Title: Observations and analysis of U Cephei
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Title: Observations and analysis of U Cephei
Alternate Title: U Cephei
Physical Description: vii, 182 leaves : ill. ; 28 cm.
Language: English
Creator: Markworth, Norman Lee, 1950-
Copyright Date: 1977
 Subjects
Subject: Cepheids   ( lcsh )
Variable stars   ( lcsh )
Astronomy thesis Ph. D   ( lcsh )
Dissertations, Academic -- Astronomy -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
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 Notes
Thesis: Thesis--University of Florida.
Bibliography: Bibliography: leaves 180-181.
General Note: Typescript.
General Note: Vita.
Statement of Responsibility: by Norman Lee Markworth.
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Bibliographic ID: UF00098861
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: alephbibnum - 000202982
oclc - 03928641
notis - AAW9748

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OBSERVATIONS AND ANALYSIS OF U CEPHEI


By

NORMAN LEE MARKWORTH















A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF TIE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PllILOSOPHY











UNIVERSITY OF FLORIDA


1977













ACKNOWLEDGEMENTS


I wish to express my appreciation to Dr. F.B. Wood for

suggesting the topic of this dissertation and for his expert

advice and guidance throughout the course of the work.

My sincere thanks goes to Dr. J.E. Merrill for many

helpful and encouraging conversations, which kept me on the

straight and narrow in the "black art" of eclipsing binary

solutions. His constant attention to detail is an example

that I will carry with me throughout my career.

I wish to thank Dr. R.E. Wilson for his advice on the

use of the Wilson-Devinney solution method. Our many fruitful

conversations broadened my understanding of the process of

solution and helped me avoid the many pitfalls in this dif-

ficult numerical problem.

Drs. K-Y.Chen, H.L. Cohen, and E.J. Devinney also deserve

thanks for their help in formulating ideas and arranging the

manuscript.

Much of the computer time necessary for the completion

of this study was provided by the Central Florida Regional

Data Center in Tampa. Without their support, as well as the

supplemental support of the Northeast Regional Data Center in

Gainesville, much of the detailed analysis presented here

could not have been accomplished.








Finally, I owe many words of thanks to my family,

especially my wife, Mary, whose patience and encouragement

provided the necessary impetus to complete this work.













TABLE OF CONTENTS


Page

ACKNOWLEDGEMENTS. . . . . . . . . . ii

ABSTRACT . . ... . . . . . . .. vi

CHAPTER

I INTRODUCTION. .. . . . . . . . 1

History of the Observations of
U Cephei . . . . . . 1

Current Research . . ... . .. .. 5

II INSTRUMENTATION . . . . . . . 7

The Eighteen-Inch (46 cm.) f/10.5
Ritchey-Chr6tien System. ... . . 7

The Thirty-Inch (76 cm.) f/16
Casscgrain System. . .... . . 8

III THE OBSERVATIONS ... .. ....... 15

Introduction . . .. .. . . 15

The Close Companions of U Cephei . .. 15

Reduction of the Observations . . .19

The Reduction to the Standard UBV System 21

The Comparison Stars . . . . . 26

IV RECTIFICATION .. . . . . . . . 64

Fourier Analyses . . . . .... .. 64

Rectification Formulas .. . . . . 73

V THE SOLUTIONS . . . . . . . 78

The Russell-Merrill Solutions. . . ... 78








Page

Nomographic Solutions . . ... 78

The i Solution. . . . .... ... 80

The Intermediate i Solution .... .84

The Adopted Solution. . ... . .. 86

The Wilson-Devinney Solution . . ... .93

The D.B. Wood Solution . . ... .. . 95

VI THE PERIOD STUDY. . . . . . . . 103

The Current Ephemeris . . . .. . 103

The O-C Diagram. .... . . . . 104

Evidence for Recent Period Changes . .. 111

VII THE MODEL .. . . . . . . . . 117

Introduction . .. . . . . . 117

Constructing the Test Grid . . . .119

The Progression of the Eclipse .... . .125

The Geometry of the Source Region ... 127

Fitting the Model to the Observations. . 133

VIII THE U CEPIIEI SYSTEM . . . . . . . 172

The Geometrical Parameters .... . ... .172

The Primary Star . . . . . . 174

The Chronology of the Outburst .. .176

Future Research. . . . ... . ... 179

BIBLIOGRAPHY . . . . . . . . . .. . 180

BIOGRAPHICAL SKETCH . . . . . . . ... 182













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


OBSERVATIONS AND ANALYSIS OF U CEPHEI

By

Norman Lee Markworth

August, 1977

Chairman: Frank Bradshaw Wood
Cochairman: Kwan-Yu Chen
Major Department: Astronomy

Photoelectric observations of the well-known eclipsing

binary system U Cephei commenced following the report of a

major outburst on the system in the summer of 1974. Approxi-

mately 3200 measurements in the standard Johnson-Morgan UBV

system were obtained from October, 1974, through May, 1976.

The reported outburst was observed as well as another out-

burst of approximately the same intensity in the fall of 1975.

The conventional Fourier analysis of the outside eclipse

variations failed to produce coefficients of the sine and

cosine terms in accordance with theory. A trial-and-error

approach produced the coefficients used in the rectification

procedure. A Russell-Merrill solution was obtained which

seemed to confirm the results of Hall and Walter. The light

curve synthesis approach of Wilson and Devinney yielded a

very similar solution, in which the observed asynchronism of

the primary component was used as an additional input



vi








parameter. The computer solution technique of D.B. Wood was

also attempted, but it failed because provision is not made

for the asynchronous rotation of the components. Residuals

of the observations in primary eclipse from the Russell-

Merrill solution revealed excess light during the outbursts.

This excess was modeled as hot source regions near the poles

of the primary star and rotating with that star. Shock

heating of the surface by infalling material is suggested as

the cause of the excess light. Excess temperatures range

from 7000-170000K, yielding mean velocities of the infalling

material in the range 12-21 km/s. A magnetic field on the

primary star is suggested as the steering mechanism for the

infalling material and conclusions based on the model are

discussed.













CHAPTER I

INTRODUCTION



History of the Observations of U Cephci


The variability of U Cephei (BD+81025, HD 5679, ADS 830)

was discovered by Ceraski (1880) on 23 June 1880. It was

the seventh eclipsing variable discovered and displayed the

deepest primary eclipse known at the time. Because of its

circumpolar position at most northern hemisphere sites

(declination +82') and its relative brightness (m = 68 9m0),

it has been frequently and faithfully observed. The depth of

primary eclipse lends itself to rather accurate visual deter-

mination of the time of minimum. This rather substantial

bulk of data represents a detailed account of period changes

extending for almost a century. Two prediscovery observations

may extend the time scale to 150 years. Schwerd observed

U Cephei on 12 May 1828, to be two magnitudes fainter than

normal. Carrington estimated the variable one magnitude

fainter than normal on 30 December 1855. The earlier obser-

vatipn seems the more reliable. Lalande observed U Cephei

at normal brightness on 30 March 1790.

Early photometric observers concentrated on the primary

minimum. Schmidt, Knott, Wilsing, Chandler, Yendell,

Pickering, Searle, Wendell, Lehnert, and Bemporad helped to



-1-








confirm that the ingress branch is less steep than the egress

branch. Dugan (L920) did the first complete photometric

study and found that the shoulder of the eclipse near first

contact was depressed, causing ingress to be less steep. He

invoked a tidal bulge on the primary star lagging the con-

junction by 240. Although Cowling (1941) later showed that

this explanation could not be responsible for the observed

asymmetry, Dugan did show that some effects were present in

U Cephei that simple theory could not explain.

The next great step in understanding U Cephei came from

spectroscopy. The first orbital elements by Carpenter (1930)

found e = 0.47, ; = 250. This considerable eccentricity was

in complete disagreement with the work of Dugan (1920), who

found secondary eclipse at 0.5P. Other spectroscopic studies

followed but all found a rather large eccentricity. Struve

(1944) was the first to suggest that a conventional approach

could not be used to obtain the orbital elements from the

radial velocity curve. He postulated a gaseous stream flow-

ing from the cooler toward the hotter star. At times the

spectrum of this stream would be seen projected onto the

primary (hotter) star, giving aberrant radial velocity mea-

sures. Such gaseous streams could also account for the

depression of the light curve around first contact.

Batten (1974) observed the system spectroscopically and

found rare instances of emission in the hydrogen lines.

These lines were red-displaced at second contact and violet-

displaced at third contact. The same sort of emission is







more obvious in RW Tauri and Joy (1947) suggested a ring or

disk has been formed around the hotter star. The transitory

nature of the emission lines in U Cephei suggests that such

a ring is not a stable part of the system.

Another observational fact from the spectroscopic work

must be considered in any working model of U Cephei. The

radial velocity curve clearly indicates that the primary star

is rotating at five times the synchronous rate. This will

change the brightness distribution of the primary and cause

some change in the light curve.

Another step toward the understanding of U Cephei came

with the resolution of the so-called "Algol paradox." U Cephei

is an example of an Algol type system, in which the deeper

eclipse is an occultation. This requires the smaller star

to be the hotter, contrary to known stellar relationships on

the main sequence. Plavec (1973) has written a summary of

the most widely accepted explanation of this strange paradox.

Kopal (1959) reconstructs the development of the Roche

equipotential surfaces, which give a limit to the size of

each star in the system. If the stars are sufficiently close,

evolution of the initially more massive star will cause it to

fill its critical lobe. Further expansion is impossible, but

it can lose its mass through the inner Lagrangian point to

the less massive star. Calculations indicate that such mass

flow is initially self-sustaining and can, in most cases,

reverse the mass ratio. Several stages have been identified

for the mass loss, but the net result is to produce a system








in which a cool, large star filling its critical lobe is

slowly losing mass to a rather hot, small, main sequence

star. This sort of evolutionary history can explain the

observations of U Cephei. The system contains a B7V primary

with a G8III (spectral types by Batten, 1974) filling its

critical lobe. The secondary star is more evolved and is

losing mass to the primary. We now have qualitative agree-

ment between the observations and the model. The model also

accounts qualitatively for the changes in period which have

been noted since the earliest observations.

The mass loss responsible for the gaseous stream in

U Cephei must be highly sporadic. Each episode of increased

mass flow should be accompanied by a sudden change in period.

Hall and Walter (1974) show several suspected period changes

separated by nine to eleven years. Walter (1975) has shown

that a precessional period of twelve years for the primary

star is consistent with changes in the slant of the total

phase of primary eclipse.

Several photoelectric studies have recently been done

aimed at determining the rates of mass loss and period change.

The once inconceivably large values of (2-5) x 10- M /yr.

are now accepted as the minimum values of mass loss for

U Cephei. Properties of the gaseous stream have been dis-

cussed in some detail by Batten (1974).

Attempts at solution of the light curve have always been

hampered by contamination of the light from the gaseous

stream. Hall and Walter (1974) have solved three sets of







independent data from Tschudovitchev (1950), Khozov and

Minaev (1969), and Catalano and Rodono (1974) in order to

compare the geometrical parameters of the solutions. Their

results are probably influenced by some special assumptions,

but their work probably represents the first major attempt

to retrieve the true geometry from the confusing effects of

the gas flow using current theory.



Current Research


The present study was initiated by the report (Batten

et al., 1974) of a major outburst in the U Cephei system in

the summer and fall of 1974. During the course of the ob-

servations another major outburst occurred in the fall of

1975. These outbursts presented an unique opportunity to

obtain information on the dynamical behavior of the gaseous

stream and its affect on the hotter star.

This dissertation attempts to bring about the next step

in understanding the physical nature of U Cephei. The

modeling of the dynamic surface brightness distribution of

the primary star should unite the previous work concerning

the shape of the components and the properties of the gaseous

stream.

'Consistent values of the geometrical parameters of the

system are essential before convincing theoretical arguments

can be made. Since variations in the light curve outside the

scope of the solution models were felt to be primarily in-

trinsic, considerable effort went into the accurate reduction








and standardization of the data. These efforts occupy

Chapters II and III. The application of the Russell-Merrill

solution method depends on being able to correct the obser-

vations to the equivalent spherical model, i.e., remove the

variation outside of eclipse. The intrinsic variation in

U Cephei causes uncertainty in this "rectification" procedure.

Chapter IV outlines the rectification procedure and results.

Chapter V deals with the solution techniques and results.

Chapter VI presents a new period study, employing all known

observations. With a new solution yielding consistent geo-

metrical parameters, a model can now be presented for nights

of high photometric activity. Chapter VII discusses this

model for the primary star. Chapter VIII concludes the work

with the model for the U Cephei system and predictions based

on the model.













CHAPTER II

INSTRUMENTATION



Observations were carried out on two telescopes so as to

cover the light curve in as short a time as possible within

the constraints of scheduling. The descriptions of both

photometric systems follows.



The Eighteen-Inch (46 cm.) f/10.5
Ritchey-Chretien System


The photometer was the same employed by K-Y. Chen and

D.A. Rekenthaler (Chen and Rekenthaler, 1966). It uses an

uncooled 1P21 photomultiplier operated at 900 volts. Two

diaphragms were used (see Chapter III) yielding diameters of

86.2 and 42.5 arc seconds. The filters were ultraviolet

(Corning 9863), blue (Corning 5030 and Schott GC13), and

yellow (Corning 3389), corresponding to the Johnson-Morgan

standard UBV system. Unfortunately, the telescope tailpiece

has a diagonal prism before the mount for the photometer.

This makes viewing through the photometer much easier, but

presents severe transmission problems in the ultraviolet.

In fact, it appears that the red leak component alone remains

in the "ultraviolet" band. An adapter was built of the

proper dimensions, so that good focus could still be achieved

without the diagonal prism. It was found, however, that the








new arrangement would not permit the telescope to reach the

declination of the variable. Yellow and blue observations

should be unaffected by the diagonal prism and, therefore,

were the only filter bands used for this system.

The signal is sent from the photometer to a newly de-

signed fast electrometer amplifier (Oliver, 1976). The fine-

gain range is such that the same coarse-gain step (5.0) could

be used for all observations. Tables 1 and 2 show the cali-

brations of the PA/10 amplifier for the nights of observation.

On two nights (10/31/74 and 11/10/74) no calibration of

coarse-gain steps was done since no procedure was available

at that time. The mean of the 5.0 magnitude gain step was

used on these nights. Although rather large night to night

variation in the coarse-gain calibrations is obvious, cali-

brations at the beginning and end of each night show very

little variation (typically < 0o001). The trend of the

calibrations suggest aging effects. A Heath strip-chart

recorder monitored the observations at inch/minute.



The Thirty-Inch (76 cm.) f/16 Cassegrain System


The photometer used was the dual channel photometer of

the Astro-Mechanics Company, Inc. Only channel I was used

which incorporates an EMI 6256B (S11) photocathode refriger-

ated with dry ice. The light was initially passed through a

dichroic filter which reflects 95% of the light between

3500 X and 6000 X into the light path of channel I and passes

80% of the light above 6500 X into the light path of channel






-9-












Lo L~ 00 on u 00 L '"
co. oo It < o n co 1





Lm o eoo co i j oO
LO be L e mt ^- O j a


0 0 0 0 0 0 0 0 0 0
C:) C) = C: C C C C C C

C; C; a; = C r; ; ; --


0
,-




















S*H
or-t

1-










L,












C
0



















0
4-






H





ID
(3l


IX,


-o
f o


.H



CL)
4-1


i






-10-


0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0
SCO OC O O CO C0









0 0 0 00 00 0 0

c cc cc c c cc cc cc






0
*H






















0 'O '"
-4 Co N N N c n U') CN ^ N Lfn C)
100 Lr 10 LC U) L U :l 10 '0 C
F cc cc cc c0 o cc cc cc cc cc


CD





S -- 0C t- c cc




N0 F t",


c- D) C C

C 0 'H



U) 0 1 0 1' c 1 -N 1-


*10 0H N C O c F- C
U0 CZ C0 Cr 0 C) 0 '.
CN U U11

0 0


rt
Cn LO F' n -)- LO
iw N I c' 'r m 0) C
'ZN cc i q = C C CC C 1 )


o- '- 4 -4 r --
U








) I r- N 0 0 i C C- 4 N

/ C 'Hl r-4 F' U) ) Cd Ch C C
C) ,- t~ -4 C C '4








II. The dichroic filter is typically used for simultaneous

photometry in the red and one of the UBV colors. It was

thought, however, that its insertion for this one channel

work would eliminate the problem of red leak which plagues

the broad band ultraviolet filter. A diaphragm of 15.2 arc

seconds was inserted in front of the photocathode. The

filters used for the ultraviolet (Corning 9863), blue (Corning

5030 and Schott GC13), and yellow (Corning 3389) approximate

the Johnson-Morgan standard UBV system.

The amplifier was the PA-1/C, the latest version of the

PA-1, designed at the UCLA Astronomy Department and constructed

at the University of Florida. Even though the range of the

fine-gain steps is 5.0 magnitudes, not all observations could

be taken on the same coarse-gain step. The difference be-

tween the largest (variable-blue) and the smallest (compari-

son-ultraviolet) deflections was typically 6.0 magnitudes

outside of the eclipse on photometric nights. The variable-

blue readings are the only observations taken on a different

coarse-gain step. Calibrations were carefully done twice a

night for both the -- and 2-magnitude gain steps and are

presented in Tables 3 and 4. The amplifier was quite stable

even in the 22-magnitude gain steps and variations during

the night were negligible.

Observations were monitored on a Honeywell strip-chart

recorder, which was replaced in January 1976, by a Texas

Instrument strip-chart recorder. Observations were also

recorded on digital tape, the tape drive of which was replaced




































a aa CaC a CrN a
ii oI oI O I CII d CI \ CT C- OT Cl OT vr iT ^i vl U V "'







0 0 1 0 o m o o. 0 n o c o~ c o 's o n C C



YJ I- c3 0 0 0 0 0 0 II t3 0 t_' O Q C; C: C= C3 n ~






a- a a a I- a a o a a a ^. aa 0.a3. a 3 = =
o o oa a a O a sOC






0 O 0 a o a O O O 0 a O a a a O 3. a -
a aa-f~~rD~~laaf a CC a aa
3- ) C-T




o aj a a a a a ai a a a a a a a ai a a a 0.



I- I-
ac -. ai a, a t a a C C
aa aa a a






C a a a a a a a a a a a o C D 0 a a a




0 a- a> a a a a a a a a a: a a a a a a; a a a
SC 1- a a a O O a a O a O a a a a O O a a Ca a




a a -o o o O c a ~ c '" 0 a a a a a o a a a a a







a i CS 0 O O a a a C a a a a a 0 0
a as a a a a a C '-3 a a a a a a a a a a a
3-3 a a a~ a a c a a a a c a a a a
*, o~ Q v o oo oo r~-i C oc^
a .c~a a a aO C OD O OJ
*~ a n a a a I a l a a n a n 3 a a a l a I a I a a a a a


a a a a a a a a a a a a a a a a a a a a






-13-


Table 4

Coarse Cain Calibrations for the PA-1/C


DJ t G a i n


--------------


12.5


C3/19/75

03/21/75

04/15/75

04/21/75

04/22/75

08/17/75

08/21/75

08/25/75

10/15/75

10/23/75

02/03/76

02/08/76

02/15/76

03/02/76

03/18/76

/4/04/76

04/18/76

04/26/76

05/04/76

05/19/76


10.0 7.5 5.0 2.5 0.0 T II

(F) (1)

20.0006 7.4993 4.9S78 2.4953 -0.0190 60 65

10.0253 7.5187 5.0023 2.5050 -0.0145 60 85

10.0311 7.5245 5.0071 2.5079 -0.0114 48 90

10.0317 7.5227 5.0049 2.50o6 -0.0126 56 94

10.0292 7.5219 5.0044 2.5064 -0.0132 60 95

10.0169 7.5160 5.0012 2.504S -0.0176 80 95

10.0211 7.5189 5.0025 2.5056 -0.0161 73 90

10.0159 7.5158 5.0005 2.5048 -0.0170 75 90

10.0227 7.5202 5.0016 2.5053 -0.0147 70 85

10.0233 7.5185 5.0020 2.5050 -0.0152 64 94

10.0319 7.5215 5.0043 2.5063 -0.0120 54 90

10.039] 7.5251 5.0064 2.5075 -0.0n95 40 70

10.0238 7.5191 5.0028 2.5048 -0.0157 68 70

10.0208 7.5174 5.0017 2.5049 -0.01,2 70 80

10.0376 7.5230 5.0048 2.5063 -0.009' 55 80

10.0272 7.5203 5.0035 2.5060 -0.01 1 60 85

10.0339 7.5238 5.0062 2 5081 -0.0118 65 85

10.0348 7.5231 5.0051 2.5068 -0.0101 58 80

10.0238 7.5192 5.0025 2.5054 -0.0154 55 80

10.0220 7.5177 5.0018 2.5049 -0.0156 55 80


12.4727

12.5021

12.5052

12.5095

12.5056

12.4925

12.4969

12.4902

12.4984

12.4997

12.5089

12.5181

12.4995

12.4960

12.5156

12.3034

12.5117

12.5119

12.5009

12.4977





-14-


in January 1976. By inspecting the i-magnitude calibrations,

it appears as if the system was noisier prior to the instal-

lation of the new tape drive unit.













CHAPTER III

THE OBSERVATIONS



Introduction


Thirty nights of observation (ten and twenty on the

eighteen-inch and thirty-inch telescopes, respectively) have

yielded approximately 3200 measurements of intensity and

time for U Cephei. Table 5 lists the dates of observation

for each telescope.



The Close Companions of U Cephei


Observations of U Cephei are plagued by the presence of

two close companions. The following table (Table 6) lists

properties of the stars of the system.

Observations of the two companions were made in 1881 and

1899. Aside from these observations, there is scant infor-

mation on these stars. The original observations by Knott

(Jeffers et al., 1963) gave AO as the spectral type for both,

whereas later photoelectric work gave color differences more

consistent with G6-7 for B. The magnitude differences be-

tween these companions and U Cephei in conjunction with their

estimated spectral type indicate that C is probably a back-

ground object, while B is probably at the distance of U

Cephei.


-15-















Date at the



Eighteen-inch


Oct.

Nov.

Nov.

Mar.

Apr.

Apr.

Sep.

Sep.

Oct.

Oct.


Table 5

Beginning of Each Night of Observation


Thirty-inch


1974

1974

1974

1975

1975

1975

1975

1975

1975

1975


Mar.

Mar.

Apr.

Apr.

Apr.

Aug.

Aug.

Aug.

Oct.

Oct.

Feb.

Feb.

Feb.

Mar.

Mar.

Apr.

Apr.

Apr.

May

May


1975

1975

1975

1975

1975

1975

1975

1975

1975

1975

1976

1976

1976

1976

1976

1976

1976

1976

1976

1976








Table 6

Properties of the Stars In and Near the U Cephei System


Star V B-V U-B Sp. PA D


U Cephei

Combined 6m83 -005 -0.40

Primary 6.95 -0.11 -0.43 B7V

Secondary 9.11 +0.88 +0.40 G8IIIa

B 11.83 +0.73 +0.29 (G6-7) 620 13'.'8

C 12.9 A0(?) 3210 21"2


aFrom Batten (1974).



Figure 1 shows the positions of the companions super-

imposed on the smallest diaphragm of the eighteen-inch tele-

scope system. This diaphragm was used until 6 April 1975.

An observing note on 10 November 1974, noted the two com-

panions with no diaphragm in place. Calculations show that

C is below the sky limit on all but moonless nights with

little haze. These conditions were obtained only on 10

November 1974, and 6 April 1975, when the smallest diaphragm

was still in use. Even under ideal conditions C is only

about 0 2 above the typical sky through the eighteen-inch

telescope system. Tests of the Fabry lens of the eighteen-

inch telescope indicate that edge effects may reduce the

intensity of C by half at its position. The maximum error

introduced by not correcting for C is 0.036 in the visual






































/








ti
Cfu










as~
cnd
cJ,
*'- 2:a e



C']a
cO

\ S 'Hi
\ *H*H/
\ +-> 0
\ !/) O


us
0
'H



Ln
4j
7C
r-
a
U
o
u

-i

-a
c
*r
*H

(D
ao
C)
U




3


CU)
C



C,





-19-


and 0" 041 in the blue. The variable must be about 70% into

eclipse before the relative intensity of C compared to the

variable could introduce as much magnitude variation in the

light curve as the typical error of the observations (0.01).

For all these reasons C was assumed negligible and cor-

rections were made only for B on nights before 6 April 1975.

After that time the next larger diaphragm was used and both

companions were used for correction. Observations in eclipse

(i.e., 10/31/74, 11/10/74, 11/15/74, 04/06/75) were checked

for abnormal fluctuations after corrections and none were

found.

No such problems were encountered on the thirty-inch

telescope, since the chosen diaphragms were small enough to

consistently exclude both companions. On moonless nights of

superior photometric quality, deflections were taken on these

companions. Table 7 shows the magnitude differences with

respect to BD+81029 of both companions in the three colors

as well as the ratio of intensities. The weights reflect

the quality of the other observations around the time of the

deflections.



Reduction of the Observations


Observations were taken in the sequence comparison VBU,

variable UBV, comparison, VBU. The typical time between

consecutive variable star measurements was four to five

minutes. Sky readings at all gains used were taken every





-20-


Table 7

Differential Magnitudes of the Close Companions
With Respect to BD+81029


Companion B Companion C

Date AV AB AU Wt. AV AB AU lt.

04/15/75 3'38 2m94 2m71 1 4 38 4m38 5m61

4.58 4.71 4.60 1

08/17/75 3.33 3.21 3.00 1 4.24 4.35 4.54 1

02/03/76 3.22 3.17 2.94 2 4.34 4.54 4.96

Means 3.29 3.12 2.90 4.38 4.54 4.86
5 8 + 8 7 + 7 19


Ratio of Intensities

0.0484 0.0563 0.0695 0.0177 0.0153 0.0114
S 3 6 6 + 2 2 4





-21-


-hour on moonless nights, more frequently on nights with a

moon, and after every comparison star observation if the moon

was rising or setting. All sky readings were taken just

north of the comparison star in a field clear of stars.

Check star observations were taken two to three times per

night.

My initial reduction routine removed the effect of the

sky, found differential magnitudes (m var-m ) and color
var comp
differences by linear interpolation. All data were then re-

checked on the strip-chart records to insure that all usable

data were free of keypunch errors.



The Reduction to the Standard UBV System


Stars in the Hyades cluster were used as standards. Two

observing runs were done on the thirty-inch telescope (one

run each on the eastern and western halves of the sky) and one

run of excellent quality was done on the eighteen-inch tele-

scope. Since the declination of U Cephei is almost +820,

good atmospheric extinction information cannot be obtained

from the comparison stars. Extinction information from the

standard stars was used to transform the variable star obser-

vations to the standard UBV system. The symbols and equations

that follow in this section are those of R.H. Hardie (Hardie,

1962). Table 8 lists the data for the standard stars. The

stars were observed sequentially as given in the table and

this sequence was repeated four to seven times as the stars

progressed through approximately 14 air masses.












Table 8

Data on Stars in the Hyades Cluster


Star BD#I V B V U- B RA (1950.0) Dec


72 150632 3'41 0179 0'.132 4h25m48s 15045'42"

71 150631 3.85 .955 .741 4 25 43 15 51 10

75 150633 6.59 .531 .060 4 26 08 16 03 00

82 150647 4.78 .173 .126 4 27 42 16 05 12

85 150640 6.51 .426 .009 4 27 55 16 02 30

83 150639 5.48 .259 .097 4 27 48 15 35 05

80 150636 5.58 .319 .009 4 27 17 15 31 49


From Johnson and Knuckles (1955).








The second order coefficients were calculated first by

least squares solutions of Hardic's equations 25,


A(b v) = k"vA(b v)X + A(b -v)0


A(u b) = kub (u b)X + A(u b)0


where small case letters refer to observed quantities, X is

the air mass, and the second order coefficients are k" and
by
k". The A quantities in the equations refer to differential

measures between the standards. The procedure followed was

to take each standard star in sequence as the base star and

calculate the second order coefficients, using all the other

stars in combination with the base star. This yielded forty-

two determinations of each coefficient. Useful information,

however, can only be obtained between stars having suffi-

ciently different color indices. Table 9 shows the combina-

tions of stars used for each color index. Weighted means



Table 9

Star Combinations Used to Calculate Second Order Extinction
Coefficients

Nomenclature is that of Table 8.


B-V U-B


72-71 71-75

71-72 71-85

71-82 75-71

82-71 85-71

80-71








were taken of the coefficients resulting from the above

combinations with the weights being the normalized ratio of

the coefficient and its standard error. The results of the

two runs on the thirty-inch telescope were averaged to give

the values used.

Having established values for the second order coef-

ficients, Hardie's equations 26 were used to determine the

first order coefficients, namely,


(b v)Jx = k'vX + (b v)


(u b)Gx = kbXX + (u b)


where J = 1 k" X and G = 1 k" X. The first order
x bv x ub
coefficients were calculated for each star and straight means

taken to give the results of Table 10. The primary coeffi-

cient k is given by Hardie's equation 22,


v = v kX
O V

The coefficient k was also calculated for each star and
V
means taken. The large differences in the standard errors

quoted for the two telescopes are misleading in that the

error shown for the thirty-inch telescope results from two

nights and that from the eighteen-inch telescope from only

one. The latter error is more typical of the single night

error for the primary coefficient, whereas the former error

is useful for showing, but not strictly indicative of, the

range of k for photometric nights.
V





-25-


Table 10

Extinction Coefficients and Auxiliary


Quantities


Telescope
18 inch 30 inch

k 0.26 0.32
v 2 + 9

k 0.15 0.18
bv I 3

k" -0.050 -0.033
by
bv 9 8

kub 0.36
+ 5

kb 0.001
b 12

S1.039 1.185
11 + 16

e 0.019 -0.144
+ 10 + 11

1 0.986
+ 20




Three auxiliary quantities are needed before the trans-

formation to the standard system can be done. These quan-

tities, j, e, i, are found by linear regressions of (V vo)

with (B V) for e, (B V) (b v) with (B V) for p,

and (U B) (u b) with (U B) for 4. Upper case let-

ters here stand for magnitudes in Table 8.

The differential magnitudes for the variable were then

transformed to the standard system via Hardie's equations 28,








AV = Av k AX + A(B V) ,
V

A(B V) = jA(b v) jukvAX wkvA(b v)X

A(U B) = iA(u b) PkLbAX 'kubA(u b)X


Here A stands for differential quantities and X is the mean

air mass for the variable and comparison. The differential

air mass, AX, is given by,


AX = (Psin h + Qcos h + R)X2


where h is the mean hour angle between the variable and com-

parison (positive east) and,


P = Aacos4 cos ,

Q = A6cos sin6 ,

R = -A6sin4 cos&


Here Aa and A6 are the differences in right ascension and

declination in radians, respectively, and is the observer's

latitude. P, Q, R are constant for a pair of stars and at

Rosemary Hill Observatory have the values,


P = 0.00248 R = -0.00169

Q = 0.00046 BD+81029, Q = 0.00130 BD+8130.

R = -0.00004 R = 0.00011



The Comparison Stars


My original choice of a comparison star was BD+81030.

This star has colors strikingly close to the variable and its







proximity was also alluring. An early paper on U Cephei

(Dugan, 1920), however, indicates that this star may be vari-

able. It is interesting to note that most observers of

U Cephei still use BD+81030 as a comparison despite the

warning. This may be due to the difficulty in finding a

nearby star of substantially the same colors or finding any

color information on the nearby stars. All data until 15

April 1975, used BD+81027 as a check. On 15 April 1975, and

for all subsequent data, BD+81029 was used as a comparison

and BD+81030 as a check. This change required that data

taken earlier be transformed to the new comparison star.

Table 11 gives the pertinent information for all stars used.



Table 11

Data for the Comparison Stars


Star RA (1975.0) Dec V B- V U- B


BD+81030 1h03m06 81049'43" 7m89 0m019 0 80

BD+81027 1 01 30 81 58 12 8.41 \0.6 '0.1

BD+81029 1 02 17 82 06 58 8.58 0.49 \0.0




Figure 2 shows the differential magnitude (BD+81030 -

BD+81029) versus time for the three colors used. This figure

indicates no trends or unusual fluctuations. Nights on which

BD+81030 was used as a comparison required the addition to

the data of the magnitude difference between BD+81030 and













































I.-
t'
-j
C)


T~i~"' '"'8~7~1"' r~9~"f~~i' "'1~"~'"T '*


Sw
_J -J





L LL L CJ
Q3 Z3

> 3



0 vr < co 0 o to l

d o d -
I I I I I I T I
3DN31~1j-rt0 .301.lINOV IN


r 1""9'^ "' IT~~P







1'

.r


I

ts
i :.1
I-I


-:
C:
O
O L~
O .s


J
i-
;r
a

O ~:


:j
1.1




O
f~








BD+81 29 to transform them to differential measures with

respect to BD+81 29. These magnitude differences on the

natural photometric system are listed below.


Telescope
eighteen-inch thirty-inch

v -0.55 -0.65

b -1.13 -1.17

u -1.47


After all corrections were made, an inspection of the

light curve reveals no systematic deviations between nights

on the eighteen-inch and thirty-inch telescopes with either

comparison star. Thus all transformations are compatible.

The data reduced to the standard system were thoroughly

scanned and compared to the chart records to eliminate data

suspected of being affected by weather. These scanned data

are presented tabularly in Table 12 and as the light curves

of Figure 3. The phase has been claculated using the ephem-

eris Min JD = 2442352.6999 + 2d4930709 E (see Chapter VI).

Two major outbursts were noted, one in August 1974, and one

in September/October 1975. Data around these dates were

eliminated from the initial analysis in an attempt to obtain

less perturbed light curves for solution. Specifically, the

nights of 31 October; 10 and 15 November 1974; 18 and 25












TALL 12
STANL)AkU DIF-- i-LNTIAL i.AGNI TUL)LS
U CEPHLI


HLL J4
(2440000+)


2352.542t65
23L2. 04980
2352.55484
2J52.55935
2352.56309

23b2.508035
2352.57443
2J c. b9o5 b
23z2. 60151
2352 .61180

23o2. 01911
230bz 02444
302. 63, 15
2352. J549
23J2.04242

2352. 64823


c352^ t7201
2352.51O3oJ
352b 1 72 1




23o2. 52401
200u2.52707
236 .534036

23602535 17
230L53b91 1
23o2.542b3
23o2.54594
362. 54 936

2362.5490
2362.55081
2362.56301
235 50325
2362.50439

2342.56 /53
23o2. 50628
2362.57060
2362.57130
2362.57394

2362.537499
2362.57928
23b .58025
2jo2. 58426
2362.58471

2362.58541
2362.58 343
2362 .58922
2362.59015
2362.59420


VISUAL

DMAG-H. tiLL JL)
(2444000 +)


-1.,4J
-1. 09





-i.370
-1 .cj7



-l. lb7

-0.700
-C. 700




0. 07
-0 .c 2
-0. 430








-1 cVb

-0.077


-i 3bC
-0. o91
-I.004
-1 104


-1 02

-1U 0









-0.9 oc
-61 .< 29

-1.197





-1. -10



- 0.0 u
-0.930





-C b o
-0. o b
-0.o* 3
-0.7c,)
-0.7o9
-0.707

-0.7j3
-0.057
-O.(J9 o
-0.005


-0.59


-0.403
-OIqO3

-Lb, 03


2 30 2. 5950



2352.0 61 0
2362.0 950t0
Jo2c .c995
Liou2,bO-z>7
2362.l003 3




20o2.00L,
23o0.o0b19
23od.ul007


036 c.i) 403
23o2.ul 1o
246.c191,'
23to2.u c 1

2302. 42070


ZJuid.ull.3
2c cJ2105
230b .La321

232b .Loo-)9
23Cu .)7 744
23Lc.O 030

2Jo62.9lol
2302 0400

-3od.u4UC j
236c- .04 0




nab2 .o795(
2o02 .c8CJ7
23o2.*Odoco
230,J.c7u73
2 c02. c92 Co

c362.t9 106
236?.69o975
2302.099B2
J3i..700b9
230.7u7o02

2Jo2.70872
2302.71253
230J.71380
2362.71 89
2362.71931

2302.71994
2302.724 87
23o2.72555
230C.72931
23o2. 7303o


OMAG.



-0.372
-3.lt8

- .224
-0.100


- 1 C o

-0.043
- ,01.


0.132

0 .2c~L
o .29v
I.,3 3
0 2. 4 1
O.J J
0.421
4 .443

0.51,
0 500




u .o74
0.098


0.c'7
;.72o
0.7o3
0 .740
3.7Jl1

0o. /4o
C. ?744
. 737
S.732
0 .uu7



1 5.)2
0.093
0.4 71

O .4b2
0.300
C .35o
0.241
0.221

0 .2;0
0.090
0 L 70
-0.007
-0.034


HLL JO
(2443000+)


doJu2.7j1lb
?JbZ.73Lbt,
2J&2.7?'., u
23J 2.7oJ1 I
2J"o2. 7, 2" i
2362.74424

, 3--u 74 i-j o
23o i ,. 7.'. j
2302. 7.90
230 7 Coo -
?c2..7lGoo



23o2. 7 ~u

2 .,b2. 7( c4
2?302.70lb




2 3o02. 7i2 .1'
230d.. 7jc'

ZJo3 c7, 13

2Jo2. 4'4


2j 7.3> 7 j3
23071.53 74
2327. b5l
2 -3o7 *35.' c
2307. 51 77

2367. 5177
23, 7. lo 1 6_LI
SJo7. i 14

23o7.5'o5>
3567.5270 3j,
2 J bc 52i 7, c
23o7.o3029z

2 36u bJ.I )s
2307. 5310 ,



2'-67.53874
23(T .5394 I

2 3b7. 54 uj
23o7 *S5j 11j
2367.54-32_
2367.b4714
23o7.549o;,
23o7.50C091
23L7. 50038
2367 o5b4
2367. 5'4o3C


DMAG .



-0.0 3
-0.133
-'.1 J'2

-3.324


-0'. )

-0. 402
-0. 1 1







- .11j
- .JJ I
-0.5o
-0 *.c3





-1. '1
-1.0- .

-1.*1

-1.2-i
-0. 2 2







-1.1-
-1. 1 -
- 1 *





-1.3 7 '1
- .31
- 1. 327 S
-1. 1_5

-1.219
-1.21$
-1.1/2

-1. 141

-1.131
-1 .0 9r'
- 1 329
-1.-')]1


-0. 9
-0. J92
-0. 32
-0. 911
-0 37o

-C.6d71
-0. 7c2

-0. 771












TAdLE 12(CUNT'D)


HEL JD
(2440000+)


2367. s5791
2367.55t 71
2307.5o279
2367.0u347
23o7. 567 ju

23o7. 56'21
2Jb7., 7110
2357.57198
23u7. 57544
2367. 57G69

2367. b7938
2367. 0 018
2367. 5dj07
23t7. bb56s9
2367.b8429

23b7.58713
2367.5877o
23b7.5o06?
2367. 5do80
2367.o91 /7

2jb7. b9262
23o7. 59J07
23t7. b59a
23o7. 5'9992
2j37. 01 .

23J7.0C790
2,b7. 60, 95
2307. 01757
23u7.61924
23o7. 0236

23o7.6245.S
23o7.62922
2367. 63066
23o7. 63637
2367. 3789

2367.6417o
2367.64260
2367.64749
2367.64db4
2367.65235

2367. o0500
23o 7. 653'2
2367.ob033
2367.66115
2367.6b041

2367.60729
2367. 73899
2367.674tb4
2367.67840
2367.67933


VI s UAL

DOlMA. hML JUL
(2440vC0+)t


-0.737
-0.7 lI
-C. 91
-L.uwbC
-u l:

90
-0.539

-0.4 7


-O.Jb
-o..J7.





-; .1 99
-0.107
-0.1 99

-0.113



-0.0 i
-0. 7
-O .




. 4 bo

C.099





0.721




0. /7J
0. 7J3
u 7 2


U.7L7
0.713

0.754
0. 7 1
6./2 7
0.721
. 1720
G. 77..

0 7ol
0. 745
0 7/-3
0.719
0.7e/3


23067.1-o3 14
2367.604 29
2307. 89o3
23o7.0902o
2i/6 L l361

2367.a9443
2307.9j33
23o7. o99o
2367.7025D4
2307.705339

b367 /06&d
230o7.70760
2J 7. i 034

2307.11303

2317.713 o1
236/7*./l o9
<307.7174?
2367.71971
2367. 72041

2367. 72>02

2307.7292U
23u7.72 99o
2do7.734Cb

3w/7.733000
2367.7j304
23u7.73074-
3oT7.73940
23o7. 740Cj

2367.74280
2307.74.134
2367.74ob2
2307.74714
c367.7 7322

23o7.75375
2307.75o 69
2e67. 70773
ito7 /o J 2
23o7.75083

2307.7.6452
23o7.7o466
2307.76.074
2307.768b5
2367.70973

23o7.79017
23o7.795 U
2307.60221
2491. 0o832
2491.570 34


DMAG.



0.71 1
6.71)

0 .u0




0 J 3b
0.j'o1

0.502








1 1
*).418
3 .4 17
O.J3b
0.2933
0.270

$.l1,
0. 140
0.12 J
0 1 0
0.0t4


-0. 04L6
-, .Oo9
-u.12
-0.149
-J.1-0


-0.2 2
-0.20s


-0 .303

-0.433
-0; .44
-0.5064
-U.3149

-0.,54
-0. 659
-J 6 7Vo

-0 704
-S.754

-O6.o26
-0.17
-0. o0
--0.d70
-0.678

-1.149
-1. 1 1
- 1.239
-1.64o
-1.668


rlEL JOL
(244 30.0+)


2491.57121
2491. ,t j.)
2491. 0"2 oi
e491.nl,711

2 4 91 1
2491 .0 1,
2491 .*, 3'-




2 4 9 1 c 1 7 1 .
2491.ul I .I




2491.6174s9
2491.o1 ~3
4Y1.16230
2491 0J:4J








2439 1 6.2 -,
.491.OS/ 1
2491 642 '1
2.491. 04 /1


24' 1. 03- J
2491. 05:' i



249i1 ,. 5
2491. .7 t .







2493.5 19

2493. 573 /

2493.568/7
2493. -9704
249JJ..).5 72'.
2493. 5* 270.1








2493. 5403l



249o. o4 noI

2493.561371
? 4:03. b i 40.'55
2493.533 61
24 93. 634u2



2493.S5I 7s
2493.Q 7231




2493,00J.b 0
249 3.o637
2497.577J7


DM AG.






- 7. o' :
-1.0 43

-1.031
- 1.o27
-1. c3:




-I .o? c
-1J.o27

-1 .02

-1 oj i
-1.'..
-1 30
-1 r, 3




-1. 2


- 1.o
-1..I'l
-l.3,
- 1. ,'' *


-I.0s3
- 1.J <
-1. ,o
-1 . 11

-1. JA-
- 1.31,
-1 .97






-1 .:3J
-1 .* D2J
-1 .* 4
-1. i-






-1. 5;72


-1.003


-1.572






-32-


TAbLL le(CUNT'D)


VISUAL

DAAUl. rlL Jo
(24'.O000+)


hEL JO
(2440000+)


2497.57389
24'97.08400
2497.58764
2497. 58689
2497.b9303

2497.59407
2490.b?97oq
2497.59o7o
2497.002z3
24Y7.603t5

2497.60034
2497.b0901
2497.ol008
249?7.614C4
2497.61349

2497.02451
2497. 62bt4
2497.63140
2497. 63279
4497.63ull

z497.6.o 0
2497. t41u9
2497.04291
2497.647Z0
2497. 04992

2497.obl39
2497.0037C
2497. 63 70
2497.6uJ74q
2497.60o14'

2497. oo214
2497.6oov42
2497.o l974
2497.67373
2497.o7b12

2497. o781
2497.olCd0
2497.68642
2497.b0859
2497.69IbO

2497. 9282
2497.o9ub4
2497. 97b9
Z497.70 180
2497.70307

2497.70d93
2497.71524
2497.71051
2497.72154
2497.72292


-1 o54
-1.002
-1.usd
-1.04


-l1ub4
-1 u 4
- 1.04c
-l.016
-1.6o1


-1.0u
-1 .o49
-10.47
-i.000
-lo.Ol


-l.o10
-1.031
-01.4 l
-1.01-


-1 034
-1 0l
-1.0u39
- 1 .034
-1 o

-1 039




-l.0Ul
-1.oJ9


-1.0U7



-1.o73


-1.04/
-1.0*79
-1. o'00



- 1.072






-lubb7
-1.oS7
-1.009


-1.0734
-1.049

-1.041



-l.ob2
-1.u89
-1.091


2497.70920
2497.7?399v
2497.74057
2497.7412/
2497.7419o

2497.742 u
24I7.74o0'
2497.74909
2497.7b02 J
2497.? ,9

497.7J77 7
2497. 7tf
2497.70o54
219l.77juD
2497 .7717- 9J
2497.77393

2497.7750,9
2497.7/ ,*( o
2497.70357
2:s97.d4b14
2497.7dbL;l

2497.70971
2497. 97: 07
2497. 79 14
24 97. 799
2 ,97. LbkO u


2497. 00od
2497.bOb97
2497.60j9?7
Z497.01471

2497 .1093
497 .02304
2497.b2478
2497.62900
2497.03048

2497.642o6
2497.64371
2497.047u0
2497. 4674
2497.Bb2bo

2'+97. ob373
c249 1.db629)
2497.obu9o9

<497.07obo

249 7.7774
2497 .oai77
2497.6o399
24.97.099?7
2497.90082


DMA,.


-l.D61
-1 .681

-1 .77

-1 73

-1. 72
- 172



-1. -7
-1 .s'.1





- 1 7
-1.037L
-1.031



-1.647
-1.u37





-I .o356
-1 .3,3
-1 .u o
-1 .017

-1 .647




--1.b2
-1 .o27
-1 .937




-1.01
-1 .o0
-1.0.,+

-1.b17






-1 .L.27
-1.64I
-1 .u47



-1 .*(4(
-1.032
-1 .642



-1.05
-1.045

- 1.obo


-1.-Iu4
-1.009

-1.6j1u
-1.0o35
-1.04

-1.-41

-1.o35


HEL JJ
(24430L 0)


2497.93743
2497.9C5S4a
2497.91o u
2497.91713
25C7.5 790.



20 7. 6,- L7 7
23C7. zi7u 1
2507.o;941

2-507. 012
25~ 7.61 27c

507. O,1S
2507.o197:J



2 C /O oc. 71
2O0 7. z-.71
2507.6 -1 u

2b07.6o 7l
2507.0 o 7

SOj 7. uS4 3

2 6i 7. 64(4

253 7.0, 7.,

2507.0 31



2 C00 7. coj L-,
2507. O 1 L,,
2507.o0,31
2LJ7. o64^

2b07. o9 tIc

25 7. 731 23
2b507. 7uo27
2507.73914
2507.7o6o0
2507. 7;u77




2b09. b70 21
250b9. 530

2509. 'St 0
2a09.508194
2bO9. bb6u4
2509.57191


D94 Au.



-1 .* 3
-1.- sO


-1.c, 5


-1.ul
-1..0 2

-1.3<(


-1.o 2
-1. -o?
- .* 0


-1.5 I





-1. 11
-1.'-1





-1. .. i
-l . 1
-1 1

-l.o11



-1 6


-1. i:2




-1 .* 7

-1. a'4
-1.05
-1 .* 1



-1.340
-1 3l 2
-1. 1s 1


-1.538



-1. *i31
-1.:,22
-1. l 1












TAbLt 12(CUNT'D)


VI UAL

DMAG. HcL JD
(2440000+)


HLL JLO
(2440000+)


2509.572U61
2509.575J7
2509.b7o07
2b09.57o32
2bG9.57901

25C09.5o12d
2509.54198
25'0 5i+* 22
2509. bt549
2b09. 5c839

2bC09. 5=909
2509 59135
2i09.5922 j
2oC9.94053
25C9.59634

2509. 59779
sgCS.60.so5
2509.b0125
2509.60419

25C9.63 34
2o09.307 34
2509. t;C 1I
25L9. ul 63
cbs y.61132.

2509. lb144

250C.ul72J
2509.61791
2509.62127

2509.62192
250 9. 22451
2509.02521
2 C9.627u3
2509. 02833

2509.03199
2509. 632oo
25bL0.6 857
2oC9.b3927
2509.64171

2509. t4239
2509. 04553
2509.04 623
509. 4 84 7
2509.04917

2509.05161
2509. o0229
2509.65455
2509.6 s525
2509.65750


-1.500
-1.492
-1.*49>

-1.474
-1.4/1
-1.4)7/
- 1.o7
-1.4494

-1.400
-1.404

-1.41,
-1.411

-I. 4 -
-1 .417
-1.957



-1.307
-1.o5o1
-1 .04.



-1.- 9 9
-1 .2's
-*,30



-l..
-1 o74
-1.<72



-1.199

-1 190
-1.2*07

-1.111



-1, *

-1.Co0
-0.97G0

-i l ^1
-0.915

-0.bd7
-L.0.74
-GC.tbb55
- 1640
-0.01J


lObQ.obbiv
2509.o5619


5sC9.ooLoO
9bub.bbl5
.,bboo 13



2bu9. ubd l
LO09.Y07171
2o09.57240
Zo09 .0 74 5)

0O09.o7o20
2oO9.u67779

20,9. tO 9 b
2 50 9.0610

2o09.092 l
2009.o9o3o

2509 .SYo 0
2~09.7014 C
2o09.(7021O

20J9. 7027
o09.71> o
2009 71o53


2o09.71912
25 9 7 1 1 2;4
2u9. 72Doo3
2509.72uo3
2509.73174

250U.70o.3d
2509.73909
25Ov. 1t-.73
2009.74700
2569. 75-31

2509.75924
oO09.772o j
251b.l 5L90
2b1idolbbO


251.02b 2 6
251 ,.0 309 c





251lb. oo7J
2t16.o7.247

2018.o70,25
2518.07528


UMAG.



-0.801
-0.745
-0 7o
-0.70u
-:.*o97
-3 c)9 7

--0 D2 7

-0 .u7

-0. o 4
-3.5092


-0.478
-0.463J
-0 13
-0. 97

-0 1 1
-0. vJ
0. 075
O.140

0 1 tu

S.1 7d


O. dol
0.47.

0.5/9


0. 1. 3
> .7C7

0.74.3
S.754
0.7u5

0 .7.0

0.b63
0 .7 7
-1.515
-1.5so
-1 .5*o


-I .067
-1.7o0


-1.5o7

-1 .uLC
-1 .5b1
-1.025
-1 .301
-1.540


HbL Jo
(244C00C0+


251 .6.- 724
2 1 .71-) o,
2513. 71~ ;i
2>lo. /4 30O>





2513.77 901


2 1 d. 7d C,

26316..77 7
~518.7 l- j




o2bild.7/ :
o218. l0,
S1 H -lc 7 7



42510.133.


2524.683: :











bZ 24. ..-
201 d.80 0




















2 5 4. c 1
25l2.oo L
92a4 or









2524. 71 37a








2 24. 71 9 '1
2524. 7.l -5






25 2 4 7 1 3






2524.~3,i7
2524. 1
2524. 7Si-,j

2524.500743



2024. 5o79


--1. 40
-1. .14



- 1.114
-1 .00
-1. -
-I 2,,2

-1."12
-1 0

-1.213



-1... '1
-1. .!


-1... -1

-1.7-


-1. .3
-1. .e

- .1 I






- ... 15







L 'i 1'
3. 1~1







C0. 702
-1./4'



I- / ('

C 7o o
.* 0053
-L.o'Z












TAbLe 12(CJNT'D)


VI UAL

LMIAG. htL JJ
(2440000+)


tLL JL)
(2440000+)


2525.57154
2525.d8119
2255. 5b65
2525.5926J
2525.595O72

2525.25 OC 11
2525 6323
25c5. 00o99
2525.61275
2325.63162

2325.6 tjolo
2525.03988
2b25.o43Ja
2525.04334
2525.o51u7
52525. 575

252b.o2 lo
2525. 03 :dd
2525. c6b79
255 074 J5
252b. 7765

2525.66779
2525. b'0141
202.5 9532
2255. 70736
252 71 1 7

252t3. 71o1l
2525.72J1I
2025.72o34
2525. 7,900
2525.7347/4

2525.739b
2025. 74299
2b25. 790t87
232b. 801bo
2525. 8103

2525. 81741
2525.82115
2525. 825uo
2525. 83237
2525.83010

255.684049
2b252. 44bo
2320.85214
252b.85712
2025. 86542

2525.87393
2042.63104
2642.63964
2642.64613
2642.65089


-I.bvO
-1. 41
-1.5o7

-1I.ut?
-1 71
-1.507
-I do




- 1. 5o 7
-1.o21



-1.590u

-1 .Dbb

-1.0 6
-1.755
-1.5/C




-1. 390
-I.oio
-1.0731
-1.u9/
-1. 46t
-1.0u1




-1.033
-l.udd









-I t.)0
-1 .050
- 1 *.a4
-1 51 0
-1.3bb
- I. oil7




-1.022
--13093
-1.5304

-1. 00C
-1 .*91
-1.504
-1 .031


-I.U75
-I .oC
-l.173
-1 .u70
-1.o57


-1.5792
-1.070
-1.509


2o42.o0503
204Ez.3)40




264 .06387

2420 /7 1 63
2o4c.oo000

2042.70133
2o42.7/G04
264c.71D42

<:o42 1 4
2o42. 10 12
2042.72450
2642.7207J
204c 7.-19

2042.74373
2c42.7461
co04.73i 18
4642.7(301
eo42.7uo3C2

2u,4 7700

2o0 .77 l10
uo4-.bo7J
2o4c. 7o- 07

2042. / 7304
2o4c.79J~27

2o42.50,9"
204 c .7 62I


2 o4 2* 6 1 9S
2o4e.bCdd2

2642S1954
2042.O3200
2o4c .o4 97d
2u4 .600024

204 c.0717 /
2o400.9940
2640.7C009
2o46.71023
2o40.71b45

2040.72070
i24o.750uC
co4 .761 71
2o40.70030
2o40.77140

2040.77744

2040.7051U
2o4O*76109
2o46 7o5 1
2o4o.78L6o9
2o4o.790 11


UMAG.



-1.5d4
- 1.u2d
-1 .j0
-1 91
-1 .Uo4

-1.o03

-1.uvO
-I o.7
-1. 10
-1 oLO1!
-i.00

-1.5'37






-1.013
-1 .ul3
- I .,dl
-1 .o00

-1 6 18
-1 .u29







-1.021
- 1 .022

-1.021

-1 .u3

-1 .oC
-1.o53
-1.0 39

- 1 .L. 11
-1.0 45

-1 .u-72
-1.59 1

-10 .362
-1.591
-1.5db
-1 .590
- 1.589



-1.bd5
-1.442
-1 .j99
-1.390
-1.341

-1.J47
-0.999
- .921
-O0.d52
-0.79
-0.60o

-0.072
-0. 3/
-0.492


HEL JU
(2443000+)


2o4 6. 7o4

2040 ;t7 7
2040. 87 7Jz,
e.r40 .8 117




20o46. 6 48. 1

2640. 5 3-3I
2 64 B3,'i 1.
204u. 3,11

;'t040. 8r .-


2 oL 4 Du 1i3.
'A3c. j3 7
2040. :5.. 4.,
2 4 L. 7 i
2.4o. 0/8 Sn


2o0o.667J2

J2000. 7 1
Jo20.o4 ui


*,OD3 ] 7J.,..,
,: 6. Z, 7 1 1 :

L2i, 0. 71 30
2 503.71,1 .





2o0S,. 740,
20 0. 7304




L00 7.':2
2,650. 73571
2o33.7346W
265C. 71 .J





;,030 46#J1961
20 So .7 44.

2uoO. .8 71


6530. 7'45.J

2o50. 87odd7
2 5 0. di Cj 3







2050. 839o1

2o50. 04 24-


DM A .




-0. 2'
- 1 2.





L.uS
U. : )4



0. 22
3.z17
C. .-40




0.14
. L.1 4







. 'n
0. 701


-1. 7









-1. 2

-1.-3

-I..i3
-I . 53





-1.a 2
-1.ul3


-1.0 7
-1 .57.3

-1 .-1

-1 .* 7
-1. 3-'i


-1.015

-1.J393
-1. 3 1
-1.0'1
-1.314
-1. 39

-1 2
-1. "1 3
-1. 01 7
-1. o2u











TABLt 12(CONT'D)


VI UAL

DmiAb. HEL Jo
(2440003+)


tILL JD
(244G000+)


2650.84955
2674.71 o74
2074.72230
2074 72o 14
2o74.73175

2674.737e4
2674.7397J
2674 .740o2
2074.75321
2074 .76119

2674. 76703
2074.77161
2674 .77764
2o74. 78490
2074.79033

2674.79520
2b74 .60 33
2674.00781
2674 81 1
2o74.61004

26/74.82C90
2674 b>o7
2074.82992
2081.68430
2061.69170

26O1 .69 19
26bl. os91
2cbl.70225
20ol.70537
261l.70560

2681.713b7
col 1.71J92
2o81.72035
2681.82lIo
2681.83072

2681.8346o
2698.51804
2098.b243C
2696.5240o
2t98.b3322

2b69.b3 18
2698.54J04
2698.54790
20698.5t237
2698.50007

2698.5645o
2698.57J9o
2690.57812
2698. 58228
2696.5b810


-1.03b

-I.joo
-1 01


-1.000
-1.070
-1.000
-1 .091

-1.370

-1.087
-1i. 9
-1.750
-1.50 0

-1.001




- .o12

-10. 575
-l.oo4












-0..49
-0.400
-0.5o5
-0.0 1.






-0.4oJ



--1.4b4
-1.424
-1.414

-1.443

-1.454
-1.441
-1.4o4
-1-.398
-1. 44 1

-1.439
- 1.455
-1 .450
-1. 459
-1 .470


2698,5902z
2o09.io 9 17
2o98.01917
2o9o .60u1o
2o96.60o54
209b.61 74





2o98.0o3o81

269o.04307
20o9.047bb
2096.651474
269bo. u590

2u90. 7000
20.90.u0u0
269od. uS9

2o9o.6972o

2668.7027/

2o9o. 7009'.
2o98.7103c
o209 .722>9
O6b. /3072c

2o90.735Cd

* (b.7t9204
2o98. /47c3

2o9o.7bl61

2uG9.7o737
2o98.774o7
2o98.77'9o4
20vo.784l0


206o.7941
2ov.79861b
2l9o .o04 14
2b98.bo79o
2701. 53914

2701.34o7o
2701.t5048
2701.5u40
2701.b900o
270 1. 6402

2701.b6653
2701.bo474
2701.b5920
2701 .b993
2701.5976J


DN AU.



-1.477
-1.451
-1.47o
-1 .477
-1 .491

-1.4/3
-1.47D
-1.475
-1.4o ,
-1.504

-1.4200
-1. 523
-1i.517
-1.501
-1 ul

-1 .5.b
-1.512
-1.503
-1.519
-1.020

-1 * 9
-1.535


-1.905

-1 .b 5
-1 .0o(
-1.0
-1.517
-1. 25
-1.53 4

- 1 .451
-l.bso7
-1 .549
-1 .567

-1.573








-1.513
-1.5/3
-1.50*o
-1.5494
-1.550
-1 .593

-1.574
-1 .508
-1.536
-1.000
-1.488

-1.452
-1. 3o
-1.3jo
-1 .719
-1.195


HLL JO
(2443000+)


2701.6, ; 5
2 701 .61 02 :
27C1.ol o3,
: 7 C 1 o 0 o i
2707.01ol :

27C 632
S2707. 0b6 30 4
2707.64 2,2
2707.64 o0>
2707. 652o -

270 7. 64-, o-
,. 707.b I o7
2 707. 6 5 6S4
2707.o07/jl
2707. o783.

2 707. 6. 3 -1
e707o.os7-3,
2707.uj,3ul
2707.o6777
27C07.7. 1

27C7.71, ,
'7 7. 71 .,1 q

2707. 7 i7 7
,7C7 7 -.2

L7'7.7_--7
2727. 7- I D5-
27C7.7 _.4

2707. 7 1 .
707.74Sb




2/ C 7. 7-,35.,
2707.7727 7
2707.773 50
2707. 7c23l

2707.76737
2707.7. 1 o
2707. z 7 3 ,
2707.6072 1
2707.81230

e707.7817o
2707. S6: 2;
2707.83323

2707.844J3

2707.8.l9e

2707.8644t
2737.8o9 JC
2707.874 43


DMAI .




-1.2.'4 u
-1.lA<



--1 3'5

-I 43

-1. P j
-1. C.

-1 .u7


-I1.o2
-1 .'7 7
-1 3
-1.,2'
-I.o2C



-1.02



-I .: 4

-1. i'
-1 .* 7



-1.07!,

- 1.olu





-1.0.4
-1.0 2"

- 1. 0*L3


-1. 317
-1 .* o>2
-1.o32


- 1.01 2
-1 623

-1.0* o
-1.u32
-1.594
-1.607









-1.573
-1.J 3






-1 .* 7
- 1.959
-1.o07

-1.5-7
-1*b72
-1.047



-l.o/]












TABLL 12(CUNT'U)


IEL JO
(2q4C000+)


2707.87902
2707. 884s ,
2707.89o37
2707.90123
2707.9102

2707.91442
2709.ol j7
2709.old873
2709.6028C
2709. 62700

270i. b3344
27CG.6 37 o0
27C9. 4272
27C9.6474J
2/ 0.65152

Sl70. 5jbo92
2709.00124
270 oo 1
27. ,. 70o32
270 9. bd60 So

27609.6d04
2 7(,9 0q699
2 7039. 6077
27C9. o d26
2709. 709<'o

2 709. 71 4o4
2709.71801
2709.72247
2709.72o4,
2709. 73107

2709. 74152
2709.74550
2709. 74b967
2709. 7o 4o
709.7379C

2709. 7o368
2709.7o772
2709. 7679o
2709.79288,
2709.80G00

2812.522b3
2612.52719
2812.53111
2612.53329
2612.54160

2612.54b94
2612.55005
2812. b5041
2812.5b247
2812.56580


-1.bbo

-I.t4:
-1 52/


-1.347
-1 5
-1.55o



-1.567
-1.,vC

-i.ol7


-1. !* o




-I.1 *
-1 *


- 1.074

-1.t94
-1 330


-1.bod
-1 .oC9
-1 .oC9
-1.0oo
-1. 79



-1.017
- 1 i
-1*0300
-l.abo7


-1.56730
-1.6 3




-1.* <
-1.506



-1.540


-1.590

-1.b*599
-1 79
-1 b 02


VISUAL

HLL JJ
(244000J+)



2bl8.3740bb
2b12.Dbu89
4bl.tbbilbb


to 1c. .3jou
201t.59ioo
coll.89770

2712!)2397
1i .v 6 1 41



2o 12.u31 t.


2012.o4t 7



tz 12. o0tS2
b3 Ic. o03 9)
d12. .oo743
261 .0 7 1 1

col 2.0o77d
2Cl ,. b~ 9
2817.bl u1 3
217.bb53

17 7.5295 i
S617. as j
cdl 7.5.ijd3.
cdl 7..,4, 7


217.573197
o17.ubo97

617. 0J 74
col 7 bo79 1

2817.5,7172
cdl7.57038
zo17o.5219
817.489u00


2817.59o7d
2017.000 9
617 o.0037
281 7 .. 1 ld

2017.01377
8d17.o2039
2817. tc'10
L817.t2782
2817 .b3Y2


MA(G.



-1.0768
-1.30S
-1 .b 72

-1 .57-

-1. 79
-1 .b7-
-1.574
-1 .53
-lool
-1.a31


-1.5o1


-I.sdo
-1 o9
-1 .bo3
-1 b3

-1 .734





-l.s33
-1 .005
-1. ,J2








-l .o3
- *OJJ





-1.s09
-1.045
-1 .030





-1 *3o3
-l.o9d
-I. 37






-1.017
-1.307
-1 .00 u
-1.364
--1.092













-1.57o
-1.uu7









-1.507
-1.507
-1.379
- 1.7(0






-1.57o




-I.07;
- 1.*00 u


HLL u L
( 24400 J)


2817. c. 740
o 17.b64 12
S817. 04 7 0 ;

2317 ... b .,

2817.o0J c.

i (7.0,77 ,
"31 7. t 7I2w-
0- 17.*0c34
81 7. tb 7o.
6 1 7 u ,t


tr1 7. 7 1 -'1
2c 17.71 1:




2~17.7js4



'2 1 7. 7 7 7L9
51 7 7,1,


Z,1 7. 7.. ;
n 17. 77.9
2- 1 7. d 7 i
>17.7C..,

2S17. bl

c 17. 16 1

2 al 1 7c.. 7
2 17 tU I


2 24. l5 ,i

2 624. 5 0-8
2d 24. 5b..b

2jE24.53 o3
ci 24.8'l327
2o24. 54-7l

2824.572 31




2'L24. 54 b 3



282 4. -t, 2 3 b
2 2 4 b ,C 5
2 24. 0. 7> 4
2o24. 5(62
2824.57231

2824.577 7J

2924.89.43o

2a24.09420


D' AG.




-1,567
-1. ''-
-l.eas






-1.7
- 1 o._.7
-1 04
-1.144



-1..15
- 1 ..
-1. .11





-1.5 14
-I L .





-1.5;72


-1.037
-1.,..

-1. 1.


-1..- 1
-1 OJ41


-1.*..l
- 1 .l77



- 1.015

-1 .o2u
-1. 0241

-1."2 7
-1.032
-1 ..30




- I '- 2 4
-1 .2.30
-1.021

-1 3* 10

-1.l24

-1.c15
-l.L 14
-1 0J











TAUL 12C(CUNT'D)


VIS UAL

UMA,. HEL JW
(244 OC0G +1)


HLL JD
(2440C00+)


2824.60375
2824.00739
2824 .t1076
2t24.61415
2824.02051

2824.02407
2824.62d3J1
2824.63210
6b24.635b1
2624.05204

2b24.6bb703
2824.06092
2824.6oS54
2824.0090C5
2824. 6733o

2824.67737
2640 b584
24C0. C5599

2U40.b0718

284L.57407
2t64.6o96d5
2o40.67484
2840 .o7S95
2o4C.od271

2840.75920
2640. 70409
264 0. 7o8C
2640.77231
284C.7761

2840. 77982
2840. 7oo35
2840.78942
2840.79261
2840.79017

2640.62419
2840.82856
2840 .8320-
2840.83654
2840.83973

2840.84294
2840.846C6
2 40 .8 8oo
2840.6o8005
264i. 8631

2840.86728
2840.87156
2b40.87470
2840.87780
2640.88712


-1.081
-1 (- 2



-I.oC1
-I.020


-1 .t20
-I.0so
-I .041
-1.019


--Io14
- 1.b2C
-1 0'+



- I. ct6
-1 .7 2
-1.542
-1.6ol

-1.0 20
-1 .0t
















--I.t374
-1.027
-1 7 53
-1.094c




-1.054
-1.00 /

-1.373
-1 ci0
-1.c013
-1.000
-1.547




-1.5537
-1.003
-1.672

-1.642

-1.671
-1.5286
-1.o09
-1.049
-1.074

-1.000
-1.630
-1.629
-1.019


-1.691
-1.047
-1.-*54
-1.540
-.bb 6/


204C.69046
2o40. 68950
2830 .b oD I

2800.012 24



L250 .o3bO)
28b. 0413 o
2600 .0074

28s36. 6213 I J


286o. o0 0
2850.00c.dJ

650 .o763I
2bo,.0o CO2
2 086.0c037
e 86(-. 0,7, 72

a doo. uo 02

2bu .09 od 8


2o5o 7 0/ j J
aC30. 7092j


2S8o.7426 a
2-oo 7.3. 57
2ob6.73891


2806 751 00
o5t6 ./o5o4
So65o.7uaoo
2dbc.7719 b
206b./7531
2850o. 7692

2800.79341
28b6.79912
b850.U0912
6)b .6b1233
2650.61607

286db.61934
2850. 620375
28D .029 01
2806 63290
2856. 3079

2 d5o .84093
2856.64694
28506.85048
2 850. 85394
285o.8b746


UM4AG .



-1.537
-1.6 1


-I (, w
-1.5917
-1.633

-i .51j6
- 1.bo2
-1 .b :
-1 .5o,

-I.601

- 1 .lo02














-I .o05
-1.58o

-1.-7a)
-1 .b 6





-I .OC

-1 .6t,


-1.I 33
1 '0.7
- 1. .59




-I .3l6
-1.sSI
-1 ..ob
- .0c3



-1 .u24
-I .b O
-1. 4

-1 633

-1.c22
-1.629
-1.021
-1 .Sbo
-1.i 83

-1 .01l
-1. 76
-1 .c.90
-1 .W9o
-1.613

-1 t 12
-1.034
- 1.o2d2
-1.003
-1.605


HEL JL)
(24400CS+)


266.8 O 97
2 830. 67207
2866.67o2s
280o k.21 7.+
,3So. tOc5Z


283t o.6) 43

2b8o.9.124
2856. u 00o

28360.31 *4
28o6.91077
2 6 u
2dbo. o 2 3q
2-73 574

2 373. 57 8.
23/ .i(;l .
2J/ 5 569c





2Z7J. 0. i
2c73.t Ia 1
2 73. ,12 ,

2873.0 3o .

2 73. 07 7 .
2 :/7 o 1 7
C/J. 64,t
2373.0.3771


26 73. Usit.1 ..
2 d173. o I o 3
2873. 134 .

1O73.7 LL'
2 8 7 3 d .) j

2373. 79O7

2873. 68Cl .
2873J.8132
2873.8140c

26 73. 81934
2 b73. 623 ,23
2673. t8227
2873.83 11
2873. 83341

2873. 43J0,
2873. 8400u
2887.77501;
2887.77909
2887. 775o0


D' AG .



1 c2 1
-I.021
-1. 12




-1.o01
-1. .13

-1. 7






-1.olo
-1 ,21




1 1 01-5


-1. 13
-1 3


-1.33,
-l..i

-1.1 i

1. 1
-1 .02o






-1. 23
- .1 .3
- I 31









- .1 1 ;'

-3.1 J
-C.71,



-0. 2l








-C. 7ti
C.7 23









0.381
-1. 7 1
-3. 1o3

C-. 152




-0. 171
-0.7. 10



-1.0386
- 1. 91
-1. 119












TABLL 12(CUNFrUJ


VISUAL


U.MA. hcL JO
(2440000+)


HEL JD
(2440000+)


2887.78929
2b67.7926o
2887.79b27
2d87.800u38
2687. 80492

28t7. 80833
2687. 2304
2887. 3489
86b7.042371
287. 4905C

bd7. 8, 29 1
27b7.B5600t
2La7. 8399
2887.a6715
2887.873buJ

2887. b679J
2887. F02U0
28o7. 9390
2E87.897t30
2895. 74406

269S. 740C4
2896.75241
295t.75659
2695. 7t0C50
2o95l.7u382

2895. 70749
289,. 7704.8
2d95.7601o
6895.78571
2895.78o90

2895. 79222
2890.790o3
2o9b. 79 402
2d9b.80301
2895. 80638

2t95.80957


-1. 09

-i.uio
-1.030
-I.c~uo
-1 013



-1.ol0



-1.o19



-I .007



-1.071

-1 7
-1 L.37U













-l.s75
-1.013




-1. 0574

-1.00-7
-1.071
-1.* 04
-1.073

-1.373

-1.507
-1.570


- 1.*40
-1.334
-1.0000
-1.00 9
-1.b79


-1.572 <903.75992 -O.b06


2895.810oo
2o4ob.e lubd2
o69b.b27b25





2895.24970
2rdb.b3Jul






2095. *cs4

289b.6.5143
2890.90477
2tl90 .aoU4o
209.C7o03J4
2d9b.b7970

290 .5o04/4
29C3.00810
29Co.o71 o
2903.oY7Iy
290.0s6110
29C3.06041
290J3.b7o3
2900.L9005
i0J3.0937,
290.09022

29C3.0o154
2.90.uJ4o0
2V3. bob92
c903.729.v-
29C3.73327

2903.735974
2903.74 1 o
2903.74402
c90J.74770o
2903.737 -1


JMAA.



-1.534
-i t57
-1.591
-1 .*72
-1 c-7


-1 .b3
-1.b3a
-1.53t
-1 .bt.9
-I.bo7


-1.715
-1.5I4

-1.143

-0.7783
- .713

- .Lt7
-,ull
- .4 7B

-0.433
-u.Jd7


-4. C 93

0 7 0 7a
-0. D.
0.07






-0.139
-0.42d


Ht"L JO
(244COCJ+;


2903. 70f- .
2903.771 I
29L3.774715
2 9C 1, 7' 1
)9C 3. 7-. .7

d 9 C 3. -"I -.,
20 1 03 1 04
29GJ.7o IJ,
2918.ubo4.1

2918 7-4

2918. 5'.2 22

291 7c L,71
2915. 11224
291 8. 71 585

6291 .71 7 J4
- 91 -. 7 31 I 4,:
- 1 .72 31



291 7. -j -
291.,R 7 ..
291 7. 7c 3
c910.7tl.
2 9 1i 8. 7u'* 1
291 7 l4cc

2918.l77.
291 7 1- -/ 1
91 d.7-/l s
291 3.791 ;7
2918.79, (S

291 .9T354l

e29 18.dI
2918 .& L,2
2 91 6. t2 2> _*


DMAG.





-C.753





-C. u-
-0 7.J







-0. 1;


- 0.












-1 .44
C 4.3t


-0.241




-1 *, J





-1.

-1. -4

-1.*24
-1 .3-v7
- 1.44C

-1 .4(
-1. '+o3
-1.3C4
-1 .1 0
- 1.52


2918.82o3a -1 32












TALL 12(CONT'D)


HtL JD
(2440000+)


2352.53,417
2350. 53 790
2352.54285
2352 .5 4 g
2352. 564U4

2352. 55 935
2352. bot369
2352.506d53
2352 &57440
2352.59u5b

2J52 60 1 b 1
2332.011dC
2Ji2.61911
2352.62444
2352.03015

2352. 6354 9
2352.64242
2J52 .6 6823
2J52. 65s'9
2.-52 cc,4J6
2352. o72C6


2362,o0832
2362.*51319
2362.51 22
2362. b2401

2Jo2. 527C7
2302. 5J30o
c 362. 53 17
2362.5391 1
2Jo2.b4 253

2302. 459.4
23o2. 549J.,
2302. 549t3
2302.55081
2362. 5001

2362. 56325
2362.5o439
23u2.5o753
2362 .560<8
2362. 57065

2362.57130
2362. 57394
2J62. 57499
2302.57928
2362.58325

2362.5642o
2362. 58471
23b2. 58541
2362. 58 843
2362.58922


UMAu.



-2.C12l
-2. 20
-1. uol
-1.901-
-1 .915
-1.oc9

-1.023
- 1.7 7
-1.703
--1.uJb
-1 293

-1.191
-0. 9 1
-0.7o3
-0.s7L
-G.a 345

-L.1 b
C.. 155

L. 71



-I .96

-1 9tj1
-1.901


- I. c I b
-l.oba







-1.04o
-17* coo

-1 .0 05


-1 .4ol
-1. 139.
-1I J I
-1.407
-I.1 *9


-1 .310

-1.2/9

-1.J-J
-1. 94
-1.2 93
-1 .223

-1.119


-1.012
-. 997
-0 .9 03
-0.903


HLUL

HIcL J-
(2440020+


2J32.99013
2Juc: .094
CJU2.595-6
23, .bUoob7


2J02.objoj
2302.U04bl
2Jbc cjuo
i230u. 000



230 .0 UI914





Loo2 071
L 342 33 1- 4

2Jo2.uOb70J




2o.o- .09o I

C 3 02 < ,021
23(IC.030741
Zc:.c.09sc:7







j2. cJ 74
2502.o900 d





230bc .0U4 C'
23J. u4c877t


230L L+9


5o2.c80J7




2JO t. 93 1
2320.o90o70
2302.0990.:

23oD.70309
230c .707t02
Jo02.70672
23- 125
2302.7100O

2302.71839
23 2 .71 9 1l
230o2. 7194
23o2.72467
2302.72Lbb^


oMAG.



-0. o2
- /7 A

-S .016


-0 .523


-J. j



-L c0
-0. 11


-O.IlC
-C .3 j



0.421
v.4 7b5

0 33

C.o .



SO.* 3
0 o
3.49 Ia
U. 9 4 4
0.977

0. 9o7





v.47o
0.962
C. 981
C .9o3


0 .6o0
3.6;41
0 .782

0. 7 Z
J *. 5L 2
C.478
0.-50
3.303

0.1 30
0 .986
0.072
-0.141
-0.141


HLIL Jj
( 2443CC0+}


2302.7 931
2302.7323
2 3 o 7 1
2362. /lL34
2 302 7J -3, )
2302. 7 -- 1


23j 2. 7', ,
2 3t 2 74 l53
c3o2.74nbOi
2362. 7-4 9V,

23o2. 7zO ul
23o2 /o4ul

J23o. 7 0>:


2,j2 7 7 cJ.
2Ju2.77^cr

23o2. 77 c


2362. 7ct, i-.
23o2. 79',

SJu3 2 / 1 7.
t302o. r, o1


2 Jo 2. 1 o'
230 2. Be. .1
23o02.ti ,
237 7.* - C.
'Jo7. SC7.
2367. 5-d4

2Jo7.51, 7z
236:7.51451
23c7.51775
2307. t51,u7
L307,b25l 3

2367. b,10
23C7. 5c 1I
2367.52 13:)
23,7. 5/02L



23o7.54 1J7
230o7. 5- b3
2o367.531 4



23o7. 53674
23u7. 54t-2;
23o7.5*042c

2307.54373
2do7.59t03
2307. 540 lJ
23o7. 54 9961
2307.5bsO91


L),; A,3





- 0* 2
-0.4 -2


-:. /I
-0.1 7








1377
-0.732









-1 7* 7
-I.77


-1. 13
-1.2'










-I ,ir 7
-1. -1,

-1.-i7
-1. 71.



-1.' 17
-1.24c


-1.*71
-1. 72
- 1. 91

-1.751
-1.*11
-1 79.;

-1.755

-1 /a
-1. 391
-1 a

-1 .51o

-1.725


- 1. 457t


-1.371
-1.355












TAbUic 1r(CUNT*D)


HEL JOL
(2440000+)


2307. 5385
2367. 5460
2307.b3791
2307. 5b871
2jo7.0o276

2J67. 634 7
2367.5o75to
2.o7.56d2 1
230 . s7 1 1 '
2307.57196
2Jo7.b719d

2iJ7. 57544
2367. 07u5-
23o7. 57/,3t
2367.58018
2367.58307

2jb7. bd3bi
2307.5t4'.2
2J67.58713
2jo7. td77L
2367, 562 B

2jo7.5 ,B30S1
c2307. 31 77
2ou7. 592o02
23j7. t>93 7
2371. b59o8o

2307.59-92
2367.60132
-lo07. 0079
23o7. 609 :b
2307. 61757

2.07. 01924
2307. 23Jo
23o7.624.658
2367. 0o922
2367.o-iJ2
2307. 0306bu

23t7. 03637
2j07. 06789
2307.04176
23b7. b42cd
2367. 474 9

2367.648,54
2j37.6b23b
2367.65300
,3o7. o5392
2367. 60033

2367.6611b
23b7. 66o4 1
2367.66729
2367.67J89
2307.67464


ULUE

Dl. AG HtL JD
(24400u0+)


-1.294
-1 .zo


-I 123
-1.217
-1. 1B2

-1. 1 74

-1.074
- C u C.3
-C.902

-0.911
- .8o61
-u. ?76
-(, .7 7
-0.o7o

-C.ob







u .526
-C."3 *4











4.73
-0. 7>9

-.0 19 i t
-0. 0 3

-U.OJ 3

SG.u 7
O.'u7:
O. 43L
0 .4 /4
0.4 74
O. 729

0 /9





0. 97
0 o.o
0 ',4o
C.). 9-s2

0.9,7



0. 963

0. *92
0. 977
0 .97o
C.9-78
0.944

0. 949

S.9v 4
0.948


2367.u7O40
3o7 .o79732
2307 .03 14
23c7.6a429
c0o7.u0952

236t.69C20
23u7.69Jt 1
23T7.69443
23J7.o9B37
23o7 .099C0



23 .7. 70,u&
T,07.7(.76
2367. 7Y i4
230o7.77J9

2387.71'34




13u7.7io97
2307.71747

2-ju7.71971
307.,72lJ41
2367/.7262
3C/ .72o1d
2-3o7.72 vj :

23j7.72 974
f367t .73 0 J
d3o 1. 7j.3 c
-3 7 .7 304
Jo37.73o74

230t7.7 34o
23o7.7 003
2367.74i/60
2367.74-343
2307.744 02

6Jb7.74714
62o7. 73022
2do7.7j73
2307.75609
2307.75773

2o7.700232
2307.70u86
2b67.7 ,5qB
2307.704b0
cJo7.70o74

2J07.760687
20o7.7097J
2367.790 17
2207. 79620
?3o7.tbG21


UMAU.




r. 903
C.96c5



0 .0


.C 43


0.54'3
G b6u
0.432
0.294
S.27-

0. 1 b
0. 1 us
u.044

0.607

-,0 .3d
- t-'. 1 'J
- .1 i


-0 .4 36

-.4bo0





-3 .7o3
-- -/0
- '.S74
-0.9s3



-1.11 ^
-1.1 24
-1 .170
-1 .* 7
-1 :d
-1. c: 72


-1.218

-1 .427
-1.44e
-1.731
-1 .7o0
-1.650


HrL J L)
(2 .40 C +


2491 5, dJ
249 1 L 70 1)
2491.571 2i
2491. t ".
2 4 91. 1 5' j


2491 .5j4 -
2 4 I I 5., 2.3

4'i .5, 7 17
L491
2491. toio

491 .6 62504
2491.61t

4491 t-1 7'.4
Z491.6 1


,,421 o,1s/

2'91. 6472 U
24' 1 c.-, :, 71

2491.D -






2'491 ,7'7

2491. coi-


- 4vI 54 4 ,
2q14 5L 72z.-


4 jj. l7615
249 3 0,4 '14

2"483. o2 4u
249'3.61371
2492.3 ol 45
2 493. aoL 4-







'493. 6 .'1 -7

Z493. o'49,.
2493.67u IJ



4 7 0 o4-;
2493.60971





2497 7.b407
249 7. 5b97 -,


-2;.37
2,,' 2 7
- 2 1
- '. 5


-2. 45
-2.cJq




-2. 1 o































- ; .
-2.222
-,..'q2































-2.'2C
-4.243
-2. 9I




























-2.-34
-2. '2
- .2-3




- 2. 291
-2.293









--2. D
-L.il0
-12.el7
-2. el d
-LI.=38


1












TAbLL 12(CUN D0)


HEL JO
(2440000+)


2497.59876
2497.602o5
2497. 0355
2497.605J4
2497.00901

2497.61038
2497.61404
2497. ol49
2497.62451
2497.62084

2497.6 3 40
2497.03279
2497.03 71
2497.63800
2497.64169

2497.04291
2497.64720
2497.04992
2497. 6189
2497.60570

2497.65702
2497.b0074
2497.00o144
2497.6 21 4
2497.6o842

2497.6o974
2497.67373
2497.0751 2
2497.67961
2497.6d 108

2497.6t042
2497.6duD9
2497.69100
2497. 092t2
2497.o9o04

2497.69769
2497.70180
2497.703C7
2497.7389j
2497.71524

2497.71051
2497.72154
2497.72292
2497.73923
2497.73969

2497.74057
2497.74127
2497.74190
2497.742o6
2497.74336


DMAG,.



-2.<97

-2. o0
-2.. 91
-c.277

-2 ul
-2.291

-2. 91
-2.290


- 2A* A96




-2. c 92
-2.290

-2.230





-2. -iL
-L.CU3


-2.313
-2. J9





-.*'77




-c. 10


-.2 C4
-2,.377







-2. -91

-2. 20
-2. 305
-2.290






2.352




-2. o4

-L.279
-2. 77


UL UE

HEL JO
( 440000+)


2497.74969
<497.7b529
2497.7z)09-
2497.7s777
249/.7J7b7t

2497.707L49
2,9?7 ./730
2497.77090
2497.77oLq
2t S'.7/Siy

;497.76oa71
24v7/ .7o4 10
2497.78 6l7

2497 .-tbOtO 4
z4v7.78671
2497.79387

497 .7 0 14
497 79- z,

2497. 0C 9O
2497.0 4/1<

E497 .bO ,JIc
2497.w,0,37
2497.01471
2497.01093
2497. o,2. c,4

S4 9 7 24 7 o
.497.629CC
2497 .b3) u
2497.5420oo
247" .84072
2497.647 I0
2 49z7 64? 7 9
2497 .U 0474
2497 .bD2Lo
2497. bt)- 76
2497.8t002)

2497.85909
2497 .070
2497 .678
2497.4;77 74
2497.00277

2497 .83 99
2497.o997 ?
2497 .9 J82
24 97.07 46
2497.90840

2497.911o Q
2497.91o10
2507.57930)
2507 .o66t8
2507.6b077


OMAG.



-2.290
-2.tc90
-2.289
-2.271
-c.271

-2.200

-2 .c1- o
-2.279
-2.2u7o

- .c/O
-2.2714
-2.244
-2.243
-2 .2 54

-2.255
-2 .c7
-2 .20o7
-2L.71
-2, .< 71

-2.2 .9
-.2L63
-2 2 3
-2.2o4


-2-.o4
-<-.70
-2.201
-2.270
-2.7 7

-2.272
-2.274
-2 .290
-2.22
-2.2783

-2.271
-2.293
-2.29 1
-2 .38
-2. 2

-2.2/8
-2.245
-2.24 -J
-2.271
-2.272

-2.177

--4.246
-2.2379

-2.21b


IHEL J ,
(24400CO+)


2507.oC nu7
c'bO/7. ul ;J4 i
2507.102C.1
'507.o127 .
2o07. bt 1 5
2507.01529

'507.01 91;
2507. ol '.l7;

2507. t.-_ I,
2507. 6~ s1






2507. c,-



2 7. ok.J,

2507. 7.12

251j7.ou I1o
c5O7. 7ol 7
u,3.7. 6o'.
2 ?C 7. 0, 1I






2007. 71 b .
Ih7. 7 1



25u07. 7oc 77
2.:> 0 7. 7(: I I



2o09. 2,01l
2 o09. oo,
2 509. 7. '-

2509.on'1 4


2- 09. c ,o
25 09. 7t 6-,

2009.501uu

2 5 0 9 5 72
2509.5(3.72
2309.57o071

25G09. 5703J
2509. 05',3 I

2509.51261
2 509. 5 2L
25G9.5ol23
2509.58422


UMAG .



-2.21 -

-2.514

2.22


-2.2J3




-2.23:
- 2. S. -



-2. /'
-2.252





-2. -A4



-2. 14
-2.-1




-2. -12




-2. 71
-2.15
-2.122


-2.125
-2..17



-2. 151

-2. 22
-2. 125
-2. 124
-2.1317

-2. 1 23
-2. 104

-2. 122
-2. 135






-2. 125
-2.1 j
-2.1Jl










-2. '., 1





-42-


TAOLL 12(CUNT'D)


HtL JO
( 24400000+


2509.58492
2509.58839
2509.5d3909
2;09. 39135
2509.59223

2509.5946 u
2b09. b9oD4
2509.b9779
2309. b984u
2bo9.6UObb

2509.00125
2009.0, 41 5
2309.60469

2b09.bC8o01

2309.61003
2o09.o 132
2009. 01444
2509.ol014
2509. 61723

20C9.01 791
2309.u2127
2539. 02 t92
2 -9.0/451
2'CG9. 2521

2:5(.9 o270 J
2509. 6b33
2509.63199
2u09 63200
25C9. t3857

2509.63927
2509.04171
2509.64239
2509.04553
2=09. 64o23

2509.64t 47
2509.04917
2b C9.0 51 1
2509.05229
2509.6O45b

2509.65b25
2309. 0370
2309.ood 19
2509. 0t 1 54
2509. 020 1

2b09.oo 15
2509.00 585
2509.60>792
2509.ob6db
2509.07171


BLUE

DMAG. HEL JO
(2440C00+)


-2. 74



-2.0 9

-I .9D7
-2. 0o9


-i.017
-2.0Cb
-c.LUO

-1. 995
-1.937
-1.9 5
-1.932
--1.94+

-1. 1il
-1.910
-1.090
--1. o7b
- I .o7

-1.U03
-1.0* b
-l.oC5
-1.7ri
-1.777
-1 .7ob


-1.73o
-1.007
-1.307



-1.504
-1.550
--1.bb5
-.Ibu4
- .1 40b

-1 .470
-1.4 73
-1.4 1d
-1 .-4L6
-1-.J7


-1.329
-1.3c,3

-1.240

-1 .
- 1 70
-1.117
-1. 112
-1.053


2309.07240
5C9.o7450
,b09.o7b 20
2009,a779
2309.07d49

2509 Yutl4J
2C09.o82 13
25C 9.09255

2509.,u904

25U9. 70140
2b09.70-1G
25 09.70279
2509.703b9
2509.710 1

2009.71143
2o09.71453
23,9.71912
2509.7.224
209 .723ooo0

2009.72oZ3
209.731 74
2OC9.7332d
ztoC. 739C9
25C9.7427-

2309 74 /
205 .75 15
c509. 759b4
2D09.772o0
25b10.olob

2310.01009
25 1 .olu90
2ol1.u1 20-4
2518 .cr U 03
51b0.o2392
231 l. 03090

25b1t.03926
2510.b4,7
2blo.L065rP
2510.05499
2316.06010

2bl30.ob31
2:1d.075<8S
251b.71300
2d18./2003
2516.74603

231 8.7DB80
251o.7 007
310.,7o00o
2513.77122
231 6.77395


DMAG.



-1 .C37
-0.997
-U.96 O
-3 .034
-C.oad
-0. 003

-0.79o4
-0.773
-C.77j
-3.39-j

-O.C 79

-C.Cb3
-0 .Co d
0.CJ
C. 1 7
0. b1

0.335
G.bu2Z
I0./09

1.C 7d





i 120
1.120
1.C- lo
1 20







.1 .7 9
1.C72

1. 127

-2.159
-d13G



-2.1




-2.203
-2.197
-c.179

-2.1 ?S
-2.2 3
-2.177
-2 .10 6
-2.174

-2.lob
-2.219
-2.199
-2.2Co
-2.222


HI-L JOL
(2440Ov +1


23 1 .74ri 4
251 7 -4 1j
2b1 d. 7-9J




251 b I I ,

23lc9.. 8 -?
2514.&170i
r l .'3. 22//
231'+. 3o'.
25 d. ou
23lo. 8J 23-9


251o.t3o 4
2514 7all4
251 3. 4 u.1 +
251 .8Q 9

25Id.8/52o


231-' 3' J:)t
24. 1G'-.2o


2335. 4 9379
2524. 60jld
252 4.7 77i
2524+. 7. J3b
2,,24 .'->; 2.-.j
S32.+. o u.. ,t
2324o,0. u0/



2524. ou-.
:324. u4 7-,J
2 3 4 o5 .' o

2524.0o3' ..
252 .05 1 d,'
. b24. ubda) *.
2524. 07:/ 1 7


2524.71 d2j
2524.71941
2024. 7210_
252. 721, ,

2524 73215
2 4 .7 995
2524.74414
2524. 7-0ou3
2324. 7b33

2525.5o 79
252b. 5b 7: ,
23 3 3?b71 4
2 25.5-> 11
2520. 523u


DAAG.



-2. 1 9
-2.174
-2. 1 -3

-2. 5


-2.217
-2.234




-2. 205




-2. -77
-2. 7o
-21. (7
-1.- 5?
-1. .;
-1.'"7





-. 7





-1.21
- 4


C. ': 2
C. -io2

C. 5o\




1.2/1
1.-c4




1.127
1 71







-2.1 4
-2.170

-2.2 1
-2L.19












1AULL 12(LLNT'D)


ULUE

ui4A. HIEL Ju
(2440UC0 )


hEL JO
(2440000+)


2025.59203
2b5b.59b72
22525. 001
2o25.ti 305
2525.00099

252 .61 275
2oe5. 6310 2
2523. b63Jl
02>2. 63988
32=o. o4 334

2525.64708
2b52 t;olu7
20325.6 uo7
235.6021 6
2525.oo0b3

z52j.60679
2o25. o7,35
2525.07705
tI-2b.6d779
2t25.o914 1

252b. 9532
2o2o.7C73D
2525.71175
2L2. 71 : 1
t52b.72319

225b. 72034
2t>b 72980
252. 734 74
2525.74299
22525.79587

2525. 3158
2 b.81 03 -
,2 25.61741
2525.82115
2525.8250o

2525.83237
2b25.63olC
2525.84049
2o42. 03104
2042.63964

2C42.64518
2042.ob089
2042. 5630
2642.6604o
2642.6b6706

2042.67185
2642.67804
2642.68387
2042.o9006
2642.70155


-2.171
-2.113
-2. 11 1
-2.154
-2.172

-2.127
-a.177
- 173
-2.109
-a. jo
-t .l2 d

-2.17'9
-2.173
-2.14o
-2.17o


-a.10/
-.104
- *.19
-a. 12)


-2.-3

-2. 100
- 141
-2.174

-2.1 o-

-2 17



-2. 194

-c.c I 7
-2.207
-2.-17
-a. a20

-2.229

-2. 1 V4


-2. -4L
-2 7
C* 205


-2.214


-c. 223
-2.251
-c.228
-2.232)


204a.70643
2042.71042
2o4-.714bo
2u4.7 1012
2o42.724qo

2042.72o7C
2o42.732 19J
't42. 14373
So42. 4901
2u4 .7b531

o264 .739.51
2o42 7/c3C2
20u2.7 .002
2u42.773C0
2042.77710

2b4c.7607.j

20 42.7* 7
2o4a.7 873


20'4t .6029'.
2642 0'.52
2 t4 2 c 1 9 c)4
2042.0L, c, ?


204a0007 '.
2u4(Luo4Lo

2o4 .tc7177
L046. 0994
2046.70oC9
2o4o.71009

2o40. 71457
Lb40.72570
ao46.7 /bo0
204o.7 l171

2o4o.70u30
2o06.77140
2040.77744
20o .781J23
2040.76810

2 04o. 785 9
bo640.79011
2 4L .793)49
co4c.bOjI 1
,o4 t,.t007 7

204u. t07 d
26460.ol1 7
204o.dlo0
2o40.82042
2o46.62424


jMAo.



-2.231

-c 229
-2.226
-2 .21 c

-2.223
-2.22d
-2. 1 f9
-2.1J5
-2 1 So


-2.140





-2.1 12

-a. 122
-2 1 24
-2 4
-2.242



-Z.2 2


-2. 2-3

-- .17d
-2.07
-2. o 7

-2.*04C
-I2.C74

-1 0o97

-1.400

-1.234

-1.115

-I.I01

-t.o13
-U.7Ca
-0.572

-0.407
-217
-0.005
0.214
0.40o


HtL J D
(24400COt)


2o46. 8, 2 04 8331
2o0c.8oo1%
2c4o. 0'0- .3 I


2o4u. 'oc.27
u2 c4 8,- 1 2 7

2640. 8ol 7

2040..87417

2' b t7 o 7.,





2o503. '^31 ,
o53.731 o
2650.0007 I
,o z0.71 -
2. j00.71 j

20 0 7 19
> .5.5, 7 -; .


?o3 7 0. 7/1

02 0. 76. 1.?
2 t.O. 7 o602

2 ,s0.754'


'i) 3 7 s 4 4*
2o53. 7o80:7
2052). 7817o,
20.O 81 .037

2o50. ol 3o

26bC 825t7
2t,5 C. 85a;.


2o5C. 832.o3

650. &4zq,4
2650. 84Ubo
2o74. 71o74
ao74. 720C

2o74.72014
2u74.7317-
2674.73584
2074.73j7,;
20/4.7462z,


DMAG .



C.o 0
0. 70
0.871
C. oo
C H- o
C .,1 4



C. 14





-2.234


-2.5' )


-2. 21 7






- 2 1i
2. 7




-2.2.2,
-c .2' i

-2.27.


-2.a -4


-2.2',"


-2.2 <2
-2.2 20


-2.212



-2. 2o

-2.2 7

-2. 2i
-2.212

-2. 1 2
-2. 1o
-2. L32


-2.215
-2. Ic

-2. 1 '4
-2. 147



-2.147












TABLE 12(CUNT'D)


HEL JO
(244C0000+)


2674.75321
2674.761 19
2674. 76o70
2o74. 771ol
2674.77784

2674. 7493
2674. 7903J
2074. 795s23
2074.8023J
2674.80 76 1

2b74. dl 1b
2o74.816b04
2o74. 82290
2074. 82570
2o74.8 992

2btl o43C
2uol.o9176
2c.61. 09 1 9
20c61.69918
2o1l .732;

2081. 703J7
2601.7C u*
261. 71 357
2681.71392
2o81.72035

20ol .y821o
26o1 3072
2o81.8J4o0
2986.bl604
269d.52430

209d. 52840
2698. t3332
2698. b3l1 S
2o96.54304
2698.54790

269b.55207
2090. ooO07
2698 .45u5S
2696.57396
2098.57812

2696.56228
209d. a5810
2693.59029
2696.59617
2o 98. 0080

2098.60554
2698.ol07b
209o.61631
2698.62063
2c98. 62604


OMAG. HLL JD
(2440310t)


U IAG .




-2. 14

-2.173
- 141

-. 112
-d 111

-2.122


-2. 1 1
-2. 1
-2.1u4
-c. 117
-2.111


-1 lrt-
-1..631
-I.va
-1 .Vo





-c.uoJ
-0.09

0.3ujo
0.0003
C. 90o
-1.902

-1. *.
-1.937
-1. 987
-1 .Y9
-1.989

--1.9o6
-1.996
-2.0l3


-. v44
-2. 03
-2. C27
-1.90'
-2.024

-2.045
-2.u70

-2. '.9
-2.040


JLUc

HLL JL)
(24400C0)+)


2u9b.0~1630
2olo.oulol1
209,. .0.81

2o9b .c4758
SUYb.00 1 74


0o95.00,3,j
eoLo .coD 7')
2o90 .co9 a7-

2098.0934
2 o't .oy 9d64
2u9i. 7027/
9u'2 .7Lou94
209b. 71250
2o96..71 bo

)o9o. 72292
2r -)Jo YJ j
o209 .73j00
2o96. 7 4723


S09 .e 7,2 9
o- e C./161
2o90 .7u737
2oYo.71 itL


2o9 .7o /'=b
2,o 9u8 7/ eq0 2


uo090 79.3
2u ,o.I ( u I*



701.53914
i701 .534173
271.J5j48
z IC C Zia.t 4

e701.35986
701 .to4 oC
2701 .o05
2731 .5b074
2701. 69 S

,701 .093 7
2701 9760
2701.oO35o
2701.u1J22
c 7UI .0Il30

ci701 .ulbO
2701 .0160
27C7.o1uold
d/07.u0351
2707.03 -4
2707.u4223


-2 .C o


-2.Co3
-d. C o9

-2 .0-O
-c.076
-2.0 0

-2. 009

- 1. 73
-.1 12
-2 .0 oo
-2.C9S
-2.079

-2.12C

-c .1 07
-C. 134
-d 122

- .i07
-2.120

-2. 12C7
-d .11
-2, 17
-c.126

-2.1I 4
-.127
-2.1 3J
-1. 132
-2.1 10

-2.11

-2 1 764
-2.1 7o
-2. 1


-2.14C

-1., 36
-1 .930

-1 040

-1.723
-1 .00
-1 536

-1 .404
-2.2460
- d .220
-2.243
-2.244


270J 7. 04 o
2707.,)2o 2
c 707. uou lI
2707.o001 o
2707. ou00

2707.6631,l
2 707.67/ 03
_707. u b 1-
'70C7. 0r07j7
2' 7. 7. 6 1

270 7. 6so 7?
2797. 0v777

2707. 70 19
2707.7102C,
707.71-'j,
2707.719-1

27)7.7 14 :
2737. 7--2 .
-7 7 .7-r7 7
S1/0 7. 74 1 i
2707.74ous ?
27 7 L. 7 4 7
77 .7 7: I
2707. 7 3 .11.
-/07 7-o I /

2707. 7:.
-07.7(L- .

27C 7. 77 o-
.707.7 2-
7; 7. 7" / :/
27 7. 7 /.
7 7.7 10-
27 7 7;


2707.. 123
2707.818ol
27C7.E2 .cl
2737. 1320

2707. 6, '3
- /07 8'-. 4 ,
270 /. Hfio 9
27C 7. 0 bl
S707. 80 444

2707.8 93-.
2707.874 -i
2707.67- 0.
2707.b340 o
2707.89,37

2707.9: 123
.707.91C0
27u7.9144'-
270.9.61 37
2709.61bd73


-2.1 o
-2.172
-2. lo
-2. 2C6
-2.177


-L.217
-2. c
-2.22


-2 3 1
-2. 51

-2.2-:
-2. 2 /2






-2. 3,
-2.12d














-2. 1 C6
-<;. 2









-2.210



-2. 12
-2.171

-2. 1.



-2 1'C
-2.20
-2.217





-2 2 1 1
-2. 1 7


-2.171
-21.70
-2.203

-2.206

-2-105

-2. 14 1
-2.1 t 1
-2.271
-2.217


-2.112
-2.171
-2.210












TABLE I (LUNT'D)


E L UD

3DmA4. HEL JU
(244Q0000+)


HEL JU
(244C00Gt)


2709.62280
2709.627CO
2709. 63344
2709.637uu
2709.64272

27C0 .64743
27C0.651b2
27C 4.Co5t933
2709.6b124
2709. obtlo

2709.67632
2709. 6J05o
2709.68408
2709.68899
2'709.6)477

2709.69826
2709.7095u
2709.71404
27C9.71 b I
2709.72247

2709. 12049
270,.7J167
27"9.7-t 15
2709. 7 55 u
2709. 74967

2709.75340
27C9.77790
2709.7o Jod
2709.76772
2709.78790

2709.79288
27C9.8006-i
2612.52253
2812.52719
2812.53111

2612. 3829
2812.54160
2812.54594
2812.55005
2612.55541

2812.! 6247
2812.56586
2812.56475
2812.27404
2812.58059

2812.50518
2812.58977
2812.59366
2812.59770
2812. 61141


-2.197
-2.21,3



-2.212
-. 2 1

-2.21
-c.1 9

-2.172

-2.195
-2. 176
- .172




-2. i73
-2.1/9
-- bb
-2. 172

-2.1/7
-2.132
-c.*194
-2.199
-c.1 06

-2.I196
-2.193

-2.165
-2 .12 4

-2. 1 05
-2.17
-2.2o7

-2.204

-e-.204
-2.2oI
-2. 2o15
-2.2bb
-2.255

-2.256


-.248
-2.Z43

-2.235
-2.232
-2.231

-2.39,
LJ1L~f
e-i 91


2812.62 9/
2812 .6o79u
2al2.o31o 6
L6 12 3654I
2o12 .04217

2812.04584
2812.04930
2812 .tb5a9





o012.1721 1
2d12.07578
2612.0 5 98
2817.51o33

2017 2 ,52

2617 .2051
2817.5 313
2817. -964

2617.543o0
2817. 472'9
2317.D;-o33
817.,2297
<617.35993

2817.s3374
2817. 56791
2o17.I 7172
d817. :783
2817.oe219

2817.30583
2d17.689o0
2817.05978d
2617.o0089
8d17.u0488

2817. 00837
4817.ul218
2817.o1377
2817.o2039
2dl7.b2410

2817.o2782
2817.ou290
2817..3746
8l7.04123
2817.04793

2817.00222
2617.bo 531
2817.ob392
2617.0oD28o
2817.o7232


OMAG.



-2.235


-c.234
- .227

-Z ZL_0
-2.217
-e.22a
-2. 29
-2.c2o

-c.21 /
-2. 21 1
-2.221

-,-. 42 7








2-c.7 1
-2.2o
-2.243






- .23




-c .C40
-2.2b4





-2 ,3-)
-2 .2 *
-2.271
-2.24


-2 .40
- c.4 2
-2.242
-2.247

-2.2392
22 10 36
-2.224

-2 .228
-2.243
-2.219
-2.242
-2.236
-2.2024
-2.207


HEL JU
(24400C0+,


281 7. 77u0
2617. 60o4J
281 7. b7o0
2817. b92to>
2u017.7'}17

2817. 7C212
2a17.71 1-
31 7. 7e, a
2817.7 7J9
2c17.7335-

2817. 7i du
S1 7.7u,; 2,
2517. 7,495

2317.77u--j

2 1 7. 7o 1

-a17.7u-17

2o17. 7-37-<
2017.?4 ,
L i o,7 l :.. 2
232174. 911
231 1 4. 5.9
c 7dI 7.'1j7.





2824.51 .-2
2 24. Z <' 1 S -j


2324.5 937

2824.+.507
2824 .53j1 7


2324.53277
2824 .5- 001 7





2024.57251
2824.5 0 43
2624. 5j432





2824. 5bo2c
2824 .5 902
2 o24. 594


2,24. 56J72,

2824.6 73S
2824. 507'3


2824.011 o
2624.5901412


2824.62051
2624. 640 7
2d24.02831


O1 AG.



-2.201

-2 22
-2 .2 7
-2. 17

-2.1 7.
-2. 215
-2. 15

-2.217



-2. t41 ,
-2.2 2
-2..1'f

-2.219


- 2. 1 u




-2.1 10

-2. 2cc



-2.21'


-2.221

-2. 2:


-2. 54
-2.227


-2. 242
-2.24 :

-2.2342

-2. 24 1
-2.22u

-2.210
-2.2257
-2.210
-2.233








-2. 227
-2.242
-2.244






--2.26.3












TABLE 12(CONT'U)


BLUL

LOiAo. hEL JU
(2440OOO+)


HLL JU
(2440000+)


28-4. 6321
2824. 63Db1
2824.652304
2824.65703
2624.0 o 9Oc

2824.00b54

2624 .o 7336
2824. 7 73 7
284 C,.b 84

2840. 55 9t ,
2840.bo334
2 64 6o 7 1
284C. b57+ 7
2~4C. 7484

264C .7895
6 40. 68271
2b40. 7uJ00
"84,. 77231
2o40.7761 0

2640.779i4,
2640 .7o 3
2b40.7d942
2840.79216
2e40. 79617

26d4. 02419
2 84. b2 35o
2840C.3205
2 40*.830o54
2E40. 3 6973

2840 b4c4
2040.4 4 Co
2840 8buoIJ
2 4C c. boO O'.
2840.68316t

,640. 86726
2840 .871 b
2b40.87470
2 E4 C. 67.70
2t40. o7780
2840.83712

2840. t8940
2840.8 350

26bo. uo124



285.642281 3
28650.63520
280O6. 04133
285o.04874


-2.2602
-c.216
-<.229






-2.249
-d.23d
-6.c1.C
-4.205







-2. 1 o7
-1 2'.9


-2.1o7
-.243







-2. 21
-2. t42
-. 1 74
-.-.lJ3
-2. 1 7

- 2. 4- J

-2. Icc


-2.127
-2. 1 9C

--.190

-. 130 l
- .* 90
- c.l90
-c.107
-6. I
-2.1331
-2. 135

-2.14E
-2.121
-2.142
-*.141I
-21c

-2.10O

-t.1 /4
-2.173



-d.2CJ
-2. 100
-2.-07
-c. 1'


UMAG. HEL JJ
(24400CO+J


28bo. b21u
2850.003C87

2d6u.ooul 3



adbu.t7838
2 d50 5 >c20
2ob.obZ574
2Bao.o97tbd
2a a. 7 0009



2d5o.70o23
2abo.72520

20.6.73o41
2tao.73407

2ou .703 t3

2o50.7b5 c3

2835o.* 7o04
,a3047343/oi
2do6. /ooo
203b.771950
2o5o.77331



20 0 .7 id 1
265o.769b 5
cbbo.479341

28o0.oCS I



2.-So .014c4

2050.62901
2650.63290
2b30. i3o79

2bbo .04093
85b.8 64o94
28b0.t50 4d
2b5bb.b394
2dbo,07 4o

2800.00097
2d6o.67207
26bb .b7828
2650.081 74
2 d5 .ob 20

2uo0.088672
26d50b .693
2o50.o97o0
285b.90124
2o5 .90i000


-2 14

-2. 1 3

-2.193
-241 ,


-2. 1 60
-2., 1 b
-2,-51
-e. 104

-2.19)
-2.1 1 '4
-2.147
-2. 104

-2.1024


-2 1 o2
- *-, 30


-2. 2
-2. 1 4

-2.144

-2.221


-2 .-4t
-2.2)33
- 164
-2.1 90

-2.2 Id
-2.175
-- 1 91
-2 .1 i4
-Z.21 1

-2.1 2
-2.21]4
-2.22J
-2. 196
-2.221

-2.222
-2.213
-2.215
-2.214

-2.217
-2.228
-2.209
-2. 1 90
-2.213J


2 b 91 7
250. 92 01
S60u. 9234.)
2 873. 574'. I
2873. 57340

2373.578 13
2073. b81 j4

2573.52,.-7
2 73 .o3 V 3


2 73. o'- 4
2075.0o3244
2 o7 3. t> ,3 5 1 .5
2673.ol 03
2873.Ui o1C


2675. c 31, ':.
2 373.00 1


2, 73. 7: 1 32
2673. o;4 S13

2 d7 3 7 ,
2673.7 5$ 3
2673. 7, 4

2o73. 7vd 2
2873. 3; 33,3
28 7J. rtitbO
2u73.d10C2
2 3/3.bl o

2873. 81 34
2n73.82? 3 %
2873.62o27
28/3.931 1,
2873. 8j li

2673. 43 J3
287. 84 03
28d7.771b4
28d7. 77500
2687.779C9

2867. 7d8oc
2887. 7692
2867. 7926b
2887.790o27
28 7. 830 oo

2887. 04592
2887. 80 o3
2887.d2304
2887.6348)
2887. 49J-


UMAG.



-2. -1
-2.220
-2. 21
-2.216


-I -' b 7

-1 .*1
-1.717


-1. 7,'




-1. 2
-1. 43


- 1 .34




-1.,
-1.2 4










-.313
- 1 9 3
-0. 3
C.vi

-C. 1, 7
-0.332














-2. 106
-0. 71I
-0. d30
-1 1 0

-1 .207
-1 .31J
-2.146
-2.139
-2.1 0

-2. 19
-2. 72
-2.193
-2.217
-2. 17J

-2. *1&
-2. 2 .1
-2.240
-2.212
-2. 179












[AuLE 1d(CUNT'D)


bLUE


HEL JL)
(2440000+)


2b87.d5291
2687.85bobC
2887.85999
2887.86715
2887. d70 b

28d7.87393
2887. 6283j
28b7.69390
2887.89/U6
20695. 74071

2t95. 7440B
2895.74904
289o. 75241
2695.75o59
2895.7bCbo

28d9b. 73b52
2695. 707,2.
2895.707449
2895.7704
269b5.7801
2895.7db71

2b95.78d9C
2095.79222
2o95.79o36
209o. 799c2



2895.80957
2895 8160ob
2895.8d1882
2b95.82752

2895.j3081
209o. 83od2
295. 84293
2895.40o22


DMAG.



-2.21 1
-2. c 4


-2. 1

-2.217
-2.235
-2. 1 d
-,c. cc


-2.10o9
-2. I 07
-2.1 o7
-2. loo
-2. 1o9

-2.0G9
-2. 175
-2.100
-2. 20o


-c. 1 97
- C. CJ4
-2.21o
-2.21o
-2.*20&

-2. Z02
-2.1 "-7

-2 .1 9
-2.196

-2. Li 2
-2.191
-2.1 40
-2. 1


HEL JJ
(2440000+)


209t.6b4vol
2690.05276
2895..50b34
2895.85477

2895.boou 3
2dYb.o7oo3'
2895.b7v73
29 .50474I
2900.0bu1 1

290 .5>7130
2903.5743j-
290J.bblI10
293J. 5451
cY903.b 57o

':90o .5s to
290.L;9z7-,
2900 .5b902
290V .bClo4
290o.odo .


400.L 709
290o.73o27
29C0.73597
290 .741 o

290S. 74o42
<9 i .74776
2903.750701
2903. 7'v92
2907 > 7ba

290-3.77159
2903.77c7o
e900,.71 09
2903.79o07


OMAG,.



-2.1 50

-1.i80
-2. 190
-2. 19

-2.196
-2 1 /2
-c.1 /2
-1 .2 t 7
-1.2 71
-1.210

-I 1 57
-1.137
-0."59

-. 7 5
-v.705

-' .1/4
-0.bo
-3.J31
-0 .c4
-C0.07

C .C7j
C. bO
G.025
-0 .97
-0.C,58

-0.481
-C.bd
-0.693

-1 .16

-1 2 03u3
-1.263
-1.366
-1 .o 9


HEL Ji)
(24400L j +)


2903. d63 J2

29 1d.b C-


2914 fc3 sc.

j 910 71 J13
2vl 6. 7. 4Li
291 8. 71 31o

2-:1 0./ 7 2 -u4C
2910.7;1224
291 .71jd 5


2918. 78-3s
2 1 7.;

291 3.7-939

2910. 70-30
2 91 :7)*c
2 91 cl .7 7 ,

S918. 7740,

- 18.77 1n -

2 9 1 7o' 1 /
291. 1 7c, 1 .



2913. 8 o7
291 .8 7 -
2918.bi5 -
291 d. 8, 1
291 b. 6 22~

91U.C^32
2918. B2 3L1
291 63o0.7


OMA .



-2.,J
0.900
0.935
L. 934
0.971

- .,. c2 0



-0.9:2

- 0. 99
-1 .C.;
-1 0>. 0
-1. 17
-1 39 1


-1. 01
-1.0 2
- 1 73 :

-1. 97
-1.-92

- 1. 9c4




-2. 1 3
-0.137
-2.133
-2. 115

-':. I 16
-2.13s
-2. 1io3





-48-


TABLL 12(CfNT'D)


HEL JD
(2440000+)


2491.55773
2491.50 94'
2491.57170
2491.58149
2491.58298

2491.59223
2491.59323
2491.59772
2491.59B74
2491.604C2

2491.o05b19
2491.611903
2491.01297
2491 .o13 3
2491.01 76

2491.61891
2491. 62339
2491 o2t4
2491.04C03
2491 04 15

2491.64o5C
2491. 6474i
24l1.4 o 980
2491.ooC03i
2491 .oo70

2491. t0413
2491 .0oo07
2491.6o0994
2491.67233
2491.67687

2491.67847
2491.68520
2491.6d729
249. 54326
2493.54473

2493.55173
2493.55320
2493.55849
2493.50051
2493.59708

2493.59933
2493.60451
2493.00000
2493.01329
2493.6 o 03

2493.61925
2493.62104
2403.o3411
2493.63598
2493. 4343


)MAG. HEL JyO
S24403C. O+)


D5AG.



-2.0o9
-2.044
-2.6oo5
-2.olb

-2.ob1
-2.00
-2. 00
- .obj7


-2.o63
-2. 04
-2.0073

-2. 648


-2.b o
-2.045

-2.u 5


-2.o d


-2 704


-C.oL7
-2.039
-2.021
-2. o5l



-L.602
-2.097
L. 6. 1

-2.uo4
-2.764
-c .762

-2. 2A
- 002
-2. o28
-2.ol0
-2.o40

-6.842
-2.o21
-2. o09
-2.600
-2./9o

-2.790
-2.29
-2.7 Y
-C .6 10
-2.73o


JV

HLL JJ
(2440003O +


2490.00oi30
24', 0.ood 9
2493.0o99u
2493.07090


2o 1 .59/ 4
2oIlo.ou7 1
2 1 b 01 9712
Zs1b.olb924
Zl b. 204 q

251 .03 1 bb
bl b. 49b08
251 .ob'*61
2o1 ob.Lboob
2ioo.b6459

5l1 0.6o09o2


51 b.7 1 J
col b. 7o4 I
018.73041

2b51 .7o042
51lb.70u.t


o1 8.7799 .

2i16.77(644
c I 6. 7- -4 ,
S10 .7 901 4
6o1 0. ao I0 3
251d .607 19

2018. dl93n
2 18.el0o53
010 l.631o 1
20)l8.b54 0
2o Jbo.0024

2b l o.oooo9
2&18.bo7o3
2516. o 841
21 6d.sbo91
2524 .851B

224 .o9u2o2
2t24 .bC351
2524.03742
524 12 46
2a24 ..13C

2024.01b 99
2524.u20j O
2024.602709
2524.6272o
2o24.uj03a0


- .831
-2.773
-2.715
-c .7uO
-2o .00

-2.712

-c 7.4
-c.7 30
-2.7C4


-2c.8 U
-2.b00
-2.621
-c.7 I

-2.625
-..774
-c.70 o
-c .bd77
-2.703

-2.767
-2.796
-2.7t
-2 700*


-2.733
--.72-)
-c 7 o

-2 .7 U
-2.74,3
-c.o34

- 619
-2 .40
-2.u31
- 627


-2.861
-2.095
-*.769
-2.790
-2. 9u

-2.122
-2.176
-2.C32
-1.977
-2.030

-1.779
-1 .9o
-1 .3j
-1 .287
-1.2 C00


654. o000
2524.ou 0b/
2b24 o 174
2 524. 52 74
23 4.0527
2 L324 65c- 2 7
2024.6,00C
S2524. o0(c,

524. 6-/72L
2324 .o/7d

2524 oc. 2
2024. 73 00
2524. 74.0
2524.7- '2 i
2524. 70431

2525.5- 700
o25. 571 <
2025.S;e 7u
2ols.ot u70
-525. b d c



5025. 04,

323.W17to,

32d5.07

2 23 0.0 1-,
625.7C 77-i
1525. 716o7

6520. 72o0.
5225. 7,00

52b. 7jo'
25>25. 76, 1J .

525. 7/'~9
2525.71,jo7



tb25. 780 1
225. 87(.9 c
525 b. 1566
2525.7S91bc


2525.8-016
2025. 8C990o
2525.617 (1

2525. 6215

2 525. 6r' 6 I
2525. t 632 4

2525.6400C

2525.S45oCl
262b..5l7'+o

2525. a8,433
2525.8 /12>


DMAG.



-1.34
-1.173

-C. 30
97 3

-0.o5c
-0. Jo
0. -07
3.2b

S** 4

1.198

1.230
1.141
1.243

-2 1
-2.6:1
-2.657
-2. 7o5
-2.7 5
-2. 0C

-2.773


-2. 74



-2. >0: *
-2.7o

-2.771
-2. 744









-2.72C
-2.774
-6. 71
-2.,154
-2. i 4

-2. 372
-2.972
-2.72C
-2.77
-2. 1 3

-2. .53
-2. 6 b
-2.835




-2.511
- 2.744
-2.7E9
-2.;55






-49-


TAoLE 12(LLNT'D)


HEL JD
(2440000+)


2c42.63164
2642. 3929
e0o42 64475
2o42.65144
2642 .656o7

26,2. 06086
204 z.6732
2042.67148
2642.67764
2642.d8452

2642.69576
2642. 70 15
2642.70686
2642.71 084
2642.71 416

2642.71647
2042.724 6
2642.72"32
2o42.7J 18o
2042.744035

2642.75021
2642. 753 46
2042.73910
2o42.7o2o8
2042. 7oo2

2042.77270
2042.77771
2642.73105
2642.76442
2642.78773

2642.79115
2o42.79716
2042 .797o3
2042. 0261
2642.80937

2642.81989
2642.82515
2642.63236
2642.d04933
2u42.80oo54

2642.87132
2646.70028
2046.70559
2t40.70981
2646.71407

2646.72117
2646.75348
2c46.7o24
2646.760/72
2646.7711


DMAG.



-c.797
-2.7b9
-2.791
-2. C01

-2.77b

-2.767





-,.ol I
-2.70c6




-2. b.9

-2.610
-2.oid
-2.017
-2.7




-2.7 9


-2.794









-2. 791
-Z. 84










-2.613
-.6 1 b










-2.oio
-2. 41













-9.0,3
-2.8214









-2. 0o7



-2.787
-2.794
















-2.17974
-2.013














-c. 80
-2. -12:
-2.03D
-2.6202

-2.841

-2.59o



-2.517
-2.187

-2.041
-1.950


UV

HtL JU
(2446COG+)


2640.77762
204o.7610 1
2o40U.7oo02
2040. 790o8
20o4. 19o04

264o.79t03
zo40.t00 7

204ci.t071 U
o4o. 61150


2o40.020L 7
eo4o0.o2ol
2o04u. 2892
Co-46 .6330o
204 o.0J'92



2o4o. bbo0
040.o o0 o



040.670o4
204. b7,4bi

2uuj. 6C646

265L t).b (4t
265u .obo /
So .J Lo9 1 ',9
26o0 o:ooj
2050.701 04
2obj 704J

2c3D .71079
20 0. 71 17
2o5b .71921
26)0 .72467
205b.73110

Os50o.7344c
2o5 .73: 11
2u50.741o2
2o5C.74549
265 0.74978

20o0.730094
2650.707 lo
2050.75790
Z00C. 70421
200C.7oo07

2050. l11 0
2650.o15u7
26o0.bi 61 I
2650.822286
2o50 .20o54


LMAG.



-1 .621
-1.781
-1.115
-1.583


-1.334
-1.114
-3.;I
-0 *


-O.J0o
-0 86
-C.802
-0.583i

-0.340
0.079
0.360o



1.0 14
1 .139
1.1 59
1 1 39
1.109

1.i1 4

1.133
1.129
1 I :2
1.141
-2 .-4d


-2.849


-2.084

-2.8a9
-e2. b68
-2.873
-2.86o2
-2.840

-2.819
-2.049
-2.858
-2.071
-2.6d9

-2.829
-2.921
-2.847
-2.844
-2.806

-2.704
-2.775
-2.795
-2.803
-2.821


HEL JD
(24400, 0+)


2&50 nito ,3
16b0. 831 ')

26 U0. 63,>3
2650. .831





2 7' 1. 5.91
270 1. 5'43 ,
c701. 55i0-'
:70 I. 55, 0

2701.53bo'5

270 1. 54c 9
27. 01. ' / k.;
2701.5 42,o,
2701.500 ,.


2731.5 2: 4;

2 7CI .^-7 ,n
2701.56 ,j
27.1 r, ,."
27)1., .D ) -

270 1 .ol 3o
27C 01., ,

270'1.. 01
S7J09. 2,

27' 9. 1.27 I

2709. 03712
2709. 62 71 L
2739.6421 l
273 .6 407

2709.u00107
2709.0574
2739.cu15
2709.o5b53
27309.. t7-9

2/09. 7977
2709. 0 .373
2709.oov39
2709.95 1
2709. 09009

2709.791 j
2739. 715Z+
2 709. 71 890
27C9.72 2c9
2709. 7 691

2709.73207
2709.74114
2709.7451-
27C9. 7' 92L3
2709.75299


O4AG, .



-2. 5
-2. -0r
-2.017
-2.- 11
-2. 1

-2. 7D3
-2. 705

-2. t. c
-2. 7i0




-2. '57
2 .- --


-.372
-2. 274



-2. 912
-2. 4 7


-2. 7"1
-L. -41

-2. 171
- .i 1l

-'.71.
-2. 7 7



-c.703
-2. 771
-2. 771


-2.703
-2.777
-2.749
-2.764
-2. 7 4
-2. 7o -.'

-2. 74 1
-2. 71 1
-2. 73
-2. 730
-2. 711

-2.712
-2.701
-2.o 7
-2.075
-2. 719

-2. 706
-2. 74 )
-2.713
-2.719
-2. 731





-50-


TAbLE 12(CUNT'D)


uV

DMAG. HEL JJ
(2440L00+)


HEL JD
(2440000+ )


27C9.7b7n2
2709.76408
2709. 70o17
2709.788bo
2709.79345

27C9.80021
2612. 52211
2812.527o4
2812.53140
2812.537d4

2612.54 lc3
2812.5 04Lo2
2012.54963
2612.55589
2812.50287

2012.5O621
2o12.o7Cs23
2812.57451
2812. 03012
2612.56847

2dl12.58 32
2812.59311

2o12.69148
2o12.o 435

2612. 6289Y
2012.uoJ21
2612.0359 1
2612.64172
2812.04544

2812. 04893
2812.65279
2ol2.ab083
2612.o0354
2812. o703

2812.o7234
2012. 6761
2612.68550
2817.51593
2817.52246

2817.52005
2817. 52C98
2817.53J5J
2817.03949
2817.54-13

2817.54o94
2617.55280
2817. 3564
2817.56025
2817.5o42+


-2.713
-2.714
-2.707
-2. Dco4
-2.oU4d

-2. u79
-2. o04
-2 034

- C024

-2.o01
-2.007
-2. 0 7

-2. ot




-2. 024
- c. c 4
-2. 614

-2.854
-2.oC 1

-2. 00
-.790

-2.794
-2. l97
- 27390
-2.807
- .814

-c.ob9

-2.831
-2.856
-2.oob

-2.862
-e.o014
-o. 783
-2.7r(
-2.911
-2 811

-2.047
-2.o30
-2. 02



-2.858
-2.o00
-2.b74
-2.oa8
-2. or8


2817.oo6031
2d17.b57d5
2dl7.577o0
2817.0: 179
2ol 7.,864o

2017.56914
2617 *.5938
2817.59C38
2847.004044
2o17.o0420
JU17. c790

2o1 /.oI 176
2017.61c32
2817.uo08o
2017.02450
2817.02o2o

2tl7.03330(
2817.c37-69
S1 / .41800
2 1 7. 4701
2817.05177

2817. o09C0
2c 17.oo1l7
l017.3oC00
2617.07192
2817.o76CoJ

2817. 6 84 1
2017.800 3
2817 ..Usj3
2617. 9 o4 3
2o1/.702b3

2817. 711 78
2817.79290
2o17.72o0C
20o7.73 07
2017.75 u7

2817.753J4
2817.75t 19
2817.7o000
2817.7049o0
2817.7694 7

2o17.77640
2o17. / 82o6
2817.78o00
ol7.793ld
2817.79909

2817.t03J3i
2817.80871
2817.01883
2817 .2295
2817.82990


DMAG.



-2.833
-2 .829
-.621
- .b26


-2.b34
-2.000
-2.30 1
-2..010
-2 5o2

-C. 66
-c 0 be. 7

-2.797
-2.777

-2.797
-e cOb
-2 .7 7

- 2. 499

-2.809
-2.85'
-2.6u
-2.797
-e.diO

-2.810

-2.767
-2.782
-2.793

- 773
- .745
-2.811
-2. bb7
-2.793

-2.794
-2.745
-2.788
-2.7t65
-2.012

- e.6-+2
-2 .755
-2.712
-2.714
-2.745

-2.769
-2.720
-2.750
-2.700
-2.840


HEL JO
(2440C00+)


2824.51953
2824.5 I C
2824.0.247/
2b24. 5 X,

,824. 5_ 3Cu
2624. 5 37
2824.,, ,


2824. o 5' 7 i3
.2a4. 5Elc
2824.550j 9
2624. 5 7c,
2824. 00 31
2824.53Lio3
2824. b5771

2824 703
28,24.5 1t 0,l

od24. O099:
2824. 5.93o

r824 C., 41 3
2024. LC /7
2.24 1 1 15
2 9Ic 4. cr1 4 5)
8-4.6 2J1.

zo24.t.l3'.
2824.. s~2 7 1
2824.3Iu7




2 24. ij 34o
2824.65667
S62I4t. Dd
2824. UL 947
2 24. 7 )70o
2040.53 537

2840. 5.,0 3
2840.50 394
2 84 0. 5- 7 7;
284. 57 50t
284 .60 34.

2840. 67907
2540 o83 14
284 3. 75 385
2840.7044o
840.76835

2b43. 172o09
2840. 77468
2840. 78019
2040 7?0oo3
2840 .7 9-:7


-2.d;2
-2. '6 2
-2. Ooj b
-- 1 7 J
-2. 517
-2. "?o





-2. 7os
-2. C-



-2. '9;
-c.al7








-2. 1
-2. 35
-2. 1''1

-2. 6 ,

- 0. lt



-2. 77B ,

-2. 779


-2. "I


-2 3 L-
-2 7_

-d. 7 ;7
-2.o74
-C. 7l 4 '
-. 7'7


-2. 74-.9
-2.717
-2.7;9
-2.335


-2. .7
-2. 8 5




-2. 779
-2.7Ls4
-2. s71
-2.044
-2.713

-2.7l1

-2. 745
-2. 7 31
-2.098












TAdLE le(CuNT*D)


HEL JD
(2440000+)


2640.79241
2o4C.79582
2840. 62362
2840.82683
2840 .833CC

2840. 86086
2640. 64 u0
2b40. 84337
28640.84638
2 840. 85o4

2840.8597C
2d40.&6281
2L40.8o762
2840.67121
2840.87433

2640.6774'z
2640.88744
2!40. 69063
2o40 9388
28L6. Oo6C

2850.61187
2 dL .61o01
2d5 6222b
2656.03480
285o.o4178

263bb. 4 3 1
2d5o.o 0170
2B5o.o0729
2b5.o I0185
2d56. 6ob47

26b6.60891
2856.67437
2656.o770oQ
2db6.081 0
2856.68534

2856.69310
286d.70197
2856.70365
2850.709339
2856.725o2

2e56.73081
2856.73490
2650.743CJ
2 56. 7,094
2850.79944

2856.80879
2850.81198
2d56.81577
26o0.8241C
2856.84130


UMA.



-2. 711
-,.uo8
-C.048
-2. 744
-2.7ja

--2. 7 30
-2.794
-2.077
-2 .oo8
-2.706

-2.091
-2 .oc3
-L.7 s7
-2 7ob
-2.72b

-2.714
-c. 70C

-2.710
-2. 700

-2.717

-c. 7o9
-c.79 1
-2.013

- .7to4
-2.779
-2.774
-2. 04cc
-e. 794

-2.729
-2.762





- .770
-2.77o
-2.752
-2. ?Oc

-2.7o4
-2. 783
-C.77T
-2.7 11



-2.747
-2.757
-2.700o
-c.091

-2.722
-2.709
-2.749
-2.721
-2 .79o


uv

HEL JU
(2440000+)


25b.o b061
265o.d50 13
2dSc .86J57
265b. 67Co'
25db .boJud

Z6bo. b7773
2bbo.6dd13'
2dbo.d8 doJ
850o .0 3G
3d5o.u947L0

2tu .6uo 14
2b6o .9ultc,
cSo. 900 97
28b6.91383
2bbo. l11l4


Obbo .23o 2
7d7j. -7Jb2
d7>.b7,01
2ci73j .b6L7

267-3.5097
2o73.9,ylo7
S87j .*90jC
eo7.L5s gG
2073 .toS j


2o73.0t bJ

:d73 l u3
2673.o2oCj
2873 .t3 66

d273.o0 u5b
2 73. 04059
26773.u440
267s .03001
2d7-io5^J^
267J.ot433

273.tb0 186
2875.7b005
t73j. .7940
267j.79703
2873 .b628B

267 3.bCb2b
267J.blOoo
2673.814395
287tj. l1902
cj7j uo22oo

2U87-.0292
2t873 .J1 bo
267o .6Jd81
2673.6433J
287J.6470o


IDAG .



-.78uS
-2.837
-2 .20
-2.82d
-2.740

-e.798
-2.7 u
-2.777







-2.791

-2.601
-2 .783
-2.510
-4.4o5
-c. .5

- .4 75
-6.434
-2.40-4
-2. 37
-i.351

-2.204



-1.9ob

-1.910
-1.7o0
-1.050
-1 .401
-1.25ss

-1.244
S1 703
C .084
0.371
-0.112

-0. 73
-0.b92
-0.o27
-0.989
-1.210o

-1.331
- 1.389
-1.632
-1.783
-1 .600


HEL JL
(2440200+)


2887.7710 O
28d7. 775 -'7
26b7.7/947

2a87.7;u9l7
2897 7-:l : 37
2 8 7. 79
266?/7. 7-_v .7
2o677 SC 1
2067.C d2ol


6d87. 8c22u
26d7. 1352Od
2 P'd 7. .6,S 1 17
2867.8641 7
2od7.852 .1
286-7. 4, 3

2;i7.85-3
2.w 7. I 7* 7
2 5. 7 5; / i.
2s57. b6'7
28 7. 87j, -

26di7. 873
2 3 r 7. b .-,
20, 7. Li 3 i
2093. 7-i '1
2895. 7 4 L

-'"Y.7.*o/
289P .74. 17
2 d 95. 7l> 1
E289. 7o0 )'
2 95. 7:.

2895. 7 4 17
2595 75 7-
2893.7730),
269s. 750S,
2895. 7 -o3,
269b. 73J -,


2695. 78O i
28 791 -L

2 895. S "C
2595.S5;33.

2 895. : 73
2895. 8C9 -
2895.31521
26 95 .8 Li
2895.62795

2895.8311
2895.6363 j
2d95 *. 84 3L,
2095.84002
2895.85001


DMAG.



-2.3j4
-2. o05
-2. 1 i
-2. 10
-2. 33

-2.-147
--2. -J 7 2
-2. 4
-2. s47
-2.002

-2. u7
-2. 57
-2. Lt 7

-2. -22


-2.7-.7
-2. "7
-2.1
-2. -c.l





-2. I,9



-2. iJ2
-2. 7b
-2. 7-T

- 2. n 1 -3
-2. 41
-2. 7 4

-2. 33
-c.832



-2. 337
-2.23
-2.322
-2. 6 9

-2.320
-2. 79,
-2.B30
-2.773
-2.358

-2.786
-e. 73D
-2.7 5
-2. 04
-2.79J











TALL 12(CUNTLD)


D'iAG. HEL JJ
( 44000j +)


HEL JD
(2440000+)


2695.85313
2895. 85619
2895.8610 d
2895. 66439
2695. 88000

2895.87671
289b. 88010
2,0.5b6431
2903.5 d5ol
2503.571bo

2903.57474
2S.03 58077
2903.5841o
2903.56728
2903.59030

2903.59J39
29J3.59860
290.3 60 Id6
290C3. o506
2900.60924

2903.72o76


290J.732866
290J.7o334
2903.74133
2903 .7442,
2903.7474o

293J.75071
2 903 .7 9 o u
2903. 703o
290J.7719-.
2903.77211

c90 3. 7a77
4904. 77b3J

2910.o338ti
c91 b399<9

291d8.L4300
291i.73 o14

2910.70 7 0
291o.7 11d 9


C.19o 2 ld8.71540 -1.434


-2.791
-2.b 1
-2 .021
-c..30
-2. bl0


-2.7d0
-1.a80
-1 .t37
-1.770

-1 .71
-ihood
-1* '.d
-1.376


-1 .169
-0.942
-0. 742
-0.o39
-0. 093
393,


MA G



-0."61
-0 .41
-C 791
-C.Gdl1
-1. 1 59

-1 .403
-1.433
-1.7al
-1 .72 8


-1.847
-c.c01
1.1l ^
1 .1 I
1.08 4


-j.0u9

-1 .12L
-1.J10


HEL JO
(244300CO+)


2918.7190/
2918.72 z4s
2918. 7j3b-
29 dl.7js'5r
2 91 8. 7-. )4 J
291 8.7s 3ot
291 8.73 /o
2918. 7,7 ',
291 6.733 .i
291 6. 7.4 /
C 918.773,

2 9 1 5.77:. 4
,2 91 d 7: 7
291,: 7d6 i

-2918.7j3 l4

,29 18. 791
291 .8 6 1;
2 913 .31 4
291 6. 86 1 oi


2 vl3S.8boo


L)O AG .



-1. ,35

-1. 1
-1. ).70
-2.170/

-2. 14



-2.1 3
-2. U.s.








-2. 4s
-27 1
-2 .74
-2.711














18) "(US 'U JIo
Oat or01-i- 0 ['0- 0oS0 o0 3
L -i I------- i -- -- --







I II
.-



'C. .


-,v
^ "^--: .'- *-** **


'A

ooF. 1 t~


InM 'ou jjio
c0:7 o.'o- ) ,


( S5 0








September; 12, 15, 21, and 23 October 1975, were initially

eliminated.

Data in the region 0.12-0.25P show two distinct levels,

corresponding to observations separated by about one year.

Since the transformations discussed above gave consistent

results, this separation of about 0.1 appears to be real.

The question as to which level represents the "less perturbed"

state of the system is thoroughly discussed in the next

chapter.

Normal points were obtained by taking simple means of

the magnitudes in bins 0.005P wide. These normal points,

converted to intensity units and normalized, appear in Table

13 and are plotted in Figure 4. The normalization factors

were 4.330859 for visual, 7.790122 for blue, and 13.033539

for ultraviolet. For reasons alluded to in the previous

paragraph and discussed fully in Chapter IV, the higher level

in the region 0.12-0.25P was not used in forming the normals

of Table 13. Observations on 25 March, 4 April, and 18

September 1975, form this higher level.











TAuLEL 1
, uk U CELPHIL


PHAI- lnil rl I.


0.C 0297
L,. .72b
. 21d7a
u .01771
LC. ,22u

v,.JZ7o9

u. 0 3 u : $
i 041 S e
. 04649

*, 5191
i .3o71
.vo274

S.i'7u9
,. 729'i

L.07 ?94
*is21ob
.J 1271 b
0.1 34,o
1. 514 30

,.143L4
. 1 47 o
I. D44
-.Ib944



u.17 7 i
. I u I7
0.190 53

L l 97C u
C .d01214

0.211 J4
i.2171

0.222o3

0.23321e
S. 2-oo.3
0.24e1 7

C .24731
0. .217

C,. o02ob05
0.i-o727

S.27231
0.277o1
0.2081 9
0.267ol
0.29J1 i


0.11o9
al. el
*1 7 11
v. lil




0r1717

-s11O4o

31,0






S .9 1
0., 1ri

'10091)










11109
v- aco



I. 111.0






I l i. L/

1 U Gj W


I*lL) 1
i o ^v4




loweb *
i j





1 dJ


V 1 ,UAL

PiHA-.' INILN .


0o.c770 1.ulo2
>.U L 2 0 .9-844

O.j31e4 1. Cli
O. 1Doo I .Ool


0.J IJ o IJil3

.J7 Oil 2J. vVu
.' -C o i 1. 'oL7

'. -4 7d o 1.01 47


S.J. )-/J l.u1 1



y.07-oJ7 0.99-..
0.7/1h i.97dji
vu o7. u 71 '4
vJ j7 r I 9do

U -uJ.a J 1.l ,Ii
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S.aS7,Lr1^ J.^. .,

-.4-,j0 I.ULUO
0.42c l.1 Q '7
L. 4 el 0l )
J .4'+7 1.0'0'o
4+u7i 1 1.




.rs 4b173j V. 9o1
v.4927U1 9dow

ui.s+47 44 L9.1




v. a177 0.97o3

t.oI2iu i. 947uj


.4cJ7od 0.9/ocs

0.vdleu C.v9lu


0 'L. >4, 'Lu.v9ob


u.uo7.5 i.ttc3L
L 0 2 1.0003 o


PHAOL INTEN.


. 'o72 j
. /7 22
-.3 770

0 .d/31
O. 0' 14J
1 .00411
u.u4v1 1
", 2 1,
0.03 2
5. 7 i o 1









..u7ol
J u 0<.. 8









,u jo u
J oJ4 1o





O. > 7
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.J .u ,7


J.7 o 1 7
j.70// ,
0 .1o 71
U.,7172
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6. ,* 41 7_ 0
0.s70010
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0.771I0

0.7811

0.70IJ I"
2.)797O 2

J oz74.'





0 .d 4 1 7C
J d 7 t(
0.-37/4
0. Jb4L2
0. i 3t J


1 31,


1. 0.c 7
1. 2o4

U. zo9L7
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1.0214
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0. 9 74

1 077
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1.3 J d


1.3J77
1."036
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1.3i27
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1 J,


1.vI22

1 0 1 72
1.J I141

1 Y71
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3.997o
1.3307



3.. y 9
S0.95o




0 .9 d a'


2. ,ios

2. -9 o 1
0.9001
j.soZu
. 0, 91
0. I 71
0.9803
















PHASE INicl l


j .85212
O.bb717
0.86240
S.86o7i
0.872ebo


uv.717
avO43
u Ybb2
U*MbSh


0,d75o1 J.4oju4
U.9177b v l,


-57-




ITAbL- I (CUN T" L)


V I ,AL

PHAQ-L INILN.


U. 97773
0 .'JiL 44
(0. 37, 7

. 9'47 4


v. o7o9
0. d&Y1

U.7,,.o
S0.0*27


U. ,7t0 u0.4 U:i


PI'r A, L INTEND.


0.97317


S.96d/7 7


0.J 171
0 2241
2.1724

0. 110
5ol.7i


3.9lcll 1 1
2.997S7 3.114u











AuLcL 13(CLNIT'U


PHASE INILN.


0. 0297

0.0127t
G. 0 1 7 7 1
C*01771
0.022o0

0.02789

C.u3odo9
C.0j4192
0.04o049

L.05191
C.0b712
. 00274
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0 C, 729

7.0783'
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i. 1271b
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0,.1094
0. 1724J
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0.1 9003
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u .20214
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0.21713
L.22283
.< e2939
0.233J2b
C.2JOO3

0.24217
o .2471i1
0.*2 21 7
0 2'-177
0.t0265

o1.20727
S.-272331
0.277ol
0 .28 cL
* .2;701


0.0313
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0.9710









us u707
0.90 u
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u. 9750
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1. 90/0
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1. 'y*C

1 v03o
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bL UE

PHAzLE


0.^y^1 u
0.2977
L 012/2
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Ui lbo6
L .n198
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i0. Jtco
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0.00712


v ^ ob
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IN LI4.


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10.9 1 7











U. 9967
0. U9 7 o


S. 90 0
0.97790
Li 99/4
0. 9o 2












0. *979o


PHAoE INTEN.


S4302030
u.0jo767
0. 7 04
;0.0377100
282 4 Zd


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0 *7058 d
0.77c71L
6.737024
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0 01 1 7
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C. 7147-1
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0.79obd

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0. 749 7

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Li. 9o.4
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PHASE IIENI l


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uiol o-


-59-




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bLU l c

PHASL INItN.


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iJ. y7~7
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1. 7J-1
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PHAbL INTEN.


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C *67t61 i*.U v u.,4b2V U.493oJ
3. 177c v. t44, j .u73z>b v.4 /lO


u.6d777 0.3J12
J. '97 1 O.7o.i
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lAdLL 13 (CUiT*" 0)


HAS E ItilLh.


U. 026b
0 0023b 4
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r.* 122 l 1
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-61-




TAULl Io(CUNTI')


uv

PHAbS INTC,4.


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u. u771I


0. I


C.c *i D


PtiA oE INT, N.


C.98e .


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41
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CHAPTER IV

RECTIFICATION



Fourier Analyses


The initial Fourier analysis was obtained for normal

points taken as described above, including both levels in the

region 0.12-0.25P. These coefficients are labelled Fl in

Table 14. Data in the regions 0.90-0.10P and 0.40-0.60P were

not used in the analysis in order to avoid the eclipses. The

defining relationship is


I = A0 + Al + 1 A2cos2 + A2cos Acos30 + A4cos4t + B sint

+ B2sin2i + B3sin34 + B4sin4p


The phase angle inside eclipse was rectified by



2
1 zcos


with z = 0.021. The result of this Fourier analysis was un-

satisfactory since if left secondary eclipse inverted in all

three colors and put the bottom of primary negative in the

blue and ultraviolet. This was anticipated since the

shoulders of primary eclipse do not match for this choice of

normal points.




-65-


Next a graphical analysis was made following the pro-

cedure of J.E. Merrill (Merrill, 1970). These coefficients

are labelled F2 in Table 14. This gave reasonable results

for A2, but positive values for A,, contrary to theory. The

B1 and B3 coefficients in the ultraviolet are quite un-

acceptable. Higher order harmonics may be present in the

blue and ultraviolet, which leads to the suspicion that even

the data treated here, considered to be less perturbed, are

somewhat affected.

At this point a reexamination of the data seemed appro-

priate. Two regions in particular were felt to be responsible

for the peculiar results of the first analyses. The region

from 0.75P to primary eclipse has a noticeable downward

trend, seemingly indicative of gas streaming effects. This

downward trend is a permanent feature of the light curve.

Likewise the dual level of light from after primary to 0.25P

requires resolution.

Consideration of the Roche model in conjunction with the

hydrodynamical gas streaming models (e.g., Prendergast and

Taam, 1974) indicate that during periods of increased activ-

ity, the light level just after primary should be slightly

depressed. Thus it was assumed that the higher level repre-

sented the "less perturbed" state of the system. The Fourier

analysis of the tops when the higher level is used, however,

again yielded aberrant results for Al. This result can also

be anticipated since no equivalent "less perturbed" level

exists prior to primary eclipse. A short region analysis of





-66-


the same data excluding the points greater than 0.75P did not

improve the situation. The cause of this failure, however,

seemed to be that the solution was not sufficiently con-

strained to yield valid results.

Similar Fourier analyses were run to test the results

when the lower level after primary is used. These coeffi-

cients came closer to matching theory, especially in the

ultraviolet. The same short region analysis indicated in

the previous paragraph seemed to improve the situation in

the visual and blue. The same caution concerning the con-

straint on the solution should be emphasized here, especially

in view of the very unusual values obtained in the ultra-

violet. These coefficients are listed as F3 in Table 14.

A recent solution of the U Cephei system by Hall and

Walter (1974) obtained Fourier coefficients closely matching

theory by means of a very short region analysis in the region

0.25P to just before secondary eclipse. This is probably the

only region of the light curve relatively undisturbed by gas

streaming effects. When this approach was applied to my

data, completely unrealistic results were obtained. Hall and

Walter's analysis is on only five normal points to find A0,

Al, and A2. It would seem mathematically fortuitous that

such, a procedure would yield usable results. In addition,

these three coefficients cannot account for the harmonic

variation of the tops of the light curve. At this point

numerical least squares methods were abandoned since con-

sistent, realistic results seemed impossible to obtain. The








lower level following primary eclipse yielded the most en-

couraging results and will be used in the next analysis.

The data for the graphical solutions were now reexamined.

If only terms up to 24 are important, then linear solutions

result from the graphical approach. Data points which

deviated significantly from the apparent linear solutions

were temporarily ignored, in order to obtain better approxi-

mations to the more important terms. This approach appeared

to reap better results. Using the linear coefficients thus

obtained, a process of trial-and-error was followed to obtain

the other terms and better the approximations to the linear

terms. During the trial-and-error process, the following

guidelines were helpful:

1) Fix the coefficients to approximate theory;

2) Produce flat tops to the light curves tempered with

knowledge of known nonconformity from other solu-

tions;

3) Fix the sine terms to reproduce in shape the total

portion of the unrectified primary eclipse;

4) Produce secondary eclipse of roughly the same depth

as the original data.

The following comments need to be emphasized about the

above guidelines:

1) Number one may not be important since U Cephei is

known to be highly unstable. It was thus accorded

low weight in the trial-and-error process;








2) Number two was adhered to so long as terms up to 44

were still apparent in the residuals;

3) Number three is valid only as long as the sine terms

affect both eclipses in a similar way. The secondary

eclipse was sufficiently shallow in the blue and

ultraviolet to render it useless for this determina-

tion. The shape of the rectified curve in the total

portion of the primary eclipse is somewhat sensitive

to the sine terms, and the complete symmetrization

of the eclipse cannot be achieved without doing severe

damage to the resulting total portion. Assuming gas

streaming to be important in the observed asymmetry,

such gas streaming cannot be rectified by any simple

combination of sine terms. Guideline three was used

as a compromise, realizing that the full asymmetry

could not be removed by the sine terms;

4) Number four was very difficult to achieve, especially

in light of guideline three. The resulting secondary

eclipses were shallower than have been previously

reported. A possible explanation for this will be

discussed in a later chapter.

The resulting coefficients of the trial-and-error process are

labelled F4 in Table 14. These will be used in the initial

solution, with the caution that rerectification may be

necessary. A word about the errors of the coefficients

listed in Table 14 seems appropriate. The machine solutions

do indeed yield errors for the coefficients. These errors,
































000 0a. a 0 0, a 0, 0




0 0 9 0 0 0t 00 0a 0






OH0 00 41 0 0,4 00 04. 0
: :N 0 0 0 +1 1


a4 0' 4 4


NO Q T)r( (4
I~r( ~C~O) mN o
O 0100 o

0'100 Orl O







(101
I IO NY)
D rrO~l o

D II 00 OII D







~O 0 L*
e- rC, coh O
O Ou3 3 O

O I: UC Orl O







)10 D
- HO ny) CI
O Y701'0

O Il 00 o+I O


044









0 4]
co



















CO
u



















444,4 4.
.4,44.,
44 U,
0 ^1UI




































4,.-~ '4
*^ U
































o )4]
.44-C o


C:L1 111










































444 CLC
00444

4'4*-
r *U1l
N6







(LO
e".)







Cra
-uI -



O*
nIS
.^ roe*
0 0O
C.0
L C
rd -r.
l-< I-
EO *i




0 :3

0 OV
"- c





,3 L
o aj





c >-





r ru
Eu -1-
F: *-
ll .M v
^.j^- r

.- I i..
*~L LC t
EIi *- l *


0 ,


.4 "4 44 a


o a o 0 0 Ca 0 0 04- .. 0

OI1 00 0 0 o0 sO' a 04 0O
c; .1 1 or c; I 31











.11 00 0.tl 0 Ct 00 01 0 .1 ; 00 .'1 c;
Vill 011 Y13 P -C; C


m In
9N1 U~01 Q
- PIC nN Cj
O CN C O

O11 oO O11 D






e


II
DrD

"""I;"
u.rrrr







however, are mathematical in nature, dependent upon the num-

ber of coefficients to be solved for in the solution. They

are important as an indication of the mathematical signifi-

cance of a given coefficient, but beyond that their use is

illusory at best. The errors involved in the graphical

approach and especially in the trial-and-error approach would

be difficult to evaluate and have not been given. The coef-
2
ficient of determination, r is given to indicate the

"goodness of fit" for the graphical solution.

Listed in Table 15 are the theoretical reradiation

coefficients, CO, C1, C2, obtained from the theory of Russell

and Merrill (1952). Using their equations 107 we have


C1 = -A1 = 0.40(Gc Gh)sin i

SG + Gh
CO = -(0.75 0.25cos i)Gc h A1csc i (4-1)
c h

G + G
C2 = -0.25 c-- A1sin i
2 c -C_ h G
c h

where i is the angle of inclination. In order to obtain

Gc + Gh and Gc Gh, use Russell and Merrill's equation 104,


Gc/h = Jh/E / J/E 4-2)


where J's are apparent surface brightnesses and E's are

luminous efficiencies. If we assume that the J's can be

given by Planck's law, then the E's are given by,


EA,T = J,T/T4





-71-






Table 15

Reradiation Coefficients

Russell-Merrill Theory


CO C1 C2 Xeff ()


Visual 0.040 0.051 0.013 5500

Blue 0.020 0.024 0.007 4300

Ultraviolet 0.010 0.011 0.003 3500








where a is the Stefan-Boltzmann constant. We can write JKT

as,

-5
clA
A,T c2/AT
e -1

-5 2 -1
where cl = 3.74 x 10 erg cm sec ,

c2 = 1.439 cm deg,

A is wavelength in cm,

T is temperature in degrees Kelvin.

Letting,

B = c2/T


and expanding (e 1) by the binomial theorem, the maximum

value of B can be obtained by differentiating EX, with re-

spect to B and setting the result equal to zero. This results

for a specific wavelength in,


5max = 4

Letting,

E = ET/Emax

and

S= ,T /max

it can easily be shown that,


log(J/E2) = 3.0946 81ogB + log(e ) 1) (-3)


Using approximate values derived from my Wilson-Devinney

computer solution (Wilson and Devinney, 1971) of U Cephei,








equation (4-3) is solved for both stars. Then equation (4-2)

is solved and the results are inserted into equations 108 of

Russell and Merrill in the form



G + Gh + h) r ,
cLh Gilh ic G c h




Gc Gh = i ] (IcIh)2 r rh


where the I's are the specific intensities of the stars and

the r's are the fractional radii of the stars. Equations

(4-1) then yield CO, Cl, and C2. In the absence of effects

other than reradiation, C1 equals -AI.



Rectification Formulas


The following formulas were used to rectify the inten-

sities with the coefficients F4 of Table 14. The formulas

follow from the Russell-Merrill theory.

Visual

I' = I 0.040 + 0.023cos# + 0.013cos2t

0.0065sin0 0.0122sin24 + 0.012sin4

I" = I'/(1.034 0.003cos2 ) ,


Blue

I' = I + 0.030 + 0.020cos4 + 0.010cos2o

0.023sin4 0.005sin24 + 0.010sin44

I" = I'/(1.006 + 0.004cos2 )








Ultraviolet

I' = I + 0.010 + 0.018cos4 + 0.003cos21

0.020sino 0.005sin21 + 0.008sin4(

I" = I'/(1.014 + 0.0007cos26)


where 0 is the phase angle and I the observed intensity.

Notice that in the blue, the constant term on the right hand

side of the first equation is not CO. Some adjustment was

necessary to obtain a better fit. The coefficient of cos24

in the same equation was also adjusted from the model value

C2'
Table 16 shows the mean intensities after the above

rectification taken over quarter phase intervals. No obvious

residual periodicity remains that can be accounted for by

sine or cosine terms up to 41.

In light of the guidelines stated earlier for the trial-

and-error analysis, it would be instructive to examine the

relative success of each before proceeding with the solution.

By carefully examining Table 14 it is clear the size and sign

of several of the coefficients (notably Al and A2) are

critically dependent upon the particular set of data used

in the Fourier analysis. This seems to support the observa-

tion made earlier that some residual perturbations remain of

which proper account cannot be taken by the standard analysis.

Indeed, these perturbations should be expected since the

observations were taken during a period of increased activity.

It appears doubtful, in fact, after examining light curves

of U Cephei from past epochs, whether anyone has observed








this system in a state when problems with the formal rectifi-

cation do not arise. It would seem advisable, therefore, to

find a set of coefficients which closely resemble what one

would expect from the Russell-Merrill theory. Table 16 shows

that some nonconformity exists of which gas streaming models

can, hopefully, take account. Overall, however, no severe

damage is done to the light curve by achieving the first

guideline. Table 16 also shows that the second guideline

was reasonably successful, especially in light of the pre-

ceding discussion.



Table 16

Mean Intensities Outside of Eclipse After Rectification


Interval

Color 0.0-0.25 0.25-0.5 0.5-0.75 0.75-1.0 0.0-1.0

Visual 0.9885 1.0093 1.0004 1.0005 0.9997

Blue 0.9709 1.0047 1.0198 1.0075 1.0007

Ultraviolet 0.9591 1.0003 1.0367 1.0037 1.0000




The eclipses are rather insensitive to any sine term

adjustment since the functions are smallest here. It was

noticed, however, that changes of 20% in the coefficients of

the sine terms had a perceptible effect on the slant of the

total eclipse. The maximum effect of the sine terms occurs

at quadrature points. Close examination of the residuals







near the quadratures show a much better fit of the rectified

curves to unity than even Table 16 (which has already been

noted as being passably good) reveals. It appears, then,

that requiring the sine terms to preserve the shape of the

observed total portion of the light curves favorably affects

the rectification outside of the primary eclipse.

The agreement in the depths of the secondary eclipses

between the rectified and original data was the most dif-

ficult of the guidelines to achieve. Any adjustment of the

cosine terms to achieve a reasonable fit in the secondary

eclipse resulted in a deterioration of the fit outside

eclipse. Examination of Figure 3 shows that only in the

visual is the secondary well defined. The blue data show

only a hint of the eclipse and it is invisible in the ultra-

violet. Notice also that the scatter (presumably intrinsic

to the system) is much greater in all colors just after O.SP.

The visibility of the eclipse makes it difficult to work with

in all but visual light and even there the scatter during

egress causes additional problems. Less weight was placed,

therefore, upon achieving the final guideline.

The difficulties presented here in achieving a workable

rectification of U Cephei seems to have been minimized in

previous attempts at solution. These difficulties are pri-

marily responsible for the noticeable lack of all but

preliminary solutions of the system. We have in U Cephei,

however, an outstanding opportunity to gather information

about the geometry of the system. The total primary eclipse





-77-


is the ideal case for eclipsing binary systems. A patient

approach to the solution should reap rich rewards, not only

in the geometry of the system, but also in further under-

standing of the dynamics of these unstable close binary

systems. Three solution procedures were followed in order

to compare the geometrical properties of U Cephei. The

Russell-Merrill solution is presented first. Following this

solution are the computer approaches of Wilson and Devinncy

and David B. Wood.













CHAPTER V

THE SOLUTIONS



The Russell-Merrill Solutions


Nomographic Solutions


Figure S shows the normal points inside of primary

eclipse from Table 13. It is obvious that some asymmetry

exists and that this asymmetry is.not simply a result of an

inaccurate ephemeris, since the asymmetry increases with

decreasing wavelength. This asymmetry has long been known

for U Cephei. For solution purposes symmetric eclipses are

required. Assuming that gas streaming effects are responsible

for the lower level of light on the ingress branch, only

egress was used for the Russell-Merrill solutions. Egress

is presumably less affected by these perturbations and the

resulting solution should be much closer to the true geo-

metrical parameters.

The previously noted difficulty with the secondary

eclipses made it very difficult to get a reliable nomographic

solution. From the estimated depth of secondary eclipse in

the visual (0.9595), a nomographic solution was done. Solu-

tions only exist for limb darkening coefficients greater than

0.4. This solution fit the primary eclipse rather poorly.

It was discovered that a good fit to the primary eclipse


-78-




-79-


could not be obtained using the observed depth line. In

fact, acceptable solutions for primary eclipse could only be

obtained for a secondary eclipse considerably deeper than

that observed. Due to the observational scatter after 0.5P

and the poorly determined eclipse in the blue and ultraviolet,

less weight should, perhaps, be given to the secondary

eclipse. Little information will come from the secondary

eclipses and progress beyond a preliminary solution will be

difficult by the Russell-Merrill approach.

Some further attempt was made to restrict the area of

the nomographs in which solutions lie. Since the depth line,

given by,

-tr / oc
(1 -1 )/l
0 0


has a very well determined denomiantor due to the deep total

primary eclipse, any range in solutions can be ascribed to

variations in the numerator. Values of tr consistent with
o
the observed depth of secondary eclipse give depth lines 50%

less than values of 1tr giving good fits of primary eclipse.

Values for the depth line vary from 0.2 to 0.4. There is

insufficient constraint here to make the nomographic solution

meaningful.

Perturbations in secondary eclipse are probably respon-

sible for the unusually shallow depths. The assumption was

made, therefore, that the observations do not show the

expected variations due to the transit eclipse and the

secondary eclipses were not used for solution purposes. An





-80-


important constraint on the problem is thus removed and great

care must now be taken to insure physically reasonable re-

sults.



The 4 Solution


The P function is given by,


2 2
sin2 sin2 0
S= (a 0.6)2 (5-1)
sin (a= 0.6)- sin (a = 0.9)


where subscripts refer to the value of the phase angle, q, at

the specified value of a, which, for an occultation eclipse,

is simply the fractional light loss. Values of 4 for a =0.6

and a=0.9 can be read directly from the light curve and thus

t can be calculated for any other value of 0. Tables of the

$ function have been provided by J.E. Merrill (Merrill, 1950a)

in terms of x (limb darkening coefficient), k (ratio of the

radii), and a. It is more convenient, therefore, to choose

values of at tabular values of a. Having calculated for

each of these phase angles, the tables are entered to deter-

mine k for some particular value of x. Table 17 shows the

results of these calculations. Also given in Table 17 are


Aoc = sin 2(a = 0.6)


Boc = sin2(a = 0.6) sin2(a = 0.9)


It is clear from the mean values of k and their standard

errors that the coefficient of limb darkening is a difficult





-81-


Table 17

The i Solution

Visual


AOC = 0.06578 BOC = 0.04187
oc 2k 4k 6k 8


0.05 3.42860 0.5941 0.6115 0.6319 0.6566

0.10 2.81238 .5768 .5991 .6235 .652-

0.20 1.93215 .5410 .5671 .5968 .6290

0.30 1.34707 .5524 .5818 .6146 .6511

0.40 0.85283 .5752 .6076 .6426 .6811

0.50 0.41043 .6115 .6458 .6823 .-221

0.60 0.0 -- --

0.70 -0.35397 .5321 .5684 .6071 .6490

0.80 -0.67122 .4442 .4832 .5240 .5686

0.90 -1.0

0.95 -1.17667 .4580 .5002 .5455 .5932

0.97 -1.26050 .4389 .4832 .5309 .5818

0.985 -1.32301 .4649 .5086 .5565 .6081

Means 0.53 0.56 0.60 0.64
6 6 5 5





-82-


Table 17

(Continued)

Blue


Aoc = 0.068609 Bc = 0.043689

oc 2 4k 6k 8


3.036005

2.493034

1.780613

1.286613

0.840886

0.405354

0.0

-0.364472

-0.686530

-1.0

-1.177947

-1.263844

-1.332027

Means


0.4930

.4690

.4568

.5004

.5577

.5939



.6076

.5603



.4489

.4473

.4356

0.51
6


0.5077

.4882

.4812

.5288

.5896

.6275



.6430

.5960



.4913

.4688

.4809

0.54
6


0.5247

.5097

.5089

.5607

.6242

.6645



.6811

.6355



.5370

.5173

.5308

0.57
+ 7


0.5455

.5346

.5386

.5956

.6655

.7044



.7219

.6778



.5851

.5688

.5843

0.61
7


0.05

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

0.95

0.97

0.985










Table 17

(Continued)

Ultraviolet


Aoc = 0.070688 Bc = 0.045591
oc 2k 4k 6 8k


0.05 3.268129 0.5551 0.5714 0.5909 0.6136

0.10 2.743888 .5558 .5768 .6016 .6290

0.20 2.015237 .5818 .6090 .6390 .6728

0.30 1.431369 .6171 .6482 .6816 .7191

0.40 0.893857 .6317 .6644 .7008 .7399

0.50 0.422534 .6519 .6853 .7223 .7630

0.60 0.0

0.70 -0.373223 .6632 .6983 .7357 .7760

0.80 -0.699236 .6422 .6761 .7137 .7556

0.90 -1.0

0.95 -1.162504 .5537 .5901 .6314 .6761

0.97 -1.235766 .5456 .5828 .6255 .6712

0.985 -1.311870 .5004 .5415 .5870 .6366

Means 0.59 0.62 0.66 0.70
5 5 6 6








quantity to determine. Some idea of the limb darkening

coefficient may be obtained by comparing the p solutions with

another solution approach described below. Notice the trend

of calculated k values in Table 17. A peak is reached in all

colors midway through the eclipse branch and the smallest k

values occur near the bottom of eclipse. This shows the same

trend evidenced in many eclipsing systems, i.e., that the

same value of the ratio of the radii cannot be applied through-

out the eclipse. Superficially, this states that the ob-

served light curve is not of the simple shape predicted by

the Russell-Merrill model. It is tempting to make further

conclusions, but this would not be fruitful until some of the

perturbing effects are better understood. Upon calculating

some sample light curves using the mean k values of Table 17,

it was found that limb darkening coefficients near 0.6 fit

slightly better in the visual and blue, but x values near

0.4 fit slightly better in ultraviolet. These values will

act as a guide in selecting the preliminary solution in con-

junction with the results of the next solution attempt.



The Intermediate Solution


This approach was suggested by J.E. Merrill and is out-

lined in Princeton Observatory Contributions Number 26, page

59ff. The procedure is to take weighted means of sin2 for

three groups of light levels taken at the top, middle, and

bottom of the eclipse curve. Group I includes sin2 values

corresponding to a values of 0.05, 0.10, 0.20, and 0.30 with








weights 2, 2, 2, and 1, respectively. Group II uses a values

of 0.50, 0.60, and 0.70 with unit weights. Group III in-

cludes a values of 0.95, 0.97, and 0.985 with weights 1, 2,

and 2, respectively. The weighted means of sin2 for each

group are designated MNl, M2, and M3, from which a value, R,

can be calculated as follows,

M M2
R M2 M3
2 3


Using equation (5-1) in the form,

oc (. 2 Oc oc
oc = (sin2) A )/BOC


it is clear that,


Ml[sin24] = A + BM1[]

M2[sin24] = A + BM2[)] (5-2)

M3[sin2 ] = A + BM3[]


where M1[ p], M2[i)], and M3[ip] are the weighted means in p

corresponding to the previously calculated Ml[sin2 )],

M2[sin2 ], and M3[sin 24]. From equations (5-2) it follows

that,

M[] M2[]
R M2L] M3[TI


Merrill has provided tables giving values of R, MI[4], m'2~],

and M3[p] as a function of k and x for both occultation and

transit eclipses. Given the value of R, inverse interpolation





-86-


yields k, M1[], M2[)], and M3[LI]. Then equations (5-2) are

used to obtain values of AOC and BOC. Table 18 shows the

results of this procedure.

The usefulness of this technique is that it produces

values of k which fit the entire run of the eclipse. In that

sense it is probably preferable to taking means of the k

values obtained in the i solution. Examining the results of

the two methods shows that k is the same for x = 0.6 in the

visual. The comparisons in the blue and ultraviolet are not

as good, but the differences are smaller for small values of x.



The Adopted Solution


The real test of the solution is the fit to the observa-

tions. Many of the above solutions were plotted against the

observations, including some past solutions by other authors.

The intermediate 4 solution given by x = 0.6 seems to be the

best fit in the visual. This solution was clearly preferred

and an attempt was made to keep the value of k sensibly

around the value for this solution (k = 0.6026) in the other

two colors, without doing damage to the fit of the observa-

tions. This lead to the i solution given by x = 0.6 for the

blue, which fitted the observations quite well. The preferred

solution in the ultraviolet was the intermediate p solution

for x = 0.2. This seemed clearly better than the others.

The solutions seemed close enough to assume that they

should be identical for wavelength independent parameters.

The average value of k for all the previously mentioned




-87-


Table 18

The Intermediate p Solution


x
0.2 0.4 0.6 0.8

Visual

k 0.5522 0.5757 0.6026 0.6335

MN [P] 2.461235 2.468497 2.476399 2.485532

M2[1] 0.012488 0.012228 0.011904 0.0116737

M3[f] -1.250531 -1.256798 -1.254265 -1.266061

Boc 0.042679 0.042477 0.042573 0.0421879

Aoc 0.066033 0.066046 0.066059 0.066073

Blue

k 0.4677 0.4901 0.5157 0.5455

MI[4] 2.2494 2.25457 2.264007 2.27531

M2[D] 0.00927 0.00905 0.00885 0.00869
M3[P] -1.26939 -1.27597 -1.28516 -1.28630

Boc 0.04399 0.043775 0.043471 0.04344

AOC 0.068797 0.068808 0.06882 0.068827

Ultraviolet

k 0.5556 0.5792 0.6060 0.6371

Ml[1 ] 2.470583 2.477731 2.485738 2.494611

M2[] 0.012625 0.012368 0.012041 0.011799
MN3[ ] -1.243159 -1.247021 -1.251722 -1.256867

Boc 0.046034 0.045903 0.045744 0.045567

AOC 0.070856 0.070870 0.070887 0.070900








"best" solutions was taken and p values calculated for this

value of k. The Aoc and Boc parameters were then adjusted

until the internal and external tangency points agreed for

all colors. This insures the same value of inclination.

These new "mean" solutions were then plotted against the

observations. The fits in the visual and blue remain quite

good, but this approach failed for the ultraviolet. The

solutions listed in Table 19, then, are the "mean" solutions

for the visual and blue, but the previously determined "best"

solution for the ultraviolet. In Table 19, i is the

"rectified" inclination, r the fractional radius, and L the

fractional light, where subscripts g and s refer to the great-

er and smaller stars, respectively. Also listed are the

geometrical parameters when the effects of rectification are

removed. Figure 5 shows the normal point intensities during

primary eclipse along with the eclipse curves given by these

solutions.

The values of Table 19 represent a good preliminary

solution to U Cephei. A procedure has been outlined (Russell

and Merrill, 1952) to proceed to a refinement of the solution.

Unfortunately, the procedure requires some good light values

inside of secondary eclipse. As has already been noted, this

cannot be done for the light curve of this epoch. The pre-

liminary solution is, then, all that can be reliably obtained

from these data. As a comparison to past solutions, Table 20

lists elements, authors, and epochs.


















VI '0 rc 10 co r
> * tU co w- N[
S- co c o-i








t- CO CO rC- C) N-
c; ,- cc 0 c ,


0 0



Ui) v
0 N- C \o r- r-1 \'
> CD r t- I-
S c cc -t J u co cc co
0 C) 0 3 I- l-


00 oc -- .:t
U .U

0 4-' 4J


3 .- H ,H.

O 0c O C') C') 0
0 Cc CA C0 )- cc-
C') T U) cc cc cc '0 N N C)







'A 1 cc cc 0 0 -
c C ) to ct ca




00 00


) U)l C N cc cc

r-H It C O
T- m m o



















U) L/) It) C 0.
4 O 4 0

S0 0












.,-I -H
(D a









C) $C 0 C -
::D =o N
U CO CO U) 0 -
O t0








eI i > *





























Figure 5. The primary eclipse is plotted in the three
colors from the data in Table 13 as points.
The solid curves represent the Russell-Merrill
solutions given in Table 19.





































2-












0.05 .
k-


PHASE







-92-














44 4~- ..-

C)



0 n t+04 mo 0>0
i ni-< rt

> -0
04 ' 044 --
44 04 0 0, J 0'- Th
4) d r3 CJ'~ n'-< C -
r: I c3 o^ *'- ijn^ -H c
0 04 04 00: *: t0
4- 04 .>4 >. >.
11 *- to >.ro 44.0 0.0^
4 4 o 0
CJ 0 .- > 00T1 > 0o0
C4 0 O-.- 0 0 i00 O
4 b0Cs- N C.> 44> 40>
t Or- 0 U0 4 044 44r,-
3 t-.(7 J= V, 0 O X0 o



04 0- >4 HO 00 03
0<


4 0 04 4 04 04 0
l to o 4 to
to o 44 0 o o








03 l- Q t
l/ i- 03 0 k0 CO
0 40 .0 .*: -
0 04



'0 0 -41 04 0 4 ^


44 44) t .
o 04
.0

H- 0
) 0VI 4 '0 44 44 4t

44 0~ 0 04 0 04

0.

0 to 4 0
U I e co u2 03 Uj i r- 3

\o '0 '0) z] M


44 40 to CO 04 04


44 4 to 4o to to

04




4-, '0 44 '0
0/ 04> '0 44 44
.0 44< 0- -l

04


04 04 40 '4
0 40 Ito o4 4 04
o/ 04 04 44 4- 4- 4*-



0 < 04 44' <7


00 04- t- 04
tO o ^' "' 4
.0 40l 40 11 "







44 tol I NO 44i to 44
00 44 44 4
0 0 4 4 *








The Wilson-Devinney Solution


The computer solution technique of Wilson and Devinney

(1971) was employed using the same normal points as the

Russell-Merrill solution. The Wilson-Devinney program

(hereafter called W-D) uses the Roche model described in

detail by Kopal (1959). Since U Cephei is an example of an

Algol type eclipsing binary and mass flow has been observed

spectroscopically, the assumption was made that the secondary

component fills its Roche lobe. This assumption corresponds

to mode five of the W-D program, which also couples the

luminosity (L2) and temperature (T2) of the secondary through

the Planck function. Provision has very recently been made

to allow the user to employ the model atmospheres of Carbon

and Gingerich (1969) for either star. This provision, how-

ever, was not available at the time of this solution.

A new feature of the I-D program allows the user to

specify an asynchronous rotation rate for either star. Since

spectroscopic evidence indicates that the primary component

of U Cephci is rotating five times synchronous, this value

was used for that star while the secondary was allowed to

rotate synchronously.

The free parameters were divided into two sets. Set one

consisted of i, the inclination; T2, the polar temperature of

the secondary; U1, the potential of the surface of the

primary; q, the mass ratio; and L1, the luminosity of the

primary. Set two included G1, the gravity exponent for the

primary (this is 1.00 for the von Zeipel law); G2, the same




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