861
0.627991
0.633480

0.639007
0.644b53
C.650710
0.657266
0.663738

0.668857
0.674948
0.681375
0.686752
0.691187

0.694314
C.700709
C.7C7931
".713465
0.715666

0.719808
0.726144
0.731258
0.737339
3.743584

0.751750
0.76110
0.771179
0.7UC883
0.790924

0.801032
0809899A
0.814616
0.818999
0.825071

0.833367
0.839352
0.844532
0.849592
0.855115

0.862114
0.866620
0.H88923
0.873299
0.874104

0.878847
0.89C128
0.883539
C.885575
0.839955


NORMAL POINTS
(MAGNITUDE)

0.120
0.133
0.116
0.109
0.121

0.124
0.109
0.115
0.127
0.120

0.107
0.110C
0.121
0.110
0.12c

0.119
0.109
0.119
0.111
0.107

0.399
0.119
0.100
0.117
0.112

0.112
0.111
0.090
0.108
0.097

0.121
0.114
0.111
0.109

0. 137
0.116
0.123
0.130
0.127

0.129
0.118
0.110
0.127
0.127

0.133
0.132
3.143
0.140
C. 130


RESDLUALS
(MAGNITUCE)

0.0o00
0.015
-0.002
-0.008
0.006
0.009
-0.004
0.002
0.015
0.009
-0.004
0.000
0.012
0.002
0.018

0.011
0.002
0.012
0.004
0.000

-0.008
0.012
-0.007
0.010
0.004

0.004
0.002
-0.020
-0.004
-0.016

0.005
-0.003
-U.C08
-0.011
-0.017

-0.31f
0.002
-0.003
0.003
-0.002

-0.002
-0.014
-0.014
-0.007
-0.008

-0.03
-C.C04
0.305
0..3?
-0.013