Mass and momentum exchange in close binary systems

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Title:
Mass and momentum exchange in close binary systems
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2 v. (vii, 786 leaves) : ill. ; 28 cm.
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English
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Rafert, James Bruce, 1950-
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Subjects / Keywords:
Double stars -- Masses   ( lcsh )
Mass loss (Astrophysics)   ( lcsh )
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theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Thesis:
Thesis--University of Florida.
Bibliography:
Includes bibliographical references (vol. 2, leaves 780-785).
Statement of Responsibility:
by James Bruce Rafert.
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Typescript.
General Note:
Vita.

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Full Text











MASS AND MO".:,;'i.,i EXC!i\NGE
IN CLOSE BINAiRY SYSTEMS












By
JaAMES 3BRUCE RAFERT


















A DISS"RTATION IPf"SENT ) TfIE G::TATEr COUNCIL OF
TAE UN 1,, T:, fdF.I] OFRlA

IN PARTIAL FULi. LLMNQT COF ';.i -, .?,'.'S :OR THE
DfGr:E OF DOC'OR O' PHILOSOPhY





3 q78,






























For Donna
















ACN OWLEDGMENTS


I would specially like to thank Dr. R. E. Wilson

for his help and comments. I would also like to thank

my other committee members, whose suggestions helped me

define this study.

I owe an extreme debt of gratitude to Dr. F. B. Wood

for use of the University of Florida Card Catalogue of

Eclipsing Binaries, as well as to the curators of this

valuable resource,

Great amounts of computer time, provided jointly

by the Astroinomy Department and the Central Florida

Regional Data Center, made the implementation of this study

possible.

Finally, I wish to acknowledge suggestions and

comments, made during discussions with other graduate

students and faculty at the University of Florida, with

special thanks to advice given on numerous occasions by

Dr. J. E. Merrill.


















TABLE OF CONTENTS


Page


ACKNOWLEDG' . .


ABSTRACT

CHAPTER


INTRODUCTION .

HISTORY . .

2.1 Introduction .
2.2 Early Concepts .
2.3 From Speculatio- to
Calculation .

PRELIMINARY CONSIDER RTIONS


3.1 Theoretical Context of
Mass Transfer .
3.2 The Roche del .
3.3 Evidence for icass
Loss/Transfer .
3.4 Some Relation .


SELECTION OF T'HE SYSTEMS


4.1 Selection of Stars .
4.2 Absorlute PCarameters
4.3 Parameter Corre nations


S. 25
. 27
. 35


DETER MINATION OF THE PERIOD
AND ITS -C ANGEi ............ 47


5.1 Basic Concepts .....
5.2 Causes of VariatioEt .
5.3 Progjram MC .. .
5.4 ei htin ..
5.5 Philosophy of Curve
Fitting .
5.C Least Squar-e Parar:clers


iii


ONE

TWO







TIIREE


. . v i


3

3
3



. 13
. 13


FOUR


13
14

16
20


FIVEr


. 25


S 47
. 48
S 52
. 55

59
71











SIX



SEVEN






EIGHT







NINE






TEN


AN EPHEMERIS FOR EACH SYSTEM .
6.1 Basic Considerations .
6.2 A Word on Each System .
THEORIES OF MASS EXCHANGE .
7.1 Introduction .
7.2 Particle Trajectory Models
7.3 Hydrodynamical Models .
7.4 Evolutionary Models .
EJECTION OF MATTER FROM
UNSTABLE COMPONENTS .. ...
8.1 Non-synchronous Rotation
8.2 Matter Ejected from
Unstable Components .
8.3 Program ORBIT .
8.4 Efficiency Tables .
LIMITING VALUES OF dM and dJ .

9.1 Limiting dJ .
9.2 Calculation of dM/dt for
Each System .
9.3 Lmits on dM .
CORRELATIONS OF COMPUTED AND
ABSOLUiT PARAMETERS .
10.1 Period Correlations .
10.2 dP/P Correlations .
10.3 dP/dt Corelations .
10.4 dr N C correlation ...
10.5 Sumary Tables ..
10.6 Statistical Trends ...


APPENDIX
ONE
TWO
TH REE
REFERENCES


OBSERVATIONS .
O-C DIAGRAMS .
BIBLIOGRCPHIC ATERI AL


blU()al ~J ~ SKETCII


CHAPTER


. 93
. 93
. 94
. 146
. 146
S. 147
. 159
. 161


. 171
. 171

. 172
. 175
. 178
S 229
. 229

. 233
. 235


. 240
. 240
. 250
. 254
. 264
. 272
. 275


. . 278

.. 601
761
.. 780
. . 786


I














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



MASS AND MOMENTUM EXCHANGE
IN CLOSE BINARY SYSTEMS

By

James Bruce Rafert

March 1978

Chairman: R. E. Wilson
Major Department: Physics and Astronomy

Aspects of mass and momentum exchange for 188 close

binary systems, chosen primarily for their rapid period

variations, are presented. All available times of primary

and secondary minirria for these systems were assembled and

subjected to least squares analysis. Linear, parabolic,

cubic and periodic/parabolic representations were computed

for each system. The most reasonable representation of the

period variation for each system was combined with other

systemic parameters to yield conservative case values of

the period change with time dP/dt, and of the mass flow

rate di/dt. ,Nonconservative effects. were investigated by

a part cle trajectory model in which material conforming

to a Maxwellian velocity distribution was allowed to leave

the vJc.i2;ity of the inner Lagrangc:ian Point L1 for a wide

3ra:ng of systemic parameters. Particle integration yielded









the amount of orbital angular momentum which could be either

temporarily or permanently stored as rotational angular

momentum. Subject to these non-conservation assumptions,

new values of the period change (dP/dt)NCJ and the mass flow

rate (dm/dt)NCJ were calculated. Correlations of (dP/dt)NCJ

and (dm/dt)NCJ with other systemic parameters were generally

better than those for the conservative case for the range of

parameters of these systems. As even a high thermal boil-

off velocity coupled with a large degree of non-synchronism

fails to allow mass to escape from any of these binaries,

it is suggested that the conversion of orbital to rota-

tional moment should not be ignored for the calculation of

accurate mass transfer rates as deduced from period varia-

tions. However, it usually cannot affect the order of

magnitude of computed dm/dt values, as shown by Wilson and

Stothers.















CHAPTER ONE

INTRODUCTION


The period is perhaps the most fundamental property

of an individual binary system, as it can be determined to

a degree of accuracy which far exceeds that of the other

parameters. This high degree of accuracy can be attributed

to the repetitive measurements of successive minima, and

the accompanying decrease of error which additional observa-

tions provide. Over a long interval of time, the period

or the change in the period of a close binary system can be

determined to a very high degree of accuracy.

The fact that not all binary star systems display

a constant period has been known almost as long as times

of minima have been recorded. Today, it is generally

believed that an increase or decrease in a binary period

can be related to the way in which the system is losing or

transferring mass. If we assume conservation of both

mass and orbital angular momentum, observed rates of

period change can easily be used to calculate the mass

transfer rate in a close binary system.

Natura] ly, more accurate transfer rates could be

calculated if these two assumptions could be relaxed, or

elimi;n Ltd entirely. It is the aim of this study to analyze





2



the character of as many binaries which exhibit period

variations as possible in an effort to find correlations

between the rate of change of the period dP/dt, the mass

transfer rate dM/dt, and the rate of change of orbital

angular momentum dJ/dt.
















CHAPTER TWO

HISTORY


2.1 Introduction

Although mass loss from single stars has been

observationally detectable for a long time, the theoretical

ramifications of mass loss or exchange in a close binary

system have only recently been explored in detail. The

advent of photoelectric photometry has provided very

detailed information on period variation and thus some

ideas about the mass exchange mechanism. However, the

questionable validity of the assumptions of conservation

of mass and angular momentum has cast doubt on the

calculated values of mass flow rates.

In the remainder of this chapter we will attempt

to lay the historical framework for the ideas of mass

exchange which have led to our present understanding of

the subject. It will become apparent that a study of a

large number of close binary systems would be highly

desirable in Lhat it could allow us to obtain an overview

of the situation.


2.2 Early Concepts

Certaiilly onoe of thi. earliest results of the

study of eclipsing stars was the fact that some displayed









variable periods. This fact was apparent to many observers

around the turn of the century and shortly thereafter.

Aside from the well-known variability in the period of

Beta Lyrae, discovered by Argelandor (1855), several other

stars had been noticed which displayed prominent period

variations. S. C. Chandler (1887a) noticed the presence

of a secular term in U Oph in 1887. During this same year,

Chandler (1887b) also discovered Y Cyg to display a variable

period. In 1889, Chandler (1889a,b) found a non-uniform

period in U Cep, as well as a period variation for U CrB.

The period of R CMa was discovered to be variable in 1924,

and U Sge in 1930. Even in early works, a variety of

different types of period variation was noticed. Nijland

(1916) found the period of RW Tau to follow a sine term.

Dugan and Wright (1939) confirmed this result, and predicted

that RW Tau would continue this sinusoidal variation.

Kordylewski (1931) discovered a quadratic term in XZ And.

O'Connell (1935) suggested the possibility of apsidal

motion for V526 Sgr, and in their work of 1939, Dugan and

Wright confirmed this possibility and discovered period

variability for several other stars.

These period variations were usually explained by

some periodic function which would reflect either the

presence of apsidal motion or light-time effects (Chapter

Three). However, evidence was rapidly mounting that some

systems displayed seemingly random, non-periodic variations.









Several concepts, upon which our present understanding of

period variations is based, were put forth at this time.

Kuiper (1941) calculated the so-called "mechanical

radii," based on the equipotential surfaces of the Roche

model, and compared these values to observationally

calculated radii for the system Beta Lyrae. He noticed

that due to the size of the larger star, mass transfer

within the system was a very definite possibility. Kuiper

stated that this mass transfer occurred because the

pressures of the two components, which he had shown to be

in contact, were different. Although we are dealing with

a fine point here, it does not seem that Kuiper put forth

the general concept of mass transfer as a dynamical con-

sequence of size and mass ratio, but, at least for this

case, that it was due to an equalization of gas pressure.

In his pioneering work, Wood (1950) suggested that

if we were willing to assume only the universal law of

gravitation and to use the Roche model, a non-periodic

period variation could only be due to either a change in

the masses of at least one star, a changc in their

separations, or both effects. Based only upon knowledge

of mass loss from the sun, it is then possible to conclude

that a change of mass is indeed possible if either of the

components fills its Roche lobe. While such a stability

criterion had been taken to limit systemic size prior to

this time by Wood (1946), this publication gave the first









indication that mass could be transferred in a non-contact

system. Although Wood put forth the general idea that

matter could be lost by stars whose components were near

their limits of stability, he suggested that this activity

was probably due to some sort of prominence-like activity.

While this type of activity undoubtedly occurs, it probably

has no significant bearing on the evolutionary history of

close binary systems, nor is it the dominant mechanism of

period variation for the vast majority of the systems we

observe.

Observational evidence for "spare mass" in a close

binary system had already been furnished by Joy's (1942)

observations and analysis of RW Tau. It seemed highly

probable that a gaseous ring surrounded the primary com-

ponent of this system. However, Struve (1946) suggested

that this ring might be a break-up remnant of the initial

system, and not a mass exchange phenomenon.

The inception of the contemporary concept of mass

exchange due to the evolution of the system can be

credited to Crawford (1954). While studying a list of

secondary components observed by Parengo (1950), in which

Parengo made the observation that 42 out of 54 secondaries

were over-luminous, Crawford demonstrated that these stars

either fill their Roche surfaces, or come so close to

filling their surfaces that for all practical purposes

they can be assumedC to fill theor.. He explicitly stated









that this behavior is completely unavoidable if we assume

that the more massive star will evolve until it acquires

an unstable, hydrogen-depleted core, at which time it will

move to the right in the H-R diagram, increasing its

stellar radius. Eventually the expanding surface of the

star will reach its Roche limiting surface, if the two

components of the binary system were initially close enough

together.

At about this time Kopal (1954) began a systematic

investigation of the properties of the Roche model. He

also tabulated the absolute dimensions of some close

binaries as a function of Roche lobe size. He concluded

that there are essentially three main classes of close

binary systems. First of all, we have the so-called

detached systems. Neither of the two components is in

contact with its Roche limiting surface in this case. In

semi-detached systems, one of the two components fills its

Roche lobe, while the other does not. Finally, we have

contact systems, where both components fill their Roche

limiting surfaces.

However, virtually all of the observed Roche lobe

filling stairs were dri scoverr'd to bp the secondary, or less

massive component. If this component were to have been the

initially more massive star, huge mass flow rates would

have been necessary. Were such large transfer rates of

stellar matei- al possible? In an effort to avoid Lhis









problem Kopal (1959) altered Crawford's concept slightly,

making the need of mass ratio reversal unnecessary. Kopal

stated that only the outer layers of the expanding com-

ponent were transferred to the other star. Hence, the

mass ratio need not be greatly altered.

The correctness of either of these approaches

seemed equally possible, as well as equally speculative.

It remained for the theory of stellar evolution to be able

to treat stars beyond the main sequence before these

diverse premises could be fully resolved.


2.3 From Speculation to Calculation

The next important step was taken by Morton (1960).

Although his approach was rough (Mcrton used a series of

stationary models with time-dependent terms disregarded),

the fact emerged that large-scale mass loss was not only

desirable, but unavoidable. It might be mentioned that

Morton assumed a constant period throughout the mass loss

process. This assumption is certainly inferior to the

assumptions of conservation of total mass and angular

momentum. Smak (1962) made the appropriate corrections to

Morton's results, but arrived at essentially the same

conclusions.

Also during this time, decisive steps were taken

in regard to both particle trajectory models, and simple

mathematical relationships of period change and mass









transfer. Kopal (1954) showed that an extremely large

variety of particle trajectories is possible for matter

ejected from the inner Lagrangian point, as a function

of mass ratio, velocity of ejection and angle of ejection.

The ejected material can fall back on the ejecting star,

on the other star, or be entirely lost from the system.

So, in principle, the period variation caused by mass

transfer or loss can be rather intricate. It should be

noted that Kopal's particle trajectory approach was rather

limited. Only single particles ejected at some mean

velocity were considered.

Kruzcwski (1963, 1964a), Piotrowski (1965), and

Huang (1963a, 1963b) derived formulae to express the mass

transfer rate in terms of the period variation for all of

these cases, subject to the usual conservation assumptions.

In some cases, mass was also allowed to escape from the

system. The next decisive step occurred when Iben (1965,

1966, 1967) examined evolutionary sequences for single

stars in a systematic, detailed manner. Figure 2-1 and

Figure 2-2 show graphically some of the results which Iben's

calculations made available. Perhaps the most important

quantity was the variation of stellar radius with time.

Further refinements, most notably by Plavec (1968),

now made it possible to tell at what point in a star's

evolution that its size was great enough to fill its Roche

lobe. Its subsequent evolution would now deviate greatly




10









LOG (Tc)


0.8-


1~l
0.6 s



0.4 \






Sj
0.2 2
LOG L\


1 2 3
1 (!C" yoe rs)


Figure 2-1


/


I .2.5. I'A i


^












tixanjLi-. .. s...A I -- a ....... tn T -:iYv -i.. a ..alsrlr i$aeIt.. ... r .....W C



0.8 1. 2-5 IM 0
I I

/ /
I


// / I


/o / c


/


I 23 '4
i (x ?0 ye rc .)


Figure 2-2





12



from what it would have been, had it not filled its Roche

lobe. Progress now came rapidly. The more important ad-

vances will be discussed in a later chapter, after we have

put forth the basic concepts and assumptions involved in

mass exchange, as well as some requisite definitions.















CHAPTER THREE

PRELIMINARY CONSIDERATIONS


3.1 Theoretical Context of Mass Transfer

Observed rates of period change are used to calcu-

late mass transfer rates in close binary systems. For

the most part, conservation of the total mass and con-

servation of total orbital momentum are assumed in such

calculations. The resulting values of dM/dt, which are

used to fit the observed system to theoretical models,

most likely suffer from errors introduced by these assump-

tions.

Henceforth, the term "mass loss" will refer to

mass lost from the gravitational potential well of a single

star or binary system. "Mass transfer," on the other hand,

will refer to exchange of material from component to

component in a binary system. In addition, "close binary"

will retain its usual meaning: a binary system which at

some point in its evolution has either one or both of its

stars exceed their Roche limiting surfaces. Furthermore,

we shall confine ourselves to use of the Roche model in

our study of mass transfer and loss.









3.2 The Roche Model

The classic Roche model gives surface of equal

potential around two stars in a binary system, subject to

the following assumptions:

1. The stars revolve about their mutual center

of gravity in circular orbits.

2. The density concentration is sufficiently high

so that the gravitational potential can be approximated by

that of mass points.

3. The stars rotate about their axes at a rate

equal to the orbital velocity, i.e., synchronously.

In a rotating cartesian coordinate system with the

origin of coordinates at the center of M1, the more

massive component, with the x-axis joining the two compo-

nents, and the the y-axis in the orbital plane, the surfaces

of equal potential given by


2 1 + 2q 2 2 q2 (3.2.1
C- x + x + y + ( 3. 2 .1)
Srl +q r2 +q2



where q = M2/MI (3.2.2)


Figure 3-1 shows the location of the two mass points, as

well as other parameters in Equation 3.2.1 for a Roche lobe.

Of particular interest for: this study is the point

labeled L1. The L1 point, or inner Lagrangian point, is

located at a point on the x-axis where; surfaces of constant

potential first become small enough so that surfaces





















I
I
I




I P















I
I I













Fioujre 3--].


~-jr ~IISi~U~F ~IMW-a L~IE~.Za~l~i~Orl~%~P1)6~)L~M-J*-d~C15~0


pakvBlrwfirwma lz arn -Iw sM x A= jw -l ii- -in I -ws ie >jmr,'









surrounding each star merge at that point. This point is

an obvious site of mass exchange between the components

in a close binary system, as material leaving the L1 point

with some initial velocity will continue in the direction

of ejection. If we allow the potentials of the equipoten-

tial surfaces to decrease further, we eventually come to

the L2 and L3 points.


3.3 Evidence for Mass Loss/Transfer

What evidence do we have of mass loss from stars?

In even as unspectacular a star as the sun, Parker (1958)

pointed out that the solar gravitational field cannot

retain coronal gases which have a temperature on the

order of one million degrees. It is presently thought

that all stars later than spectral type F5 possess well-

developed convection, and hence coronas and stellar winds.

Even though the average rate of mass loss from the sun is

exceedingly low, on the order of 10-1-3 M./year, the

"requirement" of mass loss is met. By stellar winds

alone we could expect intersystemic mass flow in close

binary systems.

We also have reasons to expect more spectacular

mass loss from stars. Despite the wide range of masses

to be found on the main sequence, the white dwarfs which

constitute the end product of stellar evolution all have

masses less than 1. 2M,. It appears thatI the precursors









of these white dwarfs lost a great deal of mass during some

phase of their existence, possibly in a cataclysmic event.

In a binary system, mass loss from a star normally

occurs much sooner and faster than in a single star. As

the initially more massive star evolves, it will reach a

point in suitably close systems, at which it will fill its

entire Roche lobe. Kuiper (1941) suggested that mass flow

could develop in a binary system in which both of the

components fill their Roche lobes.

Measurements of stellar polarization also give us

reason to expect mass loss in close binary systems. Pfeiffer

and Koch (1977) has recently outlined much of the current

thought on this matter. They reach the conclusion that

"ordinary" stellar and interstellar polarization may be

noticed and removed from a system. The interstellar

polarization component of a system, as seen in any given

direction, will remain more or less constant over a long

period of time, while it is unlikely that polarization

resulting from sources such as the Chandsiekhar Effect

will be sufficiently strong to influence the net polariza-

tion of the system. The only remaining source of variable

polarization arises from scattering due to circumstellar

envelopes of gaseous streams.

The system U Cep is again a prime example. Polar-

ization estJimtes have been negative for this system most

of the time. However, polariz.aion is detectable during









periods of high photometric and spectroscopic activity,

which probably signal a mass transfer event.

Spectroscopic evidence for mass loss from close

binary systems can be divided into two main categories:

1. Evidence for gaseous streams between the

components

2. Evidence for circumstellar envelopes

around either or both components

Evidence for the presence of gaseous streams can

be inferred from either emission or absorption lines in the

stellar spectra at crucial phases. Well-known examples of

this phenomenon include the "satellite" lines observed

before and after the mid-point of primary minimum for Beta

Lyr; emission during quadrature in Beta Per; and recent

observations of emission in U Cep during its 1974 outburst.

In all these cases, the phases at which emission or absorp-

tion is seen indicate that the region of activity is not

located on either of the stars, and hence must be caused

by a gasecus stream.

Observations of other systems, such as AO Cas or

U CB1 display broad emission features through which an

inspection of their radial velocities indicates that

these lines are produced by an envelope which surrounds

both stars.

Spectroscopic observations of several systems

also reveal the presence of non-synchronous]y rotating









primary and secondary components. As tidal forces in a

close binary system tend to synchronize the rotation of

the components, some driving mechanism is required to

produce the non-synchronism. Probably the best explanation

is mass transfer, on a time scale which is short compared

to the tidal time scale.

There is also a great deal of non-spectroscopic

evidence for mass transfer or loss in close binaries.

1. The presence of "extra" light at discrete

phases, usually before and after primary eclipse: Markworth

(1977) has accounted for such an effect in the close binary

U Cep by assuming the presence of a polar hot spot induced

by mass transfer.

2. Variation in the depth or width of the bottom

of primary eclipse: A large number of binaries display

this effect, which could be due to a change in the physical

dimensions of the star, or as seems more likely, it could

be due to mass traveling from the secondary to form a ring

around the primary.

3. Depression of the descending branch of primary

eclipse: This well-known effect could be due to a gaseous

stream impinqing upon the leading hemisphere of the primary

star.

4. Parabolic period var.iatiion: as is shown later,

this effect may be explained in terms of mass loss or

transfer.









5. Periodic variation in light level caused by

non-synchronous rotation: this is really the photometric

analogue to the spectroscopic effect mentioned previously.

The binary U Cep displays prominent light residuals which

reflect the five-times synchronous rotation of its primary.

There is unquestionably a wide and diverse body of

evidence which supports the possibility of mass loss and

transfer in a close binary system. We should like to

develop a theory of mass exchange which not only explains

these features, but also relates observed quantities to

theoretical predictions through these features.


3.4 Some Relations

A relation between the orbital period P, the orbital

angular momentum J and the mass transfer rate dM/dt has been

derived by Wilson and Stothers (1974) for conservative mass

flow. To be completely general, we may write


2 4 7T2
(M1 + .2)P G a3 (3.4.1)

where a =- semi--major axis.


Setting the orbital angular momentur'i equal to


aG 1/2
J = M.M2, i/2 (3.4.2)
1 MI 4+ M 2,

and sub.sti.tutinc Elquation 4.2 into ELquation 3.4 .1 we obt-ainr

27 (M1 --2) J3
P -. .--- (3.4.3)
C (Ml 2)









which, upon differentiation and some rearrangement yields

dJ kMIM2P1/3 -dP 3dM2
= + (MI -CM2)
dt 3 (M +M) 2/3 Pdt M 2dt



2dM2 1-C
dt (MI+M2)



where k = G2/3/(2n)1/3 (3.4.5)


and dMI = -CdM2 (3.4.6)


Here C is the function of mass leaving the secondary component

which falls upon the primary.

We may also write


Jtot orb rot rot2 lost (3.4.7)


So dJorb dJrotl drot2 dJlost (3.4.8)


as dJtot = 0 (3.4.9)


The change in orbital angular momentum must be accounted for

by either a change in rotational velocity of one component

or the other, or it must be lost to the system.

We may also express the ro.tati on. angur momenta,

and their changes, in the following fashion:


dJrotl --2 d"ri1 (3A410)









dJ _2 -- dM2r 2 (3.4.11)
dJrot2 p 2 2 2


Here e is the efficiency coefficient for conversion of

orbital to rotational angular momentum of material leaving

the secondary and intercepting the primary. If this

material falls tangentially on the primary, e assumes the

value of unity, while if material accretes radially, there

would be no net transfer of orbital angular momentum to

rotational angular momentum, and e would equal zero (to

the first order). The efficiency coefficient will be

discussed in detail in a later section.

If we now set C=O (no mass transferred) both dJrot

and dJrot2 equal zero and dJorb=dJlost. We have


dJlost klM1M2pl/3 dP 3Md1M2 M2dM2
d + c-2-K2d 1 (3.4.12)
dt 3 (M +I )2/3 PdLLt + (M(3.4.122)dt


while if we set C=l (all mass transferred) dJiost=0 and


dJrottot dJrotl r+ dJ o2 (3.4.13)

giving

dJrot kMIM2P1/ dP 3dM2 -M)
St 3(M+M) + .1Md4 (3.4.14)
3t (M P 4 MI'



In both Equation 3.4.12 and Equation 3.4.14, dP/dt is an

observational quantity. The quantity dJlost can ba calcu-

lated by either partic.le trajectory or hydrodynamical









models. For all cases in this study it is assumed to be

negligible and is set equal to zero. The quantity dJrottot

can also be calculated by either of these causes, or

measured directly on systems which have a non-synchronously

rotating primary star. It is tacitly assumed that the time

scale of mass transfer and the accompanying change of

orbital to rotational moment, as in the Biermann-Hall model,

proceed on a time scale dynamicall) which is short compared

to the tidal time scale. As such, synchronously rotating

primaries probably display a large fraction of the actual

orbital-rotational. momentum conversion process.

To be general, we may write


dJrot2 kMiM2P1/3 dP
dJ + ----" + dJ -
Jrotl lost 3 (MIM2) 2/3



3dM2 (M1--C2)
+



2dM (l-C)
S 2) (3.4.15)
(M1+1i 2)


This is just the previously derived equation without any

simplifying assumptions.

It might be argued that by computing the magnitude

of the orbital--rotal-tional conversion process frorm non-

synchronously rotating :tars, we are neglecting the fact

that at the point in time wh en they had synchronous rotation









velocities, the (then) secondary actually had much less

mass than the (now) primary does. Thus, the error intro-

duced will tend to underestimate the total transfer

process. Nonetheless, the method used in the previous

section can be taken as a generous lower limit. A more

definitive value could be obtained by a "two-dimensional"

evolution program. This process will be covered in a later

section. An approximation to this "definitive" value can

be calculated in the following fashion. Initially,


Jrot2i = M2i i 2i (3.4.16)



while eventually


ot2f M 2 0 (3.4.17)
rot27 2 f 2 2f


Subtracting to get dJ we obtain


dJ = -Jrot2f + roti (3.4.18)



where we assume values of initial and final sizes from the

Mass-Radius Law.
















CHAPTER FOUR

SELECTION OF THE SYSTEMS


4.1 Selection of Stars

The stars used for this investigation were selected

from several sources. As a rough initial indication of

which systems would be most suitable for a study of mass

transfer, the quantity dP/P was calculated for the 333 Wood

and Forbes (1963) systems. This quantity, the observed

rate of period change, has generally been used to calculate

mass transfer rates in the so-called conservative case--a

case where total mass and orbital angular momentum are

conserved. To the first order, it can be assumed that the

greater the quantity dP/P, the greater the mass exchange

rate regardless of what is happening to the angular momentum

of the system.

In addition to the Wood and Forbes material, the

Graded Photometric Catalogue (Koch, Plavec, Wood; 1970),

Rocznik Astronomiczy and an article by Kreiner (1971) were

utilized to develop a working list of systems which have

suspected or known period variations. These references

were cross-checked to provide a preliminary list of over

750 objects. As it is the aim of this investigation to









accumulate all available times of minima for the stars

under consideration, this rather cumbersome number of stars

was pruned down to a workable number of more or less

high-grade systems by several methods. In fact, very few

of these 750 systems were suitable for this study. In

general, to be of high value for our purposes, a system

should display a long history of recorded minima obtained

at a large number of epochs. Thus, both the quantity and

quality of the observations are important parameters. As

discussed by several authors, photoelectric times of

minima are preferable to visual or photographic timings

of minima. These particular criteria were difficult to

fulfill for most of the 750 systems.

The 100 close binary systems which displayed the

greatest variation of period, dP/P, in the Wood and Forbes

article (1963) were given a preliminary inspection. Each

of these systems was cross-referenced to the literature,

mainly through the Catalogue of Eclipsing Binaries which

is available here at the University of Florida.

For each system, the card file contains cross-

references to the literature for a broad spectrum of

material. It was apparent that a large percentage of

these top 100 systeirs were so poorly observed that they

would be of minimal -use for this study. In many cases,

the star had not been observed even once since 1963 when

Wood and I'o-rb made their sLudy. On the other hand, some









systems which had a sufficient number of observations

had insufficient quality of observations. Low quality does

not necessarily mean that the individual observations were

substandard, but that, for example, they might have a

tendency to clump together at two discrete intervals. This

introduces ambiguity into the nature of the period varia-

tion and leads to minimal information content of the

variation.

A "final" list of 188 stars which were to be used

for the correlation study was created by a combination of

several diverse criteria.

1. The object was restricted to known-contact or

semi-detached type systems.

2. Interesting objects which did not necessarily

fall into these groups, but which displayed a large-scale

period variation, were used.

3. As a final criterion, sufficient data for each

system were necessary. Except for very interesting objects

this means, at the bare minimurn, ;-t least a dozen observa-

tions.


4.2 Absolute Paramreters

The value of the mass flow; rate, dM/dt, can be

calculated from the period change by Equation 4.2.1 under

the assumptions of consei vationo of mass end o-rbital angular

momentum. In order to perform










dP 3(1-1) dM (4.2.1)
P p M


where i = M2_
M1 + M2


this calculation, the values of P, M1 and M2 are necessary.

To make the best possible use of the data assembled thus far in

this study, a knowledge of these parameters, as well as other

"absolute" parameters of each system, is desirable. Table 4-1

lists some of the "absolute" parameters for each binary used in

this study. Column one lists the star and columns two and three

for its spectral classification. The next two columns contain

the masses of the secondary and primary star. Columns five and

six contain the radii of the stars in terms of the orbital

separation. The final column contains an integer which speci-

fies the configuration of the system. A "1" signifies a

detached system, a "2" denotes senmi-datached, while a "4'

signifies a contact system.

The classification of a system as detached, semi-

detached, or contact was performed in the following fashion.

1. The position of the inner Lacgrangian point was

calculated by numerically solving Equation 4.2.2 to an

accuracy of about one percent. In this equation, which was

derived by Kruzewski (1963), q is the


S1 (1+q) ( f 2r = 0 (4.2.2)
r 3(1+r)









'TABLE 4-1
ABSOLUTE PARA!;MEiTERS



SPL CTir.L IY4S ,ASMOfM'


kT AN .
1 ,', Ai,,o
, Z A ND)
X/ AlND
At; A;,D
x AiND
QIY A;
SU AI'.
C x I "A


,.G AL.L

i) .A uL
*LV /,vL
QY AwL
V23 7 :, L
V3\&' A .
Rz ALu-:
TT A'

Z/ Auk




I'. 'kL i.3\
? Au R-.
1 u Au%
SU "LU
Iy" -C,

Y CrA.
35 CAM
&V C -.',,
SZ CA m
A5 C AM
S CNO,
'Y C (-C
TY CC -,
v S C. i
R C(' A
A CAP
Ot CAS,
'-X C 0,S
SX C/ A S
TV (.CAS.
T. C ,b
S,4 C AS


C* 'C ,


L Co s

' .. t i
V ,'. C t.
I' CLIN


C.' C'. "



U Cl

S C Y,
w UP.


f-
F 0
AO


A3
A,0
A
f 2
A 2
S- <


A 1 V



f-

A 7






Li
AL.










G 3
A O
f* 1 V











I 2.i
;..4









1,4
A 7










A '3
. i
1 .7







A 7
A -


r-.C


C 9 -30
0.4c 0
I : CC 0


2. 4*200


1 O -0


. 2 4 U 0

.3000
. 44G
*, 30 O
. 2400


A 3V

u ^
t, 2


0. 5.C 2. 900


* '; C 0
.1 5H
S.3600
12930
. 36000
. 3 .20
. 2070


. 3! 4 1 .2030


23 C r'3
0. 70- 2. 7C0 .24j)0 1 8M


9, MC; 3 4.900 3 c
1-c00
r- C7 1' -" > ,

0.4801 2. 200 2 40
9.3 9. i30 1-:
9 700 6.'0 0 3
?? C) 3

0., 0 *. O
0.3 C (. 1 0 c 0


0 .7 oC 1. .5 0 ,. ,-O


: C 0 ( 0 1 ,, 3


kOi V
r. C3
A4



F
o j
,o



Or-
L I

C,3 3 1
rsb.


.2310



. 293




.377



0 1 7'

.03 >7


1 -3 C 1. 0 C I 2 3 0940
06. 1 C 30,0 .2:9 c. 0

5,770 5. o90 ,v1u0 .?160
.2CO0 C. 040 0 2. 1 70
2?00 Q 0 C0C
I. 000 1 700 211 .2 C70
1 1 0 2. C0 0 o-o .2 210 0


0.640 2.040


1. 10 9I. 3c

1 ,SCG 7ICC
0. 7' C C
0C).2 0.01 0


0. 70
CI .. C





1 ., 00

C. 7 (C


1. 7 ;: C,
1 i ., i' C



1. 750
2 9 (. 0

2. ., 0 0


.57w2
,o500

* l 100
. .> C

. 4I j C

. ? C., I"


.4270
* 2 0 0
S. 2 7 03



1 2 0

* :2 1 C' 0

.1 Y.',0


,. .uo0 2 O C*I 2.
. 2 cO l -30 I.


NA& L.


r.AD) I I


TYPL


.21a0D 1 ,30










TABLE 4- l--Continuced


bPEiCTrAL 1YPE'!:S


M ASSET. S


63
64
65
66
67
( 3
69
70
71
72
73
74
75
76
77
7b
79
80
81
82
83
S i
L'

87

c'S
90
91
92
G J
94
9Q
96
C97
956
99

1051
1 02
10
1034
I DS
100,


110
11
120
1 09
1 1 0


114

11 5
316.
137
11 24
111



1: 1
121

12 4


V., CYC
SY CYG
t. ,* C Y I
vZ C G
7Z C.YG
CV CYG
E M*I C o
Grj CL(
GU C. C1

V 453 C Y G
V 4,

Z D) A






S Gl~
*< ( i FZ O.AL
S K D rA
T/ DU-A
T/ t-. A


R U EL I
U. i i
U ,G M
YY GcEM
AF CEN ;
AY G -,
DEX ,C^

FY CG !


! L r
IT h;;lk
TU i'L. .
T. HR
UX Hr :
AK HIR.
CC H LR
C T !Wt_
DL) HL.
V33. !' i
.S\ i 'A
S,\ I_ A(



VS tA (
AISo L AC
T o L AC
Us' LAC
vx L ..



DO LAC
Y LK ..
RT L **C

KY LW,
I L l


rV l Ii

TT 1. Y'


KO

GM IV




F 0
09
- 1


0.700 2.500 .22D0 .1 110

4. 300 7. O 0 .274C0 .21 90

0.700 1 2.'0 r 0 35, 0 0


. S30 (0. 9 0 4540 .4080

13.000 17.000 .2900 1780
0okO 0 '/;40
0 .4 0 2. 13 27 370
0 3 0 1.400 2 0 2570


.3270 .2050
0.620 2. O0 .330C .2100


0. 3. 2. 0O 2230 2310

.5050 ,4000

0.5o 0. B 0 .1 5 1 t 60


A3
A3
b7V,
- 0
b-7
t d

bLy
il I


A0


1,
A7 v
AL2




AL



A
AO




A 7
A c

AO
F -'






A 2


A 3
AO





A V
A 9
A

A 3
AO













6 0 V

Lbi V
A 3








AC 0

GI U V
A O
A.,O


A U
LA' i'
1). 1i
I'-
1.:U
p.0


0 C- 3 0
0. t2
1 .1.. 0
2.30 0



1. 7,0
0 1 4 90





0. 3sO:

0 71
0. C


2 c 0
L. 700
3 .:- :
2. 7- .

3.310

2.050
1 .Y00
1 *3 0





1 700

0. ;0


19-'30




3 c c 0
320




5340




291 0
1730








2270
S1 10 0


1 l 0 3 2 0 .1 1-.0



0, ,;0 0 1 100 2, 3

0.9350 1 .020 3')o0


.0700
.3 270
.1130

.3109
..3 1 0 ?
.2720

. 1 3 C

.24 0 0



.2270
51 00

S1 300
.330


.1940
. 1 i :0
S2 1 600

.2300

. 3010 C


S26L- .2170


4. 4,. .2 6?40


S29.', 0 .1 P O


RADIO 1


1YPL


G 3
/ 1
,<0


A '
io
to


SL) I I


N 3
K0

Go-




r. C* 1 v'





I V


INAVE










TABLE 4-1-- Continued


SPE C.T'

125
126
1 P?7
1 25
129
130
1 51
132
133
1 3 4
135
1 3
1 37
338
139
I (
14C
141
142.
143
11. ?

145
140
147
1 4&
I C
144
150
15 !
1 '52
15 .




1 5
1 59
1to

1 62
163
1 64
165
160
1 67
169

171
172
171
1 7 b
1703
177

1 7
3 7 U
I t 0
1'F
1 !'
1e ;
1! C-L
35 *


TZ L.. Y
UZ L Y N
E.I LY" 1
!' ,; 1-U.N
n ;ULN
U C[.PI-I
- V L 'PG




V1. t&lP i
V4I0 *'.Pth

V5C lPM


CL 'M
E(. ( i 1
Si'' '. r I
F 1 C 41
F-H CP I
F .;,

G ( I. I

U PL TY P[.G
UX Pi.G
A 0 I;. (.
,0 pn6
r1u PL.
[:'< 'tc (
1,1 ;'' L
Z P :_.



X/ P k
6 't k
il ir F I-
XZ PEr

Y p SC.
SZ P.C
U S(, :
I S luk
"uX Su .
YY LoAr
V 3' SG4R
V ?: oi.
v ?QF 50K-.






x, 1


1? U.iA-
r., ii'.
lx UOAA

AG V 1 i
AH \ i.
. VUL'


.4180 .3140


67

Pc5
G5


AZ

G2V

" 3


O 0
A2
FO
A 0
A2
Cb4
AO
AD

;,5
AO C




cl\/
pi0 9V

f V'

GO


Gi
G3
Al
A 7
AO
A i
A2

1- (I
A R


AC
A?.
A V
Fb
KO
AO
F2V

Abi l>
A;> I I I
A 3
l1
t-'9
A S








I- i ;V
Ub7V









I L'
A)
AO
uj'V
Al V
A.1
AO

A
F :-

h9




wV
A.-
. 0 'V
bny


1.300 3.700 .3000 .2170

4. Cb0 .30C 0 .. 2520
.2070 14 6
0.420 2.760 33,0 1970

2.60C 3. 000 ,2I o) 173C


0.490


1 .220 .4.0 .3200 4


0.440 1. 300 C 54 0 .2500 4
j. 3 C 700 .3 4 3 0 .1 790 1



0.2_90 0.4C0 4000 3400 4


1.070 1.3 30
0. 30 .1 500

0.0C0 2.5C 0


0 3? 0
CC 4 v C' i


CD, 41C)

1 ,: 170

C!. 9~ Lv


1 .320
2 C2 C
2. 000C
2.40 0


2. I0C 0
1. 370
1.400
1 400


2 290 2.P (' 0
4 700 1 2 3 00
1 .210 2. 30

1 .:0 0 2. 11 0


0. 2 0

I 000

0. .20.


0. 39.


3 c I C 0

5 720 C

0 9' 2 0
0. 0


1'3 ". 0
> 300


* 4 '3 0






.22 0
*27, 0
. '7<.0O
. 1700
. 23 0







, :, 9 }

. 150
, 31M0
.3100


. 3400
. 2170

,0230





* 1 3 (.. 0'
.1 1 0
.1900




.21 10
.2610

. 1 5c 0
.1100
. 3 4 0


. 150O 1


. 4 i4 .3300

C. 2-3"0 1 850

32 8t< 2631
. 3300 .1 (600
S,- C0 2 7^' 0

, 21" 0 1 0 00
. -, b .2300
. "0 0 ;t'0
. 33C.0 .31 00


NAM .'


MASSi S


m

TYPE


t 3
nO
tGO



G0 1 V
r. 1

S00
lV





A3

A 0
A 21 1I
AZII


A, i

F 3

KU IV
A0
G-A

F t
f 2


A .*













NAr :
N A,;" .


1U7 PS VULt
I 18 bC Vu L


TABLE 4--l--Continued


SPE C' AL T YiL MAS,.LS


ObV 6 1 V 1.400 4.6
AO /I V


t ALDI I YT L


00 .23<0 .2000 2









mass ratio, r the distance to the inner Lagrangian point,

and f a non-synchronous parameter.

2. Inspection of Table 3-1 in Kopal (1959) shows

that for a range of mass ratios from 0.1 to 1.0, the average

value of the average of the top and side radii of the Roche

limiting surfaces, as compared to the radius to the inner

Lagrangian point, varies less than one percent from a value

of 0.75. As such, a "mean radius" of a star which would

fill its Roche lobe was taken to be 75 percent of the

distance from the center of the star to the inner Lagrangian

point.

3. Naturally, not all stars which have previously

designated as "contact" or "semi-detached" will necessarily

have radii equal to this value. The difficulty here seems

to be the fact that no matLer what scale height we adopt

for the atmosphere of the component .in question, its

atmosphere theoretically extends to infinity. Naturally,

the amount above one or two scale heights is insignificant

but at what point do we make the cutoff? At what point

of the star's atmosphere do we observe to determine a value

for the star's radius? If we examine this question from

an evoliut ionary viewpoint, an answer seems possible. During

the evolution of the more massive component a point will

be reached where the star: initiates its ex-pansion to the

red giant tip. (We have heie disregarded the possibility

of Case A mass e:xcl;ange.) The ratio of time spent on the









main sequence to time spent in the expansion phase is very

large, so from an observational viewpoint we would expect

to find virtually all of the stars which have initiated

their expansion to lie at or near their Roche limiting

surface for close binary systems. To allow for some minor

departures from the Roche surface, due to perhaps some sort

of instability or pulsations, we shall assume that a star

which has an "observable" radius of more than 95 percent

of that of the mean radius, as calculated earlier, actually

fills its Roche lobe for purposes of identification.

4. If the combined radii of the two components

exceed 0.75, in terms of fractional radius, the system will

be designated as a contact system.

In large, initial values for the period, initial

epoch and spectral type were found from either the GCVS

or the Graded Photometric Catalogue (Koch, Plavec, Wood;

1970). Relative radii were also obtained mainly from the

GPC. Rotational velocities were obtained from a list

furnished by Stothers (1973) and a list by Levato (1974)

Values for the masses were obtained from a much wider

variety of references, although Batten (1962) and Giannone

et al. (1.967) provided the majority of the entries.

Whereas the value of the period can be determined

to a high degree of accuracy for each system, numerical

values of other systemic parameters are less well-known.

Kopal (1959) and Giannonne (1967) have outlined procedures









whereby a binary system's absolute parameters can be deduced

by photometric methods, spectroscopic methods, or a com-

bination of assorted scattered data from both methods

simultaneously. Unfortunately, reliable values of MI, M2,

Li, L2, R1 and R2 are available from primary standards for

only a limited number of systems. The need for continual

expansion, revision and reassessment of the available data

cannot be over-emphasized.


4.3 Parameter Correlations

Some important relationships exist among the

parameters listed in Table 4-1. In all graphical repre-

sentations of parameters to be found in this section, the

usual designation of a system as detached, semi-detached

or contact will be performed in the following fashion:

Detached = *

Semi-detached = +

Contact = x

Uncla ssif ied -

Criteria discussed in Section 4.2 have been used for this

subdivision. It can be noted in most cases that there is

an overlap of the different types of systems on the various

diagrams,

Figures 4-i and 4--2 contain Maiss-Radius plots for

the primary and secondary c omp:onc:nts respectively. It can

be not c o d th::t an inspe cti con of detached systems only


















1.OCO +----- --.


I

I
I
I
I
1
I
I
C.(;C)G ~--
1

I
I
I
1

1
I
?IT -I--
1
I
j
I
1
1
1
J
i
--Cc3'vl. ~...-
I
I

I
I
1
1




I
f
1
i

I
I
I
1
~ --


1 1 1 1. 1
I I I I I
I I I I I
J 1 1 + 1 1


I
I
I

I

.4

I
I
I
I


Ix



jI



I

Ix
I


I

Ix )

I





I

I



I


I


!


I I I
I X < I I


1. I
I I
-f I, I

I I


I
I

I +
-- ----

1 --

I
1 +

I 4 iC:
1 +





-1 X
i I
1







I





I
I
1





1
I


1

I
1





1
1
1

r-- "i *'- --


I

I

4--.


I

-4
1
I

I




1
1

I


I

J

I

I

I

I


I
r


J o -'i ,. ''


1. i ;. ( %, 1 "


F-gure 4-


I. & i; u *.


1 1
I I
I I
I I



1 1
1 I
I I

1 1
II
I I


I I
I I



I I

1 1
I- I
I I

1 1
I I

I 1
I I

I

I I

I I
I 1
* -I


] *" "


'
I


















--4---- ----- -


I

1 +


I

I .


I


I
X I. 1


I
'-
--4 .. .. -- -- .. ......


C.?(K ~-:-

I 1

SI-
I
I

I

-C.:!C<



II

I







it I


I 4 .~
1


1 Y4





I X:
-f I
I )>.

x


SI





I


I

I


- C 4. (


Ir '
1K 4


4 X I
I
'; I

1
+ I


I



I
I


I
1
I
I

1
1
I
I
1
I
1
C


I
1


1
I
-1--
1
I
I.
I

~1
4-

I










I

I


I
I
I


1
I
I
I
I



I
I
---- ------~- ------I-

I
I
I


I

1
I
I
I.
I

I


1 4 0 .:'I


I ( : .," )


1
I
7
I
I


I
+ 1
1


I
I
i
I
I
I


I
I
I
1
I
I
I




2 C C00


1.00C .------ ------ -----+--


- -------i------~- -


'*^*'


.. -- .- -- 4-


*" 0


)"igcare 4-2









reveals an M-R diagram similar to that given by Kopal

(1959). Some other general features are as follows:

1. The primary component of the semi-detached

systems behaves essentially as the primary in a detached

system, at least as far as its behavior in the M-R diagram

is concerned. To a certain extent, primary components of

contact systems also display this behavior.

2. There appears to be a change of slope in the

M-R diagram for the primary component at log(Rs)=0.5.

3. The secondary components of semi-detached

systems deviate from the M-R relationship given by the

other types.

4. The secondary components of contact systems

strangely obey the same M-R relation as given by the

detached systems.

Linear regressions performed upon the entire

ensemble of data give the following two relations:


Log(R ) 0.156 + 0.647o09(M ) o = 0.240 (4.3.1)
S42 83


Log(1)s) 0.534 + 0.3091og(Mc) a 0.312 (4.3.2)
38 87


Here s and g refer to the smaller and greater radii, re-

spectively, while a is the standard deviation.

Figure 4-3 shows a plot of Rs versus R Tis

diagram serves mainly to provide an apprecit ion of the















1 .OCO -..----------
I
1
I
I
1
I
I
1
I
C.6Cc ----------.
1
I
I
1
I
I
I
I

I
C.?CC: 4 -- ------ -- -


I

1
I
I
I


1



1

i
1
I
1
1
-- C ,, (, f : .' .. ... .... .. .. .
I
I
I
1
1I

I
1


;. --------- j --- ------- 4 ~ ------ --- --~ -----
I I I I I
I 1 I I 1
I I I I I
I I I I I

I I I I 1
I I I X *
1 I I 1
1 I I I + I
I I 1 I + + I
t.-------------. -------- -- -..-.----. -- .. .+-+- --
I 1 + 4' I
I I I 1I I

11 I 1 1 I
I I I I I1

I 1 1 + +- I I
I I I *+ I I
I I I I 1I
1I 1 X+ x I I
.. .. .... ...... +. ..... .. --- : 4 % --y .. ..+ -'' .. .. .- +
I ) 1 I I
I I + I I
I1 I I I
1 I X I x I I
1 I XX I I
I1 I; 1 1
I I X- 1I I I
I I1 I
I 1 X i I I

1 ) X: [ I I
I 5 I I I
I I I I I

II X i I I
r 1 1 1 I
I i I I J
I I I I
I I I I I
I I I

1 i J I I
I I I I I
1 1 I 1 1
I I I I I
I 1 1 1 1
1 ) 1 1 I
1 1 ) 1 1
1 1 I I I
; 5 1 i


C c GC 0


1 C r.





40



dispersion of radii for the different types of close

binaries. A linear regression yields


Rs = 0.181 + 0.649Rg a = 0.788 (4.3.3)
114 25


Figures 4-4 and 4-5 display the percentage of total

systemic mass contained by the primary and secondary com-

ponents. With few exceptions, a linear relationship

adequately fits both cases. Again, it is to be noticed

that the exceptions are the semi-detached systems. In the

Log(M1+M2) versus Log(MII) diagram, we notice that not

only do the semi-detached systems display a grouping toward

lower masses than the detached systems, but they also

contain a lower fraction of the systemic mass than the

detached systems. It is possible to conclude that these

systems have lost a relatively large fraction of their

systemic mass during their evolution as close binary sys-

tems, if we are willing to believe that this distribution

is not greatly altered by selection effects. We have


M1 4 M2 = -0.4864 1.695M = 1.284 (4.3.4)
1919 42


M1 + M.2 1.280 + 2.3.20M_ a 1.629 (4.3.5)
211 65


Suhtracting Equation 4.3.4 from Equation 4.3.5 and rearrang-

ing somewhat yields


Mg i. 2771M + 1.065


(4.3.6)





41








2.00C ---------+--------- ---------4 ---------+---------+
2.000 ------ -- -- -- -
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I + I I
1.400 + -- ------- +-- ---- -- -----+--- -----
I I I I 4 I I
I I I I I I
I I I I + I I
I I I I I
I I I -4 + I I
I I I X I I
I I I +- I I
I 1 1 I I I
I I I +I I I
C.80C --------- +- ----------+--- ----- -+ ---------+--- -------
I I I + I I I
I I I ** + I I I
1 I 1 XX I I I
I I Ic-+4+ I I I
I I I'4 ++' I I I
I I 'IIX +X I I I
I I xtx4 + I I I
I I t 1+ 1 I I
I I +XXX+ I I I
C 200 -------- -----X -+--- + +---- ----+------ --+ ---------
I I I 1 I I
1 I I I I I
I I X I I I I
I IX 1 I 1 I
I IX I I I I
1 1 + 1 I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
-C 4 0 ------ ----.-.----.-- -. ---- -- -.--..-- ---- --- -----------+
I I I I I I
I I I I I I
J I I I I I


I I I I I I
I I I I I I
I I I I I I
1 1 1 1 1 I
-1 2 30 C ------------------------------ ---------------------------------- -


-1 00


- 0 & I


C .2 0 )


0.800G


1.4 00


2,CO0


"i'jri, .-A


LOGCC ( T w,)0















2c;D s 00 0 ---------+ --- --- ----+-r------ ---+-(--- --- ---+-----------
I I 1 1 I I
I I I 1 1 I
I I 1 1 I I
I I I I I I
I 1 I I I I
I I I I I I
I I I I I I
I I I I 1 I
I I I I .44 1
I 0 +-- -------.----- --- -- --- ---------------+---- ---- +
I I I I '; I I
I I I I I I
1 1 1 I I I
I I I I I
1 1 1 I- 1 I I
I I X 1 I I I
r' i I -4 < I I

L. I 1 1 I I I
O C t C C ------- i ---- ---- -:-. -;.----.- -- -.- -- ---- ..-- -:- .---- _..---- -
C- 1 I + : I I I
( I I 1 1 1 1

C I I > ,: 1 I I I
I 1 + I I
E I 1 I 1 1
4 I I + ,: X i I I I
.' I + 1 1 1 I I
T 1 X X I 1 J I1
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C I i 1 I I 1
)I I I I I I
I X I I I I
I >: I 1 I I I

I I 1 1 I I
I- I I 1 I 1
I 1 1 1 i .


1 i I I I I
1 1 1 I I I
I I I ] I I
I I I I I

I 1 1 1 I I
1 I 1 1 I

1 I I ] 1 I
.L I

L -C 1 ., '." -- ,, / )' +: i ,, i ': cPy. c "j

L t-: ( l; : : r )


P" q-r 4-3









The spectral class of each component is plotted

against its mass in Figures 4-6 and 4-7. For a given

spectral classification, we can again see that the detached

components have a generally higher mass than either the

semi-detached or contact systems. Likewise, the semi-

detached systems have systematically greater masses than

the contact systems for a given spectral class.

Finally, Figure 4-8 shows the run of spectral types

for the hotter and cooler components. We see that the semi-

detached components lie above the locus of detached components

and contact components. To a certain degree, the location

of the contact components in this diagram is to be expected,

as they are presumed to share a common envelope. However,

four (about 25 percent) contact systems display a lack of

similarity in spectral types for their components.

The abscissa of diagrams 4-6, 4-7 and 4-8 can be

related to the spectral type in the following fashion:

1. Spectral types 0, 1, A, F, G and K

begin at decimal 0, 2, 4, ;, 8 and

10, respect ively.

2. Subclasses are plotted as the

appropriate fraction Co the int1er-

val between spec:tral types.





44










2C.C000 +---------- 4-------- --+---- ----+ ------------- ------- -
I I 1 I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I I I I
I I I 1 I I
I 1 I I I I
1 I I I I I
I I 1 I I I

I 1 I I I 1
1 + I I 1 I I
I 1 I I I I

I I I I 1 I
r Xi I I I
I 1 I I i 1 I
, 5 + -. .. + .- + ...-... .. ... .. -- -- ... .... .. ..-- -. .. .. ..


1 +1
I 1
1 14d
I 1
1 1
1 >: j':'
i I
1 1
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Sr ,


-i -?

















2.000 C --------.---
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C (0 C -:---
1
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- --------- --------- +---------+.---------I.
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I 1 I 1 I
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1 I 1 1 1
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41 -r I


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^-


0


SF CT "AL 1 (.Y R LLP

















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4


+-I I
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+ 4 1

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+ + 1
4 1I
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>'( j -4


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+- I
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- ~tJtt


1i C C


* ,* *
















CHAPTER FIVE

DETERMINATION OF THE PERIOD AND ITS CHANGE


5.1 Basic Concents

As stated before, the period of an eclipsing binary

can be determined to a very high degree of precision. This

is accomplished by recording observed times of minimum

light, and fitting these observations to an equation of the

form


T = TO + P-E (5.1.1)


by either the method of least squares or some other criteria

of fit. Here, T is the time of observation, TO the initial

epoch from whence the cycle count E is determined, and P

the period.

Each observed time of primary or secondary minimum

which is obtained is plotted on an O-C diagram, where the

abscissa represents the cycle count E, and the ordinate the

deviation of that particular observation from the expected

time of minima as given by Equation 5.1.1, Deciphering the

exact functi onal form of the resulting curve is not nearly

as easy as merely noticing if the period is variable. Sub--

jective crite-ria of period variat:icn rest heavily upon the









individual researcher, as well as what sort of variations

we are willing to allow.


5.2 Causes of Variation

In general, there are many reasons to expect the

period of a binary system to vary. First of all, if the

binary system is revolving about another relatively

distant mass point, the times of both primary and secondary

minima will display a periodic variation in phase, caused

by the light-time effect of their orbit about the third

body. Of course, a strictly sinusoidal variation will

occur only for perfectly circular orbits. The presence of

eccentricity in the orbit about the wide pair will lead to

a distorted sinusoid whose shape depends upon the elements

of the elliptical orbit.

Secondly, we can also expect a periodic variation

of primary and secondary minima, but with secondary 180

degrees out of phase with respect to primary, caused by the

rotation of the line of apsides. Wood and Sahade (1977)

have recently discussed aps.idal motion as a cause of period

variation. Essentially, the density distribution of non--

spherical bodies wil. lead to an advance of the line of

apsides. The ratio of the period of rev'\olution of the

orbit to the rotation of the apsidal line, is given by

Equation 5.2.1, due to Cox] ing (1938) Here, k- and k2 are

constants which (depend upon the actual










R1 5 fR2 5
e = kj -5 (1416M /M2) + k2 (1+16M1/M2) (5.2.1)


density distribution, R1 and R2 the stellar radii, M1 and M2

the masses, and r the separation of the centers of the two

stars. Although a long interval is needed for a complete

cycle of apsidal motion, its presence o. absence can be

inferred by only a few strategically placed observations of

secondary minima. Even a single observation of secondary

minimum can be used under optimal circumstances to infer if

the variation of the period is due to apsidal motion.

Several observations of secondary minima can substantiate

that it is not.

As the calculation of the mass flow rate in a large

number of systems is one of the prime objectives of this

study, the following procedure will be adopted: both light

time and apsidal motion effects will be "rectified" from a

system's O-C diagram, so that tle resulting variation can

be carefully inspected to see whether any of the remaining

variation can be attributed to mass loss or transfer. In

practice, this will be done simultaneously in the least

squares process.

Mass loss or transfer) from ei the star will also

lead to a variation of period for an eclipsing binary.

The exact functional form of this variation will be dealt

with later as we are now concerned on.ly with the qualitative

shape of the O-C curve. Depending upon the nature of the









mass exchange, the period can either increase or decrease.

Several different types of curves can be expected. So-

called "abrupt" changes, or rapid variations in the slope

of an O-C curve, indicate a discontinuity in the behavior

of a star's O-C curve. For example, this effect is illus-

trated by a system whose behavior can adequately be

represented by a linear fit, undergoes a discontinuous

period change, and then can be explained by a slightly

different linear fit. Such a change could be caused by

mass loss from the system, or mass loss or transfer from

either star, as long as the event occurred quickly. If

sufficient time resolution of successive minima is available,

the discontinuity in slope may exhibit some detailed

structure. To find out just how quickly such an event can

occur is one of the objectives of this study.

Mass transfer can aJ.so produce a parabolic shape

in the O-C diagram. We can write


dTmin PdE (5.2.2)


integrating 5.2.2 yields:


Tmin d (5.2.3)


Assuming thai-


dP
P : P + --I E (5.2.4)
0 d+ a i L












f1] .dP E 2
Tmin = TO + POE + (JdE E





dP dP dt
dE dt dE





dt
= mP
dE


Ti T + P I + (1p ][F 11 F2
min 0 0+ +2 dtj


The shape of this parabola can be either upwards or down-

wards, depending upon the sign of tile coefficients.

In theory, even more intricate curves are possible.

Instead of assuming a linear change in P with E as we did

in 5.2.4, we could have included an "acceleration" factor

as in Equation 5.2.9. Assuming


dP d0P ,2
d GE dE


(5.2.9)


we arrive at



Tmin --TOE T+ ( +-) d E2' t (
TP-OET + _1T ---j E~+ ,1 I T- (5.2.10)
C, ~ dtl 31j'dt3


yields


but


and


(5.2.5)


(5.2.6)


hence:


(5.2.7)


(5.2.8)









In practice, the so-called "acceleration" term in Equation

5.2.9 has been more or less devoid of relevant physical

meaning thus far,

These effects are in no way mutually exclusive.

Apsidal motion, light-time effect, parabolic mass exchange

and abrupt changes could all occur simultaneously. What

were more or less physically unrelated processes for the

binary star system could become analytically intertwined

if insufficient data were available, or if the analysis

were carried out incorrectly.


5.3 Program OMC

In order to retain as much flexibility as possible,

analysis of the available data will be done so as to

provide several different types of fit. As we are ulti-

mately interested in the quantity dP/dt so that we can also

obtain dM/dt and dJ/dt, special attention will be paid to

variation in dP/dt from one type of fit to another. The

following least squares fits will be calculated for the

ensemble of data for each system:


I. A linear fit of the form rT=T+PE (5.3.1)


2. A parabolic fit of ihe form TT -i-PE +AE2 (5.3.2)


3. A cubic fit of the form T--TO+PE'+AE-2+B (5.3.3)


4. A comhbinatiJon parabolic/periodic fit of

the form T=TO PE+AE -Din (' (E--E ) ) (5.3.4)









where D is the amplitude of the sine term, o, its frequency,

E1 the time of periastron passage and E the time of observa-

tion. Within the framework of the weighting scheme used

in this study, the calculated values of the coefficients

of these equations corresponded quite well between the

present study and that of Wood and Forbes (1963).

Solution of method (4) will be carried out in the

following fashion. We can write


T TO + PE + AE2 + Dsin(m(E El)) = (5.3.5)


and by Taylor's theorem


dd de,
( = f(TO,P,], D,(, E,E.) + d, A'0 + d0 AP
d d0 d dc



+ A A + F AD Ad +j AE
dA dD de dE


dq
+ d- AEl 0 (5.3.6)



here f(T0,P,A,D,w,E,E]i) equals calculated time of minima

based on initial guesses for TOg ,P' g ,Dg g,Eg, and E.

Eventually we obtain


O-C = ATO APE -+- AAE2 + ADsi n(o)(E El)


+ Ai(E -- E 1)Dcos (- (E E1)(5


AE I D; co ((, (EC El ) ) (5.3.7)









where O-C = T TO PE A E2


DgSin(ag(E Eg)) (5.3.8)


In all cases, values of Tg P and A were taken from

previously computed parabolic parameters from method (2)

Using Equation 5.3.8, as many equations as there are obser-

vations may be constructed and solved by the method of

least squares. The corrections to the initial guesses will

be applied, and the process repeated. It was found that

systems which displayed a noticeable sinusoidal form

converged to a solution quite rapidly through this technique--

usually within six or seven ite rations. If a particularly

good set of initial parameters was used, convergence was

even more rapid. Those systems which did not display

noticeable periodicity were found to converge more slowly,

if at all.

One interesting consequence of using method (4)

in relevant systems was the removal of incorrect values

of TO, P and A as supplied by the parabolic fit only. This

consequence is analogous to errors introduced into lower

order terms in a Fourier analysis when higher order terms

are ignored, and the dat.a are non-uniform]y distributted.

The p rogramA 1 develop d for thi s analysis a lso

performs the following operations.

1. There is separate least: square s adjustment

of primary and secondary m inima. Sufficient deviation









of corresponding coefficients by each method gives informa-

tion in regard to apsidal motion.

2. An O-C diagram is furnished for all the input

data.

3. Corresponding residual maps for methods (1),

(2), (3), and (4) are created for each system, and will be

made available upon request.

4. A list of observations, type of observation,

O-C value for an initial linear ephemeris and a computed

value of the residual from the theoretical fit as given

either by method (3) or (4) is provided. These data are

available in Appendix One. Appendix One also contains the

key which identifies the various plotting symbols with that

observational technique.

5. An 82 by 11 inch plot showing the general

nature of the O-C diagram and its residuals is produced.

This material is located in Appendix Two.


5.4 Wejighting

A generalized system of weighting, able to take into

account different types of observatjonal techniques, such

as visual estimates, wedge photoimeler observations, photo-

graphic observations and photoelectric observations for

a wide variety of eclipsing systems has: not yet been

developers Such variables a. instrumental size, depth and

width of primary eclipse magnitudec of minima and any types









of random or systematical variation in technique for differ-

ent observers have conspired to give such schemes a dubious

value. However, it cannot be denied that some types of

observational technique can more accurately determine the

mid-point of primary eclipse than others. Duerbeck (1975b)

finds the mean error of a visual observation to be six times

that of a photoelectrically determined observation. He goes

on to state that this is probably a lower limit, valid only

for certain well-observed systems. Although there is no

conceivable method of weighting which will give optimal

results for all data sets, the immense quantity of observa-

tional material used in this study virtually precludes

intimate familiarity with each observation, as well as its

weight relative to other observations of that system. In

view of these factors, a relatively simple weighting scheme

has been constructed. This scheme wiill hopefully reflect

the actual accuracies of different types of observations

to a sufficiently high deg ree for the effort of finding the

period change parameters. Table 5--1 lists the different

types of observations, and the weights assigned to them.

If the user wisles, he may insert his own weights directly

and eas5.ily into the p:rogrcami used in this study.

How good is this weight tingC scheme compared to

other possibilities? t mgt ih be argued, for example, that

there is a large difference in accuracy between v a visual

estimate and a goo;' series of visual o1)s rv:ations performed











TABLE 5-1
WEIGHTING SCHEME


Decimal
Type Weight Designation

Single visual observation 1 1

Single photographic observation 2 2

Single photoelectric observation 36 3


Visual, 2 or 3 (normal) 1.5 4

Photographic, 2 or 3 (normal) 3.0 5

Photoelectric, 2 or 3 (normal) 50.0 6


Visual, 4 or more (normal) 3.0 7

Photographic, 4 or more (normal) 5.0 8

Photoelectric, 4 or more (normal) 100.0 9









with a wedge or polarizing photometer. Certainly these

observational techniques cannot be placed in the same

weight classification. It would certainly be desirable

to subdivide this group of data into weight sub-classes,

were it indeed possible to hand cull and examine every

observation right back to the original set of measurements.

This could certainly be claimed as a valid reason for the

purposes of period studies. However, as this work is

concerned primarily with the values of the period change

parameters, it would seem that it would take a very ill-

conditioned set of data such that a failure to subdivide

this weight group would influence the results even

moderately. Sufficient reference material will be made

available so that each observation can ]be traced, should

a reader who is interested in a particular system desire

to implen.ct his own weighting scheme or perform a period

study.

Using our reduction teclhnlique for a detailed study

for the period of U Cep, the author has found that by

changing aI. of the weights to unity, the final least

squares fit deviates a v.,ry small amount from the fit

provided by the- initjal schreme Although this maiy in part

be due to the relatively deep primary eclipse of U Cep,

our weighting schema st ill scos an adequate scheme with

respect to the obsrvational mater i a. In fact; the system

of U Cep has sevelr:: feature :; which parT.-ial)y offset the









advantage provided by the deep minima. Most noticeable

is the severe asymmetry of the eclipse. That this weight-

ing scheme should suffice for all of the systems used in

this study may be seen from an inspection of the O-C

diagrams in Appendix Two, or from the larger residual maps

which are available from the author. The scatter of the

different types of observations conforms tolerably to the

square roots of their weights.


5.5 Philosophy of Curve Fitting

Residuals of a system which has a variable period

will deviate from a straight line fit as calculated from

some initial cphemeris. It would seem rcar;onable to

expect deviations from the "best" least squares line will

display any auxiliary behavior inherent in the system.

If a constant period change is present, parabolic residuals

will occur while an inherent periodic variation will produce

some sort of residual sine wave. It is in this fashion that

contemporary investigators generally deter ;ine what sort of

period vrii; ti.on is pricscet However, what happens when

both parabolic and sinulso.id1 variations are present? This

situation undoubtCdly ours ur often in n :ature, such as when

a mass transferring systi. o rbits a disti a third body.

A good examp 1-of t his i S tuaLion is the systemm I of RT Per.

A highly disLorted sjin wave is in evidence on the linear--

fit O-C diagv:;m. The rather noon-}::rionc appearance of this









curve has even led some investigators to state that a light-

time effect is impossible for this system. Obviously, a

different approach is necessary. Examination of residuals

from a linear fit introduces an unfavorable shift in per-

spective for real data.

The researcher might be tempted to try a parabolic

fit, and then examine the accompanying residuals. In this

case it could be argued that not only would an uncomplicated

parabolic variation be fit exactly, yielding residuals in

a straight line, but that the absence of parabolic variation

could he inferred from the error of the parabolic coef-

ficient. Such a system also offers the advantage over the

linear approach that any system with both parabolic and

periodic period variation terms will now in principle dis-

play a non-distorted sine wave. However, this will only

occur for a data set w:i.th a uniform, equally weighted

distribution of data. Figures 5--la, 5-1b, and 5-cl show a

parabola, a sine term, and their sunm. Now, to approximate

the situation with regard to actual observation, a segment

of data has been removed simulating a time when no one

observed the system. Figures 5-2a, 5-2b, and 5-2c show

the residuals of these observations from a linear fit, while

Figures 5-3a, 5-3b, and 5-3c show the re..iJdu als froi a

parabolic fit. It can Ie noticed that, .in neither case,

are the originally knon parabola anrd sine term recovered

from the data.


























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Only by including a full scet of adjustable

parameters as in method (4) can the parameters be given

an "equal chance." This system employs the advantages of

the parabolic method, plus the added ability of the machine

to tell. when a periodic representation of the data is not

possible. This is again done by comparing the magnitude

of the periodic term with the size of its error. As the

method is set up in this study, only the lowest frequency

sine term will be removed. Conceivably other higher order

terms will remain, as in the case of Algol. This method

has the disadvantages of gre-ater length and computational

time. If used properly, its versatility easily outweighs

the disadvantages. The unjustified stigma which seems to

have become attached to any attempt to display light

residuals -in terms of a periodic representation is placed

completely to rest., ovic.ng to the flexibility of this tech-

nique.

Occasionally, a cubic fit will be of intLcest. Wood

and Forbes (.963) utiljzed cub; fits ft or all of the stars

in their study. The justification for this technique was
that not only does a cubic employ all the power of a

parabolic solut ion, but that it ofj. : '_ tvo addiL ioal

advanta ge.s. F first of all, it allows us to calculate any

secular acccelrcatiion in a binary systcie due to Jmass t rans-

for. Secc:ondly, it provide, a fit for periodic data w hh

is almost as good a' ac ptrio.i c _repr (:sE:a' tati on for a short










section of data, on the order of perhaps up to three-

fourths of a cycle of the periodic term.


5.6 Least Squares Parameters

Values of the least squares parameters calculated

by Program OMC for the data incorporated in this study are

presented in this section. Due to space limitations,

least squares values for periodic representations are not

offered here, but are presented in Chapter Six for systems

which display this tyoe of behavior. The first column of

Table 5-2 contains the star's number, which cross-references

the system to Table 4-1. The second column contains a

value of the initLial epoch from which the cycle count is

computed, as .well as the error of that initial epoch for

cubic, parabolic and linear fit.:, respectively. The next

column contains Jsim il ar infori: mation on the value of the

period and .its error for the different types of fit, while

columns four and five contain the values of the A and B

coefficients and their errors. Finally, the last column

gives the value of the standard deviation for that repre-

sen station.








TABLE 5-2
PERI OD PARAMiET'ERS


LPUC I


*1 2424 11 9. 2Ut30)
7d3
242.4 1 J9. 3 "0
7I
2424 i i 1 '49,'i
84

2 24 3:, i b..-) o ,

24J3 48J., ij)97
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3 2 5 o291,:-.3. l

24 '; 1 t:, 1 ';, i
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242591) t-b 3
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7 2P 434


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TABLIE 5-2--ContLi inued


L[PO'L i


10 2435370'.C00o4

243b37C 0iU o'

0 2 4 3 7 tI) i- tb(.,
I t- .
2 4 32 f7.-,. ]1 7 cCU O
0
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4. ,- 'C

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1 2 2433 cfC. C, '

2433c 00. 'J c,,7






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15 24331 -i -.1 07

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2. 3v2 0021

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74




STABLE 5-2--Cont inuaJC


EPUCLH


19 24212q .-u c 5. -

2421242.k t,





70
24 329455> ..4 ] 7a.
77
243294 1, t, 1 "
73
21


Pt K, I ~U.


1 .J3720 76
7, ;
1 3. 327oJ 1b)
1 ,
1o .* Z'7 v;J J :,
2'.




0:3Y
2. ; 170

50


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C K I: 't -- ?


-. 5'43.' L-1 A
.3271t -13





-. *2 [ ,.-i 1
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22 243'q 4ib ,,3o,07

24 344 5b. .Jh771

24 34 L. b : i
1 -'

23 ;-24 3' t' :(. 7.,:
19C
24 3S44 6 .0i .- o
I L.




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24 38q-- L 4%, C, L. a: C,4
22 3 !, i., -4 '. t.. ,', 7o''l
~24352,? .7 .

0(1


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243413L.n ) I,
I W
24 3 .,, S .- .3, % i (* ;








1 :- .'. :
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17

5


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( -1 )AL-C9 0,. 2'.9.O ) .
. 1 1 001--C;9 1 C0 1 L- 1 3
O- n .- 1] '
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- 7 .; 7:-- :9 C. 73 7 --1 '
, 1'. ,. 7 111. '-11






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- 7 7',7 i --C '.










C. ,O :.. .--. ,. ,. 71L- i7

C <,c 7 .i *- 1 ".'
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- ]' 171 "4 914 ,"-:- C9
, ,.>' .' _-- ';. 9 4 C '.*
, :/ i ,>


SJ GC4A



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.00 19

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IPL i, I i


29 2?'33777 .i, -.cl ,
I )3
24J 3 7 7 7, 27

24337"77- :
lo


33 ?4295.5-, 2_ic ,',,
2 & .I U


242933-scol o;i


I'i~
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0 ?>113u 7;:J
I7-
G;. "-?L' 71 7'.
0
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6


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TABL]E 5--2--Contin.ue.c


S l ,1M! A


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76





TABLE 5- 2---Continnued


LPUCL.H


37 241 tIl>7, 7 405

241 b.o.', Qtu o
0t, '99J
241 3~,4 ~ 1
1.1)


P' LFI OD


3. 073"2 o 07

3,3917Lo.0o
-) O4) .'1
i J. 4 j_, 12.'.
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0 zj ;t .' .7 7 7
. 1 ''t .L- 07
.1o O]L-C 7


40 24330 : ,'L ) j
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2433o3 000.104 I ,

2, 5o it,

41 242M 2 ,- ). ,-

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241.2;: 0 ,7




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412.






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TABLE. 5-2----Conli. inu.d


EPLCtI


46 2 44 3< 4 u 97

24 4 L 3 20 914 9,






1










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S 2421 ,, b. c.*; 7
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53

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1 91C 1 ., 9 -
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1i 91* vi (-'JV.' .10
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* ."2 ,.( L. C 3 :. G -- 12
C .,d : ,1 t 1 0
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-. 1 1 CL-,07 -,732'.E-12
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0 ,e ) <'t C(-*.' 0 7 O. '.,*'L {:- }1 r:
, 1 ,,i> 0 'L- _9 1 5 9L- 1 3
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. L', ,L-- 11 1 '72:E-1 b
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Sl yMA





. 0 -' ;.7 1

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.0 .0 o(.
9c o ^'










TA7BLEF 5--2- -Cont inucd


EPOCH I


55 24349609.,_0003

2434 90", 4 37

1 4 3
24'353o73'. 4




24 3 3 .3 7, .. ,4 9:


21
57


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77 / ', 1 01 .
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77 1 7 .'.-i> i
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7291-370.


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0. 4'- /( .- 0 0. J377--12
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60 241 7, L7 :4o ',.a;'.',a*

24 1 <7-7 1. j
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TABLE 5- 2- -Conti Pnid


PLk I Lu


S 464 t& I .:9 0 r.S2


2436 1 3 o 7 2 '4

SoY769

65 243 19i0.~u14 1
7 *',<-.
2431 9o0 ;.7: to )
709
2431 oC 9. 0 U870
64 u

60 2414 9:Oi C'./ -
7,.9oCn:
24 1 4 9, .A ?:i 1 i a
04 679
24 1 *, 1'. o ,u
1 7 q 4

67 2421 13/. .3,o
10 .
2421 137. ',, 071

24 1 I.,7 ,i '.o ;W











-li
S .. 19
6 to 2 /. A :. ,>. 5 A & 6.






6 24 3 74 ,- o O aY9w
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70 24339LC .w C(

24239,>0.<. 51
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21 339.:>. On Ot 7
p I
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24 15 i *.3 .L

7 1 15 3:., 0 b
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71 ev I 5 ,'- 00 .70D

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i 71 ;


3,jl4dw+dt3P
.0 1 7 9-..
3, 1 to']'-'0,.

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3..1 02 t' ] 0 7


3. 31 772 32 '
497



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C c .)< 4 o _, 2 <_
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C <,; >;* 1 7 70o
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80





TABLE 5-2--Conti. nud


PL I UD


2427 o

7 24 87U 0 U,01 7
3o-y

74 ?''1 '.4 t, 7/>(013




1 Vc: '
1. 77.-






S24332(,b 7C? 09

jI SL
?24 .332_ 75,. JS





7o 24 333; S ', C,9 C ..


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24 333 0 7 t'. '



1 -r




24 70o1 ,/
24'374'il:0 c
J l.



78 2l421 -';,a l ,o
3, i '

242 7 j; ,. t .:', ."



79 74.3Cob C, .o7
2 33 ,.. i .
77,
24 33G d ,, 7 ,- 9 t...
?/, 2;4 8t!.7ie o m.: G,.. ,7




j W..
H 3 P 3'M, ,.Y! :'..,, :1
24) 3 v3 Qi, s7 .-;:





S1 ..,,


2. 1 ] 7 ,01 31






1. 357 >C I o
21 7C o0 2 1 /
99I
2. 11 7,772.







93
3 6.6




1 P 1 7I. :,nn,
(t3.-
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2 (33 1 1 -. 2







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4 4



( t,* : ..'7 1
12



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2. I 2. 2 .' 17 ',

h, ] (,":..'( (:( ,'3 .>
11 b72:


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35 (_,


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(23
St.




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.2000 -00





-.3 770L t-0 :
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C.1(i, I F-C' -
* .1 0WN C3 3
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IABLE: 5i-?--- -Coni ti rl'd~


PLkP 1 D


83 24 19(,C09,099
ev ?: .- T. >*
241 b6 ., 49
1 .4 1 4 '9i

I j L)S-y
24 19o .;. o 7 I3

84







85 2 437l3(-. ./73
21
25 4 .;3 71 ( L; -. r j 9 1

24 3763>. 0 b + o-,:
17

Su 2 4 '> 42 .,, 0 1+

2.4 2 62 : c o i )
(> < l ''. '-'a



2. O O :C ,
7 24;')7L2. C'-, :c;^.
2.00

2420Y 7"'a .0:;o 7
24 2. C 7 < 7
..21



S24 a 33 'L.> h '. )
2 4 .3 3$ 3Z .c


5t,1






'9 /-
9 (


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94G4
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0. 1 J 7o~ O I


O, 31 ] 7 (/'
1] '; Z 9<
C, ,] 4 .. b 7L.
/-,09
C 0 I2"- 30
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31:.
. 243: 5 .0 3
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'>* 0!j. C,' / :, (9j)
303
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'', t Lit (> J',* '' .'
390;


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S1 0C !- 0,o
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0. (>' CE:-12
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- 70C' t 1
. jb37 -l-3


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0. 3'C 1j 't -- 1 '


0, .37 9::- !
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"'1 241 C


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1.! ). I
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S./, t-. i


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.1 39




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TABLE 5-2- -Cont in.ued


fIPU' LI


9 224 I3C1 t'., C ; 2o
2 '4 6 .t,O
2 4 130.o .., ) C
1I I '. 0.
2, 1 b30 ,.43 o t4)3


93 24323 (.- 0 Y1 0',

243232o. 7 41 ;.
77
2433 > /(e / 1 '.;
72

9 4 2 434i 7,. 3 ,': 'I


24O 4-'/ 3 /(.4

141

95 2 34 3; Lt u 30
I 0c:-
24 34 3 v
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24 .34 ;3.'> 'A j 1









9Y 2424430' C') 7-


1 1. 4',
97 243C3 .9,37 ,) 71



07


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67-





96 24 2, 9o 2' Y ,. 75
9 2429C,7 1C.'>
731
2q.!290:,7 7 >,9. 32

24 3 6Y)7, 0 ( '-,'7






1 'O 2431 ,*. > .., 7
2'sil~ b.:oc .<


PI'-L-.I


3 9-. .r,' '<,C3b 1 L>
3 1 1 1
3,9927711 ,

3. 0'.'92007C i i
02

I e 7/ 0tb;22
3',
1 7733 7 1 13
2C
1, 77357j 13.


S.* 1 ,090'- ;-



C ...A LOS I 1 9
C 2 ) U Y 97~ .5
7 c.

0. 9) 20 :-03
21
0. 912{ C7 93 07
1)
0. 91 "(. 7 .^
20

I, Y 7! 7799

2.2 7 7- 3/4,5
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- .: ,9>0'. >3 7


2 L,
2 ( b39 373
'1
] .3.

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O 2) 971-'.- 0-
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C. C)2 ." 1
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0. 1 499L 1
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3. 93 C L I
0. obi LL--l3
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-. 77(C 5- I 3
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-, 61 7i L--3 4







* 10 .' .,;> 1 1





0. 1 1 b,_L-i 2
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TABLE 5-?- Connti :ued


LPDLt I


101 24 309C04 ) o02:
,G 99
24 0904 c*:o/

243090i 3701-
1954








103 2423377190 .47

243377J 30 7

2433J77) I o i.,
000

1 C4 2 '42031 0. i. 7 07
> ::.+++ ., y ). ) :
24202s7.:, 7 .-i. l 33
1 b :.,C3
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46j 91
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31 )..9
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) 7 30 8 0 ) )

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4 l2

S.* 7 9 t : -:.




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1 3',572 /790
J .05 ,4 7.

1 -3




'. 9 4 7 9
2 9 0 i4 :
2 .t I u --i .
0J i


- 47 1 L. -C 7


* -b 0 *. -- C' j


-. I W 2 --07
0- 07L --

0.1 O c i.- (
.1 20! L-00



U, 3'1 1 -- so
. 1 1 I t C. ;
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0 28'-i.41 '-'
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0. 37 97L- 1 1
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0.1, 1 :-i [L--.9
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-. 27I t.- 12
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. bb)33 lf- IC


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)G:J 2434 0 .,-: 9, 97. 1

2434 04 9< *7 0( 17.


3j,7
109


17' L.I


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1 1777- ~ -
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TABILE 5- 2---ContI ind


LPUCIH


1 1 p24L26 2 24...7 ,. ;.)/
20
24266< .J3 1 4 1

24206,.: .37141
3

1 12 242702o. 373.:
2 t4 70 4070 .,
2I 1 702o .:,_:.; 0 Y

24270 .o .' '. .-:1 *
124



113 2427 /., o1 771 6
3.3




242 734 <1 7 7u
370
24 27 ,4 0 / 1 3



219
1 4 24 219J&4 .1/*

2421 91 .1 1770
24219 1 / 3 ,. ) 9., < 9.--.



















i 1
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24 22372,_ : .,27

24 2237,.', g b : l -):
91,


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24 3h44 0, F/t,/ 1

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b7

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7
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0.A 339i.--1 3
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0. -87 : -1
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TABLE 5--2---CUni nued


P L< I LI)


121 2423;53.31 1.

2 4 2 6 -3 31 5 '
o 7 C
22 42 ,t' 3 <.>23 o
40

122 243K .1 ,.3c.'i,.7
130
24 32 6 03 .> / /,

24 3- 4 (, .. J 7 -' j
210

123 1 23 1 1 8j 6

2 42 b K .- -'.i ';





124 246 o'- -2 o .7
2420'o C7! i

1, 2 *'
bJ

319




2 2,i 2 ,6 ,. t..C3 1



128 2' 4 .c,;;o ;.o
3v

2434 i.' .t a

201
127


C .004 .C 09-4

0 0 oct 7 0 i L

0, L (. 0 ,.:-3
2O

3. 53 oc (.7.ol




61

0, 0
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0 1 3,':0 1 b 2.,


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61
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94 7190.A

94071 0- I


73


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1 7 23'i ] L-0-
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-11




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1. 906U 10O2 7
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120


SI GMA


.0332

.03Y90

.034 1


.01 +73

S0 237 '

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* 0r4, C (3




.007bC


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TABLE 5--2---Conti1"u"c'


PLk lOD)


129 2? ?26211. l i bb
't04
242821 uj ,0j7w
.390
242821 3U4al


1 30 24002 .: 7 ,3Y>






131 24239 7/.4 u9I
les's
24/23o:,-7/ 3 7 ,-
32^ '
2^ 2 %3 2)7 4 3 2
O'D

132 24M 3 3. .1' ,. bO, 52
1003
247 3 6 03 I o19 a



1 3 3 2 1 27 s 1




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24 7 1on. u2) 0
02,1







243427 i () :.. .u Z
2 70 2 ,5 217)










I M)2) 24 3 DO 1 1 ,Ii jQ7
134 24 34 ; ,o, 7.,- D '.5
3,016300
243 103,. 7'/923





243 3 4' i o ,: 3c 33
133 243091 i t.>73C7
oll
2 t3C l 1 1] 7 .-, 7 7 7
24JOll 0. .5
4i23

13 2 ; 37. .A > .,, b '0.
91
243 74 773
1 K'
24 37,1; ,-. '," 2 7
77

1J7 2142 3.)1 !.:i.). 3 8s

24 bL ..
011
24200Leano;b


2 .) 1 )1 4629
809
2. 2,^-33l b 9
323
2. 2251 9393,







4 3,.
13 Ut753L.1 J ,
I 77,) J



1.0b77y 1 7j.
t,l7 b ,1 I





I7 i i.
S 1100. ,
3.bE71JW4o,

3 u.7 1I 1 :>o
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31.
2 t q4 ubk, 4., "
549



) A




1 2 '-- <.. 1,
244
1 20









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0 ,. 3 .39
12

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C,. ..> .7'i 7 9.,39
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'3,u L- 07




--,270.t3 --O' o
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- 2,0 /..(

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0. 0 .,,.
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3 ~ 3 --~3,3
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.4 05E- 1 2


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C 7 E 1 2
C / I '.-; 7 -1
, 57 Ilb,. -!3


- 271 ; L- 1C
. ) 270 <*('-- C 9


- 20i<-:' 13
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-.1 C. 7 ot-I I
I L7=-I11


51 o:.MA



.310; 21

.0 10 C2







.0 3121








1 0C-'
. _) j103

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1 I/i v 3C




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.10550



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.0 C;:. 1 /

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87





TABLE 5---2--Cont inuod


EPUL Hi


138 243524d.0 9'4ji.
62
24Jb245. 5. Ma31

2435245-., 0 3
79

139 2i4200f,. 1. 7W
1 070
2'4 ? 095 c ,r-i 78 t
1 023
21204'25 7 w /

14C '4-22 97u7%'74
4 'it,'
24 2"}7. 7 7, ,: 3:3'

2422-707. ^ I 17 1
JJ13

141 2427 4 .) o00
13, 7


66'-
2427 b- e .,5












(4. ) 1',
47~9

141' 20 6250,5 7: 6. 1C
















2Y ^(,o .0 .;**)' o.,
243650m., u lb


14i


1 2044 24 C- .: ,'4
Z24
2920 0:.. 0c1 /
t-^^ ~ ~ 36 '('JlI ; >}-^
24~~J 2f.n *4ubb


PLK 10


0 c4, 9 9u 4( b



0. 4C09041 ( .,
7









71 o(.
b 4 L.j 3 709 P




1 '!3 ,0 7 .3.





71 S ..,
71 W;

2.0 1(





1 o : i '. j ; 1.
I 3;

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44




1 i



1,
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4) r .-0 / C
0., .0 9'.. C .:."

1i


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. 12'Y7L- 10



- 3 A u F C'

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. 1 '- -'




c I 29 F -- C:.u
* 1 '. i.-. 4( 1


0 1 '. C" {


. =>7 70L


U. 1 .3 K. -
. 1 u L..

S.. 1 ( : -
. 74 C. (I


1 0 .-.c .

1 397i.


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0. 75o. -- 1
. 201- 0L-'34


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* Jl 31 -1.3





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* '-0/7) L-1i 0





-- .2t 3. -' 1 3-1~
. 20k-f4 -12






--. 737Ci.. -1






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- <.23 -- i3
. 9:* ',, "- 1 4


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2 742 7P..-. I.9

t-_'.'; ;.' '. /, ,,7
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L
4. 4 4 -'-, I .~'1 (4;~
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C. 29.,.91 ~ 4~.,
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C-~~~~;. '-. -,' C


SI GMA



S0 1 i 82.

.01I59

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. 00-. 3C'C






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* 0 0 u I: o

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F88





P1I! 2-< '1 i~r~~


' EPu C I


147 24232 ,L c; Iz

2 4 23 I .
2422601 4/, 4. I .t',



24 230(~~ ,


148 242 i:C,.i 1 '-. -?

S42 t. ~ ,: c, .




CI 2 3 b I

24i3 : 7 t. b;,.'.
1C : .,,
24 :32 Wo7 -. -J 2 ,


151O 2430'l J ,O .::



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152 :'.'-.>0 30, :., 7.2 ,,
3:>;
2"< 3oC-.3, /* "i <.j. ,/:
153.3 7







2'43 .; .:;' b 7 ..-
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153 24I ,4:,1C, !-,:'3

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.21 C


PL. il (j)


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3 c 1 2., >'. : .
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1 bi-,v







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0 o7. 7 ..,/ "1 3 7


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19'L
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* 0 1 0,*,
, 0i I


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:ITGUifl; 5- 2---C~o; t i~nulc


[-I'PbCtI



156t 2'!4324441 ,43:00
134
432 4'i 1 */. 4410
72
24324 4 1 4'4 4
bi9
157






153 2421 :,' 5, 77 ( 7

2424 55 *.. 4'91
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224 45.3 .;. r

159


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0. 192 71 -09 -; 1 BE 1 .
2-1i E-- 10 197 -1
0. ,9 i.. 10











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160 ( 29'' ? '- 17 ,/ b ,: .4
S,77
2420217. d'.1,
22 9:! 1 7 t '..1 l

92 o

161 .24J. .1I, < .-O ,:o
2 i
243;01 71 (

4. 39';91 / .- ;-I
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2a 04 .;- (j 99


1 1 /
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1., 1 841
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S.2 7.L 07


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- 3 .', I 1C0

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0 3071 L.--2(
. 1 ) --07
(;, .o",6,._[ -CV


0 ,. ;G ] F- L-









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0. C 7-11-
O, I 7 h -- I -






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-< 2 7,a 1,i-10


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. C00 90

.00795










.3 341;



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TARLE 5-2---Conti _n.Iu(cd


LPUCH


165 236 11- 73 I 9. ., .
1 2 / 6

I !/ 1
1 C<3;^



16t 24'2911 ..3cC2b
101
242 911)1 .. ..

2q291 J 7 q'.%
167
1 6 7


I I i_ 1 u .)


.3,9'0o U'00
4- : 9 :. '

3 57 U 1 ,-.
1 -q 1




3 o.,L 1 q rb
8}.

2;.

27


SI67uL--07 0 2. ,2400'E- I 0
S. 1 1 E- ,' :-. -. 4 I
-, J J- :.l-- CO





S12? L-0.; 69/ c' --13
C .ul C L-C
Sl1 C L- C9


1 C9





170


171 24.35 15.0 97C
Q
2Y 33"w.i-.57a
99

1 u.)

172 4 f'. < 2, 4 500





;: 0



24. J ,:'; .
I30 :~~


i c i t, .' u: '
I) -:

37
1 1 -;- U7 ;3
30

(,.7(,DJ.? .:L,
S 7(.i 2j .' C C






ODJ
0 i. /'!. 9 7





1 1 n1 I; 0;> N<.

1 (1 -/
I wlw i473
4 L1


--e 3'; FE- oy
< /'.I 7L- 1

, ro C(. --i 0



0 i 7 <; 2 [ .- 0;1 ,


SC.t f L.- C.0




S. C t L .; ,'./
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. 3039;i- .3






S271 L I _--I





0. i 23 HL 1I
. 7.3Y.Au -- 3 ,


SI OqA






,1 4'1 '

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. 0 3 -,

* 2) J 4 1; ^


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* 1(- 1

S31171




. 10 L.

. 1 01 3 ;
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TABLE 5-2----Con_:inued


PL Rk IL OD


176 2429' -.0 5b..;07
17

1 I.J
242913O.4,ub0

177
242930 4, uo u
lo)

177 2427 1 vJ. 07d
230
242 7?] ;9j, 9 ',L2

24271 79.:. 20 l

178


L



IC.:


(.o 87 ...,i.- ) -- ? 1
2 13/ i-- W ...2 4
0 1 i r -'I 0
. ,1 y it-- IC



-- 0 *L --C C< / .j E--12
* 1 .,'_. L- 00 217.5 '- 1 2

. 24 7/,L- 0.(


17 243 7 2 <, t.,'.,
2i-
2437L72 c/: v0J
20
24 37b 72 .2u j
.19


180 243 3' t '.. 29
L(3







24 33 -.)& 1
tIi'
24 3. 33': i, *;) 1 i


1 1 24 3G77 2 7 Y,7J


; ,



62 2,:,30 1 ':.; .. 3J1 :


o
0. 0971 >j 2 Yu
C)
0. 9y 1 :. .. 2 ,,
7
0 .,7 J t,327
7

0C 231i a3. tn
I 'n



2 C.3 :7 2
l '



C ., 0 .':- 2
C :.;) 2 ,3 (.C 7 .


1 4





3 e u 3 ;-


-. 991, i 4 -:-" in


S10 ..'.. 1 0



0. 1 ) 3 '1.- I1
J 3'&.L- 1 0






W. 1 2' 1 0
. 3.5 -' i I0


- +. :.".3 07


I I' -. i-'-1] '-







*I (i ?r1-- I 4










( C. '.C-t- i I


ELPUC


17 4





1 7,


SI G(NA


.2 0802



S.O -v41
.00730


.3-907
r C 0- L.-, b

* C0 34

* '2) '.7


*. 0 9 7 ,)







,b 0047















LPUCt


183 24 293J', ./, L 9
u.j

2l.




G 4

22.354'i .,tl,1 i_ ,







1 9 9-5(..
j2,3
24252' 7,1.



187 2432bv.1-18055
1Ob 2 5 ;:. 2 Y 7, 4 1 i ol c

1 991
24 2':F.4L- u1C 7:




1 1
2 4 7; ^*i 4L i :, 277








23
1 7 243?b7 c./ I .1. V,





12.

24 3 37,. 9.0-20
EU


P <-k I J J)


3
0 1 950 7 1 j

O
0

( o' _oSOu
29
C 0 ct:,L t :C7
i 3
0 4 2 0.. 1 Lj'


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S.* 07 1 I -

ti'

















LC
S, 4520 9 :
099



/4 ,, '. '', 7 j ,

4 47 7331 '. ..








2(1

21


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. i3 9 1 1








, 9 ,'>(7 3 -- 1
. 17- t .- 1


I-- i i
S. 1 1 1i:


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-- ri


G.3097L-07
. 17!01 07
W O i -- 1' '
68w nj..-M


.39131.
(.. 2-', '3 .






S2: 1 ri

9 ;C


-- C.
--4 't



--07


- IC
-i1
I- C


C.1 3 0-13



. I S A/L- 1 o






.4 I4 IL- 14
-, 1 2 1 -






S0. L 327 -- i /
S1i ; i 1 .






. 121 2 L.-I 1







1 ; 3 -( ,




- 1 4', 12 -
. 2/ L- 1 3


S! IOMA








* 0 j 3 1) 0)














,03CIG


.01 1 7









.0 u 1 0 ;-






,.Q 9732
. C 1j3
.'0 1u42

.03C' .o 9

.C. l73

. 0I / c)


TABLE 5-2---Continuod















CHAPTER SIX

AN EPHEMERIS FOR LACH SYSTEM


6.1 Basic Considerations

As discussed before, a large variety of causes

contributes to period variation. It would be desirable,

whenever possible, to eliminate some of these sources as

causes of variation from the data better to understand

the effects of the remaining sources. For example,

apsidal motion may be inferred or rejected on the basis

of a few observations of secondary eclipse. An unnoticed,

or even a prematurely rejected possibility of apsidal

motion, will lead to some degree of error for our final

values of the period change. We have other basic consi.dera-

tions to contend with when we decide upon some representa-

tion of a system's change in period. Primary considerations

are listed belo.;w:

1. The value of sigma

2. Cornel3ation coefficients ,which:

have 1been calculat-ed for all

the least squares; fi ts

3 Theri re a.ive errors of the coef-

ficients for: each fit