Citation
Mass and momentum exchange in close binary systems

Material Information

Title:
Mass and momentum exchange in close binary systems
Creator:
Rafert, James Bruce, 1950-
Publication Date:
Language:
English
Physical Description:
2 v. (vii, 786 leaves) : ill. ; 28 cm.

Subjects

Subjects / Keywords:
Apsidal motion ( jstor )
Close binary stars ( jstor )
Data visualization ( jstor )
Mass ( jstor )
Mass transfer ( jstor )
Orbitals ( jstor )
Parabolas ( jstor )
Stellar masses ( jstor )
Variable star period change ( jstor )
Velocity ( jstor )
Double stars -- Masses ( lcsh )
Mass loss (Astrophysics) ( lcsh )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

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

Record Information

Source Institution:
University of Florida
Holding Location:
University of Florida
Rights Management:
Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
Resource Identifier:
026385709 ( ALEPH )
AAX6862 ( NOTIS )
04168404 ( OCLC )
AA00004913_00001 ( sobekcm )

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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,




MASS AND MOMENTUM EXCHANGE
IN CLOSE BINARY
SYSTEMS
By
JAMES BRUCE RAFERT
A DISSERTATION PRESEN
THE UNIV
IN PARTIAL FULFILLME
DEGREE OF
TED TO THE GRADUATE
ERE ... TY OF FLORI DA
NT OF THE KEQUIREME
DOCTOR OF PHILOSOPii
COUNCIL OF
NTS FOR THE
UNIVERSITY OF FT. OKI DA
197 8


For Donna


ACKNOWLEDGMENTS
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 Cara Catalogue of
Eclipsing Binaries, as well as to the curators of this
valuable resource.
Great amounts of computer time, provided jointly
by the Astronomy 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. Merri11.
iii


TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS iii
ABSTRACT vi
CHAPTER
ONE INTRODUCTION 1
TWO HISTORY 3
2.1 Introduction 3
2.2 Early Concepts 3
2.3 From Speculation to
Calculation 8
THREE PRELIMINARY CONSIDERATIONS 13
3.1 Theoretical Context of
Mass Transfer 13
3.2 The Roche Model 14
3.3 Evidence for Mass
Loss/Transfer 16
3.4 Some Relations 20
FOUR SELECTION OF THE SYSTEMS 25
4.1 Selection of Stars ....... 25
4.2 Absolute Parameters 27
4.3 Parameter Correlations 35
FIVE DETERMINATION OF THE PERIOD
AND ITS CHANGE 4 7
5.1 Basic Concepts ..... 47
5.2 Causes of Variation 48
5.3 Program CMC 52
5.4 Weighting ...... 55
5.5 Philosophy of Curve
Fitting 59
5.C Least Squares Parameters .... 71
i v


CHAPTER
SIX AN EPHEMERIS FOR EACH SYSTEM 9 3
6.1 Basic Considerations 93
6.2 A Word on Each System 9 4
SEVEN THEORIES OF MASS EXCHANGE 14 6
7.1 Introduction 146
7.2 Particle Trajectory Models ... 147
7.3 Ilydrodynamical Models 159
7.4 Evolutionary Models 161
EIGHT EJECTION OF MATTER FROM
UNSTABLE COMPONENTS 171
8.1 Non-synchronous Rotation .... 171
8.2 Matter Ejected from
Unstable Components ..... 172
8.3 Program ORBIT 17 5
8.4 Efficiency Tables 178
NINE LIMITING VALUES OF dM and dJ 229
9.1 Limiting dJ 229
9.2 Calculation of dM/dt for
Each System 233
9.3 Limits on dM 235
TEN CORRELATIONS OF COMPUTED AND
ABSOLUTE PARAMETERS 240
10.] Period Correlations 240
10.2 dP/'P Correlations 250
10.3 dP/dt Correlations 254
10.4 dP.._ T Correlations 264
NCJ
10.5 Summary Tables 272
10.6 Statistical Trends 275
APPENDIX
ONE OBSERVATIONS 278
TWO O-C DIAGRAMS 601
THREE BIBLIOGRAPHIC MATERIAL 761
REFERENCES ...... 780
BIOGRAPHICAL SKETCH ...... 786
v


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. Al] available times of primary
and secondary minima for these systems were assembled and
subjected to least squares analysis. Linear, parabolic,
cubic and per iodic/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 dm/dt. Non-conservative effects were investigated by
a particle trajectory model in which material conforming
to a Maxwellian velocity distribution was allowed to leave
the vicinity of the inner Lagrangian Point L} for a wide
range of systemic parameters.
V
i
Particle integrations 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/dtl^cj and the mass flow
rate (dm/dt)j,^j were calculated. Correlations of (dP/dt)
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 momenta 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
Sbothers.
vix


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.
Naturally, more accurate transfer rates could be
calculated if these two assumptions could be relaxed, or
eliminated entirely. It is the aim of this study to analyze
1


2
the character of as many binaries which exhibit pex'iod
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 that it could allow us to obtain an overview
of the situation.
2.2 _Early Concepts
Certainly one of the earliest results of the
study of eclipsing stars was the fact that some displayed


4
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 change 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


6
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 fil] their Roche surfaces, or come so close to
filling their surfaces that for all practical purposes
they con be assumed to fill them. He explicitly stated


7
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 stars were discovered to be 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 material possible? In an effort to avoid this


8
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 (Morton 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


9
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 inass
transfer or loss can be rather intricate. It should be
noted that Kopal1s particle trajectory approach was rather
limited. Only single particles ejected at some mean
velocity were considered.
Kruzewski (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 show7 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
Figure 2--1


11
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.
13


14
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 M-, 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
C
2_ 1 + _2q_ f_l_
1+q 1+q l'r2
x
J
2 2
x + y
(3.2.1)
where q = M2/M^ (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 L-j The L-j 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


15
Ficjure 3-1


16
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 L] 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 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~~^ M@/year, the
"requirement" of mass loss is met. By stellar winds
alone we could expect intersystemic mass flow in close
binary systerns.
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 that the precursors


17
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 Chandasekhai* 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 estimates have been negative for this system most
of the time. However, polarization is detectable during


18
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 ait 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 gaseous stream.
Observations of other systems, such as AO Cas or
U CrB 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-synchronously rotating


19
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 impinging upon the. leading hemisphere of the primary
star.
4. Parabolic period variation: as is shown later,
this effect may be explained in terms of mass loss or
transfer,


20
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 d.M/dt has been
derived by Wilson and Stothers (1974) for conservative mass
flow. To be completely general, we may write
(3.4.1)
where a semi-major axis.
Setting the orbital angular momentum equal to
(3.4.2)
and substituting Equation 3.4.2 into Equation 3.4.1 we obtain
2 71 (M^ 11*2) d 3
G2 (MgM^)) 3
P
(3.4.3)


21
which, upon differentiation and some rearrangement yields
dJ kMiM2P1//3
dt 3(M +M ) 2/3~
dP 3 dM 2
+ (M-i-CMo)
Pdt M^dt
2dM2
' >
1-C
dt
(M,+M9)
L J
where k = G2//3/ (2 it) (3.4.5)
and dMj = -Cdf^ (3.4.6)
HereC is the function of mass leaving the secondary component
which falls upon the primary.
We may also write
So
as
13tot dorb + drotl + drot2 4 ^lost
dJ^.K ~dJ ... dJ dJ.
od rotl rot2 lost
dJtot = 0
(3.4.7)
(3.4.8)
(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.
Vie may also express the rotational angular momenta,
and their changes, in the following fashion:
dJ
rotl
2 rr 2
1 dMlrlwl
(3.4.10)


22
dJ
rot2
2ttc
p
dM2r2u2
(3.4.11)
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=0 (no mass transferred) both dJrot^
and dJrot2 equal zero and dJorb~dJ^ost. We have
dJlost = kM1M2P1/3
dt 3(M1+M2)2/3
dP 3M1dM2-tM2dM2
Pdt M9 (M-i +M9) dt
(3.4.12)
while if we set C-l (all mass transferred) dJlost=0 and
^Jrottot ~ ^Jrotl + ^Jrot2 (3.4.13)
giving
dJrot = kM1H2F1/3
dt 3(M^+M2)
dP 3dM2(M1-M2;
Pdt
M-, M2dt
(3.4.14)
In both Equation 3.4.12 and Equation 3.4.14, dP/dt is an
observational quantity. The quantity dJ]_ost can be calcu
lated by either particle trajectory or hydrodynamical.


23
models. For all cases in this study it is assumed to be
negligible and is set equal to zero. The quantity <3Jrottot
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 momenta, as in the Biermann-Hall model,
proceed on a time scale (dynamical) which is short compared
to thetidal 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
dJrot2J.,T kM1M2P1/3 fdP
f!J Jit + Q.J '-WS-
rotl e lost 3(mi+m2)^/3 P
+
3dM2(M1-CM2)
2dM (1-C)
(m1+m2)
(3.4.15)
This is just, the previously derived equation without any
simplifying assumptions.
It might be argued that by computing the magnitude
of the orbital-rotational conversion process from non-
synchronously rotating stars, we are neglecting the fact
that at the point in time when they had synchronous rotation


24
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,
Jrot2y = m2y r2j_ (3.4.16)
while eventually
Jrot2f = M2f r2f u2f
Subtracting to get dJ we obtain
dJ = -Jrot2f + Jrot2i
where we assume values of initial and final sixes from the
(3.4.17)
(3.4.18)
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 casea
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
25


26
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 systems 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
Forbes made their study. On the other hand, some
Wood and


27
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 fincil criterion, sufficient data for each
system were necessary. Except for very interesting objects
this means, at the bare minimum, at least a dozen observa
tions .
4 ,__2 Absolute Parameters
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 conservation of mass and orbital angular
momentum.
In order to perform


28
dP
P
3 (1-y) dM
y M
(4.2.1)
where y =
M]_ + M2
this calculation, the values of P, 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 semi-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 Lagrangian 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
r
1
2
3(1+r)
2
2
(1+q) (1+f) r = 0
q
1
(4.2.2)


29
TABLE 4-1
ABSOI.it!TE PARAMETE RS
! 3 A M L
GPi.CTr
A L 1YPLS
MASS
3
r\ A L) 1
T
TYPt
1
NT AND
f- G
= 0
C 9 ¡3 0
i. :> c o
. s s 0 0
. 2 4 C 0
1
ci
1 V A; \ S
f 0
K 3
0.4 c 0
2. 4 0 0
. 2 4 o 0
.15 7 0
2
3
V. Z A ND
F d
0 3
) C C 0
1 .40 0
. -V!1lC
. 3 6 0 C
4
4
X/ A f-, Ij
AO
Kt
. 30 >0
,2 930
5
A fi niiLi
GG
C _>
1.130
1 8 2 0
. 4 4 C C
. 3 60 0
4
6
3 X AND
g0
,4340
. 3 8 2 0
7
NY AuK
A 3
IS 1
. 2 4 0 0
.2070
3
S U A U is
Ad
o
CX ADA
f 2
1 C
X /. A uL
A 2
1 3
r'K AL
LlW
.3541
. 2 03 9
1 2
N A u L
A 1 V
A 3 V
0.360
2.900
1 3
KP A l. L
AO v
1 4
' j A C, 3
O G V
Ci '.i
. 4 2 j 0
. 3 30 0
1 3
J Y A U L
1 0
u 4
0 7 a 0
2.700
. 2 4 j 0
. 1 63 0
)
1 C
V 5 3 7 A u L
LjO
1 7
V 3 4 3 Ai_
Ac.
O 3
. 27 0 0
.2310
3 E
NZ A 0-<
f 0
3 v
TT ADR
Gg
u 3
5 o 0 0
o. 7 C 0
t J D vs "J
. 3 2 C C
1
s o
v. A ur,
A 7
A 7
1 7s0
1.510
1 L' D c.
< 1 o2 5
1
2 1
2 3 AuN
A 7
. D 1 O 0
. 2 94 0
22
3! AuN
O G V
c o
9.3 3 C
9.35 0
. 4 1 0
.3770
*4
23
1 U A uA
Li 2
Li .
9. 70 C
! 4.9 0 0
. 5 3 0 0
.230
C.
2 4
SU 3L
A J.v
. 2 2c 0
.2130
2 3
4 Y -s.C
.4100
. 4 y 4 o
? c
44] uuj
G 2
C 4
0. 3 S C
0. cl 0
4 c ,, 0
7 0 0 .N
4 .0 W U
4
27
Y C A
^ 7 7
y ^
0.48 0
? 2 0 0
, 24 0
.2350
(-
2 o
3 3 C A M
F i>
3 1
. 4 4 S ?
.13 2 0
2 9
5v CA'..
G 3
G3
0.7 o 0
1 15 0
. 36m 0
. 2 4 7 0
1
5 0
SZ CAN
9
b0
. C 3t 4
.18 7 7
5 3
AS C AM
i_ 9
3 2
S CNu
A 0
G G
2.400
6 b 0 0
. 1 9 m 3
. 0 32 7
1
3 3
AY CNC
34
TV CNC
35
3 C. V Is
41 V
K0 i V
1 3 5 C
1.40 0
. 2 6 s 3
,0940
1
3
R C M A
M 9
N 5
o. n c
(' 4 9 C
. 3 0 2 0
. 2 4 60
2
3 7
A /. C A P
A 2
A 4
3 8
Cl l C A K
;3G
L-> 4
3.770
5.8 9 0
. 2 1 s 0
. 2 1 6 C
1
3 9
SX CAS
G 3
A o
(j < 2 C 0
G. 4 0 0
. 93 0
. 1 97 0
1
4 0
5x CAS
Au
uo
.22u 0
.090 0
4 1
T V C A .3
P. CJ
F u
1.000
1.700
.3110
, 2 3 7 0
2
4 2
T.': Cm3
L3 1 1 V
o o
1 1 3 C
2.9 C 0
, 3 s 7 0
.2100
2
4 3
7 4 CAS
D .3
4 4
AC CAS
A 3
, u 6
. 2 1 s 0
. 1 8 3 0
4 5
D i C AS
A 2
C.0
0.6 4 0
2.040
4
MS CAS
. w
4 7
r-i- ClN
2
F 9
. 5 7 c 0
.2200
>4 B
S V C N
i; --i
l. 1
1.10 0
9 3 C 0
. 4 3 0 0
. 4 2 7 0
4
4 V
V 3 A A CLP
Li 9
EC
U C FI P
G o
0 3 i !
1 i 9' C 0
4.7 0 0
.310 0
= 1 <>C0
2
5 3
PS C !_P
/\ :j
N 3
0. 7s 0
2 < 0 C 0
.21.1 0
, C o 8 0
j
3 2
V,. Ct.P
L " 0
i\ 3
0.2 5 0
0.61 0
< 3 4 0 C
. 240 0
4
5 3
V. Y Cl P
A 7
. 4 j s 0
. 320 0
3 4
XX CLP
A ¡3
Ci 4
0 2 7 0
1 7 2 0
~ l r\
; > \a
, 2 1 5 0
2
3 5
A li Cl P
l>3
L- i
1 3.9 0 C
1 L-. 1C1 C
t * L 0
. 3 4 0 0
3
5 c
C .s C r P
GO
¡1 0
1 1 C 2 C
j 1.740
. 2 6s 0
. 1 49 0
1
5 7
5 S C t_ !
AO
S c
I-:. Cl.M
u2
G .9
5 9
2 CoM
n0
K 0
0 7 3 0
1 3 8 0
30 0
, 2 6 0 0
0C
S C i L>
i o
f\ 0
1.400
2.9 0 0
2 6 c 0
. 18-30
G.
3
1 U C N, j
62
3 C Yu
A 2
K *
0 7 (. C
2. 30 0


30
TABLE 4-1Continued
N A M L
bLE CT RAL 1YHLS
M A S
SEL s
k A D 1
1
1 Y PL
3
V iv
C YG
A3
K 0
0.700
2.50 0

2200
*
15 10
1
6 4
S V
C YG
A3
6 5
V. i','
C Y G
137 7
G8 1 7
4.30 0
7. 60 0

2 7<+C
c
2 190
3
6 6
V\ /
C 1 G
f 0
6 7
77
C Y G
1- 7
k 3
0.700
3 2. a o 0

333 0
t
0
2
6 6
cv
CYG
r 3
6 0
r f.'
C VO
70
GU
C V G
bv
r 0
0. 83 0
0. 96 0
*
4 5 4 0

4 0 6 0
4
7 1
V4
6 C Y 6
111 1
09
7 2
V 4
5 7 C Y
b 1
F 1
13.000
'3
<*
o
o
o
t
2 90 0

1 780
1
7 3
V 4
5 2 C Y
l- 6 7
F- 4

0 o 0
c
0 5 4 0
7 4
W DU.
A 0
G 3
0.4 4 0
2.100

27 dO

1 3 70
9>
_
7 5
z
LX- /v
D
K, D
0,3 6 0
1.40 0

26^0
#
2 570
7 6
K><
r t< a
A 2
7 7
r z
i? k A
A 5
7 6
sx
Dh-A
A3
Fx 0

3 2 7 0
.
2 05 0
7 9
i
D A
Ad
KO
0.6 2 0
2. 20 0
*
3 3 o C
t
2 1 OC
8 0
TZ
l>(, A
A 7 V
81
UZ
[' A A
82
5
E: 0 G
69 V
G5
0.36 0
2.9 6 0

22 3 0

2 010
2
6 3
RU
F-l.I
A0
64
UA
! 1-
GO
G!

50 0 0
t
4 0 0 0
8 5
U
GF M
b0
F, 0
66
1' V
GEM
Ei 1
M i
0.5 G 0
0. 58 0
*
1 b 6 C
c
1 56 0
1
6?
A F
C f: M
AO
6 6
AY
GEL
AO
6 9
6 C.
6 L M
A 2
90
4 X
Gt_ !;
A 4
GO
0 5 3 0
2.4 0 0

1 9 0 0
*
0 700
1
01
r y
Gtr.i'i
A 2
k 1
0.62 f
2. 70 0

2 9 o

3 2 7 0
(t
0 2
7
HE i.
r 4 i v
H 5 3 V
3 1 0 0
1 2 2 0

8 6 0
*
1130
\
03
H X
Hi: i<
69
A 1
2.30 0
2 7 5 C
>
2 3 c C
#
1 8 BO
1
9 4
SZ
1 L K
AO
G 0

3 25 0

3 10 0
o 5
1 T
H;;k
A 2
0.95 0
3 3.1 C
t
50 4 0
4
2 72 0
4
9 6
TU
r(t_k.
F 6
9 7
TX
11 cJ<
A 4
A '+
3.76 0
2 0 5 0
4
1730
4
1 3 8 0
9 6
UX
H E k
A3
K 3
0.49 0
3 86 0
t
DO 1

2 3 10
2
9 9
AK
HE!<
t- 2
i D
0.4 3 0
1.430
.
5 4 0 0

2 4 0 0
1 0 0
c c
HEk
AO
1 0 1
C T
HLk
A 0 7
1 0 2
D I
HLk
o4 1 1 I
6 5111
1 03
V 3
3 8 Hi:
K
A9
F, D
*
3 0 1 0
C
2 2 7 0
1 0 4
r x
111 A
A O
A 0
0. 380
1.500

29 1 0
c
1 800
f-
1 0 5
s \
1 i Y A,
A3
D
0.5 9 0
1 7 0 0
f
3 6 4 0
r
1 1 8 0
)
C
1 0 6
11
H Y A
A3
G:
0.71 C
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31
TABLE 4--l~~Contirmed
NAM c
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32
TABLE 4-1--Continued
N A / '. L
SHE CU AL TYPES
MASSE
r-
_ o
kAUI 1
187 PS VUL
oSV 021 V
1.400
4.600
.2800 .2000
188 bt V UL
AO ASV
TYPE
2


33
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 matter what scale height we adopt
for the atmosphere of the component :i n 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 evolutionary viewpoint, an answer seems possible. During
the evolution of the more massive component a point will
be reached where the star initiates its expansion to the
red giant tip. (We have here disregarded the possibility
of Case A mass exchange.) The ratio of time spent on the


34
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 Leva to (1974) .
Values for the masses were obtained from a much wider
variety of references, although Batten (1962) and Giannone
et al. (.1967) 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 Giannone (1967) have outlined procedures


35
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, re-liable values of M^, M2,
Lj_, L2, and 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 Pai'ameter 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
Unclassified = 0
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 4-1 and 4-2 contain Mass-Radius plots for
the primary and secondary components, respectively. It can
be noticed that an inspection of detached systems only


36
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Figure 4-2


38
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 ) 0.15G + 0.6471og(M ) o 0.240 (4.3.1)
9 42 83 9
Log (Rg) 0.534 -I- 0.3091og(Mj a = 0.312 (4.3.2)
38 87
Here s and g refer to the smaller arid greater radii, re
spectively, while a is the standard deviation.
Figure 4-3 shows a plot of R versus R This
a g
diagram serves mainly to provide an appreciation of the


39
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40
dispersion of radii for the different types of close
binaries. A linear regression yields
Rs = 0.181 -l- 0.649Rq 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
LogiM^+l^) versus Log(M^) 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
Mx + M2 = -0.4864 + 1.69 5Mq a = 1.284 (4.3.4)
1919 42
M-i + M0 = 1.280 + 2.12 GM a = 1.629 (4.3.5)
J. / b
211 65
Subtracting Equation 4.3.4 from Equation 4.3.5 and rearrang
ing somewhat yields
Mg = 1.277M, + 1.065
(4.3.6)


41
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43
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, B, A, F, G and K
begin at decimal 0, 2, 4, 6, 8 and
10, respectively.
2. Subclasses are plotted as the
appropriate fraction of the inter
val between spectral types.


44


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CHAPTER FIVE
DETERMINATION OF THE PERIOD AND ITS CHANGE
5 ]_Basic Concepts
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 TQ + 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, TQ 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 0~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 functional form of the resulting curve is not nearly
as easy as merely noticing if the period is variable. Sub
jective criteria of period variation rest heavily upon the
47


48
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, Cciused by the
rotation of the line of apsides. Wood and Sahade (1977)
have recently discussed apsidal motion as a cause of period
variation. Essentially, the density distribution of non-
spherical bodies will lead to an advance of the line of
apsides. The ratio of the period of revolution of the
orbit to the rotation of the apsidal line, is given by
Equation 5.2.1, due to Cowling (19 38) Here, kq and are
constants which depend upon the actual


4 9
f
Rg1
5
V
e = kg
. r .
(1+16M1/M2) + k2
r
k 1
(1+16M /M ) (5.2.1)
density distribution, R-^ and R2 the stellar radii, Mg 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 or 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 0-C diagram, so that the 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 either star wi]1 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 only with the qualitative
shape of the 0-C curve. Depending upon the nature of the


50
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 0-C curve, indicate a discontinuity in the behavior
of a star's 0-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 sy'stem, 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 also produce a parabolic shape
in the 0-C diagram. We can write
(5.2.2)
integrating 5.2.2 yields:
T
mm
/ Pdli
(5.2.3)
Assuming that
P
32
(5.2.4)


51
yields
Tmin T0 +
E
2
but
dP dP dt
dE dt dE
and
dt
dE
P
hence:
T
mm
T,
+ P0E +
The shape of this parabola can be either upwards
wards, depending upon the sign of the coefficienl
In theory, even more intricate curves arc
Instead of assuming a linear change in P with E c
in 5.2.4, we could have included an "acceleratior
as in Equation 5.2.9. Assuming
dP dP
P0 + dE E + dE
.2
we arrive at
T
mm
T0 + P0E +
1 } dPl 9
5P 3t B +
(5.2.5)
(5.2.6)
(5.2.7)
(5.2.8)
or down-
- s
2 possible,
is we did
i" factor
(5.2.9)
E3 (5.2.10)


52
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.
53 Program OHC
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:
1.A linear fit of the form T=Tq+PE
(5.3.1)
2. A parabolic fit of the form T^TQ+PE+AE2 (5.3.2)
3. A cubic fit of the form T^Tf)+PE+AE2+BE3 (5.3.3)
4.A combination parabolic/periodic fit of
the form T=T0+PE+AE2+Dsin (m (E-E, ) )
(5.3.4)


53
where D is the amplitude of the sine term, w, its frequency,
E-^ 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 = Tq + PE + AE2 + Dsin (w (E E^) ) = (5.3.5)
and by Taylor 1s theorem
(5.3.6)
here f (Tq,P, A, D,m,E,E-^) equals calculated time of minima
based on initial guesses
and E.
Eventually we obtain
0-C = AT0 H APE -I- AAE2 + ADsin (o) (E Ex)
+ Aw (E E1)Dcos(w(E E-j ) )
- AE-jioDcos (w (E E -| ) )
(5.3.7)


54
where 0-C = T ~ TQg PgE AgE2
DgSn (a)g (E -Eg))
(5.3.8
In all canes, values of Igg' Pg
and A were taken from
g
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 iterations. 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 T 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 data are non-uniformly distributed.
The program developed for this analysis also
performs the following operations.
1. There is separate least squares adjustment
of primary and secondary minima. Sufficient, deviation


55
of corresponding coefficients by each method gives informa
tion in regard to apsidal motion.
2. An 0-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,
0-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 by 11 inch plot showing the general
nature of the 0-C diagram and its residuals is produced.
This material is located in Appendix Two.
5.4 Weighting
A generalized system of weighting, able to take into
account different types of observational techniques, such
as visual estimates, wedge photometer observations, photo
graphic observations and photoelectric observations for
a wide variety of eclipsing systems has not yet. been
developed. Such variables as instrumental size, depth and
width of primary eclipse, magnitude of minima and any types


56
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 will hopefully reflect
the actual accuracies of different types of observations
to a sufficiently high degree 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 wishes, he may insert his own weights directly
and easily into the program used in this study.
How good is this weighting scheme compared to
other possibilities? It might be argued, for example, that
there is a large difference .in accuracy between a visual
estimate and a good series of visual observations performed


57
TABLE 5-1
WEIGHTING SCHEME
Type Weight-
Single visual observation 1
Single photographic observation 2
Single photoelectric observation 36
Visual', 2 or 3 (normal) 1.5
Photographic, 2 or 3 (normal) 3.0
Photoelectric, 2 or 3 (normal) 50.0
Visual, 4 or more (normal) 3.0
Photographic, 4 or more (normal) 5.0
Photoelectric, 4 or more (normal) 100.0
Decimal
Designation
1
2
3
4
5
6
7
8
9


53
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 implement his own weighting scheme or perform a period
study.
Using our reduction technique for a detailed study
for the period of U Cep, the author has found that by
changing all of the weights to unity, the final least
squares fit deviates a very small amount from the fit
provided by the initial scheme. Although this may in part
be due to the relatively deep primary eclipse of U Cep,
our weighting scheme still seems an adequate scheme with
respect to the observational material. In fact,- the system
of U Cep has several features which partially offset the


59
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 0-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 ephemeris. It would seem reasonable 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 determine what sort of
period variation is present. However, what happens when
both parabolic and sinusoidal variations are present? This
situation undoubtedly occurs often in nature, such as when
a mass transferring system orbits a distant third body.
A good example of this situation is the system of RT Per.
A highly distorted sine wave is in evidence on the linear-
fit 0-C diagram. The rather non-periodic appearance of this


60
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 be inferred from the error of the parabolic coef
ficient. Such a system also offers the advantage over the
linear cipproach that any system with both parabolic and
periodic period variation terms will now in principle dis
play a non-distcrted sine wave. However, this will only
occur for a data set with a uniform, equally weighted
distribution of data. Figures 5--la, 5-lb, and 5-lc show a
parabola, a sine term, and their sum. Nov/, 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 residuals from a
parabolic fit. It can be noticed that, in neither case,
are the originewlly known parabola and sine term recovered
from the data.


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70
Only by including a full set 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 greater 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, owing to the flexibility of this tech
nique ,
Occasionally, a cubic fit will be of interest. Wood
and Forbes (19G3) utili zed cubic fits for 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 solution, but that it offers two additional
advantages. First of all, it allows us to calculate any
secular acceleration in a binary system due to mass trans
fer. Secondly, it provides a fit for periodic data which
is almost as good as a periodic representation for a short


71
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 type 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 initial epoch from which the cycle count is
computed, as wel1 as the error of that initial epoch for
cubic, parabolic and linear fits, respectively. The next
column contains similar information on the value of the
period and its error for trie 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
sentation


TABLE 5-2
PER10D PARAKKTE RS
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73
TABLE 5-2-
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TABLE 5-2--Con tinuo d
EPOCH
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TABLE 5- 2--Con tinued
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TABLE 5-2--Continued
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I 9099/OO
9
b
9(Ob/ObORlb2
bZ
9 6
CQp
6 r /pc

9 2 99 c / T T
*
P
99 29 C* 09/9 2b2
6 0
- 9 6 6 6 I

6 6
66 C
roopc
60
-oc/r

-
0 2 b 0 9 / [ (
a
P
<'09 9 9 09 6 2 b2
2
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a
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9
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9
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n
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/ 0699*09/(32 b2
9 Z
v!*:0 is
n
V
ani22?
mod:-)
pouT9I00 -~z~ci
aiavi
03


81
TABLE 5-2-
--Continued
02
EPOCH
PlRID
A

bl 0i\A
05
24 1 9(>o 9 0 99b >
0u33t>2bC4
- b 2 0 r. 0 y
0 b 6 4 C 6 1 2
1 b 3 9 0
V A' V ->¡' V £
352299
< 1 0 'i C t i 0 6
.11731-11
24 1 9 6 6 4 9 9 1 4 9
0.03 22 31 12
- -i O 2 9L- 09
.13200
i \ (> \j )
94 C 1
, 1 o 1 CL-0 o
i ^ i b9 9
94 0 i
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241oo^7j
Of o5 2 2 04 0 o
. 1 o 1 4 C
4 2bb
1 o 2
8 4
8b
24 5 7030.715
0
f
1 7 6 9 C o 6 9
C

4 3 2 7 L
-
3 0
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9 70 26- 1
5

C t 7
21
1 1

1 4 U 1 .

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3 b 3 7 6 1
C-.
24 3 705 0.t 2 o b1
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04 07
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1 t 57 8

1 1
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1

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82
TABLE
5- 2- -Con! iiiued
EPOCH
PLk1
A
8
9 2
2 4 1 3 C 8 6.0 4 2 6 6
3.9 9 8 9 5 6 1 6

2o 60 6 0 7
0 1 4 6 9 E 1 1
2. 4 <3600
1 i 62 99
e
1 6 65 1 -0 6
9 3 9 0 L 1 1
24 13 08o. 4- 8 ) 86
3,992 77 1 1 5
0 .
24 .9 L 0 8
1 1 4 o 9
.3 3 4 2

3135L-06
24 1 30ot>. o 4 1 63
3. 992 80167
4 1 1
02
93
24 32 3o0 71 690
I 77b569 2 2
0 .
4 b 2 4 h 1C
0.6551 E 1 3
1 3 b
b2
*
4 i 69L-10
. 257 CE-13
2432360.71402
1 .77 6 6 7 i 1. 3
0 .
67 >> 5 5 3 0
7 7
2 0
<
1 0 1 t 1-10
2433bC* /i <4 /2
1 77 8 5 71 1 8
72
3 o
9 4
24 34 6 8 7'. 8 6445
C. 61 00 90 50
0 0
1 3 8 E 0 8
- 7 7 0 5 E 1 3
1 4 8
) 2 7

31 23E-0 9
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24 3 4 9 6 7.6 8 6 6
0e 6100 C 4 b b
0 .
23971:-09
1 99
o 6

53 o 1 ti 1 0
2 43 4 97 > j7oub
G 8180 9 7 65
1 4 1
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24 3 4 62 6. 2 5 8o8
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160
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4376E-10
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Si GMA
O 0846
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ponu cnuco -z -S liaVJ,
es


84
TABLE 5- 2 Continuc d
l 1 )
J 12
5 l 3
1 1 4
1 1
1 1 6
1 ) 7
LPUCH
P
6k 1 D
A
Li
SI GMA
24 26G 2 4
36 736
) .
963 2 34 4 1
*
7 6 6 81: 0 9
>
1 2442-
1 2
.62912
200
1 4 /
c
3 2 1 O C 9
33532-
1 5
2426624
3o 1 4
3 .
963 2 3 34 6
-
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1 6 3 7 l 0 3
0 jj 0 i> 2.
1 63
t
3 8 o / L 0 9
24 26 6,r 4 .
3 7 14 1
) .
663 2 2010
04316
34 0
67
242702O.
3 7 3 3 3
1 .
OC 46 62 90
0
f
4 2 39!:- 0 6

2 0:>7 E-
1 2
01151
4 0 76
1 9 9o

3 0 0 7 - 0 6

1614 L -
1 2
24 2 7026
5 2 0 0b
3 .
0 6 4 u 86 6 4
0
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1 7 0 7 L 9
. 0 1 3 5 5
1 ICO
8 64
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2 7 6 7 L 0 'V
2 4 2 7 0 ^ 6
3 1 322
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01152
1 24
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2427526.
0 o 7 7 8
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350
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86
TABLE 5--2--Continuad
L'CM
HLklD A
b
SI oMA
129
24 2021 1 (> 1 4 00
2.t 01 4 629
Oc
99 3 C i 0 6
-.42010-12
.01021
4 0 4
609
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vi 9 } : i 0 c3
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390
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4 70 7E-09
2428211.08421
2. 2 51 93 98
.03121
993
21< 4
1 30
2408291k7080
1 .u 7 0 3 0.3 0 5
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11 1 U 0 6
2 066b- 1 1
0 vj 4 b b
< < V V v t
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t
99026-0/
. 1 83 7 E-1 1
2408279,7 o 74o
1.67 7332tt
0 ,
3t 718-0 9
0 o 4 j
t 9 3 0 9
90 4 f
c
2 6 6 4 L 0 8
2 4 C 0 2 7 9 0 j 7 3
1.67734026
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.07202
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87
TABLE 5-*2-~Continucd
EPULH PLKJD
J 38
243524534483
0i 4 0 9 o4 0 8 3
6 2
27
24J524G, 54 883
0.4G9u4007
63
2 1
243324534231
0. 4C9641 56
79
7
139
2 4 2 0 0 5 5 i. 2 1 782
1 4 6 3 3 8 4 9 4
1 C 70
4 o 1
2420093.2789
1 .4i>37731
1 023
2 4 9
24200952907
1 < 4 653 76 36
237
26
1 AC
24 2 5 70 7.48763
5 .820 4 8 3 99
4 932
1 6 8 7 9
24 2 5 7 0 7 4 3693
5.8 8 C 6 O' 5 3 9
4 1 39
71 Go
2425707.31139
5.32051277
331 3
20 10
1 4 )
2427432.24800
1 5 5 0 9 7 o 7c>
1 33 7
1 1 80
2427452 4 44 ).
1 3 505 7o66
830
4 7 0
2427432.23594
1.330 9 8348
479
1 38
1 4 3
24 2 620 9. 34 680
1.74594298
2 4 5 2 9
1 28 C 9
2 4 2 6 20 9.8 3 308
1 74 6 0 3 9 3 9
3 4 29
1 0 o 0
24 26 20 9 3018o
1 74 60 55 0
293
4 4
1 4 3
24 3 050 3. 7319 0
0* 9 3 4 3 3 3 0
9 8
1 6
24 3 6 50 3. 7o918
0 4 234 0U34
1 4 2
1 4
2436508.77262
0.4 2 o 4 00 58
1 14
1 4
1 4 4
24 8 6 6 3 4 2 6 4 4 7
0.9 0> 0 9 84 83
1 t>4
1 2 0
242668.4.28806
0. 93 093 7C 9
1 4 4
03
24 2633 4. 2 6 3 33
C 93C93546
1 24
1 7
A

SI GMA
-.733CL-
1 0
C.75363-14
.01828
4 6 3 O L
1 0
. 201 CL- 1 4
0.93048-
1 0
.01939
. 1 2 6 7 L-
1 0
0 8 4 62
- 13 265
- 0 8
C 1 D IS 3 rl [I ~
. 0 1 0 64
. 7 1 o C l;
-09
.31o2E-1 3
'.151 3 L
- 1 0
.01681
* 1 2 6 2 3
-0 9
.01794
0.18 546-
Co
- 6 6C4 6-
1 0
.05338
13406-
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.40716-
1 0
-.69 8 6 5-
0 7
0 jO7 ^
. 27 71
0 7
Du3 Jb
0.
1 0
0
0 6
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-. 2 38. 1 6-
1 3
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2 68 4 6-
1 2
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60
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. 0 1
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t>7
7
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0 L
-09
. 0 1
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1
8/
1 i"'.
fc
O
- 7 3 7 C 6 1 8
0 0t 0 c
.19 7 4 l-C 7
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0 .. 1 1 6 A c 0 6
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. 74 076 0 9
0 0 4 'i 3
- 88 9 C. if- 1 0
1 4 0 83- 1 3
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. 7 74 3l- 1 5
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0 3 cl b1 *
.13976-10
. 0 3 4 b t>
0.38!46-C9
-.20926-13
0 0 0 1 6
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- 1 09 76- 0 3
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.35336-10
.00874
1 4 3
14o 24270o2.
1
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Of ii b b 1 6 0 4 i
0.25 2 Cl.
0
l
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.00611
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c b
1 3 0 81_
- 0 ^
60j.b: -14
2427832.
1
3 2 84
0 < i 1 L) 1
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- 09
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1 3 9
t)7
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- 1 0
24 A' 7 ¡o .
1
0 2 0 7
C C- ii.) j 1 U 1 b 3
0 0 O 4 0
1 82


BB
T A BLE 3- 2--Con ti me d
EPOCH
PlKKj a
¡3
SI GMA
1 4 7
2423256 41894
J *
012 7
O
o
0
r
2529 5-
0 4
ft
1 6436-08
0 2 -7- 4 4 *
# K- v v ^ >'
1 o 9o
9 V 9 <

2 2 3 L
0 4

124 2 6 -06
2 423014, 1 3o4 6
b
1 4 6 7

c

1 5 9 66 -
Co
7 9 v- v 8 v
<- >? < 2 1
o 9
1199

63 3 36
0 6
1 6 7 19 59
65
i i 5 v

0 6 3 3L-
06
24 2 300 9 > 4 2 4 02
3 c
1 b 0 4
o4 Cu
)|. Sjt $1 v ^ 4'
6 7'i 29
1
bu 3 9
1 46
24 26 00 3 15C b 7
o c
L> / C
0 2 o 6
-
c
6 3 6 C 6
0 6
0 .
15836-09
.31040
1 8 7 55

3 7 69
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c i 0 6 6
0 6

19016-09
214 2 o 0 0 5 2 b 6 5 J
O f.
O j 1 j
1 2 64
-
*
7371L-
07
. 3 0 6 1 C
1 3 o 0 y
el
4 5 1 9

1 0 90 6-
06
24 260 u3. 31 J t >0
(v e
o 3 1 J
69 1 0
.30420
1 0 4 7 9
9 0 6 0
1 4 9
2 4 3 2 9 c 7 < 2 5 7 o: ¡
2 .
3 b 3 4
89 6 1
0

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0 7

3 7 50 6-1 1
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304
1 2 1 5
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0 7
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2 27 16-1 1
24 329b7( 2 y b61
2 .
3 ti 3 b
OooB
r
L-

b 2 ti b L -
0 8
. 0 0 o 4 2
2 98
4 7 3
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1 4 1 0 6
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24 3 2 9 o7. 2 4 9 12
2
3 6 3 b
2 t 9
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342
2 6 1
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1 i 24 6-
1 0
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37 4 7 61 4 9
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68786
1 1
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3 7 4 7
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24 34 220

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b
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91
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1
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24364ob
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1 559
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89
. FIGURE S-2Continued
EPOCH
P Lf< I D
A
6
S oMA
156 24 32 44 1 .43500
1 34
2432441*44410
7 2
2432441 o44 4 91
t>-J
157
0.71161774
1 3
0 71 1 t 1 o 66
9
0 7 1 1 t 1 7 C 3
7
0 19 2 71-09
* 22 3 1 E~ 1 0
0 6 9 36- 1 0
. 92 790- 1 1
152 8 E- 13 00690
197 5 6-14
. 0 0 795
.00613
153 2424552.77057
1 5 5 3 9
24 2 4 553.3 C 491
2 3 7 9
24 24 333 3 3 442
304
1 5 9
64950322
60 93L08
0 1 17 4 E- 1 2
v) J 4 c>
29 6 3
.1790L-06
. 3 34 3 0 1 3
64 9403 1 7
-. 13942- 0 9
e 0 0 4 4 9
290
. 6 1 3 5 : 1 0
04 93 9 931
0 0 4 c 0
2 7
1 6 0
16 1
1 62
163
16 4
2429217.56354
13. 19841365
c
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0.2 0.)
4 e: : 9
. 0 3 1 7 9
67 7
17 79
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13. 19 66 1 7 9 4
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0 <_> 3 6 O'
1. 137
15 10
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22 7O L 0 7
2429217.34631
V ac V V V V v- -V- V ;.c V
. 0 7 34 7
956
1 3 4 o
24 390 91 71226
2 64 6o 39 27
0 .
1 1 7ft:- 0 7
0. 2 c, 3
0E- 1 1
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c
OL> ] 4 l_* 7 6
. 1 7,
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24 3 90 91. 71640
2. 64 63 4 4 99
0.
3 74 0 E 06
7 7 C d 7
4 63
5.; 1
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2 4 3 9 091 7 2 2 o a
2 < o4 o3 34 60
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d ii 0
1 89
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1 15 15 6 6 4 1
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3542L-C8
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9 0-1 8
24 23 160.4 1 .5 1 4
1 1 6 1 o 33 1 7
c
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7 2 7 d
d 77
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7 4 2 CS 1 L> 0 t '¡a O4- Lr
1.151 0 3422
0 0 o o o
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2433343.1 3 733
1 7 4 5 o 4 o 7 5

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0.391
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1 14 39
c
12620-07
24 3 3 d * O c 4 i J 7 7
1 74 5o 7 o 10
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254 GO-07
.00967
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3 7 3 o J % J17 7
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o92
3 76


90
TABLE 5---Continued
J 65
166
1 6 7
EPUCM
t*
LkloU
A
3
SI ON1A
24 36 1 }. 4
.67923
3 ,
9 6 6 u 0 9 6 6
c
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2. 9 o o 6 1 0
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12 7 6
4 6 6 9

1 2 7 2 8--
0 f

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24 36 1 1 4
i66612
3
9o 5 7 16 jt;
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1 1 3 9 l -
06
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1 6 33
124 1
7b756-
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24 36114
* 4 6220
3
96 372 7 61
. 7S3oC
7 4 21
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2.4 291 1 1
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3
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0
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3967L-12
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1 21
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24 3 3 51 6
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91
TABLE
5-2--Continued
1 74
LIJCH
PLk¡ D
A
i>
Si C>MA
1 75
1 76
24 29 loG. 4 b.iO 7
0 .
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0.
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24 29130c4o310
c.
03 42 3 76 6
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24 29130.45 036
c .
6 6 4 2 39 2o
.00941
lol
1 0
1 77
2427199.30784
2
76 8 8 2 2 0 7
6
4 0 3 0
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C. 923 StfE 1 2
007 J 0
3 0 0
3 50
0
168 6
u-08
.21732-12
24 2 7i 99. 4 9 o u 0
. 6
7 O 8 8 0 9 3 2
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1 3 4
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24 74
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2427i99.301 io
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7 3 8 o 2 7 4 4
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2 60
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1 79
243 73/2 .. .'0063
0.
9 71o8274

9 9 I 4 i; -
1 0
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2
0
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24 37 37r: c 2 0 0 3 3
0.
97 1 5 326 4
t
4 7 0 7 L
3 C
0 096 0
20
7
0
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1 0
24 3 7572.2 0 0 03
0 .
37153270
5* 6 9 7 5
1 9
7
180
2 43 5 3 90. 51293
0.
2318 83 05
0.
12690-
1 1
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.00465
00
80
t
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243539o.31307
0
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MASS AND MOMENTUM EXCHANGE
A DISSEI
IN PAR'
IN CLOSE BINARY
SYSTEMS
By
JAMES BRUCE RAFERT
STATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
MAL FULFILLMENT OF CUE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY of FT.(.'RIDA
197 8

For Donna

ACKNOWLEDGMENTS
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 Astronomy 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. Merri11.
iii

TABLE OF CONTENTS
Page
ACKNOWLEDGMENTS . iii
ABSTRACT vi
CHAPTER
ONE INTRODUCTION 1
TWO HISTORY 3
2.1 Introduction 3
2.2 Early Concepts 3
2.3 From Speculation to
Calculation 8
THREE PRELIMINARY CONSIDERATIONS . 13
3.1 Theoretical Context of
Mass Transfer 13
3.2 The Roche Model 14
3.3 Evidence for Mass
Loss/Transfer 16
3.4 Some Relations . 20
FOUR SELECTION OF THE SYSTEMS 25
4.1 Selection of Stars ....... 25
4.2 Absolute Parameters 27
4.3 Parameter Correlations 35
FIVE DETERMINATION OF THE PERIOD
AND ITS CHANGE 4 7
5.1 Basic Concepts ..... 47
5.2 Causes of Variation 48
5.3 Program CMC 52
5.4 Weighting ...... 55
5.5 Philosophy of Curve
Fitting . 59
5.C Least Squares Parameters .... 71
3. v

CHAPTER
SIX AN EPHEMERIS FOR EACH SYSTEM 9 3
6.1 Basic Considerations 93
6.2 A Word on Each System 9 4
SEVEN THEORIES OF MASS EXCHANGE 14 6
7.1 Introduction 146
7.2 Particle Trajectory Models ... 147
7.3 Ilydrodynamical Models 159
7.4 Evolutionary Models 161
EIGHT EJECTION OF MATTER FROM
UNSTABLE COMPONENTS 171
8.1 Non-synchronous Rotation .... 171
8.2 Matter Ejected from
Unstable Components ..... 172
8.3 Program ORBIT 17 5
8.4 Efficiency Tables 178
NINE LIMITING VALUES OF dM and dJ 229
9.1 Limiting dJ 229
9.2 Calculation of dM/dt for
Each System 233
9.3 Limits on dM 235
TEN CORRELATIONS OF COMPUTED AND
ABSOLUTE PARAMETERS 240
10.] Period Correlations 240
10.2 dP/'P Correlations 250
10.3 dP/dt Correlations . 254
10.4 dP.7_T Correlations 264
NCJ
10.5 Summary Tables 272
10.6 Statistical Trends 275
APPENDIX
ONE OBSERVATIONS 278
TWO O-C DIAGRAMS 601
THREE BIBLIOGRAPHIC MATERIAL 761
REFERENCES ...... 780
BIOGRAPHICAL SKETCH ...... 786
v

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. Al] available times of primary
and secondary minima for these systems were assembled and
subjected to least squares analysis. Linear, parabolic,
cubic and per iodic/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 dm/dt. Non-conservative effects were investigated by
a particle trajectory model in which material conforming
to a Maxwellian velocity distribution was allowed to leave
the vicinity of the inner Lagrangian Point Lj for a wide
range of systemic parameters.
V
i
Particle integrations 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)N{-j and the mass flow
rate (dm/dt)were calculated. Correlations of (dP/dt)
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 momenta 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.
vix

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.
Naturally, more accurate transfer rates could be
calculated if these two assumptions could be relaxed, or
eliminated entirely. It is the aim of this study to analyze
1

2
the character of as many binaries which exhibit pex'iod
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 that it could allow us to obtain an overview
of the situation.
2.2 _Early Concepts
Certainly one of the earliest results of the
study of eclipsing stars was the fact that some displayed

4
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 Ü 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 change 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

6
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 fil] their Roche surfaces, or come so close to
filling their surfaces that for all practical purposes
they con be assumed to fill them. He explicitly stated

7
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 stars were discovered to be 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 material possible? In an effort to avoid this

8
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 (Morton 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

9
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 inass
transfer or loss can be rather intricate. It should be
noted that Kopal1s particle trajectory approach was rather
limited. Only single particles ejected at some mean
velocity were considered.
Kruzewski (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 show7 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 (1.968),
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
Figure 2--1

11
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 suffer1 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.
13

14
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 M-, , 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
C
2_ 1 + _2q_ f_l_
1+q 1+q l'r2
x
J
2 2
x + y
(3.2.1)
where q = M2/M^ (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 Lj . The L-j 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

15
Ficjure 3-1

16
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 L] 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 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~~^ M@/year, the
"requirement" of mass loss is met. By stellar winds
alone we could expect intersystemic mass flow in close
binary systerns.
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 that the precursors

17
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 Chandasekhar 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 estimates have been negative: for this system most
of the time. However, polarization is detectable during

18
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 ait 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 gaseous stream.
Observations of other systems, such as AO Cas or
U CrB 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-synchronously rotating

19
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 impinging upon the. leading hemisphere of the primary
star.
4. Parabolic period variation: as is shown later,
this effect may be explained in terms of mass loss or
transfer,

20
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
(3.4.1)
where a - semi-major axis.
Setting the orbital angular momentum equal to
(3.4.2)
and substituting Equation 3.4.2 into Equation 3.4.1 we obtain
2Ti (M^ +M2) J^
G2 (MgM^)3
P
(3.4.3)

21
which, upon differentiation and some rearrangement yields
dJ _ kMiM2P1//3
dt 3(M +M ) 2/3~
2dM2
' >
1-C
dt
(Mi +Mp)
l J
dP 3 dM 2
+ (Mi-CMo)
Pdt M^dt x
where k = G2//3/(2tt)-'-/3
(3.4.5)
and dMj = -Cdf^ (3.4.6)
HereC is the function of mass leaving the secondary component
which falls upon the primary.
We may also write
So
as
13tot dorb + drotl + drot2 4 dlost
- ~d.J - dJ - dJ.
oíd rotl rot2 lost
dJtot = 0
(3.4.7)
(3.4.8)
(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 rotational angular momenta,
and their changes, in the following fashion:
dJ
rotl
• 2 rr 2
T~ dMlrl°'i
(3.4.10)

22
dJ
(3.4.11)
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=0 (no mass transferred) both dJrot^
and dJrot2 equal zero and dJorb~dJ^ost. We have
(3.4.12)
while if we set C=1 (all mass transferred) dJlost=0 and
^rottot ' ^rotl + ^
rot 2
(3.4.13)
giving
dJrot
dt
(3.4.14)
In both Equation 3,4.12 and Equation 3.4.14, dP/dt is an
observational quantity. The quantity dJ^ost can be calcu¬
lated by either particle trajectory or hydrodynamics!

23
models. For all cases in this study it is assumed to be
negligible and is set equal to zero. The quantity <3Jrottot
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 momenta, as in the Biermann-Hall model,
proceed on a time scale (dynamical) 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 , kM1M2P1/3 T dP_
dJrotl e lost 3(m1+m2)2/3 P
+
3dM2(M1-CM2)
2dM (1-C)
(m1+m2)
(3.4.15)
This is just, the previously derived equation without any
simplifying assumptions.
It might be argued that by computing the magnitude
of the orbital-rotational conversion process from non-
synchronously rotating stars, we are neglecting the fact
that at the point in time when they had synchronous rotation

24
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,
Jrot2y = m2y r2j_ w2j, (3.4.16)
while eventually
Jrot2f = H2f r2£ “2£
Subtracting to get dJ we obtain
dJ = -Jrot2f + Jrot2i
where we assume values of initial and final sixes from the
(3.4.17)
(3.4.18)
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 materia]., 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
25

26
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 systems 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
Forbes made their study. On the other- hand, some
Wood and

27
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 fincil criterion, sufficient data for each
system were necessary. Except for very interesting objects
this means, at the bare minimum, at least a dozen observa¬
tions .
4.2 Absolute Parameters
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 conservation of mass and orbital angular
momentum.
In order to perform

28
dP
P
3 (1-y) dM
y M
(4.2.1)
where y = ¿
M]_ + M2
this calculation, the values of P, 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 semi-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 Lagrangian 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
r
1
2
3(1+r)
2
2
(1+q) (1+f) r = 0
q
1
(4.2.2)

29
TABLE 4-1
ABSOI.it!TE PARAMETE RS
! 3 A M L
SPLCTi
-At. 1YPLS
LASS
Í 3
re A L) I
T
TYPt
1
AT A AS
1 3
r 0
C . 6 ¡3 0
i. •:> c o
. s 0 0
. 24 0 0
1
cl
1 «V A ¡ \ s
f 0
K 3
0.4 l 0
2. 4 C- 0
. 2 4 o 0
.15 7 0
2
3
V. Z A ND
f- ‘5
03
i . e c o
1 .40 0
. 4 3 c. C
. 3 6 0 C
4
4
X/ A f-, Ij
AO
K«t
. 30 > 0
,2 930
3
A f i AGO
US
c-s
1.130
1.8 2 0
. 4 4 C C
, 3 60 0
4
6
3 X AND
uO
,4640
. 3 8 2 0
7
AY AuK
A 3
IN 1
. 2 4 0 0
.2070
b
S U A A A
As
o
CX AG-A
f 2
1 C
X A ul
A 2
1 3
r'K AL
L 0 Y
.3141
. 2 03 9
1 2
A ¡J A u L
A 1 V
A 3 V
0.560
2.900
1 3
KP Ai,.i_
AO
1 4
t;j A C L
u s V
O Ei
. 4 2 j 0
. 3 30 0
\ 3
J Y A u G
1 0
GO
0 . 7 s 0
2.700
. 2 4 j 0
. 1 53 0
1
1 C
V 5 3 7 A u L
Li 0
1 7
V 3 4 3 A^L.
At
O 3
. 27 0 0
.2310
3 E
2/ A O-c
F 0
3 v
TT A 33
t.s
Li 3
5 • o 0 0
u. 70 0
t J VS J
. 3 2 C C
1
' o
Vi A Ur,
A 7
A 7
1 , 7s0
1.510
« \ l ' J, C,
. 1 o2 5
1
2 1
2 3 As A
A 7
. S 1 O J
. 2 94 0
22
3! AuFv
use
s s
9 . 3 3 C
9.3 5 0
. 4 1 0
. 3770
*4
23
1 U A u A
Li 2
u .
9. 70 C
; 4,9 o o
. 5 9 0 0
. 2 3 0 r'
2.
2 4
8U 3L Ü
A Jv
. 2 2c. 0
.2130
2 5
4 Y
.4100
. 4
? c
44] uuj
G 2
0 4
0.3 S C
0 . L- 1 0
« 4 c -i 0
~3 o O -3
4 .0 _ Vs u
4
27
Y C a
m7 Y
y ^
0.48 0
? . 2 0 0
» 24 „ 9
.2350
(-
2 o
3 3 Cm M
2 3
0 1
. 4 4 S 9
.13 2 0
2 9
8v CAM
G 5
G3
0 . 7 Lv 0
1 . 15 0
. 3 b c. 0
. 2 4 2 0
1
5 0
SZ CAR
GO
G'0
. 4 3 4 4
.18 7 7
5 3
AS C AM
i_9
3 2
S CNu
AO
V.J L>
2.400
6 . b 0 0
. 1 9 3
. 0 32 7
1
3 3
AY CAC
34
TV CKC
35
3 C V In
Í 41 V
K0 i V
1 . 3 5 C
1.40 0
. 2 6 s 3
,0940
1
3 {_
re c a
M 1
a 5
0.1)0
('. 4 9 C
. 3 0 2 0
. 2 4 60
2
3 7
A i. C A P
A 2
A 4
3 8
G t. C A k
:iS
1-1 4
5.770
5.8 9 0
. 2 1 s 0
. 2 1 6 C
1
3 9
SX CAS
0 3
A s
(j < 2 C 0
G. 4 0 0
. 29 SO
. 1 97 0
1
4 0
5X C AS
As
OS
. 220 0
.090 0
4 1
T V C A .3
r li
F o
1.000
1.700
.3110
, 2 5 7 0
2
4 2
T.': Cm3
Li 1 1 V
o o
1 .liC
2. GO 0
, 3 s 7 0
.2100
2
4 3
/. 4 CAS
E_i3
4 4
A A CAS
A 3
. u Li
. 2 1 s 0
. 1 S 3 0
4 5
D ) C AS
A 2
0 0
0.6 4 0
2.040
4 ó
.''34 CAS
A .<
4 7
ClN
i 2
F 9
. 5 7 u 0
.2200
>4 B
S V C M
14
L 1
1 . 10 0
9 . 3 C 0
. 4 3 0 0
. 4 2 7 0
4
4 V
V.UA CL . <
L, S
5 C
U C FI P
Lj o
0 3 ! i }
1 i 9 C 0
4.7 0 0
.310 0
= 1 <>C0
2
3 3
PS l -_P
A :j
ic 3
0. 7s 0
3 < 0 C G
.21 s 0
, C o?0
j
3 2
V,. Ct.P
L-0
r- 0
0.2 5 0
0.61 0
« 5 4 0 0
. 240 0
4
5 3
V. V Cl A
A 7
. 4 j s C
. 320 0
S 4
XX CLP
A í¿
0 4
0.2 7 0
1 . 7 2 0
j r\
* ^ \s
, 2 1 5 0
2
3 5
A li CL P
L>]
L- 1
1 3.9 0 C
1 o . 1C' C
t 5 • ♦ y 0
. 3 4 0 0
1
3l
C .s C r P
i.»0
S 0
1 1 . C 2 C
1 1.740
. 2 6s 0
. 1 99 0
1
5 7
5 S C l 1
AO
3 c
I-:. Cl.M
Ug
o .9
5 0
> / CuM
n0
K 0
0 . 7 3 0
1.3 8 0
« 4 5 u 0
, 2 c. 0 0
i'C
U C i- i_>
i c>
f\ 0
1.400
2.9 0 0
« 2 Ej s 0
. I860
A
ó 3
1 CJ C A. j
62
3 - C Yu
A 2
K *♦
0 . 7 (. C
2. 30 0

30
TABLE 4-1—-Continued
NAME.
bí'E CT RAL TYRES
M A S
Sf. s
RADI
1
1 Y PL
ó 3
V IV
C YG
A3
KO
0.700
2 . SO 0
•
2200
*
1110
1
6 4
S V
C YG
A3
6 b
4 <\
C Y G
Ü7 7
G8 1 7
4.30 0
7. 60 0
•
2 7^0
c
2 190
1
6 6
V\ /
C 1 G
1 0
6 7
7 7
C Y G
1- 7
K b
0.70 0
12.^00
•
353 0
t
0
2
6 6
cv
CYG
e 3
6 9
r f.'
C YU
70
GO
C 1 G
bv
r 0
0. 63 0
0. 96 0
«
4 54 0
«
4 0 8 0
4
7 1
V4
4 H C Y U
111 1
09
7 2
V 4
5 7 CYG
b 1
E 1
13.000
'x!
<*
o
o
o
t
2 90 0
•
1 780
1
7 3
V 4
G 2 C Y o
1- b 7
F- 4
•
0 o 2 0
c
0 5 4 0
7 4
W DEL
A 0
G b
0.4 4 0
2.100
•
27 dC
«
1 3 70
')
«_
7 b
Z
J[- n
ÁD
K, E
0,3 6 0
1.40 0
«
26a0
t
2 570
7-
7 6
RA
r t< a
A 2
7 7
rz
O K A
Ab
7 b
sx
Dh-A
A3
Ex 0
•
3 2 7 0
.
2 03 0
7 9
i
0 Í - A
Ad
r',0
0.6 2 0
71. 20 0
*
3 3 o C
«
2 1 00
8 0
T /
IX, A
A 7 V
81
UZ
['• A A
82
5
El 0 G
b9 V
G3
0.36 0
2.9 6 0
«
22 3 0
«
2 010
2
6 3
RU
El . I
A0
64
UZ,
! I- I
GO
G!
•
50 0 0
t
4 0 0 0
8 b
U
GE M
6 0
NO
66
V' V
GEM
Ei 1
M i
0 . b 6 0
0. 58 0
*
1 but
c
1 36 0
1
6?
A r
C f: M
A0
8 8
AY
GEM
A0
6 9
Í' C.
u L M
A 2
90
•9 X
Gt_ !•;
A 4
G3
0 < b 3 0
2.4 0 0
•
1 9 x) 0
*
0 700
1
91
r y
Gtr.M
A 2
E, 1
0.02 0
2. 70 0
•
p ^ D
«
12 7 0
2
9 2
/.
HE is
E 4 1 V
A 5 1 V
3 . 1 0 0
1 . .2 2 0
•
’ D 0
e
1130
)
93
R X
Hi: i<
69
A 1
2.30 0
2 . 7 S C
>
2 3 C C
#
1 6 B0
1
9 4
sz
E í L R
A0
A 0
•
3 2b 0
€
3 10 0
c 5
1 T
hi: A
A 2
0.9b C
3 . 2.1 C
»
50 4 0
4
2 72 0
4
9 6
TU
Ht_R
E 5
9 7
TX
11 cJ<
A 4
A '+
3 , 76 0
2 . 050
*
1730
4
1 3 B 0
9 6
UX
H 6 k
A3
Kb
0.49 0
3 . 86 0
t
2 D o 1
•
2 3)0
2
9 9
AK
HER
t- 2
1 D
0.4 3 0
1.430
t
3 4 0 0
«
2 4 0 0
¿\
1 0 0
c c
HER
A0
1 0 1
C T
h Li;
A 0 7
1 0 2
D I
h'L E
u4 1 1 I
Ü J 1 I 1
1 03
V 3
3 6 h;; «
A9
Ex D
9
3 0 1 0
C
2 2 7 0
1 0 4
r v
i • A
m a
A D
K 0
0. 380
1.500
«
29 1 0
c
1 BOO
2
1 0 b
s \
1 i 1 7,
A3
rx D
0 . b 9 0
1 . 7 0 0
*
"¿ S ¿4 0
r
1 1 8 0
E>
í-
1 0 6
11
H V A
A3
G.
0.71 C
2,0 0 0
4
2 2 7 0
r
1 00 0
1 0 7
5 *
L AC
G 3
G 3
0.61 0
0. 33 0
«
4 10 0
♦
3 30 r'
o
(-
103
T 4
L A C
A 2
1 09
U *s
L AC
A0
1 1 0
vx
LAC
1- 0
1 1 1
AR
L AC
Gb i V
E, 0 1 7
1.310
1 . 32 0
«
i 1 C 0
«
19 4 0
3
1 1 2
C M
LAC
A 2 7
-,V
1 .47 0
3 o 86 0
*
i y d o
c>
1 6 5 0
i
1 1 3
CO
L r‘.C
LEÍ i V
u V V
*
2 3 3 C
4.
2 1 G 0
1 1 4
Ob
LAC
l>2 1 1 1
1 1 b
Y
. C L.
A 3
0.4 C 0
1 . 10 0
»
2 80 0
.
2 30 0
4
1 1 6
l?T
C. i.:L
A 0
1 1 7
UV
l. I U
G ü V
G 1 v
0.9b 0
1 .020
•
30 oO
c
3 0 1 0
1
1 1 6
K Y
Li.U
1 1 9
T
L M 1
A0
*
2 6u 0
«
2 1 7 0
i; o
99
1 1 L!
Ab
1 2 I
Si
LIN
12
M 2
»
4 4 0 0
4
2 64 0
1 2 2
! L Y K
AO
1 7 3
l< z
l. YÍ.
/ J
I
124
TT
1. YK
AO
G‘.'
*
29 4 0
«
1 5 90

31
TABLE 4--l--~Continued
NAM t.
SPECIF.
\ L TYPES
MASS!
s
PAD I
J
TYI
3 25
TZ L YP
K0
. 4 i a o
. 3 3 4 0
1 2 6
U Z L Y k
A 2
1 2 7
E .v L Y tf
FG
1 2 b
!' MLN
A 0
G 7
3 t- 3 0 0
3 .
70 0
. 30 0 0
.2)70
2
129
FH3 ML N
A2
150
U CPn
L¡ 4
Bb
4 .650
r.
^ *
3 0 0
« 2 6 o 0
. 2 520
3
1 3 5
f-.V CPU
A0
G b
. 20 7 0
« 1 4 6 0
1 32
SW OHM
A0
0.42C
¿ o
7 (: 0
. 3 3 9 0
. 15 7 0
2
1 3 3
V 4 49 uPtl
1 3 6
V 4 5 3 LPl i
A0
A 2
2.600
3 •
0 0 0
. 2 1 o 0
. 17 3 0
1
1 3 5
V 5 0 3 0 P M
A zi
1 3 6
Vb02 CPn
Í- 9 V
G 2 V
0.490
3 .
220
. 4 c Q o
. 3 2 0 0
4
1 3 7
V5 0 5 LPH
1 3 >3
V .â– > 01 0 P11
Í 4 V
6 3
0.44 C>
1 .
3 0 0
« b 4 0 0
. 2 5 0 0
4
1 3 9
VV Hi-. I
D 3 V
t> 4
3. 35 C
6 .
70 0
.33-10
, 3 79 0
3
1 6 C
CP Cb 3
GO
3 4 3
EL f'P j
M 3
1 4 2
r:o c p i
A0
1 4 3
E i\ G r; i.
G 3
G3
C . 29 C
0.
4 60
. 4 0 0 0
. 3 60 0
4
J 4 4
FI CP 3
G3
1 4 b
t H CPI
A 1
1 4 6
f P Ü ;< 1
A 7
3 47
FT Civ I
AO
1 4 b
G r, c: k i
A 2
1 4 9
L S LI í v 3
mO
3 50
LI P L b
P 3
6 3
1.07 0
3 .
330
. 4 60 0
.34 00
u
1 3 1
T Y P L. G
a2
K 0
0. 23 0
3 .
50 0
. 2 2l 0
. 23 70
2
1 52
UX f't.G
A 2
1 33
AO Pt.G
A 2
k 0
0.590
id •
SC 0
. 12 4 0
« G 2 3 0
1
1 54
E. G PL G
A 2
3 bb
b’< Ptb
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32
TABLE 4-1--Continued
N A /• '. E
SHE CU AL TYPES
MASSE
r-
. o
kAUI 1
187 PS VUL
aSV G21V
1.400
4.600
.2800 .2000
188 bt VUL,
AO ASV
TYPE
2

33
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 matter what scale height we adopt
for the atmosphere of the component :i n 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 evolutionary viewpoint, an answer seems possible. During
the evolution of the more massive component a point will
be reached where the star initiates its expansion to the
red giant tip. (We have here disregarded the possibility
of Case A mass exchange.) The ratio of time spent on the

34
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 Leva to (1974) .
Values for the masses were obtained from a much wider
variety of references, although Batten (1962) and Giannone
et aL (.1967) 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 Giannone (1967) have outlined procedures

35
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, re-liable values of M^, M2,
Lj_, L2, and 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 Pai'ameter 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
Unclassified = 0
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 4-1 and 4-2 contain Mass-Radius plots for
the primary and secondary components, respectively. It can
be noticed that an inspection of detached systems only

36
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rigure 4-2

38
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 ) - 0.15G + 0.6471og(M ) o - 0.240 (4.3.1)
9 42 83 9
Log (Rg) - 0.534 -I- 0.3091og (MJ a = 0.312 (4.3.2)
38 87
Here s and g refer to the smaller arid greater radii, re¬
spectively, while a is the standard deviation.
Figure 4-3 shows a plot of R„ versus R . This
s g
diagram serves mainly to provide an appreciation of the

O n r
39
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40
dispersion of radii for the different types of close
binaries. A linear regression yields
Rs = 0.181 -l- 0.649Rq 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
LogiM^+l^) versus Log(M^) 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
Mx + M2 = -0.4864 + 1.695Mq a - 1.284 (4.3.4)
1919 42
M-i + M0 = 1.280 + 2.12 GM„ a = 1.629 (4.3.5)
J. / b
211 65
Subtracting Equation 4.3.4 from Equation 4.3.5 and rearrang
ing somewhat yields
Mg = 1.277M, + 1.065
(4.3.6)

41
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42
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Figure 4-5

43
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, B, A, F, G and K
begin at decimal 0, 2, 4, 6, 8 and
10, respectively.
2. Subclasses are plotted as the
appropriate fraction of the inter¬
val between spectral types.

44

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CHAPTER FIVE
DETERMINATION OF THE PERIOD AND ITS CHANGE
5 . ]_Basic Concepts
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 - TQ + 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, Tq 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 0~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 functional form of the resulting curve is not nearly
as easy as merely noticing if the period is variable. Sub¬
jective criteria of period variation rest heavily upon the
47

48
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 apsidal motion as a cause of period
variation. Essentially, the density distribution of non-
spherical bodies will lead to an advance of the line of
apsides. The ratio of the period of revolution of the
orbit to the rotation of the apsidal line, is given by
Equation 5.2.1, due to Cowling (19 38) . Here, kj and are
constants which depend upon the actual

4 9
f
RX]
5
V
e - kg
. r .
(1+16M;l/M2 ) + k2
r
C 1
(1+16M /M ) (5.2.1)
density distribution, R-^ and R2 the stellar radii, Mg and
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 or 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 0-C diagram, so that the 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 either star wi]1 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 only with the qualitative
shape of the 0-C curve. Depending upon the nature of the

50
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 0-C curve, indicate a discontinuity in the behavior
of a star's 0-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 also produce a parabolic shape
in the 0-C diagram. We can write
(5.2.2)
integrating 5.2.2 yields:
T •
mm
JPdli
(5.2.3)
Assuming that
P
32
(5.2.4)

51
yields
Tmin T0 +
• E
2
but
dP _ dP dt
dE dt dE
and
dt
dE
P
hence:
T â– 
mm
T,
+ PqE +
The shape of this parabola can be either upwards
wards, depending upon the sign of the coefficient
In theory, even more intricate curves arc
Instead of assuming a linear change in P with E ¿
in 5.2.4, we could have included an "acceleratior
as in Equation 5.2.9. Assuming
dP dP
P0 + dE * E + dE
.2
we arrive at
T •
mm
T0 + P0E +
1 } ÍdP) 9
H at B
(5.2.5)
(5.2.6)
(5.2.7)
(5.2.8)
or down-
- s
2 possible,
is we did
i" factor
(5.2.9)
E3 (5.2.10)

52
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 OHC
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:
1.A linear fit of the form T=Tq+PE
(5.3.1)
2. A parabolic fit of the form T-Tq+PE+AE2 (5.3.2)
3. A cubic fit of the form T^Tq+PE+AE2+BE3 (5.3.3)
4.A combination parabolic/periodic fit of
the form T=Tn+PE+AE2+Dsin (u (E-E-, ) )
(5.3.4)

53
where D is the amplitude of the sine term, w, its frequency,
E-^ 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 = Tq + PE + AE2 + Dsin (w (E - E-j_) ) = (5.3.5)
and by Taylor 1s theorem
(5.3.6)
here f (Tq,P,A,D,w,E,E^) equals calculated time of minima
based on initial guesses
and E.
Eventually we obtain
0-C = AT0 H APE -I- AAE2 + ADsin (o) (E - E±)
+ Aw (E - E1)Dcos(w(E - E-j ) )
- AE^'wDcos (w (E - E -j ) )
(5.3.7)

54
where 0-C = T ~ TQg - PgE - AgE2
Dgsin (o)g (E -Eg))
(5.3.8
In all canes, values of T0g' pg
and A were taken from
g
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 iterations. 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 T , 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 data are non-uniformly distributed.
The program developed for this analysis also
performs the following operations.
1. There is separate least squares adjustment
of primary and secondary minima. Sufficient deviation

55
of corresponding coefficients by each method gives informa¬
tion in regard to apsidal motion.
2. An 0-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,
0-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 8h by 11 inch plot showing the general
nature of the 0-C diagram and its residuals is produced.
This material is located in Appendix Two.
5.4 Weighting
A generalized system of weighting, able to take into
account different types of observational techniques, such
as visual estimates, wedge photometer observations, photo¬
graphic observations and photoelectric observations for
a wide variety of eclipsing systems has not yet been
developed. Such variables as instrumental size, depth and
width of primary eclipse, magnitude of minima and any types

56
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 will hopefully reflect
the actual accuracies of different types of observations
to a sufficiently high degree 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 wishes, he may insert his own weights directly
and easily into the program used in this study.
How good is this weighting scheme compared to
other possibilities? It might be argued, for example, that
there is a large difference .in accuracy between a visual
estimate and a good series of visual observations performed

57
TABLE 5-1
WEIGHTING SCHEME
Type Weight-
Single visual observation 1
Single photographic observation 2
Single photoelectric observation 36
Visual', 2 or 3 (normal) 1.5
Photographic, 2 or 3 (normal) 3.0
Photoelectric, 2 or 3 (normal) 50.0
Visual, 4 or more (normal) 3.0
Photographic, 4 or more (normal) 5.0
Photoelectric, 4 or more (normal) 100.0
Decimal
Designation
1
2
3
4
5
6
7
8
9

53
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 implement his own weighting scheme or perform a period
study.
Using our reduction technique for a detailed study
for the period of U Cep, the author has found that by
changing all of the weights to unity, the final least
squares fit deviates a very small amount from the fit
provided by the initial scheme. Although this may in part
be due to the relatively deep primary eclipse of U Cep,
our weighting scheme still seems an adequate scheme with
respect to the observational material. In fact,- the system
of U Cep has several features which partially offset the

59
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 0-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 ephemeris. It would seem reasonable 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 determine what sort of
period variation is present. However, what happens when
both parabolic and sinusoidal variations are present? This
situation undoubtedly occurs often in nature, such as when
a mass transferring system orbits a distant third body.
A good example of this situation is the system of RT Per.
A highly distorted sine wave is in evidence on the linear-
fit 0-C diagram. The rather non-periodic appearance of this

60
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 be inferred from the error of the parabolic coef¬
ficient. Such a system also offers the advantage over the
linear cipproach that any system with both parabolic and
periodic period variation terms will now in principle dis¬
play a non-distcrted sine wave. However, this will only
occur for a data set with a uniform, equally weighted
distribution of data. Figures 5--la, 5-lb, and 5-lc show a
parabola, a sine term, and their sum. Nov/, 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 residuals from a
parabolic fit. It can be noticed that, in neither case,
are the o/riginally known parabola and sine term recovered
from the data.

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2000.000
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70C0 .C0C
soco.
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o
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CYCLE COUNT
Figure 5-3c

70
Only by including a full set of adjustable
parameters as in method (4) can a 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 greater 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, owing to the flexibility of this tech¬
nique .
Occasionally, a cubic fit will be of interest. Wood
and Forbes (19G3) utili zed cubic fits for 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 solution, but that it offers two additional
advantages. First of all, it allows us to calculate any
secular acceleration in a binary system due to mass trans¬
fer. Secondly, it provides a fit for periodic data which
is almost as good as a periodic representation for a short

71
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 type 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 initial epoch from which the cycle count is
computed, as well as the error of that initial epoch for
cubic, parabolic and linear fits, respectively. The next
column contains similar information on the value of the
period and its error for trie 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¬
sentation „

TABLE 5-2
PER10D PARAKKTE R S
LP6C11
P 6 k 1 o D
A
b
31 uMA
24 24 119.
6.’ 3 036
C• 32 0 9 32b 4
«
27 3 7 6-
1 0
- . 701 C 6 - 1 3
. 0 0 7 7 o
70
1 4
€
13646-
1 0
. 3 1 2 C 6 - 1 3
2424)1 9.
2 3 o 9 0
C . o 2 0 9 o2 7 7
~ <
b 9 0 3 6-
1 C
. u 0 7 ti 2
7 b
b
«
24 2 0 6 -
1 1
24 2 4 1 i V.
¿4 924
0 e u2 39 30 b 1
.0 1 1 72
04
'Í
24 33 4 63.
6 3 o 2 0
4 o 12 2 7 4 4b O
0 .
1 ooí. L -
0 7
—.24036—11
» J C c. 1 4
b 1 4
3 9 7 7
C
‘i 3 4 0 6-
0 7
. 1 2 1 0!: - 1 0
24 33 4 03<
b 3 b 9 7
4 « 12 2 7 321 9
0.
3 0 3 6.6 -
00
« 0077o
7ov
1 1 9 7
c
4 7 7 4 l —
06
24 33 o. 33 «
wi /J OUU
4* 1 227 / lot.
4 0 003i•
ObJ
3 9 4
242591b.
3 2 o ¿ 1
0»69bo0230
0 .
76936—
1 o
-. 13 2 0 6- 14
g L tí
bbL
92
c-
4 3 2 2 F. -
i o
. 3 2 1 3 6 - 1 4
2 4 2 3 S> 1 6 <
3 2 901
0 . o9 oo 02 0 2
0 .
o 3 1 1 f-
1 c
e C «L. 1 0
4 23
b 1
C
30 to!.-
1 0
242591bc 00^33
0 • 6 9 3 o ó 2 c.
. 0 49 74
4 20
4 4
242 397 7. 32 02 0
1 . 6 3 7 1 O 4 „} 6
; . 1 0 4 1 6 - C 7
* 2 v j 2 c. O " 1 2
.: 2 o o i
1 61.19
4 1 1 6
, 4 C 02 6-Co
. 1 2 6 1 U — 1 2
2 4 239 77 . ¿ 1 l.Cb
1 4 ob 72 bo 6 9
0 . 1 3 99i -0b
4 0 2 0 9 7
2 2 1 3
4 33
.2 09 76- 09
2 423977»t 7 3 77
1 . 3 3 7 2 b 3 0
. C 2 b u 7
6 52
bo
2436169.bb235
C • o* Jld9 o 32
C . 6334 f. - 1 0
0 . 3 hi C 6- ) 5
.0 09 70
4 V
b
. 23o36— 1 1
4 l. 1 ) OL-lu
24 3 b 1 0 9 » 3 o». 9 2
0 4 bol b 9 0 6 2
l./l 3 9 6- 1 0
.01612
bO
2
. 1 1 646- 1 1
24 6 ó 1 0 9 . o 0 03b
0 . 33 1 b 9 C b 2
t o7 rl! tí tí) <_
1 1 7
b
2 4 3 o 52 ci 7 76 1 b
0.610J 1446
0 . 1 3 :> í L - C 9
0.2 56 7 6-14
i"' 1 f~ s',
t V ‘t •> 6 >
29 7
6 )
. 72 2 Or..- 10
» 2 o 2 31: 14
24 3 0 52 b> 7 7 o00
0.O10114b1
0 o 0 4 6>36— J 0
c 0m vjC
23 7
4 6
. 2 C 3-. 6- 1 0
24 36 52.0*77014
0 . b 1 01 1666
* 0 .J . í VJ
24 7
3 6
24 349b9.22094
1 , 96 3 7 7 9 30
0.2 1 o 1 l - C 7
C . 2 1 G 6 6 - 1 0
, ' . ) r-
i . H w» •_> •-
6 7 6 4
1 21. 6
, 4 4 O L - 0 3
. 1 2*4 66-1 J
24 .3 4 9o o t b'. 2 00
1 < 9b 39 04 o 7
C.90936-07
.12340
3 0 6 6
19 93
.313 f.-L— 0 6
24 3 4 900. Ocwbft
1.9 o t. « c b's o
< 4 ] 3 9 C
J 1. 7o9
li U et L
9 2426329 , ijco'i)
1*0 C- 7 t 0
2•( 26330 * 0 7 b 7 0
7/92
2 4 2 6 3 b 0 » i “ OJ.3
6 9 b
0 • aboC'i / í 01
1 ¡ /v 0
0 « b o zj 9 0 1 0 o
b b V
0 * ÃœOÃœV tii.'H;
— . 2 2 16 6.— Ob
• 44b 31. — O' 6
" » 1 j 1 . O '" 0 9
.11126-09
* 2 o 6 3 6 — 1 i
. 0.0.7 /l: - 1 3
i 0 J 4 0 f
• J .* 4 O 0
0 O b 9 00

73
TABLE 5-2-
-Continued
LPüCh
PtKlUD
A
b
SI EM A
1 0
2433370v 40384
2 .
13920021
(
1567L-C7
6
o
2834E-1
1
.00320
18b
6(0
*>
o4 94 2 — 0 3
«
1 4 2 8 2- 1
1
2 4 3 S 3 7 0 . 4 0 o 7 6
2 »
13 91 09 7 <:
~~ •
33 c> 71. - 0 6
«. 0 0 b 2 0 fa
74 0
2 o 2 22 — 0 0
2 4 3 3 3 7 C n‘iG 63 (..•
7 •
13 91 7 < > 2 o
t 0 C ó It»
1 80
9 7
1 1
242689*.1jo60
2 •
01 )o90 jo
“ f
32 3 6.9-C 4
0 .
2!39L-0
Q
« 0
0
0
t
0
•
C
24 29 137.71 ¿30
¿ ♦
o 3 3 4 4 ó C o
f
30549-06
t GÍ y94 /
4.24400
17 179 9
•
1 7 37 i: — C 6
4 2 4 o V 0
17 129 9
«i
J 7 3 7: . - Oo
2429loS. 17 o 8 7
2 «
OO09227 <_•
• W. 1 JO*1
1 o 1 79
o / Oo
1 2
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2 *
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c.
1 6 ) 4 - 0 7
0«
1 0 5 9 ti - 1
1
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7o0
o t, 9
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14 7 6 8-0 <3
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L> bio 4 L: — 1 ^
24 33 8o8. 58737
2 *
G.
1 6 o O L - 0 7
«• 0 *■_ CJ o J*
80 3
) 3 7
«
66 <•> CL — 09
2433600.41947
7’ *
<18098 1 Ub
. 1 034 0
1 1 63
9 1 6
1 3
24 33 0o b. 386 ijG
) .
06,3o 9oo9
C c
1 3 G 7 o - G 7
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13992-1
1
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1 0G3
1 3 76
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8 2 â–  9 3 " v 6
t
37222-1
CL
2433856.04 CoJ
1 .
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0 *
2 4 3 9 L - 0 6
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61 0
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€
4 1 201 - 09
2433 So8. 3 1 So7
1 .
o63 7 u 1 37
c 0 1 V H O
3oS
1 20
1 4
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G' «
9 C 8 8 0 0 i .
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82 0 32-0 9
0 .
19602-1
j,
• fcJ 1 / O
1 o7
1 19
«
1 3 9 9, - 0 9
c
8 7 C9 2-1
24 33891.4 1 9 1 2
c' o
50679624
~ e
27 7 01.- 09
4 0 1 7b9
1 o3
4 C
c
22 62 L-1 C
2 4 3 3 G 9 1.4863)
0 «
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7.
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t
5 B 6 C i. - C 4
0 .
3 6 ! C. L - 0
7
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5 i C 6 9 C 0
2 6 7 7 9 9 9
«
jo 922 -C4
3 3 o 6 2 — 9
7
24 30 223 . oo 9o9
7 .
23 00 ooS)
*" e
4 1 312-OO
* 0 ‘J A S»1 O
9 9 7 2
3 7 3 69
c
3 3 7 7 i. — C (. i
24 3C 2.2 3 . o 24 oo
7 .
2,: 9 t 17 1..
.00217
975
1 7 <1 8
1 o
1 7
243132b.
O 7 O'. 8
1 .
í.í'! <; u 0 7 7 9
~~ &
3 7 7 4 t.
— 0 cj
C. 3 0 3 32- 14
*< C 1 b9k!
40)
) 7 7
•
j 7 4 6
— o Si
. 3 0 7 o 2 - 1 3
24 3 1 o.c o •
0 7 546
i .
t> 7 7 C' L 7 7 V,v
t
3 S' / 4 l
- C J
« 0 1 L) "J 1
3) 3
tiC
,
) S< 7 5 i
- 0 S'
24 3132 8,
o 7 1 7 o
) .
37 4o 0-4 7 S'
. 0 1 u o 0
2 6 8
6C
1 8
24 2 861u«
-> •
0 1 0 b 7 ;> >1 -
0.
6 ] 8 t. i.
v< b
0 . ) 2 3 o 2 - 1 )
« J ¿L ft 1 7
5 8 8
7 )
4
1 3 3i.
- 0 /
. 29 -• 32-11
24 2 2 < >) o <
4 L-1, Li iJ
3 *
0 i 0... 7 m 7
0 *
1 3 9
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* 0 4L 1 'J ¿
O 7 V
( > ,T 7
c
1 6 7 i ¡
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24 22o1o.
7 t> 97,:*
3 ••
0 1 C c J o 7
< L» j 7 b 1
oOl
b 2 2

74
TABLE 5-2-
--Continued
EPOCH
Pt-.KI üü
A
24 21 24 .r_ . 2 0 5 23
) 6 visó 3 / 0 /Ó
0.
1 b 3 4 r.
-08
o 0 L>
7 2 8
•
93 1 bf
- 0 9
24 2124 2.255oi
1 , 3 3 2 7 o 3 13
0 •
94 4 9 6
- 1 1
339
1 4 2
»
9 0 5/ L
- i 0
2421242.2b5o3
1 . 3 o 2 7 3 3J;j
29 0
24
2432945.54279
2.32 4 9 93 90
C í
1 4 8 2 5
-0 7
7 o
5 5b
t
3 o 5 1 i.
- 08
24 3294 5 ,.54 1 7o
2.525C 1 i Oo
0 .
2 1 4 bL
7 7
o3 7
«
1 ü 0 7 5
- 0 9.
2432945,54112
2.525C17/4
7 3
50
L>
.54306-13
* 3 29/i -13
.2285[-11
. O 1 6 812- 1 2
51 GHA
♦ 0 0992
• 6 10G9
. O O 9 9 o
.5 1953
. 0 20 7C
i0¿1 0 4
2 1
22
2 4 3 4 4 :> 5 <
3 o 50 7
1,55321837
0 .
3 1 i 4 e; - c y
0.20986
- 1 3
« 0 2 003
20 1
45
•
1 1 8 9 l - C 9
, 1 C 0 1 E
- 1 3
24 344 35 .
3 b 7 7 1
1.5b3¿195„
0 0
69 2 4 2 - 1 0
• 0272 (•
1 5 4
3 o
c
4 9 4 C L - 1 5
2434455.
3 59 91
1 , o5.52 1 ó 37
,027o 7
1 12
2 5
23
2 4 3 84 4 5»
4 t. 74 tí
J . o 1 ) 4 78 tí y
t
7 9 4 7 2- 0 9
0, 73 9 76
- 1 2
.0 8 o 71
190
3 55
1 o2 2.5- 05
.11106
- 1 1
2 4 3 C44 6 .
4 5 o 5 2
1 . b 1 1 4 7 O es 5
«
508CE-09
• 0 23 35
loG
2 0 9
} 3 o 7 L — 0 6
243 844b.
4 C o 5 3
1 , 5 1 1 4 7 3 ti 5
.02285
j 5 ó
1 37
2 4
24 3 52,: 2.
0 4 o 70
1 , 55 1 8 35 tío
t
4 7 6 C ;. ~ C 5
-.4310t
- 1 2
.12 1 1C
7 o 0
2 C 3 1
r
892 91.— 08
, 95 1 2 6
- 1 2
2435222.
5 4 79 0
1 » 36 12 08 2 5
- *
7947E-09
. 1 1 9 o 0
72 0
1 V 76
t
2 3 3 8 1:. - 0 tí
2435222 ,
5 4 o e 0
1 , 55 1 2 44 9 o
. i 1 7 50
o 04
3 4 4
24 34 132.51 6 o 9
c ©
o 5 7 6 1 2 7 3
c.
o 2 o 4 6- I 0
0 « 2 4 7 í. 1 7
1 1 2
4 7
/
3o31 6- i 0'
* 7 3 9 9 c. — 1
24 34 132.31oc <
0 «
u 7 6 3 8 7 o
0 «
o87ol - 1 0
V 9
1 7
o 1 2 5 L - 1 1
2434 1 32.0121 1
Oc
8( 87 5 1 4 3 8
1 ,9)
3
24 24309.31 36C
3 .
3 7 o , 9 ) -j C
€
1 4 1 7 l- - 0 4
0S 91 0 o t: - 0 9
Ó 2 V 4 \ ”
2 7 o o 9 9 v
54 ! 4 E — 0 3
< 3 ¿i v 9 L - C 9
2 4 2 4 4 o 4 i i 7 6 5 7
O r
3 0 5 3 o 4 3 o
c. <
7 0 7, !_■• 0 8
1 v 3 9 v 0
o 1 (-• 89
[, 9 4 0 l. — 0 7
13929V
ó 1 5 (.• 9
D '9 • ► t í: * J ^
24 8 4 4 o o . 4 c - v o 4
3 .
_ 5 3 3 0 7 o.
V O 7 O
1 7 o o
. 0 o 2 5 7
.012 37
, 0 1 3 ¡ib
0 1 3 .j 3

75
TABLE 5-2 — Continuod
EPOCH
PLM UD
A
it S I OMA
Zb
2 9 2 4 3377 7 .4 2o36
1 0 3
2403777.42731
1 02
24 33 77 7 . 4 2 03o
7o
30
0 . 39 3u 72 33
1 2
C . 0 93 0 71 7o
u
0 . ó 9 3 C 7 1 7 o
6
— • 2 9 4 9 L — 1 0 — * 2 0 0 3cl - 1 4 « 0 1 o 2 4
.017 11-1 1 .34 1 Oil- 1 5
— « u 84 9 L— 1 1 .013o5
. 4 9 3 O L - 1 1
• 0 1 3 o (j
3 1
32
2429334.
2.3 4 4 9
J . 0 9
3 0 4 3 3 3
1 0: 77 2-
0 7
0.3 1 1 1 E- 12
* J 0 i-1 7
0) 4 3 0
24 4 a
.34812-
“ 3
. 1310 L—12
24293o4.
4 1 0o9
1 . O' 9
el V ¿1
— . 1 1 f,oL-
C 3
. 0 0 4 3 0
1 399
4 2 7
. 2: l i C 9 2-
0 9
2 4 29 334 <
4 00 0 3
1 , 09
2 9 4 3 J 9
. 0 0 7 3 2
3o2
33
3 4
3 3
24 2 2 03 0p u 2 3 20
1 , 1 o3 9 4 1. L) 1
- . 3 3 9 C L - 0 9
0.1417 1- - 1 3
.0 1
el C* 1
4 {â–  0 3
24 2 2 03 0 c o 2'i)O
11 10.
1 „ 1 339 3 4 0.3
< 21- L — C 9
0 . 1 3 7t-.L'i9
. 2204L- 1 3
col
¿ J 7
1 t>c>3
2 4 2 2 0 3 0 r 11 2:2 9 3
o3l>
2 72
1 « 1 3 3 9 3 9 u 0
2 4
. J 7 o o L — 1 0
. 0 1
9

76
TABLli 5-2---Continued
EPOCH
P c.b i ÃœD
A
b Si uMA
37 2410987.78403
# Jf- 4 8 v (:
241092.6c C 9303
3 8 9 99
24 1 0924 i ¿SI 22
3110
38
3.37302307
2 til 39 99
3.39 1 7 6o 4 o
2 0 O 8 9
3.3924 3o9b
3 12
0 « 34 2 3 L — 0 5
< ■+ 94 0 9 — 03
C . o 91 - 4 L - 0 7
• 1 b 4 4 L - 0 7
-.2 C b7 E-C 9 .00332
. 3 0 7 9 E — 0 9
» 0 0 b 0 4
.0118b
3 9
4 0
24 3 398 3 .2 1 jjS
3 6. 38 864 1 87
G .
3b 73 L- 03
0 «
3 2 7 9 E -
0 7
. 3 1 7 8 0
1 1 o 9 9
1 2 1 3 9 9
•
39330-03
c
2 74 3b —
07
243 3 9o b« i337 4
3 8.88 7 1 D 4 4 8
G «
14901-Ob
.8221C
1 0 039
li h -> ó V
*
3 4 4 4 [_— 0 o
24 33 9o3 c 1 932 7
€ Jj 1 *4 cl G
^ jOC>
3 4 7 i7 9
4 1
3 A 2 f* *7 9 ¿1. r .9 3 ó ¿i r_
1 . o 1 2 3 J <> ] 8
0,
1 8 Q 1 L - 0 7
•
3 132 if-
1 3
.01209
844
8 9 4
»
1 3 44 L- 0b
«
1 C l 7 0 -
1 2
24 2¿1992.2 94 00
1 • >b 1 3 O 0 9 C; /
~ «
2 d 9 7 h - 0 9
.Gi47b
02 b
2 0 b
«
2 C 8 ¿3 0 - 0 9
24 2 b 9 9 2. t 3 0 0 7 c
1 < 61 2 u 0 0 o 1
.01476
3 3tj
3 o
4 2
2419323*83130
1 .4 2 632 881
«
üW OIL- 10
"" t
3 83 3b-
i 4
.01230
1 7j
8 b
•
1 3 •: L — C S>
<
36C2b-
3 4
2 4 1 9 02 3 » 8 3 2 1 b
) < 4i. ob 2 8 8 1
«
1 9 3 8L-09
.01227
1 o7
4 3
«
26 14b- i 0
24 1 9 32 3 i. o 3 61 7
1 .42832 3 7 8
. 0 1 8 3 b
1 ¿i-i
3 8
4 3
24 3 32c-2 ; 04 94 0
1 .24b3 2 4 3 8
0 .
) 7C4[-08
“ %
1 97Lfc-
1 2
.01393
40 1
r
9990L-09
r
3 68 4 b-
l cl
£ 4 3 3 3 o vd * C u> 3 19
3 t 24 jj ,-_4 8 b
O r
8 > . .. 1 L - 0 9
.014C 4
333
2 4 1
e
4 1 3 C L-09
3 4 3 3 3 c_> ¿2 « L '«> G 3 3
1 c. 3 4 3 ^ / <* J
.01427
3 20
1 0 9
4 4
24 3 3 1 3 i. j> o ¡ i 9 o
1 . 38 86 71 ¿J8
G r.
¿1 b 8 r' Í. — 0 O’
r,
V1 (
1 C- 7 C E -
l 3
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93
3V P
4ooc; --1 c
c
3 9 v o L “
1 4
24 3 3 3 1 3 t 3b 64 u
1 «88 Ob <’ 2 /’ V
0 *.
3 6 7 9i>- 09
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9 3
3 3
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24 3 3 3 1 ‘ > t J 9 « 0
1 » Jt. tib7.: 79
.02132
1 Jb
O’ o
43
24 3392 0. 3 73 9
0 > ( 4 o 88
0.
30 2 3 b- 3 0
0 *
3 0 2 0 L -
3 4
«. 0 Oob't
98
1 l, 1
23 1 CO-C 9
•
3 USE-
) 3
24 33920.4374 7
0 « 88 4 81 j 3 0 b
0 .
8 7 ? 4 L. -- ) C
.00803
6¿
í! 1
<
2 188b- 10
24 3 ¿i ‘.1 < t> * 4 ; > 3 7 4
C < 884 0 68 i 8
.6)10 0
-> U
i 1

77
TABLE 5-2--Con tin uod
EPOCH
Ptu uD
A
Li
SI OMA
4 6 2 4 4 2 32 6 « 4 o 9 7 ■'»
] jíí
2442326»4u97C
I 04
24 4 2 32 6.4 o Quo
132
4 7
1 * 9 J c* 4 2 4 4 6
OC 7
1.91692734
3 2 6
1 « 91 6 9 2o jo
1 08
1 2 7 56-06
» 52 oC L-Cb
C . 83 0 4 E - 1 0
< 62 46 E-09
-« ! 76 6 E — 12 .04210
.3960E-12
.0416)
.041 18
4 8 ¿42 1 24 o.3 0 C 3j
1 6069
24 2 ) 24 6 «■ 28 52 )
4 0 4 6
24 2 124 o.4 6 o óo
) 52 1
4 9
1.66 0140 C 4
¡314 9
1 » üOC1 ov 1 1
2 9 4 4
1 » ot>C004o2
o 0 2
- . 1 94 1 2- 07
« 8 3 1 6 L — 0 7
— • 2 6 1 6 f. ~ 0 7
* 4 9 3 2 L - 0 6
732 0 t—12 .U6 o 6 6
5 8 1 9 L - 1 1
« 0 u 4 0 0
50 243 6 84 b . 34 1 b.>
121
24 3 584 b .3c> 4 6 7
24 1
2 4 3 5 84 8 » 4 74 14
6 63
5 1
2 * 4 93 0 03
65
0 '
14 2oE-
0 7
0 , 9 A 6 3 E - 1 2
» C 7’ :>uu
4 1
*•
1 5 6 0 L —
C 9
. 1 4 50E- 1 3
2,4 9 3 0 0 o
7 J
0 , t
Jo 9 52 -
0 >3
* J ‘J 6 7 7
94
*â– 
1 4 o 6 E -
0 9
2.492 j c>264 .¿Co3C
2 1 !
52 2 4 3 6 7o Ü.4 4 016
96
2 4 3 6 7 6 8.4 4 0 v 6
ou
24 36 7o 8 » 4 0 96o
1 53
5 3
2 7 6 J
1 r>73
- . 854 2 E- 1 0
0.2 749E-15
e 0 J ¿Í 3
7’ 7 0 J»
/
3 6 90
. 38952- 1 1
- . 92 6 CE- 1 0
.1272E-13
* 0 ct J>
4
1 0 9 7
9
.2 04 5E-3 1
• C 9 3 0 b
24 2
5 096.4 6 641
¿ <
63 7 6
59 ‘i 3
-,8495L-06
0.4 27 4 t. - 1 8
.0 19 99
2 4 2
1 72 3
6 0 9 0 » 4 4 0 6 7
¿«
66 7 6
1 5 86
’J i
■» J t.
. 3 9 U1 E-06
— t 6 4 6 7 2 — 0 0
.3264C—12
•
O
O
2 4 2
1 -6 Go
5096.6.>8o/
585
7- (
66 76
:. 9 7
0 7 93
1 0 6
.571 3 L - 9
. 0 2 6 9 9

78
TABLE 5- 2--Con tinuc.d
EPOCH
P L(< 1 U1)
56
24 3 4 9 8 9 c -> 9 6 0 J
1 . 7 7 4 7
4 4 0 3
13 4 9
4 1 7
24 3 4 9o9« 4 2 535
1« 7747
3 531
1 4 9 o
6 01
24349o9. 4 2 937
1 . 7 7 4 7
JÓJ¡
1 42 3
2 72
56
24 35 373. 430 95
k « / 72 9 1
32 65
1331
1 9 0 1
2435373.9451o
k , 7k 9 1
5 7 4
1 cl/i
12 69
24 3 53 7 3 . 4 4 9 o 3
2t / 7- V 1
4 1 2 4
1 1 O')
o o 7
5 7
A
0
SI (jMA
0 « 4 2 7 Í h
- 9 0
0.337 7E
- 1 2
.02736
.16 1 3 t.
-08
. ) 1 4 oil
- 1 2
C . 2 7 9 C t
- 1 0
« 03 6 61
.698 56
- 09
» 0 3 4 y 1
0 .
1 8 7 01
-06
0. 687 4 6- 1 2.
. 1
4 0
5
0
<
5 5 8 3 L
-06
.31476-11
e.
2 0 6 86
- 06
. 1
3o
c
•
6o 96E
- 0 o
. 1
32
4
0
56
2433041
c 76796
0.
20) 72 33 6 6
0 «
299C ti¬
c >â– >
— . 2 6 2 u E - 1 3
p *• 1 9 / O
9
6 7 4 9
t-
lt 3 4 E -
‘■J o
. 1537A—13
24 3 3 04 C
* L/ *J C V Ü
G«
3 7 o 3 1 6 7
C .
5 2 . jV
0 9
< ^ 1 0 o J
7 o 0 5
4 4 1
c
o 1 9 5 o -
) c:
2433040
« 4 0 C .o j
c.
2o7 54o ! 1
«011/3
2 4
6 9
24 3 4 8. 7
• 44Cv 6
o«
3380 C 2 C 6
o.
2 4 3 4 L -
0 9
3 2 t. C L - 1 4
r 0 0 4 CJ 4
71.4
O 60
•
6 6 0 4 t. -
0 9
. 1 3 o C E - 1 3
2434637
* 4ao77
V «
33 6 3 0 o o 1
" *
1 C 7 5 6 —
1 J
« v* v d *J fc.
L U
63
c
2 7 3 41. -
1
2434 8.i7
4 ¿' 9 "J
0 €
3 o 8 50543
c 0 9 M 4 7
216
1 3
6 0
24 16 74 6
.30093
o t
4 6 2972 41
«
1 3 6 1 E -
r. j-.
0 . 7 7 7 6 L - 1 1
c 0 J 1 ó 3
o 0 0 4 9
o 0 7 6 5
t
5 1 40 6-
0 7
.2633L-11
2 4 1 0 7 4 6
*•7- ] C L)Ü
O r
4 3 2 1
0 c
■ t a 2 1 E —
C
c 0 J 6 9 6
9371
6 1 3) 7
c
23 7 C f. —
06
2416746
C v7 JO^b
O *
43219040
* 0 4 0 4
) 1 oo
1 7 7
61
¿4 3 3 1 oG .1)0433
4 < 3 7 7. V’: > 7 5 •.>
0 . 1 0 3 2 E - 0 i>
-.15o2E-10
r l) 1 0( J
119/
3o 92
.3 91 ; • L —C7
. 11 51 l -1:
24 3 3 1 o C.o3C 9 7
4.3/3 C- 1 o 1 7
C . 6 3 626-07
• 0194 6
i c: o
1 o 5 2
. 7 3 62.6 - 0 6
2 4 3 3 1 O 0 . o .â–  t: it o
4 < 373 1 O-" 2 '
• 0?¿3 90
5 1 j
o 3 4
24 20 323 «•. 4 o / 7
o. 4 36i 03 4 6
. 4 3 J o 1 — 0 3
0.9756E-C9
. 0 l 7 7 7?.
1 .79 30 3
3 6 0 4 9 V
. 2C 9 6t - C 3
S.C
!
O
24 2 0 3w 7 .7 7
6 . 5 0 o 0 4 5 8
— . 9' 9 1 7 o — 0 7
• 0 0 o J>!..>
1 3 7 1 9
1 97 5--
.O197E-C7
24 2 032 7 , 7 2 i 1 8
6< 4 302o 6 3 ]
• 0 0 í3 i) 4
1 6 4 4
3 3 6

7 9
TABLE 5-2--Continued
EPOCH
PLKiOD
A
3
51 vj.M A
0 4
24 3 6 3 5 9 < 0 2:0 3 2
3.31 4 0 O 3 2* 0
' •
1 1 7 6t - 0 4
-.7362L-
0 3
. 9 3 3 0 0
364 6 4
1 7 o 1 9 9
•
6 3 1 C L — 0 3
.64592-
0 3
24 3 615 6.91254
5.313 925 o 5
3 1 6Í-6- 0 3
. 9 0-3 1 C
4 39 4 9
16 7 3 9 3
Í
2 0 62 5-0 6
2436136c 4 5472
3 . o 1 6 2 0 3 0 7
>; 8 2 -r v
4o 76 9
6 24 9 9
05
2431960^06124
3.31 77225 2
C .
7 7 2 32 - 0 6
0.76902-
1 2
. 0 4 20 2
7 4 2
4 9o
•
1 0 90 L- C 9
.319 02-
1 2
24 3196 0 « o4 o 7 0
3.3 17 7-3261
0.
3 '9 Oc L — 0 3
. 0 4 3 9 3
7 o 9
3 10
*
89762- 09
2431960.o6070
3c 33 771469
. 0 1* 9 3 6
04 6
1 90
oo
24 14 9 0 0.0971B
0 c 3 6 7 /. 33 5 6
e
69762—07
0.5 84 l L-
1 2
. 7,5 9 o C 0
3 69 09
*
14 6 7« — 0 7
.12332-
1 2
2414 934.7 2 i io
0.3«46 bo 7 0
»
11)5o-08
.G 02 2 4
5 4 67 9
2 7 2 9
«
3 3 6 92-OS*
24 14 9o6.65930
0 » o o 4 4 o o 2 2
.0 04 20
1 4 4 9
36
6 7
24 2 1 3 3 7. 43642
0.62 6 O 1 7 03
0.
o 6 4 3 2 — 3 C
-.21442-
1 4
.04644
102
1 6
•
3 0 3 C . - 1 0
.22622-
1 5
2421137.44071
C . o2136 1 7 70
33 0 42- 3 0
.0 0790
106
) 2
«■
4 3 2 9 L - 3 1
24 2 1 15 7.4 3 99 3
0.o26610 0 9
.009 2o
125
4
0 0
2424453.3 6303
0. 963 749 39
«
32 7 3L- 0 7
0 . 112 4 t, -
3 1
.01492
3 c 1 34 0 0
1 Ü 1 0 9 9
»
10 74 2- Oo
.3 7 7 72-
3 1
2 42 4 4 3 4.3 5 o 0 3
0 . 9 6 3 -< 4 4 9
55 0 1 L— 6SI
.0 3 > 73
2 3 3 1 9
64 7 1
*
26 01-2- Oil
24 2 4 4 v/¡ tJt74 3
¿ 04 3
6 9 2437002 <,65909
29
2437062.05992
70
0 . 90 36 64 60
2 7> 6
0 « 29 0 9 3 COO
4 O
0.29090952
. 4 o C-6L- 09
< 1 7 4 7 t. 0 9
.227 12- 1 0
. 42 3 0 s:
. 1 u 4 3 t:
3 3
1 3
7 3
. 0 3 4 1 9
.00462
. 0 0 4 7 o
2 8
2 0
c27882
- 1 0
2437852.65957
0.29090933
. 0 0 4 7 3
22
6
24 3393 0c 40 64 9
0 . 71 7 7 630 7
0.11 251
- 0 9
-.33042-
1 4
* 0 0‘jOJ
66
3 4
.14262
-06
. 214 92-
1 2
24 33 93 0.4 0 051
0.7 3 77 65 0 7
0 . 7 4 2 07
- 1 0
• 0 01 ♦ o
31
4 8
. 1 383 2
-- c 9
24 33 930,4 0 63 7
0.71776323
» 0 0 4 4 4
26
37
24 16 3 61 . 0 927 d
0. 3 19 732 4 b
-.2303 2
- C 5
0.3 7 « 8 2 -
3 2
♦ U'+bb b
2 7 o 0
7 969
. 5 3 2 3 i:
- C 7
. 10 7 36-
1 0
24 i o 3 o 3 t 0 93 50
0.3197 3 0: > 7
- . 4 9 23 (.
- 0 9
* Ü • i 3 ¿ 6
2 1 < . 3
2 3 S' 7
. 0 0 2 3 i.
■“* 'J O
24 3 6 3 o J c 0 5 4 81
O c 61 972560
• v; 4 ) c: u
3 3 2 6
4 01»
72 2 415966c 697o0
<' v o 6
2415966.71110
2 6 3 7
24 3 3 956 . 0 5003
1 7 1 9
3 c 6 6 9 55 0 3 9
4 4 30
5 < «69 7 9,3 4 0
1 o 33
5.669 o 10 3o
5 C 4'
.14 0 4 6 - c 7
. 17 3 0 !- - 0 7
. 233'i «- 06
* 2 1 ol 0 6
0 . 1 7 32L-
. 1 7 7 95-
11
1 i
c 6 4 4 5
< C 94 54
0 34 5 1

80
TABLE
5-2--Co ntinued
HPÃœCH
PL2 1 ÜÜ
A
U
SI G.'mA
73
24 28760.33907
ti •
13 7 o o134
•
338CL- 09
0
*
64210—13
.02690
u7o
3 51
c
2 C 3 Oí'.- 0 9
♦
10180-12
24 2 87 3 0 , o 5 5 5 ¿>
2 .
1 17 3 0420
•
3 7 3 0109
«02662
39 v
9 9
•
1 9 9 9 L - 0 9
242 8750 c 33253
2 .
1173632o
. 02719
3o 3
93
7 4
24 1 8 <>* ü « 74 0 1 5
4 «
80 7 8 0 3 l 1
«
3 7 701.-06
0
«
33060-10
,0 13 61
] « 0 7/00
7 69 39
c
)9163-Oo
*
1 34 40-1 0
241 f i C 4 8 , 7 7 324
4 ,
8 0 o 9 4 9 2 1
ü ,
17 3ó L- 0 7
.0 1 6 6 9
2 0 i y o
99 99
*
121GL-07
24 1 804 8 , 4¿,390
4 .
«CoO 94 1 7
, 0 1 3 ó 4
1 3C2
.1 63
7 5
2 4 3323¿3 7 0 609
1 .
3 3 74 3 o 1 o
G .
1 0 3 2 í: - C 8
_
c
90730-13
. 0 09 28
144
1 O ¿i
,
35 02)1 - 09
c
331 9 0- 1 3
24332o8. 70 o 98
2 ,
ó 3 7 4 3 3 0 9
0 .
3 4 30 8-09
. 0 0 9 3 0
1 30
89
c
1 2 61 6 - 0 9
2 4 3 3 2 o b < 7 0 ¿> O 9
1 «
3 5 7 4 39 0 4
< 6 0 -y 3 0
8 9
22
7 o
2433390.49937
2 ,
83 1 14 o 24
0.
4 1 2 60 - 07
—
t
2 7 o o 0 — 1 1
,00331
100 3
2 1 9 o
•
1 1 8 312- 0 7
<
18790-11
24 3 J3í 0t 4 ¿¡034
i_ (
62 1 1 7 771
r >
V «
2 39 70 - 07
, 0 6 5 3 8
26 4
4 4 4
<
1 2 8 1 t - C 8
24 33 330 « 4 o34 0
2 ,
8 3 1 2 G 4 3 0
«0 15/6
4 61
2 1 2
7 7
2 4 3 7 4 92 , <->9 0 1 3*
0 .
53 C o 7 () oo
*“ o
62 3 70- 1 0
~
*
16820-13
,0b7 6 7
1 62
3 6
<
1 0 4 (: L - 0 9
«•
11330-13
24 37 4 92.o9XoO
0,
3.. G ¿» 7o 6 C
«•
2 1 4 30 - C 9
, 0 0 7b9
1 1 4
34
f
4 24 60- ¡0
24 3 7 492, o9091.
c.
33067331
• 3 0 8 33
1 24
1 9
78
2 42 2 74 4 , o;>4 93
¡.
t
1 6 ¿y 4 ': 0 6 8
A
*
13)1 0-00
-
*
46930-13
.01416
3 v 1 20 o
2 9 2 v 9 9
c
95801:- 0 u
9 84 30-1 0
24 2 1 74 4 , 3 0 o 21 >
c
1 6 6 3 5 4 8
0 c
1 0 6 7 0 — 0 o
.01283
32 7 29
2 0 4 2 9
0
3 0 ¿> 6 0 - 0 7
24 21 74 3.363 24
S3 e
1 c> 9 2 8 7 6> 8
.0205 O1.
3323
1 1 32
7 9
2433 868,30ou7
O
<6 f
80 ü 7 83 0 6
C <
2. 0 94 0- 0 7
0
e
32936-12
. 0 6 ¿1 b o
521
1 4 3
t
4 6 4 4 0-0 9
f
1 4 6 2 0 - 1 3
2433 8bó.6 602 3
^ i
ti 0 o 7 3 6 0 7
ó .
4 2. 6 í > 0 - 0 6
.2 03 0 0
772
7 7 7
í
13 730- 09
24 33G8 8 c 7 3 392
¿ ♦
t > í. 0 ¿> 52 2
.4114 C
1 4 85
368
8 0
2433 87} <• 3 90. >7
0*
boo 0 38 42
0 ,
66 0 30 - 1 0
c
•
7 4 15 0-14
.01005
2 4 3
64
.
32 4 (,[. - 1 0
<
4 34 2 O. - 1 4
2 4 3 3 8 7 J , 3 o 3 v C
CJ v
86 6 03 3 63
0.
3 9 3o 0 - 1. 0
.01026
1 4 0
23
0
192 3 2 - 1 0
24 33 57 I â–  obt 39
0 0
801)0 314 0 3
c 0 i C o 3
1 4 .V
2:1
a i
24 ) 9425 i 4 oo J9
-j) *â– 
2o 1 7( 4 9.4
t
«3 9 31.- 0 7
0
•
4 99 0 0-1 1
.01191
8 7 2 4 9
4 7 o O1 9
«
8 5 1 91.-- 0 7
*
6 C G 9 1 - 1 1
24 1 9 429, ,->oo61
«
2.6 ¡ 2 9.- 4 6
0 <
72390-09
.01)73
o 7 1 0
1 3 73
f
2 0670- 0 3
24 ) 9429.30 71 o
V
26130104
« 0 1 1 o 3
9 4 1
1 4 o

81
TABLE 5-2-
--Continued
02
EPOCH
P t_F< I ÃœD
A
B
bi 0MA
05
2 A 1 9 (> o 9 « 0 9 9 o >
0 .u33b2604
- « b 2 0 ó i-.
- 0 7
0 ♦ 3 6 4 C £ — 1 2
. 1 b 3 9 0
V V V V- V £
3 5 2.299
c 10 91 i:
-06
.11731-11
2 A ) 9 6 6 4 c S/ 9 1 4 9
0.03223112
— . •! O 2 9¡_
- 09
.16280
i « ^ 1 b 0 0
94 0 1
, 1 o 1 C L
- 0 0
1 ^ 1 b9 9
94 0)
. i o i o i-:
— 0 8
24 1 9obb•Job/j
Of o5 2 2 G4 0 o
. 1 8 1 4 C
4 2bb
1 o 2
8 4
8b
24 3 76 3 fci c
6 ¿ 7 1 3
0
«
1 7 6 9 0 - > 6 9
0
«•
4 3 2 7 6
-
1 0
-
•
9 70 26- 1
3
•
0
C 5 9 7
21
1 1
«
1 4 o 1 6
-
¡ e
»
33376-1
C-.
24 3 7 ó8 8 «
0 2 66 1
0
C
17696399
c
o
b 0 b 61.
-
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•
04 07
20
4
c
1 6 b 7 £
-
i i
2437638.
82642
0
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1 7 o 9 0 o 1 7
c
ü
04 87
1 7
1
8 0
2426242.
5 4 04 0
0
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—
0 6
—
l>
2 9336-1
1
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0
5 3 33
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3 4 8 0 0
132999
•
794 o. i_
0 7
L
1 no96-i
1
2426228«
o b i 9 b
0
f
o J 4 o b 4 7 6
0
90 6 C. ti
-
0 9
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0
5 6 7 1
8 6 94 9
7 8 09
2 2 2 o 6
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9 1 2 98
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2420732.
0 *4 0 6o
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t
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0
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-
1 0
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í137L-1
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16/1 £
0 9
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09 1 4
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2 4 2 C732 c
0 b 1 b 7
1
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24 ¿8 01 o 6
c
0 12 1o
32 1
24
0 8
24 33 38b o
5 0 2 .5 9
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03365464
-
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8 o o 6
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2434460.
3 2 1 O 3
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Otbo 1 8 0 7
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9 1 24 1 o0i b, 1 9.300 969 1 5
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9.8 6 67 ) í 8 8
♦ 0 2 4 C.
2 J o 0
1 2 1 9
i

82
TABLE 5-2--Continued
EPOCH
PLk100
A
b
Si CM A
92 24 1 3 C8 6 « 0 4 286
2 « 4 c i 3 0 0
3 < 9 9 2 3 5 6 1 O
1i0299
■« 2o 50 5— 0 7
. i 0555-06
93
9 4
9 5
9 7
9 8
99
100
2413088.
4 3 1 ob
3.992 771 1 fa
0 ♦ 24 b 8 L — 0 b
1 1 4 o 9
. j 3 4 2
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TABLE 5--2--Continuad
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TABLE b-
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TABLE 5-2
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. FIGURE 5“2—Continued
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TABLE 5-2--Continucd
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TABLE 3-
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21

CHAPTER SIX
AN EPHEMERIS FOR EACH 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 considera¬
tions to contend with when we decide upon some representa¬
tion of a system's change in period. Primary considerations
are listed below:
1. The value of sigma
2. Correlation coefficients which
have been calculated for all
the least squares fits
The relative errors of the coef¬
ficients for each fit
93
3.

94
4. The possibility and relative speed
of convergence for a periodic repre¬
sentation
5. The existence of data on secondary
eclipse, and the implications of these
data for or against apsidal motion
6. Visual appearance of the individual
graphs for each fit
In some cases, these considerations led to some
degree of ambiguity as to the best fit, mainly due to the
weighting scheme used in this study. Occasionally parabolic
and cubic fits seemed equally possible. In such a case, the
parabolic ephemeris was given a slight priority, as it is
not only simpler, but it probably possesses a higher degree
of physical significance.
6.2 A Word on Each System
It was not judged expedient to display all of the
results obtained from Program OMC in this work. However,
it would be desirable to have a short description of the
type of variation which seems most likely for each system,
as well as an indication either pro or con for considera¬
tions (4) and (5) in the previous section. For systems
which can be represented by a periodic fit, we also include
the full periodic/parabolic ephemeris in this section.
All adopted representations are accompanied by their errors,
in standard deviations.

95
Also, comments will be included which indicate
various features displayed by each system which are not
apparent from the graphs furnished in Appendix Two.
Several are as follows:
1. Convergent periodic representation
(or fit) not possible (for these data) :
this comment indicates that for some
reason the single periodic representa¬
tion employed in this study failed to
converge on the available data. This
could be explained since terms higher
than the first order in Taylor's
Theorem were ignored. However, this
probably indicates quite well the actual
behavior of the minima.
2. Evidence for (or against) apsidal motion:
this comment is restricted to indicate
either the in-phase or out-of-phase
behavior of secondary minima, and their
implications with regard to apsidal
motion,
KT And
This system displays a variable period, whose gen
eral character is given by a concave downward parabola.
Although there are several, intervals of scanty data, it
appears that a fit by several linear segments is not

96
possible. A periodic representation also is not possible.
We have adopted the parabolic ephemeris:
T = 2424119.2369 + 0.62093283E - .5903xl0“10E2
7 4867 243
TW And
This system's variable period is best displayed by
a concave upward parabola. No evidence is present for a
discussion of apsidal motion. A convergent periodic repre¬
sentation was found to be impossible. The parabolic
ephemeris is:
T - 2420051.69483 + 4.12269969E + 0.8059xl0_8E2
9391 85 4774
WZ And
A possible abrupt change occurred in this system
at E=-14910. This could also be explained by large scatter.
The observations are best fit by an upward parabola. The
possibility of apsidal motion exists (Cook. 1948), although
with the more limited amount of data used in this study,
this is doubtful.
T = 2425918.32981 + 0.69566206E 4 0.6311xlO”10E2
42 51 3006
XZ And
This binary displays a periodic variation. The
available evidence indicates that this variation is most

97
likely to be a light time effect, as rotation of the line
of apsides seems doubtful.
T = 2433977.15228 + 1.35726070E + 0.1623xlO"8E2
3910 870 467
- 0.0124sin(0.00269(E + 11645.9))
33 29 276.4
AB And
This system displays a variable period. It was
found to be impossible to fit the observations with a
periodic fit. No apsidal motion is detectable in this
system. There is a large amount of scatter in this system.
An upward parabola fits the majority of the observations,
although some residual variation is present before E=0.
T - 2436109.58292 + 0.33189085 + 0.7159x10"10E2
50 3 118
BX And
The times of minimum light of this system could be
fit by three linear segments, with breaks at E=-30000 and
£=--12700. However, a parabolic representation is just as
good. There is some evidence against apsidal motion, and
a convergent periodic representation was not possible.
T = 2436528.77500 + 0.61011486E -I- 0.6453xlO"10E2
237 48 2304

98
RY Aqr
A cubic was found to provide a good representation
for the observed minima of this system. Apsidal motion is
a possibility.
T = 2434989.10266 + 1.96517308E + 0.4306x10"6E2
788 2016 135
- 0.4280x10”1°E3
212
The residual maps in Appendix One are for parabolic elements.
CX Aqr
Only the most recent data arc available for this
system. We find an upward parabolic fit best. There is
no evidence for apsidal motion. A periodic representation
is not possible for these data.
T - 2426330.07570 + 0.55599200E - 0.1013xlO~'9E2
7793 589 1012
XZ Aql
Not much data are available for this system. Owing
to the describution of data, it is difficult to say that
the variation is anything but linear.
T = 2435370.40836 + 2.13917839E
181 97
FK Aql
See Table 5-2.

9 9
KO Aql
The period of this system is variable. Szafraniec
(1959) has found a periodic term of 27 years with an ampli¬
tude of 0^1. We find a periodic representation possible,
with a much smaller sine term superimposed on a long-term
parabolic variation. There is no evidence for apsidal
motion.
T = 2433888.35630 4 2.86393261E 4- 0.1867xlO~7E2
142 49 28
4- 0.0016 7 sin (-0.0 084 (E 4- 19693.5))
68 5 1209
KP Aql
The period of this system is also variable. A
convergent periodic representation is not possible, and
there is no evidence for apsidal motion. A hint of multipl
parabolas exists in the photoelectric data.
T - 2426096.62002 4- 1.68370244E + 0.25 31x10~8E2
4373 1117 707
00 Aql
This we11-observed system's period variations can
be explained with a periodic fit. As there is definite
evidence against apsidal motion, we may assume that the
periodicity is due to either the presence of a third body,
or to some periodic phenomena in the system. A significant
downward parabola is also present.

100
T = 2433925.37387 + 0.50679G48E - 0.2889xl0~9E2
377 122 732
+ 0.0175242sin(0.00068813(E - 6958.4))
20420 9978 759.5
QY Aql
See Table 5-2.
V34 3 Aql
Convergence to a sine term was exceedingly rapid
for this system. There is no evidence for apsidal motion.
h great degree of scatter exists in the photographic
observations.
T = 2431328.37517 + 1.84460449E - 0.4695xlO~9E2
209 49 1139
- 0.0 0 8 7 2 8 bin(0.0 01117 5 2)E - 5446.2))
1701 9273 286.0
RZ Aur
Whitney (1959) found the period of this system
to be variable. We find a periodic representation possible.
No evidence exists for apsidal motion. Unfortunately, much
recent information v;as unavailable.
T = 2422616.48838 + 3.01057625E + 0.1307xl0~7E2
131 250 107
- 0.00767sin(0.02324740 (E - 18389.0))
204 35090 274.3

101
TT Aur
We find it impossible to say anything definitive
about the variability of the period, due to a large data
gap.
+ 1.3 3 2 7 3 3 6 8 E
2 4
T = 2421242.25563
291
WW Aur
We find the period of this system subject to varia¬
tion. There is definite evidence against apsidal motion,
and a convergent periodic representation is not possible
on these data.
T = 2432945.54173 + 2.52501137E + 0.2148x10 8E2
77 338 1007
1007
The minima of this system display a great amount
of intrinsic scatter. The available evidence tends to
suggest that apsidal motion is not possible. A periodic
representation is possible;, although it provides no better
fit than a cubic or parabolic function. We will give the
periodic ephemeris here, and the others are available from
Table 5-2.
T = 2434455.38828 +
114
40
4 013
+ 0.007648sin(0.00059214(E - 124.7))
1731 1731 134.4

10 2
IU Aur
Apsidal motion is possible for this system. The
linear fit is best, while the coefficient of the parabola
is not significant. A periodic representation is not
possible on the available data.
T = 2438448.40688 + ld.8114'7391E
156 138
SU Boo
All fits for this system are relatively poor, due
to the bad data distribution available. No evidence for
apsidal motion is present. The linear fit is probably best
for purposes of an ephemeris.
T - 2435222.54660 + ld . 56.124 54 0E
604 345
44i Boo
The minima of this variable period star system
display interesting behavior, A general upward parabola,
with indications of rapid fluctuations, fits the data
quite well. Periodic representations are not possible,
nor is there any reason to expect apsidal motion.
T = 2434132.51689 + 0.26781275E + 0.6276xlO"10E2
99 17 612
Y Cam
Only 100 cycles of data are available for this
interesting system. Plavee, Peking and Smetanova (I960)

1C 3
find a periodic representation unexplained by either a third
body or the light-time effect. We find the same: both a
periodic representation is possible, and no apsidal evidence.
However, the data available are fit by a straight line to
comparable accuracy.
T = 2424433.98964 + 3.30560793E
9377 1757
SV Cam
This system displays a nuraber of interesting effects.
A periodic representation is possible, although it appears
that the parabolic coefficient is meaningless. The system
displays appreciable intrinsic scatter. No evidence of
apsidal motion is available.
T = 2433777.42550 + 0.59307176E + 0.3001xl0_11E2
238 14 1.042
-0.0182]7sin(0.00052417(E - 4941.9))
1236 5913 874.3
RY Cnc
A downward parabola fits these data. There is no
evidence for apsidal motion, nor is a convergent periodic
representation possible. Data are also rather scarce.
A cubic fit is almost as good as the parabolic fit.
T = 2429334.41059 + 1.09296337E - 0.1168x10"8E2
1600 427 250

104
R CMa
This system is one of the few systems with a
variable period which has an indication of multiple upward
parabolas superimposed on an overall parabolic trend. A
light-time effect is not possible. There is no apsidal
evidence. Only the last 13,000 cycles are analyzed here.
T = 2422030.65286 + 1.13593527E + 0.157SxlO“9E2
1886
977
273
RW Cap
Only the last 2,000 cycles are available for this
object. Periodic representations are not possible. No
apsidal evidence is present. A parabola gives a good fit,
although the small number of observations may bias this
result somewhat.
T = 2418926.09303 + 3.39176895E + 0.5964x10 7E2
57000 20670 1844
RS CVN
The unusual variations of this system follow a
cubic fit to its first order. This probably offers no
dynamic explanation to the actual variations, however.
There is no apsidal evidence. No convergence was obtained
in an attempt to fit a periodic ter
T = 2412270.81871 +
15] 1
4.7 9 7 75 2 9 5 E
1654
+ 0.5636x10 V
50 6

105
RZ Cas
This well-observed system has been the object of
many period studies. There is evidence for a major change
of period at E=14900. There is also a possibility of
periodicity, although a periodic representation is not
possible without the change in period. No aps.idai evidence
is present.
T = 2417355.68661 + 1.19522562E + 0.5462xlO“9E2
2339 261 727
TV Cas
Friebos-Conde and Eerczeg (1973) found a possible
light-time effect to explain the variable period of this
system. Plavec (1961) found the period to be constant. A
quick convergence to a periodic representation was achieved
in this study. No apsidal motion evidence is present. The
amplitude of the sine term is very small.
T = 2436483.80604 + 1.81260967E - 0.1114xlO"8E2
32 19 81
+
0.0 0.114Osin (0.004762 (E -
176 239
1044.3))
64.8
TW Cas
Dugan and Wright (1939) suggested the possibility
of a variable period for this object. We find the period
variable, and explainable in terms of a periodic variation.
No apsidal evidence is on hand.

106
T - 2419823.65095 + 1.42832661E - 0.1735x10 9E*2
120 31 181
+ 0.00504sin(0.000608(E + 2465.7))
102 28 589.9
SX Cas
The linear relation gives the best fit.
T = 2433963.19527 + 36.5670472E
4569 3778
ZZ Cas
This system has a variable period which can either
be explained by several linear segments with discontinuities
at E=3733 and E-4800, or by a periodic representation.
Friebos-Conde and Herczeg (1973) find it impossible to have
a periodic representation for this system. However, our
study yielded the following ephemeris with only a few
iterations. There is no apsidal evidence.
T = 2433282.05053 + 1.24352455E + 0.5425xl0“9E2
270 198 3298
- 0.006354sin(0.008869sin(E - 35606.1))
2844 2103 813.2
AB Cas
Ahnert (1974) has suggested that the variable
period of this system can be represented by three linear
segments,- with instantaneous changes at JD 2432673 and
JD 2436868. However, we find a light-time effect equally
possible. No apsidal evidence is available.

107
T
2433515.40215 + 1.36686993E +
5 7 14
-0.01790sin(.00054154(E +
527 1959
0.1554x10 V
18 3
2310.8))
166.3
DO Cas
This system is best portrayed by a parabolic
ephemeris, although a periodic representation is possible.
The amplitude of the periodic term is very small, although
within its error. There is no apsidal evidence.
T - 2433926.45747 + 0.68466512E + 0.8724xlO~10E2
82 27 2186
MN Cas
A linear ephemeris satisfactorily displays the
behavior of this system. There is evidence against, apsidal
motion, and a convergent periodic representation is not
possible on these data. Early photographic observations
suffer from great scatter.
T = 2442326.46968 + 1.91692720E
132 108
SV Cen
The period variations of this very interesting
system v?as first pointed out by Dugan and Wright (1939) .
As this system is one of the two known systems where the
more massive component is losing mass, to the less massive,
we may expect a rapid mass flow rate as this evolutionary

108
phase is the phase of rapid mass transfer. There is evi¬
dence against apsidal motion, and a periodic representation
is not possible.
T = 2421246.28321 + 1.66015933E - 0.2613xlO_7E2
4044 2945 491
U Cep
The period variations of this system have been
covered by many studies. This system is the prototype
for the Biermann-Hall model of mass transfer. Numerous
upward-turning parabolas are present on an overall parabolic
variation. There is no apsidal evidence. A periodic
representation is not possible for the entire ensemble of
delta.
T = 2407890.45179 + 2.49279849E + 0.9336x10~8e2
1105 256 141
VW Cep
This system appears to have undergone sudden,
irregular variations at E=“30100, E=--20000, E=2650,
and E-10050, We find evidence against apsidal motion
and periodic representations are not possible for these
data.
T = 2436768.44098 + 0.27831594E - 0.9260xl0"10E2
88 4 205

109
XX Cep
Fresca (1956) found a displaced secondary, and
suggested apsidal motion for this system. Our study
indicates that apsidal motion is a doubtful possibility
on the available data. One possible representation of the
periodic variation of this system would be given by two
linear segments, with a break at E=312.
T = 2425096.44657 + 2.33734380E - 0.3457x10 SE2
150 597 571
AH Cep
Scant data reveal a probable revolution of the
line of apsides. A quick convergence to a periodic repre¬
sentation was made. The parabolic term is not significant.
T = 2434989.43893 + 1.77474022E + 0.1492x1o"9E2
645 313 4397
+ 0.03710sin(.00682(E + 14571.7))
570 8 144.7
CW Cep
This system clearly displays apsidal motion. Un¬
fortunately, insufficient data are available to allow for
separate solutions of primary and secondary which include
a periodic term. The following ephemeris is for primary
eclipse.
T
24:
4 4 6!
2.72914909E +
¡34
0.2085x10
5698
V

1.10
RW Com
Only recent visual observations were available for
this system. A convergent periodic representation is not
possible and there is evidence against apsidal motion.
T = 2433040.65096 + 0.237331900 + 0.2003xl0"9E2
7804 442 619
RZ Com
Broglia (1960) finds the period of this system to
be constant. Again, we have only recent visual observations,
which yield the same result. A periodic representation is
not possible, although there is a displaced secondary with
a possibility of apsidal motion.
T = 2434837.42577 + 0.33850552E
586 86
U CrB
While Friebos-Conde and Herczeg (1973) find a light-
time effect possible, this study finds a parabolic
representation equally good, but only for the data from
E=4614 to present. It also appears possible to fit the
times of minima by two linear segments with a break at
E=61G0.
T - 2416748.21256 + 3.45213248E + 0.4 821j:1C“V‘
9 372 3.1 34 2570

Ill
V Cyg
This system displays apsidal rotation. Quick con¬
vergence to the following ephemeris was achieved for primary
minima.
T - 2425508.81G50 + 2.99584866E + 0.4888xlO~7E2
2889 1351 151
+ 0.0216sin( .00-3202 (E - 2946.5))
8 125 55.16
SW Cyg
This system has a variable period best displayed
by an upward-turning parabola. Although Friebos-Conde
and Herczeg (.1973) find a. periodic representation possible,
we do not.
T - 2433160.65097 + 4.57301676E + 0.3983xl0"7E2
1006 1693 733
VW Cyg
On the limited data available, we suggest a downward
parabola for the system. Hall and Woolley (1.973) divide
the residuals into three segments.
T = 2420327.47047 + 3.430G0532E - 0.9917xlO~7E2
15720 19800 6197
WW Cyg
Our data have several crucial gaps, preventing us
from confirming a possible periodic variation found by
Whitney (1959). No apsidal evidence is available.

112
T - 2431980.64870 + 3.31773298E + 0.5966x10 8E2
769 311 898
VíZ Cyg
See Table 5-2.
ZZ Cyg
We find a periodic representation possible for the
variable period of this system. No apsidal evidence is
present. It can be seen that the parabolic term is sig¬
nificant .
T = 2421137.43999 + 0.62861735E - 0.1944x10 10E2
186 21 666
- 0.00317sin(0.00128 (E - 2197.8))
87 2 483.9
CV Cyg
This system displays apsidal motion. Again, there
are too little data to obtain a reliable periodic represen¬
tation by the method of least squares.
T - 2424451.41566 + 0.98441037E - 0.1054xl0“6E2
1566 5009 530
+ 0.374 7xl0‘"11E3
1863
EM Cyg
This system displays an almost perfectly straight-
plot of primary minima. There is no apsidal evidence, and
a periodic representation is not possible.

113
T - 2437882.35942 + 0.29090955E - 0.2271x10 10E2
29 21 2778
GO Cyg
See Table 5-2
V448 Cyg
On the data for this system, a linear fit is
superior to all other representations. There is no evidence
of apsidal motion.
T = 2416361.09481 + 6.51972902E
1329 405
V453 Cyg
This system has a very large scatter in its times
of minima. Although apsidal motion is a possibility, it is
almost impossible to see. Separate least squares periodic
fits for primary and secondary fail to yield a unique
answer. A single periodic representation of the total
ensemble yields the best results.
T - 2415985.69846 + 3.88931342E - 0.1731x10“8E2
862 651 995
-0.03355sin(-0.002946(E + 32696.9))
515 53 6650.0
V463 Cyg
This system also has a great amount of scatter in
photographic timings. There is no apsidal evidence. A

114
light-time effect is suggested, based on a quick convergence
to a periodic representation, as well as the suggestion of
a third member by Vetesnik (1968).
T - 2428750.35797 + 2.11756229E - 0.4078xl0-9E2
283 69 1130
+ 0.00406sin(0.00349(E + 16820.8))
184 38 2116.0
W Del
Only more recent timings of minima are available,
for this well-known system. Several investigators had
suggested a periodic representation, although this effect
cannot be achieved on the available data. There is no
evidence of apsidal motion.
T = 2418048.77529 + 4.80594985E + 0.1758xl0~7
.20200 10000 1216
Z Dra
Friebos-Conde and Ilerczeg (1973) find a light-time
representation possible if we are willing to allow abrupt
changes at E-270, 4120, and 5660. Our study achieves
rapid convergence to a periodic representation without
these criteria. There is no evidence for apsidal motion.
T = 2433268.71278 + 1.35743427E + 0.6955>:10~9E2
600 502 3978
+ 0.0065). s i n (0.00115 (E -
271 21
2285.5))
2 5 8.1

115
RR Dr a
There is a slight suggestion of periodicity in the
linear residuals of this system. However, a convergent
periodic representation is not possible. An upward parabola
fits the data quite well. There is no evidence of apsidal
motion.
T - 2433390.48554 + 2.83117807E + 0.2397xl0“7E2
285 444 122
RZ Dra
The variable period of this system is best portrayed
by a periodic representation. This is most likely a light"
time effect as not only is there no evidence for apsidal
motion, but Mallaroa (1975) has suggested a possible third
body in this system.
T = 243 7492.694G7 t 0.55087608E - 0.1854xlO~9E2
387 91 873
- 0.00313sin(0.00148(E - 3343.5))
179 74 3108.0
SX Dra
See Table 5-2.
TW Dra
This system displays a periodic variation with
a huge amplitude. There is no evidence for apsidal motion.
Convergence to a periodic representation was rapid (four

116
iterations). The behavior of the residuals suggests still
another periodicity with a smaller amplitude.
T = 2433889.54597 + 2.80673218E + 0.3023xl0“8E2
2082 488 131
+ 1.118819sin(0.0001912(E - 6566.3))
14630 17 193.6
TZ Dra
A periodic representation is possible to explain
the period variations of this system. There is no evidence
of apsidal motion.
T = 2433871.38408 + 0.86603397E + 0.4990x.l0~10E2
275 37 2521
- 0.00197sin(-0.00373(E + 5622.7))
91 22 772.5
UZ Dra
A linear fit is best for all the data available,
which is only from E--4306. However, another satisfactory
representative of the times of minima could be given by
two linear segments, with a sudden change, at E=6400. There
is evidence against apsidal mclJ.cn.
T = 2419429.30716 + 3.26130139E
942 147
RU Eri.
Only recent data are available for this system.
A linear fit is presented. There is no evidence

117
for apsidal motion. All other fits are equally
poor.
T = 2419685.36973 + 0.63220410E
4259 162
U Gem
The period of this interesting system appears to
be undergoing a steady slow change. A convergent periodic
representation is not possible, and there is no evidence
for apsidal motion available. This is one of the few
systems whose cubic coefficient is significant which does
not allow a periodic fit.
T = 2437638.82713 + 0.17690574E + 0.4327xlO“10E2
22 11 1401
- 0.9702x10 15E3
3537
RY Gera
This system displays an overall downward parabolic
variation. There is no evidence for apsidal motion, nor
is a periodic representation possible for these data.
T - 2418015.32554 + 9.30095557E - 0.7723xl0"7E2
3103 4735 1511
YY Gem
Only data since E=13542 wore available for this
study. There is evidence against apsidal motion, while a

11 8
periodic representation is not possible for these data. Plavec,
Peking and Smetanova (I960) find no period change. This result
was duplicated by this investigation.
T = 2426228.41293 + 0.81428454E
5516 335
AF Gem
This system has a variable period. Quick con¬
vergence to a periodic representation occurred. There is
no evidence for apsidal motion. Scatter is relatively low
for this system.
T = 2420752.04627 + 1.24350548E - 0.1987xlO_9E2
224 45 265
-f 0.00286si.ii (0.00127 (E - 10891.7))
172 9 389.3
AY Gem
Although there is considerable scatter in the times
of minima for this system, a general overall downward
parabola is evident. There is no reason to expect apsidal
motion or periodic terms.
T = 2433385.29215 T 3.05365343E - 0.2059xl0_8E2
696 316 1737
Z Hei-
On the data available, a linear ephemeris is
slightly preferable to a parabolic one. There is no

119
evidence for apsidal motion and a convergent periodic
representation is not possible.
Tp = 2413086.43135 + 3.99277210 + 0.2468xl0~8E2
11470 3843 3153
T = 2413086.34163 + 3.99280217E
412 6249
SZ Her
The period variations of this system are best por¬
trayed by a cubic fit. A periodic fit is not possible.
We also have no evidence on apsidal motion for this system
Tc = 2434S87.38445 + 0.81809051E + 0.1386xlO_SE2
193 128 313
- 0.7706x10~13E3
2212
TT Her
There are conspicuous data gaps in this system
which make a detailed determination of the variation
difficult. One possible interpretation could be two
linear segments, with a break at E--9000. There is no
evidence for apsidal motion. Alternatively, a parabolic
representation is adequate.
T = 2434525.26368 T 0.91207963E - 0.2506xl0~'9E2
191 13 286
Finally, there is quick convergence to a possible light¬
time effect.

120
T = 2434525.25664 + 0.91208071E - 0.1400xl0~9E2
359 59 479
+ 0.0114 sin(0.00055(E - 4.5))
55 3 399.0
TU Her
The period variation of this system is given by a
downward parabola. There is no evidence for aps.idal motion.
A convergent periodic representation is also not possible.
T = 2424499.57370 + 2.26707574E - 0.9883>:10-8E2
2589 866 697
TX Her
There is definite evidence against apsidal motion
in this system. There was very quick convergence to a
periodic representation.
-9 2
T - 2430325.20098 + 2.05980873E - 0.1429x10 E
74 23 442
+ 0.0068Ssin (0.000719 (E 4- 3426.4))
56 51 392.6
UX Her
The variable period of this system can be displayed
by either a downward parabola or a periodic representation.
No evidence is available for apsidal motion.
T - 2429987
41660
72 2
548 855 0 4 E
250
0.6576x10"9E2
2333

.121
T = 2429987.41487 + 1.54885864E - 0.1159x10 8E2
579 262 272
+ 0.00459sin(-0.00173 (E + 5810.6)
135 22 1622.0
AK Her
Period studies of this system have been done by Kwee
(1958) and Herczeg and Schmidt (1959). They state that the
plot looks almost sinusoidal. This system fails to converge
to a periodic representation, although every periodic
parameter except the amplitude converges quite rapidly.
The time of periastron passage and the period of the wide
orbit are well within their errors throughout the solution
process. An examination of the first data point suggests
that a periodic representation is superior to other types
of fits for this system.
T - 2436757.66142 + 0.42152339E + 0.1394xlO-10E2
128 30 2118
+ 0.0195sin(0.00022(E - 1521.0))
23 7 996.2
CC Her
This system displays a very interesting O-C diagram.
The behavior of the period could be explained by three
segments, with sudden variations at E---300 and E-2600.
There is no evidence for apsidal motion. A periodic repre¬
sentation is possible for these data.

122
T = 2431265.30081 + 1.73403549E - 0.3517xlO~8E2
67 36 77
+ 0.0248sin(0.00109(E - 1115.0))
7 1 28.52
CT Her
A linear fit is best for this system. Only a small
amount of more recent data are presented here. There is
evidence for apsidal motion.
T = 2430904.37819 + 1.78636558E
1955 432
DI Her
This system appears to have a displaced secondary.
The most significant fit is given for both primary and
secondary observations.
T.r
2429520
886
5.27507983E
565
T .. = 2429520.60201
11 889
5.27511619E - 0.2258x10 7E2
3108 2158
V338 Her
The variations in period of this system can be ac¬
counted for by either an upward parabola or a periodic fit.
However, the error in amplitude of the periodic fit is
almost as great as the amplitude. There is no apsidal
evidence.

123
T = 2433771.38391 + 1.30572414E + 0.3148x10 8E2
639
577
920
+ 0.00463sin(-0.00166(E + 17277.7))
398 46 5600.1
RX Ilya
On a rather limited amount of data, we find a down¬
ward parabola best. A convergent periodic representation
is not possible for the data. There is also no evidence for
apsidal motion.
0.1814x10 7E2
1555
T = 2420237.33013 + 2.28197324E
2 9 050
1355
SW Lac
This system displays very curious variations. For
a long period of time, the period decreased (until E=25400)
at which point it begem what looks like a parabola or
periodic representation until E-49000, at which point it
again began to decrease. Apsidal motion may be ruled out.
A single periodic fit is not possible. Three linear
segments look dubious. We present a parabolic ephemeris.
A periodic representation is possible, if we allow for both
a general overall parabolic variation with a rapid change at
about E~200 00.0.
Tp - 2424852.53739 + 0
225
12
13

12 4
TW Lac
A downward parabola Cits these data will, although
the possibility of two linear segments cannot be entirely
ruled out due to a large data gap. A light-time effect
is not possible. There, is no evidence for apsidal motion.
-7 2
T - 2434049.59766 + 3.03752375E - 0.1830x10 E
151 603 216
VX Lac
This system also displays interesting behavior.
There is no evidence for apsidal motion, as secondary has
not yet been observed. One possible representation is
three linear segments, with breaks at E-5000 and E-10500.
A periodic fit is also possible.
T = 2424791.47978 + 1.07449341E t 0.8301xl0"10E2
505 96 4962
+ 0.01211sin(0.00028(E - 10835.1))
166 11 619.5
AR Lac
There are several intriguing possibilities for
this system. Several segments with breaks at E=--30650 and
E=6600 appear to do an adequate job of representing this
system. There is some evidence against apsidal motion,
although rather weak, Convergence to a periodic fit is
also possible.

125
T = 2426624.36802 + 1.98322964E - 0.1378x10"8E2
113 63 82
- 0.01003sin(0.00187(E - 27102.3))
73 4 517.7
CM Lac
Data for only the last 4000 cycles are presented
here. Dugan and Wright (.1939) find no period variation.
All our fits appear equally good, although a periodic
representation is not possible. There is evidence against
apsidal motion.
T = 2427026.31322 + 1.60469178E
124 21
CO Lac
This system displays apsidal motion. Ephemerides
for primary and secondary minima are given below. The
parabolic term is not significant.
T i = 2427531.07535 + 1.54220676E - 0.2711xlO~10E2
125 22 3591
T
sec
+ 0.0162sin(0.000640(E - 1839.2))
11 16 176.5
2427534.07654 T 1.54220772E -
11)
22
_ i n o
0.5073x10 ±UE~
54 3 0
+ 0.0149sin(0.00059 (E - 2216.4))
19 2 176.5

126
DD Lac
A downward parabola fits these data. Scatter is
great mainly because only visual observations are presented.
There is no apsidal evidence, nor is a periodic fit
possible.
-10 2
0.1620x10 E
224
2421914.17763
4 31
t 0.19309007E
20
T
Y Leo
Svechnikov and Surkova (1972) find "random and in¬
stantaneous" changes in the period of: this system. No
evidence for apsidal motion is present. Plots of both
parabolic and periodic fits are given in Appendix Two
as both representations appear equally feasible instead
of instantaneous changes.
T = 2422372.58327 4 1.68605070E + 0.2019x10 8E2
1012
236
138
T = 2422372.58646 + 1.68604
593
LIE + 0.2134x10 8E2
14 4
88
0.0035sin(-0.006
+ 19312.9))
8 01.5
10
RT Leo
Due to the large scatter of the observations for
this system, a linear fit is probably the safest course
to tak
There is no evidence for apsidal motion, nor is a
periodic representation possible for these data.

127
T = 24238441.02610 + 7.44795864E
1190 1278
UV Leo
The variation of period of this well-observed
system has been treated by Ilerczeg (1970) . The evidence
against apsidal motion appears quite definite. A periodic
representation is possible.
T - 2438440.72633 + 0.60008478E - 0.1458xlO-10E2
82 13 1276
- 0.00114sin(0.00143(E - 176.3))
78 4 183.0
T LMi
Severe data gaps in this system again prevent a
definitive identification of the type of variation present.
There is no evidence for apsidal motion. A periodic fit
is possible, although the error in the amplitude term is
large. Three linear segments provide an equally good
alternative representation.
T = 2423856.36592 + 3.01977253E + 0.2255x10"7E2
7728 11830 1846
+ 0.2405sin(0.000553(E - 11751.1))
2677 176 2717.0
S'W Lyn
This system displays a fair amount of scatter.
However, this occurs only for the photographic observations

128
which are of a rather dubious accuracy. There is no evidence
for apsidal motion, and a convergent periodic representation
is also not possible.
T = 2425643.31443 + 0.64407017E - 0.1593x10~9E2
670 166 690
RV Lyr
The prominent period variations displayed by this
system can be explained by several methods. One possibility
is several linear segments, separated by rapid changes at
E=~3640 and E=-190. Friebos-Conde and Herczeg (1973) find
a light-time effect not possible and also suggest sudden
variations. No evidence of apsidal motion is present.
However, a periodic representation was found which fit the
data well.
T = 2434603.38522 + 3.59899998E - 0.6827xl0“9E2
46 32 564
+ 0.0256sin(0.001208 (E + 4573.1))
3 8 22.6
RZ Lyr
See Table 5-2.
TZ Lyr
Although a periodic representation is possible for
this system, a sudden change at E-25320 looks possible.
A parabolic fit also represents the data well.

129
T - 24206G9.48359 0.52882154E + 0. 6646xlO“10E2
315 26 541
T - 2420669.53454 + 0.52881783E + 0.1271x10 9E2
761 77 163
- 0.0137sin(0.000604(E ~ 1455.5))
32 32 1129.0
UZ Lyr
This is another system whose times of minima can be
explained by what appears to be several linear segments or
by a periodic representation as derived in this study.
There is no evidence for apsidal motion.
T = 2424268.47927 + 1.89126587E + 0.2889xl0_9E2
81 47 569
+ 0.0088sin(0.00127(E - 2304.5))
8 2 62.85
EW Lyr
The variations of period for this system are
represented by an upward parabola. Evidence for apsidal
motion is not present. A convergent periodic representation
is not possible.
2434988.38332 + 1
Ob
9 4 8 7 3 8 2 4 E
35 5
+ 0.2430xl0~8E2
889
RW Mon
Although the period of this system appears to be getting
shorter (Szafraniec, 1959) , no representation other than

130
a linear fit to the data has significant terms. For pur¬
poses of an ephemeris we present a linear fit. There is
no apsidal evidence.
T = 2433680.44959 â– +- 1.90609348E
52 20
BO Mon
The times of minima of this system are represented
by an upward parabola. There is no evidence for a rotation
of the line of apsides.
T = 2428211.61378 + 2.22315369E + 0.6171xlO_8E2
396 323 475
FS Mon
Although only a limited amount of data with
relatively large scatter is presented here, a parabola fits
the data well. A longer baseline is necessary for a
definitive statement, on the variation in this system.
T - 2427125.63327 + 1.90560625E + 0.3824xlO_7E2
43610 24890 3495
U Oph
Friebos-~Con.de and Herczeg (1973) find a possible
light.-time effect for this system. We have only been able
to obtain data for the last 5000 cycles. These data, while
suggestive of a periodic representation, were found to be
best fit by a parabolic. There is evidence against apsidal
motion.

131
T - 2408280.15411 + 1.67729336E + 0.1329x10“8E2
12980 1388 371
RV Oph
For purposes of an ephemeris, a linear fit is the
only representation possible whose errors are smaller than
the coefficients. Again, some early data were inaccessible.
There is no evidence for apsidal motion and a light-time
effect is not possible in these data. Koch (1962) has
found the period to be constant.
T = 2423997.39912 + 3.68711771E
640 176
SW Oph
We find a periodic representation possible for this
system, although the classic explanation of the times of
minima involves several linear segments with sudden changes
at E--3520 and E--760. There is no evidence for apsidal
motion.
T = 2436369.45003 + 2.44605732E + 0.7081xl0~8E2
333 228 367
+ 0.0627s.in (0.00.100 (B + 2489.7))
25 3 57.16
V449 Oph
There is no evidence for apsidal motion in this
system. A convergent periodic fit is not possible. This
is one of the few systems which is fit by a cubic but not
a periodic fit.

132
T = 2427612.53953 + 1.24307824E ~ 0.2073x10 8E2
559 670 1567
+
0.0347x10
872
V451 Oph
Again only a limited amount of data is presented
here. Although it is apparent that some variation is going
on, a linear fit is best for purposes of an ephemeris.
There is no evidence for apsidal motion.
T = 2434165.38686 + 2.19662885E
9287 2924
V501 Oph
This system is represented by an upward parabola.
There is no apsidal evidence. A convergent periodic
representation is not possible.
T = 2430911.37976 + 0.96795107E + 0.8348xl0"10E2
634 50 5603
V502 Oph
Wilson (1967) states that the light curve of this
system does not repeat well. Nonetheless, it appears that
there is an intrinsic period variation which can be ex¬
plained by either tvro linear segments with a sudden change
at E--40QQ, or by a cubic. There is evidence against
apsidal motion.

133
T = 2437436.45680 + 0.45339314E - 0.1283xl0~9E2
91 16 201
+ 0.1046x10 3E3
204
V508 Oph
See Table 5-2.
V56 6 Oph
The period of this system appears constant until
E=10000. Since this epoch, the system has considerable
scatter. There is evidence against apsidal motion..
-10 2
T = 2435245.54383 -I- 0.40964012E + 0.9534x10 E
66 21 1287
VV Ori
The period variations of this system seem to
follow a cubic. A convergent periodic representation is
not possible, nor is there evidence fox' apsidal motion.
Large data gaps are present.
T - 2420095.21732 + 1.48538514E - 0.1326x10"8E2
1071 461 716
+ 0.5832x10~13E3
3.15 2
CP Ori
The times of minima for this system display con¬
siderable scatter after E-1700. There is no evidence

134
for apsidal motion and a convergent periodic representation
is not possible.
T = 2425707.43693 + 5.32068586E - 0.6926x10 7E2
4140 7104 2771
EL Ori
The variations of this system can be represented
by several linear segments, with sudden changes at E=1600,
E-3700, and E=4800. Alternatively, a periodic representa¬
tion is possible. There is no evidence for apsidal motion.
T = 2427452.25617 + 1.55097103E + 0.1414xl0“8E2
871 499 601
+ 0.00760sin().00135(E - 2468.0))
190 28 200.7
EQ Ori
A parabola fits the available data best. A periodic
representation is not possible for these data. There is
also no evidence for apsidal motion.
T --- 2426209.35503 -f 1.74603997E + 0.1152x10 8E2
3430 1061 740
ER Ori
This system has a complicated set of re
There is at least one sudden change even if we
periodic fit. However, this possibi1 i.ty seems
the errors of the coefficients are quite large.
siduals.
use a
remote as
There is

definite evidence against apsidal motion. A single light
time is not possible. We could also express the fit by four
linear segments with sudden changes at E=-8370, E--100, and
E-11000.
T = 2436508.76918 + 0.42340036E + 0.5276xlO~10E2
142 14 1397
ET Ori
This system displayed rapid convergence to a periodic
representation. However, the amplitude of the sine term
is quite small., although larger than its error. There i.s
no evidence for apsidal motion.
T = 2426684.27989 + 0.95093775E - 0.1536xlO-9E2
195 66 453
+ 0.0 0 4 3 7 s in (-~ 0.0 0 0 8 5 4 (E - 9158.7))
157 97 586.1
FR Ori
A periodic representation is possible for these
data, although it hardly seems definitive for such a few
points. There is no apsidal evidence for this system.
T = 2427862.16224 + 0.88316101E -I- 0.1000xi0-9E2
170 52 480
GG Ori
This is another system with a displaced secondary.
There is no convergent periodic representation possible.

136
T â–  - 2433596.47948 + 6.63146754E
1 ' 878 1657
T
sec
2433596.00010 + 6.63148179E
1249 2080
OS Ori
This system is best fit by an upward parabola.
There is no apsidal evidence.
T = 2432987.25560 -h 2.38350938E + 0.5283xl0-8E2
298 473 1416
V343 Ori
This system is portrayed fairly well by an upward
parabola. There is strong evidence against apsidal motion
A periodic representation is not possible for these data.
Tp = 2432865,5 077 6 + 0.80911806E + 0.7913xl0"9E2
647 355 3849
U Peg
This system has a period variation fit best by a
downward parabola. There is evidence against apsidal
motion. A convergent periodic fit is also not possible.
T = 2436511.66969 + 0.37478153E - 0.1698xl0''10E2
36 6 679
UX Peg
Zessewitaoh (1957) has found the period of this
system to be variable. There is no apsidal evidence.
Convergence to a periodic fit was
achieved in 10 iteration

137
T = 2436055.74122 + 1.54462814E + 0.8542x10 9E2
534 148 3504
+ 0.02788sin(0.000580(E + 4278.0))
6106 61 393.7
AQ Peg
Whitney (1957) has found the period of this system
to be variable. There seem to be three distinct phases in
the scatter of this system: an initial period of high
scatter up to about E=-5Q00, followed by a time of low
scatter until E-500, and then another area of high scatter.
There is no evidence for apsida.1 motion. A periodic
representation is not possible.
T - 2434220.50711 + 5.54848591E + 0.2239x10 7
1104 1590 1173
EG Peg
This system could be represented by two segments
with a major variation at E---90. A downward parabola
offers an equally good representation. A periodic fit
is not possible and there is no evidence for apsidal
motion„
0.1138x10 7E2
193
BN Peg
A linear ephemeras is given for this system. There
is not possib1e.

138
T = 2427656.43825 + 0.71329802E
301 33
DI Peg
Gaposchkin (1955) found the period of this system
to be variable. Ahnert (1974) suggests a possible third
body. One observation of secondary minima falls on the
plot of primary minima. A periodic representation was found.
T - 2432441.44696 + 0.71181768E - 0.1944xlO~9E2
266 28 659
+ 0.0107sin(0.00158(E - 6495.9))
21 63 533.8
B Per
This well-observed system displays a variable
period. The periodic representation used in this study
only takes into account the fundamental term. Algol is
known to possess at least three periodicities. As such,
we only present some data and a parabolic ephemeris to
display the general character. A further study of this
system is planned, but for now the reader should consult
Kopal, Plavec and Reilley (1950) .
T = 2433283.34091 + 2.86733021E + 0
180 283
1815x10
] 29
- 8 2
ill
RT Per
The period variations of this system have been
covered in many studies. A periodicity cf 37 years is

139
generally suggested. There is no evidence for apsidal
motion. A periodic fit was found in only four iterations.
T = 2424553.25393 + 0.84940672E - 0.6932xlO~10E2
214 31 1876
+ 0.0161sin(0.000370(E + 9026.9))
12 15 856.3
RW Per
Woodward (1943) suspected a variable period for
this system. We find prominent variations, fit by either
three linear segments with suddent changes at E=-320 and
E-400, or by a periodic fit. There is no evidence for
apsidal motion.
T = 2429217.55076 + 13.1986790E + 0.8411xlO-8E2
198 381 4862
+ 0.0 9 0 8 sin(0.0 0 3 2 3 (E + 810.4))
25 5 9.9
ST Per
Svechnikov (1972) finds sporadic and instaneous
changes for this system. This work suggests that the data
be represented by two segments with a change at E=-300,
or by an upward parabola. There is no apsidal evidence.
A single convergent periodic representation is not possible.
T = 2439091.71540 1 2.64834578E - 0.3740xl0~8E2
489 522 1693

140
XZ Per
Zessewitsch (1957) finds the period to be variable.
We present a linear ephemeris for the available data, which
covers only the last 7000 cycles. A periodic representation
is not possible for these data, nor is there any evidence
for apsidal motion.
T - 2425150.41644 + 1.15163501E
322 24
IQ Per
See Table 5-2.
Y Psc
Zessewitsch (1957) and Whitney (.1957) find the
period variable. Two linear segments with a break at
l.j-4 2 0 fit the available data, although this is by no means
conclusive due to a large delta gap. A convergent periodic
representation is not possible. There is no apsidal
motion evidence.
T = 2‘
67643 + 3
3 4 3
76573903]
12 C 8
-t 0.2 54 6x10 V
64 2
5Z Psc
A downward parabola fits this highly interesting
system. There is no apsidal. evidence. A periodic fit is
not possible for these data .
T
SGI.'
58812 + 3.96571700E - 0.1139x10 6E2
1636 1242 79

141
U Sge
Jacchia (1941) and Svechnikov (1955) find the
period of this system to be variable. There is no apsidal
motion evidence. A periodic fit was found in one iteration.
T = 2429111.32111 + 3.38061714E + 0.5738xl0~9E2
335 13 424
-I- 0.00820sin (0.00125 (E + 2466.1))
32 50.83
RS Set
Whitney (1957) suggests a variable period. We find
a periodic representation possible, although i.t is probably
meaningless on the available data. Severe gaps exist at
crucial times. There is no evidence of apsidal motion.
T- 2429130.45310 + 0.66423787E + 0.6811x10“10E2
167 39 1501
V505 Sgr
These data are fit well by a downward parabola.
There is no evidence for apsidal motion. Again we have
rather severe data gaps.
T = 2433515.32S73 + 1.18286929E - 0.3842xlO"9E2
99 37 881
V525 Sgr
A parabola is presented, as an ephemeras for the
rather limited data.

14 2
T = 2429662.28019 + 0.70516194E - 0.1701xl0"8E2
8453 1859 792
V526 Sgr
This system is a classic example of apsidal motion.
Plots of both primary and secondary observations are fur¬
nished in Appendix Two.
TII = 2421862.13867 + 1.91938019E + 0.5072xl0"9E2
147 37 1304
+ 0.00422sin(0.00241 (E - 1169.9))
139 11 173.1
RW Tau
Dugan and Wright (1939) suggested that the period
of this system was variable. Friebos-Conde and Herczeg
(1973) find a periodic representation improbable and suggest
abrupt changes. There is no evidence of apsidal motion.
T = 2427199.49666 4 2.76880964E + 0.3863xl0-8
154 119 247
X Tri
Gadomski (1932), Odinskayan and Ustinov (1951) and
Lange (1957) state that this system has a variable period.
Friebos-Conde and Herczeg (1973) find a possible light-
time effect if some abrupt changes occur. Convergence to
a periodic representation occurred in two iterations for
the avai1able data. There is no apsida1 motion evidence.

14 3
T = 2437572.20101 + 0.97153270E - 0.4827x10 10E2
17 9 1061
+ 0.00426sin(0.00106 (E - 229.5))
25 3 46.39
RW Tri
Ther period variations of this system are expressed
by a parabola. There is no apsidal motion evidence and a
convergent periodic fit was not possible.
T =-2435396.51307 + 0.23188360E + 0.1262xl0“10
56 15 676
W UMa
There have been numerous discussions of this well-
observed system. A cubic fit seems to be best. There is
evidence against apsidal motion and a single periodic
representation is not possible. We suggest sporadic and
random changes.
T = 2438792.7087] + 0.33363881E
100 24
) . 17 58x10 10E2
1527
-0.4 797xl0“14E3
1533
TX UMa
Payhs-Gaposchkin (1942) suggests a periodic
phenomenon with a period of 36 years for this system. Not
only were no observations of secondary available for this
study, but we found a periodic representation net possible
in t ne ava i 1 ab
o o a t.a

144
T = 2439193.30901 + 3.06325747E - 0.2344x10 7E2
174 228 187
UX UMa
This system has been classified as a post-nova
system. There are no indications of apsidal motion. We
also find a convergent periodic representation not possible.
T - 2437428.88750 I 0.19667123E + 0.8208xl0_12E2
9 1 3755
AG Vir
Kwee (1958) suggests a variable period with two
sudden changes. There is evidence against apsidal motion.
Vie suggest a representation by either three segments with
changes at E=-14000 and E---6600 or an upward parabola.
T = 2435561.29868 + 0.64265011E + 0.9G55xlO“10E2
82 13 1782
AH Vir
There is definite evidence against apsidal motion
in this system. Its period variations follow no periodic
function. Either two .linear segments with a sudden varia¬
tion at E-38910 or an upward parabola will fit the data
well.
T = 2425245.58080 + 0.40751469E + 0.1211xlO~9E2
120G 81 132

145
Z Vul
A cubic ephmoris is given for the data available
on this system. There is no apsidal evidence and a periodic
fit is not possible for these data.
T = 2425456.41186 + 2.45474052E + 0.3697x10"7E2
14699 9054 1781
-0.2357xlO-11E3
1134
RS Vul
See Table 5-2.
BE Vul
There is no evidence for apsidal motion. A periodic
representation is possibler although the parabolic term is
not significant.
T = 2433749.52842 + 1.55204868E - 0.1894x10~9E2
294 29 3836
+ 0.00664sin (0.000849 (E -
251 177
5169.6))
1149.0

CHAPTER SEVEN
THEORIES OF MASS EXCHANGE
7.1 Introduction
We will now explore some of the theoretical ap¬
proaches which enable us to gain insight about the
qualitative and quantitative aspects of mass exchange.
Basically, such approaches can be subdivided into two
main categories.
1. Those models or theories which deal with the
position, movement and momentum of particles in a binary
system: these theories can explain what possible sinks
and sources are available for mass exchange, and further¬
more, they can predict the paths of particles from the
source to the sink. These models will be discussed in
the Particle Trajectory and Hydrodynamics! Model sections.
2. Evolutionary models which attempt to predict
the individual characteristics of each s tcii'f as well a s
the quantity of mass available for mass exchange: these
models, for the most part, make use of the conservation
assumptions mentioned earlier in an effort to trace the
evolutionary path of the binary system, subject to the
appropriate interactive effects.
14 6

14 7
Theoretically, these two approaches are comple¬
mentary, and constitute a two-dimensional approach to
the study of close binary systems. For example, the
computer running evolutionary sequences will find it
desirable to have quantitative information available
about mass and momentum exchange for the particular
parameters of his system at that point in time, for the
purpose of calculating differential changes of these
parameters for his next time step.
In practice, such an approach has not yet been
employed. Although effects introduced by such an approach
might seem minor at first glance, it is certainly conceiv¬
able that a small change at a delicate point in an
evolutionary sequence could have profound results. This
"two-dimensional" analysis will undoubtedly provide fruitful
results, and should be the object of intensive study for
future evolutionary models.
7.2 Part, lele Trajectory Models
In the absence of gas pressure, radiation pressure,
and viscous forces the equations of motion of a test
particle, released in a binary star system, were given by
Mou1ton (1914) as:
(7.2.1)

] 48
y
(l-u)
- (]
-v)
ri
r
f \
Z
z
”3
- y
rl
N
y
(7.2.2)
(7.2.3)
where r^ = (x-x^) 2 + y^ + ?/>~ and r2 = (x-x^)
+ y2 + z2 (7.2.4)
Kuiper (1941) used these equations in his work on Beta Lyrae
to deduce the motions of the gas stream in that particular
system. Kuiper performed the appropriate numerical inte¬
grations of these three coupled equations by using Taylor's
formula, up to the third derivative. In spite of the
absence of modern computational facilities, Kuiper carried
out several thousand such integrations by hand. In these
particle trajectories, Kuiper found that upon departure from
the inner Lagrangian point, that the "ejected" particle
would travel toward the other component, and either strike
it or rotate around it for suitable initial parameters.
Calculations carried out over a larger ares of
initial parameter selction space were performed by Kopal
(1959) and Gould (1958), Both authors used the method of
numerical integration to trace the path of a particle
subjecL to Equations 7.2.1, 7.2,2, and 7.2.3 in its travel
throughout a binary system. In addition, Kopal considered

149
ejection from the point at an angle relative to the line
of centers. Gould reached several conclusions drawn from
a wide variety of trajectories. Perhaps most important,
in light of the problem under consideration in this study,
was the result that unless the star from which the matter
was ejected had a mass of approximately one-fifth or less
than that of its component, mass would go directly to that
component falling on its following face as in Figure 7-1.
She found this to be true for all velocities of ejection.
Assuming that the ejecting star has a mass less than one-
fifth of its companion, Gould also reached the following
conclusions:
1. A concentration of matter between the
stars is formed which favors the
following hemisphere of the opposite
component form which it was ejected
2. A ring rotating in the direct sense
will form and be relatively small
3. A ring rotating in the retrograde
direction can occur around both com¬
ponents and will be formed in the
case of generally high mass ratios
and initial velocities
It .is interesting to note that virtually all the calcu¬
lations carried out for our investigation are for
systems with mass ratios less than 0.2--the dividing

X
Figure 7-1
150

151
line put forth by Gould for orbits to occur in binary
systems. The quantitative form utilized by Gould did not
take into account the precise impact point on the mass
receiving star. This quantity .is necessary so that we are
able to compute the change in angular momentum which occurs
when a particle impacts on the mass receiving star.
As put forth by Huang (1963a), a satisfactory theory
of ring formation must take into account collision processes.
This amounts to a hydrodynamics1 solution of the problem.
The difficulty of the hydrodynamical approach is partly
mathematical, due to the uncertain boundary value conditions
placed on the problem, and partly physical., due to the great
amount of computer time necessary for each system. Hydro-
dynamical models will be dealt with in a later section.
A more fruitful method of evaluating the angular
momentum of an ejected particle was suggested by Huang
(1963a). Huang showed that to the first order, the angular
momentum of a particle ejected to infinity can be written
cl S
3y (1 - u)
4r
Í (cos210
- CO!
2t)
(7.2.5)
where cos(t) = x/r, sin(t) - y/r
One advantage of treating the angular momentum instead of
the orbit itself arises from a consideration of collisions.
Whereas in the particle trajectory method, a collision of

152
two particles results in two completely different new orbits,
-in the angular momentum approach, net angular momentum is con¬
served in the process of: collision. However, Hunag ' s (1963a)
analysis was greatly restricted. His initial velocities of ejec¬
tion from the inner Lagrangian point were greater than four
times the orbital velocity. As such, this study could be used
only to shed light upon the momentum of particles ejected com¬
pletely from the system without resorting to other assumptions,
and not on the motions of intersysteraic gaseous streams.
In another paper, Huang (1963b) attempted to relate the
modes of mass ejection to their effect upon the orbital elements
of a system. He distinguishes between the following three modes
I. Jeans' Mode
This mode of mass loss assumes that the ejection of mass
are much greater than the orbital velocities. Huang (1963b)
shows that:
(7.2.6)
which yield
6a -6 (Mg + hi2) 6p
"a* ’ p~
eoe
(7.2.7)
where e is the orbital eccentricity.

153
II! Slow Mode
This mode allows for a gradual loss from the eject¬
ing star, with the subsequent interaction of the ejected
material with the binary system. Huang (1963b) shows in
this case that
ÓP _ 46(Mg + M2) 6M2 36h0
P Mg + M2 M-¡ + M2 hQ
3e6e
1-e2
(7.2.8)
In the case of simple exchange of mass from one component
to the other, this equation assumes the form
<5P
'P~
3(M3 + M2)
m2
^Mi 30^0
Ml 1-e2
(7.2.9)
as 6(M^ + Mp) = 0 (conservation of mass)
and 6(h^) - 0 (conservation of angular momentum)
Naturally, this situation may or may not occur in reality.
If it does, we obtain the familiar result
<5P
”p”
(Mi + M2)
M2
(7.2.10)
for systems for small eccentricities (we expect e~0 due to
tidal effects) ,

154
III ♦ Intermediate Mode
In this mode, mass is allowed to either escape from
the inner contact surface or to form a ring around the
entire system. Unfortunately, the expression derived from
this mode is of a rather formal nature only, as no observa¬
tional. method exists which allows us to calculate the
average angular momentum of a mass particle in the ejected
ring .
Huang (1963b) also dealt briefly with the highly
important concept of coupling between orbital motion and
axial rotation. This idea is vital for a complete under¬
standing of the mass transfer process. Orbital angular
momentum can essentially disappear from the system if it
can be either temporarily or permanently stored as axial
rotation. We now have the following results.
I. __Slow .Mode
Since both M. and are conserved, as well as the
L/ a. 1 J V—- L-X 1 i lj vl
total momentum Q, where
(7.2.11)
Huang (1963b) was able to show that

155
6k?
+ A21—)
where Aj = (t^ + M2)/(M2 (1-e2)l/2) (kj/a) 2
and
A2 = (Mx -I- M2)/(M1(l-e2)1/2) (k?/a)
(7.2.12)
(7.2.13)
(7.2.14)
Thus it can be seen that the variation1 of the period is
enhanced over the non-interactive case.
II. Intermediate Mode
(1 - 3AX - 3A2)6P/P = (1 + 3y2)
6 (Ml + M2)
Ml + M2
+ —? ~ 3 (1 + AX)^1
1-e1
Mi
6M2
3(1 "i ^ p ) ~}7
¿ A2
6kp
A1 k-, + a2 k
6k_2
2
(7.2.15)
where y
(Mx + M2)2
'"M' M
12
ae ~I 3/--
71^
(7.2.16)
and
he - [G(M + M,;) ae j
1/-
(7.2.17)
Huang (1963b) points out that in the event that the axial
rotation and orbital angular momentum are not completely
coupled, it becomes necessary to take into account the time
scale of both these processes. For both these modes, kj
and k2 are the gyration radii of the stars.

156
However, it may be possible to artificially simulate
the degree of coupling which exists between these two
parameters. In our study, a particle trajectory model is
used which keeps track of the angular momentum of the
particle as well as its velocity and position. Thus, for
any specified system and ejection parameters it is possible
to know the angular momentum of a particle at the position
of impact. As the coordinates and velocities of the
particles are also known, the degree of coupling can be
computed. This is the factor episolon as listed in Table
8-1. Furthermore, the instantaneous mass transfer rate can
now be inferred unambiguously from the period change rate.
The meticulous reader may also wish to examine some
results due to Kruszewski (1964a, 1964b), Piotrowski (1964a,
1964b), and Ziolkowski (1966) which will not be covered
here. These authors extended the earlier work to cover a
larger selection of initial parameters. They also discuss
factors dealing with the size of the critical Roche surfaces.
More recently, some special cases of particle
trajectory space have come under investigation. One such
case of special interest is that case whereby matter
ejected from the L-¡ point can form a ring, which forms
either about the other star or the entire system.
Kris (1970) has computed an extensive grid of
trajectories for particles with motions initiated at the
point. Krix states that, as the ring is probably a very

157
unstable formation, the simplest condition for the formation
of a ring is that the particle ejected from the inner
Lagrangian point perform at least one orbit about the other
star. He finds that this will occur as long as the particle
does not impinge upon the "secondary.''' There is some
ambiguity as to the definition of primary and secondary
in Kriz's paper. If we assume that primary refers to the
more massive star, and secondary to the less massive, we
can see that Kriz's results complement and extend those of
Gould. Kriz presents a long table of R ^ , the minimum
distance of the ejected particle from the primary as a
function of initial velocity, angle of ejection, and mass
ratio. Kriz reaches the following conclusions:
1. The larger the value of q, the greater
the probability of forming a gaseous
ring in the system
2. The radius of the ring is practically
independent of the velocity and angle
of ejection when the velocity of ejec¬
tion is less than 0.2 times the
orbit a1 ve1o city
3. For matter ejected toward the secondary
(=0) , the radius of the ring depends
little upon the initial velocity
It appears difficnlt to achieve ring status unless we have
either a very small secondary component or a very large
mass ratio.

158
Bielecki, Piotrowski and Ziolkowski (1974) have
also studied the problem of formation of gaseous rings in
close binary systems. They have shown if we assume the
existence of a ring about the secondary (same terminology
as Kriz), that it is relatively easy to allow matter to
be transferred to a ring outside the system. A change of
less than ten percent in terms of the kinetic energy of the
particle is sufficient in some cases.
Particle trajectory models provide us with
different possible types of mass flow, as well as some
important restraints on what sort of mass exchange events
are possible. Models encountered in the literature,
however, do not take into account several important con¬
siderations with respect to mass flow in a close binary
system.
1. Velocity distribution: We could expect an
egress of particles from the inner Lagrangian point to
obey some sort of velocity distribution. Failure to take
this into account will result in a different estimation
of the trajectories and impact points.
2. Collisions: The mean free path of a particle in
a gaseous stream is on the order of several magnitures
smaller than the separation of the components. Hydrcdynami
cal calculations
show,
however
, the neglect
of colli s.ions
is permissible w;
i.th in
the cent
ext of our pre
•s :nt problem.
5. The v
ranks
examined
by this study
fail to include
physical relationships for the simultaneous consideration

159
of position, velocity, and orbital angular momentum. These
considerations are vital to a complete understanding of the
problem.
7.3 H yd rod ynamical Ilodels
Hydrodynamics! models of mass flow in close binary
systems offer one conspicuous advantage over particle
trajectory models: they allow for a large number of particles
and their interactions. Nevertheless, despite this advan¬
tage, hydrodynamical models have only been rarely used owing
to severe mathematical and physical difficulties.
Prendergast (1960) succeeded in deriving approximate
solutions to a flow of gas in a close binary subject to the
assumption that the pressure terms in the equations of
motion are negligible with respect to gravitational, cen¬
trifugal and. Coriolis terms. Furthermore, he assumed that
the system was in a steady state. Even though this was
an exploratory work, Prendergast found that, in a general
fashion, the method reproduced many features of other
empirical models which had been used to explain observa¬
tions .1 features of close binary systems.
A more detailed hydrodynamical approach was
attempted by Prendergast and Taam (1974) for the close
binary U Cep. This paper examined several possibilities
for this system.
1. Synchronous rotation of both components: In
this approach, material was allowed to escape the mass-losing

160
star at the inner Lagrangian point by thermal evapora¬
tion.
2. Non-synchronous rotation of the secondary com¬
ponent: In this case material was allowed to escape the vicinity
of the inner Lagrangian point with some additional velocity.
3. Non-synchronous rotation of the primary, or
mass-gaining component: This was done by the same method as
case (1) except the primary was allowed to rotate at five
times synchronism. This is just its rotational velocity
as observed spectroscopically.
Prendergast and Taarn reached the following conclu¬
sions in this valuable work.
1. A luminous region of gas extends from the inner
Lagrangian point to the secondary facing side of the
primary, which is rotating in the direct sense.
2. A stream of gas leaves the system, but the
amount of material involved is so small that the loss of
this matter is unimportant for the determination of period
variations.
3. Most of the orbital angular momentum is found
to be transferred to the rotational angular momentum of
the primary.
4. The position for impact of the main portion
of the gaseous stream on the primary differed less than
ten percent from that for a particle trajectory model.
Due to the relative complexity of the hydrodynami¬
cs! model, the large amount of computer time required per

161
star (4h hours on an IBM 360/90), the need to proceed
system by system with a hydrodynamical model, the fact that
particle trajectory models agree with the hydrodynamical
model to within a ten-percent accuracy, and, to conclusions
(1), (2), and (3) above, it appears as if a revised particle
trajectory model would be satisfactory for our study.
7.4 Evo].rtionary Models
The particle trajectory and hydrodynamical models
discussed in the previous sections allow us to specify a
number of quantities, such as position, velocity, and
angular momentum. These parameters allow us to know some¬
thing about mass flow in a close binary system. In almost
all cases, this material is allowed to leave the ejecting
star at or in the area of the inner Lagrangian point. We
will now briefly discuss some of the evolutionary considera¬
tions which enable us to know how this material reaches the
limiting Roche surface, and how much of this mass is
available for transfer per unit time.
An evolutionary track of a five solar mass star,
taken from Iben's (.1 965) investigations is shown in
Figure 7-2. As we are primarily concerned with the size
of the. star, Figure 7-2 should be examined simultaneously
with Figure 7-3. The life of the star begins with con¬
traction (A) which ends when a sufficiently high temperature
is reached in the pre-stellar core to initiate hydrogen

162
Figure
7-2

Figure 7-3
E9T

164
burning (B). During the main sequence portion of the star's
lifetime, hydrogen is gradually converted to helium. The
core of the star slowly contracts throughout this process,
and the rate of energy generation increases. Eventually,
as hydrogen becomes scarce in the core, the rate of con¬
traction increases, and the expanding envelope loses
continuity with the contracting core. This first local
maximum of the stellar radius is usually denoted as (D).
a period of general contraction ensues, until shell hydrogen
burning is initiated. As soon as the hydrogen content of
this shell drops below a certain critical value (E), rapid
core contraction is coincident with a rapid expansion of
the envelope. Eventually the star reaches the red giant
tip (F) and helium burning is initiated in the star's core.
Core contraction and envelope expansion halt. It is at
this point that the second local maximum value of the
stellar radius R is achieved.
Let us now examine the following scenario. Two
stars of different initial masses comprise a binary
system. As is well known, the more massive component
will evolve at a more rapid rate than the less massive
component, rates going approximately as . Hence, the
more massive component will pass through points A, B, C,
etc., in the course of its evolution. Furthermore, as
the more massive component evolves at a faster rate than
the less massive component, it will pass through these

165
points before the other star does. If at any point in its
evolution, should the radius R of the more massive star
exceed the size of its Roche lobe, its subsequent evolution
will deviate greatly from v.'hat otherwise would have occurred.
Now, as the star reaches its Roche lobe, mass flows
along the equipotential surfaces about the star, and leaves
the immediate neighborhood of the star at the inner
Lagrangian point. If Rq+dR^ is greater than R*+dR*, the
star continues its expansion, and more material is pushed
through the L^ point. A considerable portion of the star's
total mass is lost during this process, which occurs on
the time scale of 10 to 10 years. We shall call this
evolutionary phase the phase of rapid mass transfer. This
phase eventually is terminated when I^+dRj becomes less than
Rjf+dR|.
In principle, the primary star in a close binary
system could fill its Roche lobe in either its slow phase
of main sequence expansion, or during its more rapid
expansion during its rise to the red giant tip. The first
case will be designated as Phase I or Type A mass exchange,
while the second will be designated as either Type B or
Phase II mass exchange.
In either phase, we need to define what is meant
by the slow phase of mass trans
the end of the rapid phase.
This phase occurs at
as the energy generation
Now,

166
rate of this star becomes adjusted to its new condition,
the star resumes its nuclear evolution resulting in a
contraction of the core and an expansion of the envelope
on a nuclear time scale. The star still fills its Roche
lobe, but two important parameters have changed. First of
all, the expansion in this case is very small as there is
only a small deviation from thermal equilibrium. More
importantly, the radius of the critical Roche lobe now
increases with decreasing mass once the original primary
has lost enough mass to become the less massive component,
instead of decreasing.
Calculations based on a wide variety of initial
parameters have been carried out by several authors.
Kippenhahn and Weigert (1967) calculated parameters for a
system with initial masses of 9M and 5M. Plavec (.196 8)
performed a similar calculation for a 9M/8M pair. Giannone,
Kohl and Weigert: (1967) systematically investigated a large
variety of Case A and Case B models. Refsdal and Weigert
(1968) examined systems with a total mass of 2.5M. More
recently, Harmancec (1969), Lauterborn (1970), Harmancec
(1970a), Refsdal and Weigert (1971), Plavec, Ulrich and
Poliden (1973), and Webbinh (1976) have performed calcula¬
tions which have extended the grid of available solutions
for close binary systems.
Of particular interest for cur study are the maximum
and average values of the. rate of mass
transfer for each

167
case of mass exchange. It is difficult to define "average"
in terms of such a widely varying set of parameters. Never¬
theless, typical "average" values of mass exchange during
the rapid phase are on the order of 10” '*MQ/yr, while
maximum rates can reach 10~‘1’M6)/yr for several sets of initial
parameters. Corresponding "average" rates for the slow
~ ft
phase are on the order of 10 MQ/yr. It will be interesting
to compare these theoretically computed rates of mass ex¬
change' deduced from period variations, for both conservative
and non-conservative mass transfer.
The theories discussed thus far rely on numerical
techniques to give an overall picture of close binary
evolution. These sequences, however, take into account
O
smoothly varying parameters of the time scales of 10 years
or more in order to compress the evolutionary history into
a tractable amount of computer time. The behavior of
period variation, as deduced from a quick glance at the 0-C
diagrams presented in Appendix Two, leads us to believe
that, while the overall period variation of an object can
be approximated by evolutionary parameters, intrinsic
variations occur for a wi.de variety of objects on a time
scale of years or less. In some objects, discrete events
can be pinpointed to an accuracy of only ten days. What
sort of mechanism(s) can be responsible for these short
time scale events?
One possibility loading to events which occur on
a short time scale can be explained in terms of a dynamical

168
instability in the mass losing followed by mass loss from
this component. Two possibilities exist here. Paczynski
(1965) has suggested that deep convective envelopes may be
dynamically unstable. More recently, Bath (1969, 1972,
1975) has suggested the role played by the ionization zones
as a possible destabilizing process. It is even possible
that single stars may be unstable when in the red giant
phase. At this point in a star's evolution, the available
energy' is sufficient to eject the entire envelope of the
star to infinity. Bath argues that in a close binary
system "infinity” can essentially be approximated by the
position of the critical Roche surface, of which the star
is either in contact or very close to contact. Thus, matter
not too far below the Roche limiting surface can be lifted
to "infinity" by a relatively small amount of energy. Bath
finds that the time between successive instabilities is
given by the thermal relaxation time of the envelope. With
Bath's scheme, i.t is possible to have a succession of more
or less evenly spaced outbursts of material interspersed
with times of low mass exchange activity. For an 0.6M model,
the complete cycle of outbursts is found to be 34 days, with
each outburst lasting 1.1 hour. Mass flow rates reach
0 9 , . .
10 g/sec. Table 7-1 contains information on other initial
parameters i
nvestigated
by Bath.
Thes
c instab!1 it
ies occur on timesee
i 1 e s wh i e h a r e
much shorter'
than evolut
ionary time scales.
Ideally, thes

169
calculations should be incorporated into existing evolution¬
ary programs, adding a "third dimension" to the evolutionary
problem. In any event, these instabilities offer a plau¬
sible explanation of short time scale period variations in
close binary systems.
TABLE 7-1
(Solar Units)
Mass
Luminosity
Cycle
Length
^\nax
2M
34.30L
10
days
5xl021
1M
2.61L
28
days
2x10 22
1M
53.30L
500
days
2x1023
In fact, these ideas led more or less directly to
the Biermann-Hall model. This model attempts to explain
short period, alternate variations in Algol-like binaries.
The Biermann-Hall model accounts for alternate period in¬
creases and decreases in the fo]lowing fashion. Initially,
strong mass flow from the secondary to the primary leads to
a time of period decrease, caused whenever the storage of
angular momentum in the primary outweighs the expected
period increase caused by mass transfer. It is possible
to gain an idea of when this situation is likely to arise
from an equation due to Wilson and Stotters (1974) which
relates the ratio of the period variation due to mass
transfer alone, to the value of period change due to

17 0
non-conservation of orbital angular momentum. The orbital
angular momentum essentially disappears from the system when
it is temporarily stored as rotational angular momentum.
We can state that a period decrease occurs whenever the
period change due to non-conservation of angular momentum
obscures the period variation due to mass transfer:
dP,
NCJ
dP
MT
(7.4.1)
For the Algol-like binary U Cep, dPN-,j=-l. 58dP^T. Now, as
this stored angular momentum is returned to the orbit, the
period undergoes its overdue interval of period increase.
We are now in possession of a variety of important
guiding "principles" in regard to period variation. Not
only do we have knowledge of where the particles are going
from particle trajectory and hydrodynamical models, but we
also know their approximate density, maximum and average
and time
mass
flow rates,
scales for instabilities.

CHAPTER EIGHT
EJECTION OF MATTER FROM UNSTABLE COMPONENTS
8^1 Non-Synchronous Rotation
VJe expect to find non-synchronously rotating com¬
ponents in close binary systems for several reasons. They
are listed briefly below.
1. Secular contraction of either component after
it has been synchronized on a tida] (dynamical) time scale
The lav; of conservation of orbital angular momentum would
lead to an increase in the rotational velocity if the
evolutionary time scale were shorter than the tidal time
scale. Such a situation could arise at the end of mass
transfer, as the secondary detaches itself from its Roche
limiting surface.
2. Secular expansion of either component under
similar circumstances: This situation could result through
the normal, expansion of a star as it evolves. Such a
situation would lead to a decrease of the rotational
velocity relative to the orbital velocity.
3. Any mass transfer or loss mechanism which
could convert orbital to rotational momentum could produce
non-synchronously rotating components.
171

17 2.
4. Any redistribution of mass within the system
constitutes a final cause of non-synchronism.
We would expect either situation (1) or (2) to
result from evolutionary considerations, and to be
applicable to the initially more massive component. Situa¬
tion (3) or (4) would be applicable whenever a gaseous
stream impinged, upon the surface of either component, or
whenever some extrasystemic mass was accreted by the system.
Lists of observational data for eclipsing binary
systems, including their rotational velocities, have been
compiled by Stothers (1973) and Levato (1974). Table 8-2
lists the masses and rotational velocities for the secondary
components of stars from these two sources. It can be
seen that stars with rotational velocities both larger and
smaller than synchronous values are observed. Whether or
not this reflects the possible sources of non-synchronism
as put forth in (1) or (2) is subject to some controversy.
8.2 Matter Ejected from Unstable Copponents
Assuming circular orbits, and regarding the mass
distribution of the two stars such that they behave as
mass points, the equations of motion for a massless test
particle in the system assume the form
dx2
. 2dy
3_U
dt2
dt
3 x
(8.2.1)

c)U
ay
(8.2.2)
17 3
where the potential U has the form
(8.2.3)
Here
^2/+ ^2)
(8.2.4)
r^ = (x - U) 2 + y2
(8.2.5)
and
(8.2.6)
These equations may be. solved numerically, and the path of
the test particle traced through the system. As discussed
in a previous sccton, we v/ill assume that the neglect of
collisions between test particles will not invalidate our
results. Several important questions arise.
1.At what velocity is the particle ejected
from the mass-losing component?
2.At what point in the system is the mass
los
3.At what point does the gaseous stream
impinge upon the mass-gaining star, and
what percentage of its orbital angular
momentum may we expect to be transferred
t o r o t a t .i o n a 1 mom e n t u 1 r
the particle trajectory as given by Equations 8.2.1 and

174
8.2.2. If we assume that mass loss proceeds through the
inner Lagrangian point, Lj, two main mechanisms govern the
velocity of ejection, these mechanisms being thermal
evaporation and non-synchronous rotation. The mechanism
of thermal evaporation results from the finite speeds of
the gas particles in the vicinity of the point: as
governed by the temperature of the star, T. This parameter
will vary from system to system, depending mainly on the
spectral type of the mass-losing star. The actual value of
the temperature of the star is difficult to compute at the
point, especially for rotationaily distorted stars. For
our purposes, mainly as we are dealing with a large number
of systems, the mean velocity v of a gas particle at the
Im point will be given by
v
(8.2.7)
where
R universal gas constant
Te = effective temperature at the pole
mo1e cular weight
Kopal (1959) expressed this as
i.
v
8 RAT
(8.2.8)
= orbital velocity
0 J

175
In addition to the thermal evaporation velocity,
non-synchronous rotation will add an additional amount of
velocity to a particle leaving the immediate area of the L-j
point. If the secondary component were rotating at a
synchronous rate, a mass point would theoretically reach
the inner Lagrangian point with zero velocity. However, if
the rotational velocity of this component were to be either
greater or less than the synchronous velocity, some non¬
zero component of velocity would remain when the test mass
reached the critical point. As has been shown, this situa¬
tion exists in at least a dozen systems. Here, "critical"
point refers to the new location of the "L^" point brought
about due to non-synchronous rotation.
For our study, we have also assumed that the gas
which is leaving the Ly point has a velocity distribution
as given by Maxwell's velocity distribution law
dN ( v )
\ v -v» /
o: v'
N
To
2 , 2
x
(8.2.9)
where a - most probable velocity
8_. 3 JPrqgram ORBIT
A power series trajectory program, written by Rust
(1973) was used. The following procedure was adopted in an
effort to calculate the effect which different parameters
would have on both the trajectory of the particle, and on
the efficiency of orbital-to--rotational momentum conversion.

.176
1. Initial values of the mass ratio, ranging from
q-6.0 to q=0.2 were selected as tabular entries.
2. Tabular entries were also constructed for
synchronously and non-synchronously rotating secondary com¬
ponents. The non-synchronism parameter f was set to 0,
0.1, and -0.1 in these cases.
3. The quantity v

thermal/orbital velocities, was used as another tabular
entry. We used the values 0.02, 0.01, 0.06, 0.08, and 0.10.
4. Gas was allowed to escape from the inner
Lagrangian for different angles of ejection theta (theta =
-4 5, -30, -.15, 0, 15, 30, and 4 5 degrees) .
5. The position of the Inner Lagrangian point for
each of these values was calculated from the following
equation, due to Kruszewski (1964).
(1+q) qf >:T
(1--XL-, )
1 2
—3—+ (i+q) (l+f) xL
X'L1 1
+
•ra 2
(1-XT
Li
(8.3.1)
q = mass ratio
f - non-synchronism factor - v /v
rot syn
xr - position of L. point
n. - 1

6„ Particles were then allowed to escape from the
point at velocities of a, 2a, 3a, and 4a. The percentage
of particles in each velocity regime was calculated to give
a more precise determination of the effect of the gas
stream.
7. The trajectory of each velocity regime was
computed by numerical integration of Equations 8.2.1 and
8.2.2 with an eighth-order Runga-Kutta routine. The orbital
angular momentum for each time step of the trajectory was
also calculated.
8. The particle's position was tested to see whether
it liad impacted on either of the components, left the
immediate vicinity of the system, or whether it was still
traveling near the stars.
9. A cartesian plot of the particle's position
was constructed for visua1 reference.
10. In the event that the particle intersected
the primary star, the angular momentum of the particle at
impact was broken into a component tangential to the
accreting star's surface, and a component perpendicular to
it. These, quantities were saved, and summed over the
a p p r o p r i a t e
tabular
e n t r i e s . T a
able 8-1 lists the
following
Q cl cl 111 -L V. IOS t
in the
first column
the quantity q, it
i the
second column alpha, in the third theta, in the fourth
column the average orbital angular momentum per particle,
in the fifth its rotational component if it impacted on the

178
primary component, and in the final column, the efficiency
of conversion of orbital to rotational momentum.
The degree of accuracy of such a computation depends
upon how satisfactorily a particle trajectory mode], dupli¬
cates the actual situation. We would expect it to give
excellent results as long as the mean free path of the
particle was on the order of magnitude of the separation
of centers of the two stars. Such a situation is clearly
not the case. Prendergast (1960) pointed out that for
10 1
a density distribution of 10' particles per cm , as had
been indicated by several investigators, and a cross section
-IS
of 10 â–  cm, that the mean free path is only on the order of
10 kilometers. He also pointed out that particle trajectory
models cannot take into account the crossing of streamlines.
Fortunately, we can be comforted by results from Chapter
Seven, which indicate that neglect of collisions will not
seriously hamper our efforts.
8.4 Efficiency Tables
The results obtained from Program ORBIT are given
in Table 8-1. Column one contains a value of the "mass
ratio" given by Equation 8.4.1. Columns two and three
contain values of the most
FM
(8
probable velocity, alpha, and the angle of ejection with
respect to the ]ine of centers, theta. Relative values

179
of orbital and rotational momenta are furnished in the next
two columns, while the ultimate column contains their ratio,
or the efficiency of the conversion from orbital to
rotational momenta. These tables assume that rs=0.20.
A number of double line spectroscopic binaries
(DLSB) which display noticeable non-synchronism is presented
in Table 3-2. This table indicates that there is ample
reason to include the non-synchronism factor f when matter is
ejected from the secondary.

180
TABLE 8-1
TABLES
OF ORBITAL TO ROTA
TO ORAL MOMENT/'
{ |A S3
fíat i^
i. i i; A
VI .1 7 4
ü/\3 1 T A L J
*© G 1 A 1 1 JN9 _ 3
11
■“ r I C 1 1 N1 G
C.
d 3 3 3
0 . V. X 0
C . U
> 3 V (.* / 9 4
. 3 3 1 3 1 C .v 3
7 4 X o 334 1
c.
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0 « (-■ •*
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3 ) 0 7 9 0
c.
j» 3 3
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0 .
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lb.0 0
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Ü
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3, , J 3 2
4 g . o o
© o 7 4 V ■_ 7 1 1
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4 3 ó 3 X3 7 o

181
TABLE 8-1"-Continued
k ASS BATH-
7, L
< i i A,
Ihtl >\
O) L5 1
T A L J
t< *v;
r
AT
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7-
r —
1 C 1 e
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C . /I U)
0 «
0,-0
c . 0
c ^>0 J
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t
o
4 V
o
o
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1 ¡j
3 5 A 4
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0 . 7 1 •' * 3
0 e
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0
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C
C 7
C > 0 4 C
c-3
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o
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9 7
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5 c * o â– :
: o ’ q t
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9 [
r • 9
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p o iv. 1 t or 1: o o - -1 - 3 [-na ViL
• W i J
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9 ¿9 5 O
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9 5: 9 ’ 9
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% 7 « 7\
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9 0 l > 9
j 6 0 * 9
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c r- ■') • 9
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r ' 9 .) • 9
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■ : q- .7) • 9
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c q 2 j * o
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9 70 * 0
C 9¿9*9
OcO '0
0 6 29 3 0
V Id IV
0! LV.J SPV
:ot

18 3
T/\ISLE' E-1 -•-Continued
f' ASS EAT IÜ ALPHA ILUTA CPI:11AL J ROTATIONAL J LFÍ ICILNCY
c.
3 3 i> 6
C .020
C .
0
. 2 3 0 (. ‘j5 0 7
. G A 12 C 5 4 7
< 17 6 6 4 4 7 4
c.
5 5 3 6
0 « C A C
C .
0
«2232363 4
. 1 2 9 1 0 1 C 7
.53372970
C V
555 6
0 . C 6 G
C .
r\
. 2 356!267
. 1 5 7894 0.6
.67C )4 40 2
c.
5 5 Of.
0 . G A C
0 .
fl
.23701533
.10200372
. 7 £ 5 3 2 9 2 0
r
â–  / t
01 > 5 (>
0 . ! 0 C
c <
0
. 2Aj3 oCJ0
, 2 0 3 774 2 6
, 8 5m . 6 2 04 3
c <
8 Í.. 0 f;.
C iC?C
1 Í .
4 . r
0 0
â–  O' 2 6 7 7 9 7 3
• C 4 6 >1> 9 il (> A'
* * 1 2 3 c.. 7 4
A'
(
i:. it 3 'j
C . 0 A 0
1 G: ,
0 0
C ’ó b l' f_>
.110 o924 3
. 4 0. 5 2 2 Oh 3
C .
3 35 .
C . 0 6 0
15.
0 0
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.¿04 3 4 6 0 2
c,
5 5>6
C . 0 3 0
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0 0
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. 692 626 48
c. *
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C , 1 jC
1 b »
0 0
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. 1 9 1 9 1 .5 4 8
.790 2 3256
c,
5 0 3 ;>
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3 0 .
00
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.02159 907
.0554 2 3 5. 5
c .
5 5 :J> 6
c . g a â– ;
3 C >â– 
r- r~
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.00633C14
. 2 8 2 4 1 2 2 3
c,
0 i > -j ( >
0 . GO C
3 0 .
r n
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.1 1 5 G 9 3 0 3
. 5. 0 3 2 P 3 2 6
c.
iz 5 â–  6
C . C ÍS c
3 C .
0 0
. 2 2 2 4 C 5 8 3
. 5 390 22 0 7
.58 81.90 1 G
c«
53 . .
C . 3 0 0
3 C .
0 0
. 2 3 c r 5 7 7 3
.14 74 AD ) C
. 6 2 2 4 0 4 9 6
c.
5•• 0
C . C 2 0
4 5 .
0 0
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. 0 A 6 1 3 4 C 9
. 2G6 2133 7
c.
5g:.g
r\ •' /, r
« O ‘ r
A 5 .
0 0
< 2 2 C 5 3 C 3 6
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. 1 -4 3 7 C 0 9
r
V *
000 6
c * cc
A 5 .
c 0
. 2:â–  a a o 3 o
. C 0 0 767C 6
. 402 G69 7 C
r
f; < £p í ^
C . G 0 C
A 5.
GO
< 2 2 1 8-342 1
< 0 90 995 21
.41018567
0,
3 55 6
C . ! C 0
A 5 .
00
*1 .5 7: ‘,1 } L 0 «J 1 ■
. 3 19 3 34 44
. 5 3 C C 6 8 1 8
f=0

104
i; ass
c.
TABLE
8-1-~Contino
ied
PA 1 10
ALPHA
t l-ETA
op.n tai. J R
DTATIUMAL J
EPF I C I t.N
5 0 0 0
p / 2 0
0 * J
, ? ] r 2 6 4 -1 2
.0801 03 8 6
.3812383
3 C00
o. * r*
U * v.' H «7
0 . 0
.21247727
. 3 3 5796 83
.6393122
5 C 0 0
0 . ?6 0
O. 4N
0 f
. 2 3 4 72 a 3
„ 1 6 1 2 6 4 3 6
. 7 5 1 0 3 5 0
5 0 0 0
C c ^ L 0
0 «• 0
. 2 3 s 9 2 0 c 3
.3 014 7 17 '0
» a 5 6 5 8 3 6
C . 5 O 0 0
0,SC '3
0.
0.
SCO 0
5 C 0 0
C. 5 09 0
C . 50? 0
c. n c o o
c. s n o o
o.seeo
C. SC 0
C.SCOT
C. S C 0 C
C . 5 C 9 0
C . S 0 0
C. SCCO
C.5000
0 . 1 0 o
C < 0 ? 0
n , 04 0
0 < 0 ñ 0
0.05 0
C . 1 0 0
G . 00 C
c. o y r
0 * 3 ! j 0
c . oâ–  o
0.0 A 0
p p , .
g . : y c
C . 3 0 0
0.0
5.00
r: r¡ n
! 5 . 0 0
1 5.0 0
0 0.0 0
c . o o
H 5 . o 0
A 5.0 C
4 S . C 0
4 5,0 0
. 0 1 o 0 V 7 4
. .? C a 1 C. C A 4
, o cosos re,
. 2 3 2 690 4 5
, 2 1 SC CO 5
. 2 ! r- 0 7 5 3 2
. 7 - 55 0 CO
. 2 C 5 5 5 5 4 7
. 2 5 2 1 6 64 5
. 2 3 3 7 5 32 2
. 2 3 60 4Co 4
.PC 566 7 01
, 2 0 C r, 3 0 7 7
.20030324
. 2 0 3 63 4 4 '<
or , ooop -
1 os53752
06133397
3 19 1
. <0 ! Co 05 9 2
. 25 4e 475 9
* 5 0 7 7 34 5 7
4 2 1 4
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2 2 4 1
4 4 7 (
, 7 7 2 3
4 4 6 1
2 7 6 4
. 0 u ‘ •'
< 6 7 9
7 9 3 C
. 17 1 4
1 0 7 0
0 5 4 7
. 4 6 9 5
!i 3 b 0
r: ' > 1
r -.1 0 p
J 3 1 <
fc . J -2
': 7 0 V.
. tore
369 7
i a 5 2
. 7 0 r t
22 0 6
2 b 9 C
.0-31 1
4 2 C 2
7C 3 9
. 3 4 6 7
7 7 2 0
0 9 7 30? ; G
3 0 3 3 42 2.2
1 2 562 3 8C
, 4 3 07 73 9 7
. 50 20 03 7 5
. C 3 3 2 " 9 3 7
f=0

18 5
TABLE 8 -1—Con t i mi ed
ASS PAT ID
31.
PH A
i i-n a
p
PI'HAL J
RUT Al 1 3!\ At. J
EFT 1C1 PNC
0. 3?33
C.
03 0
0. 0
c
1 ! 5 8 5 51 3
. C 9 9 9 5 3 3 C
.649Oh953
0. 33 33
0 .
C 4 0
0.0
c
1 3 P 7 1 7 3 1
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TABLE
8-
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M A
TABLE 8-1 -Cont i n ue d
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C. 2 C G 0
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189
TABLE 8 -1 - - Con 1 i i mecí
5S
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190
T/ill LE 8- 1 — Cor:tinued
, 66 RATIO
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T HE V 4
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0.6250
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193
T/i E LE 8 -■ 3. ■— C o n t i n u a d
MASO RATIO
A L
Put,
T HE
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192
TABLE 8--].---Continued
BASS PAT it! ALPHA
0,50 0."! r,( B!0
0.50 0
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f- 0.1

193
TABLE
f
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AGS PATIO
AL3i 1 A
T 1 f' T A
c.
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TABLE 8-l-~Conl. i nued
A 33
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TABLE 8-1—Continued
M A f,S R A 1 I CJ ALPHA 1 |-P 1 A
c.. 7i a
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.198

199
M A SO
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TABLE 8-1—Continued
c
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TABLE 8-1-
M A 5 S R A T í U
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UR ü J TO
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. 2 3 33o3 2 4
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c 0 0 0 2 3 0! J
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20
TABLE 8- 1—Co; 11 ir,ucd
M \ S
P AH 0 ALPHA
2 0 0 0 C . C 2 C
2 0 0 0
p f â– >
;:> o 4 7j
c. o â– ; (.
o . Oí) c
C t
C « ] L 0
r , r' â– > (
C .0'iC
C , 0 0 c
0 , 0 3 c
f, .IOC
C < 0 2 c;
r.. c ti o
i c c
r' -> C
o 'i :•
06:
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1 0 0
'III 1 A
C . 0
C . 0
0 . ^
0 , 0
0 . j
i :•. o o
i 2 . o "
1 f. . C 0
1 3 , " 0
1 0,0 0
jo.:)
o . o :
:: o.o?
3 C . 0 0
*0 V * V l7
(t o, : c
,-j r., r-
3 . 0 0
4 2 . C o
4 0 . 'j 0
CtíOll Al J
. 1 M 3 0 04 A
. I. Cf>2 1 6 0 0
« 0 9 3 ; J 9 2 3
. ) 0 L !_■ Í 3 7 J
.114 4 7 4 fj 4
. 4 7 - A 6 1 4 J
, 30392200
. 0 0 , 2 " 7 6 2
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. • 1 0 ! 0 1
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. C 9 0 1 ! 6 7
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< 2 9 . '/• ’0 •:
. 2 7 9 3 O 0 â– _) 3
„ 0 0! ! 3 r 4 1 2
f=-0.1
KUTATIJNAL J
. 0 C o 4 2 3 7 f'
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.071779aC
. 2 9.3 1 u 4 1 9
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«•
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. n 7 â–  9 7 2 3 3
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. C O 0 J ' ■/, j ,¡
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. 7 3 f • 6 3 4 1 o
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. 6 4 7 2 24 9 3
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0=3
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9 26 20ZZ93
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96¿9 0 109
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299890
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9 0 6 9 +’ 9 9 9 3
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Oq¿ 6 06 2+7 31
+;2Z >9 2 0 í 3
9+7/¿roye »
0 3 0 £ ~
0 LO 3 0
9 9 99 3 0
9 0 A 5 t 8 +,â–  9 3
6 o v 9 0 9+7 23
2 0+79/209 3
0 3 0 9“
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999890
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9 i 0 K 7 £ Oí3
099 ¿20 0 9 3
03 9 r -
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9 9 0 10 t P 0 *
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0 -’0
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0/999022*
9 5 9 6 9 0 6 0 *
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£9019132 3
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205
TAP.LK 8 ■ • 1-— Con tin up d
¿S RATIO
ALPHA
T Hr. 7 A
ORHi 7 AL J
ROT A1 IONAL J
EFF 1 C ILNCY
0.7143
0.0 20
0 ,, 0
301 4 3 330
. 04 9 090¿15
. 16 2 C44 4 7
0.7143
0.0 4 0
0.0
*30 41 v * 03 4
.02297575
.07334063
0 c 7 1 4 3
0, 0 b 0
0.0
.30707350
. 1 0 1 5 3 2 1 3
* 32930439
0.714 3
0.030
0.0
. 3 0 9 6930 7
o 1 3 2 294 US
.42717232
0.7143
0.100
0 *0
c 3 1. 2 19 4 a 2
. 1 7 3 29 04 1
. 3 5 5 0 9 7 3 7
0.714 3
0 . 0 2 0
- 13.0
c 3 0 3 4- 2 7 9 6
.06231043
. 2 0 0 0 1 4 1 4
0*7143
0.04 0
13. o*
. 30027090
.00965380
. 22807 73 1
0.7143
0. 060
-16.0
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o 14 3 1 r. 4 1 2
.40490 41 3
0.714 3
0 . 0 b 0
- 1 3.0
. 5 0 6 3 * 0 3 -5
1 7 2 294 1 6
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0 . 7 3
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0.
1 00
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«• 2 0 o 3 1 4 C 3
. 67 4 4 4 o o4
0.7 3
4 3
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020
-30
c 0
. 30:â–  ! 891
c 0 5601615
. 1 8 5 53 1 2 6
0.71
4 3
0 .
04 0
-30
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.0 77 72 4 42
.25249092
0. 7 i
4 3
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0 ¿'0
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. 1 5 0 3 7 4 b .>
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0.71
4 3
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0 60
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c30093040
. 1 03 142 93
.59669203
0. 71
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1 00
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f21574022
c 7 0 8 1 8 6 2 6
0. 71
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. 0 7 4 6 90 7 4
. 24 3 5 01 ;>0
0.71
/ ■-».
V *
0 .
0 4 0
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* 0 49 34661
* 3 3 95 02 27
0. 71
4 3
0.
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— 3
V
.30897327
. 1 3 0 9 3 1 5 2
. 4 2 5 7 6-/27
0.71
4 3
0 «■
080
~ 4 3
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. 1 5637 4 4 7
* 3 3 7 6 57 3 6
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4 3
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3 0 0
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< 2 321 1 76 3
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f -- 0

206
TABLE 8-1
•-Cont.inuc'd
A A 5
PA1 ] 6
ALPHA
7 Mí. I A
UkBi TAL J
K OI ATI 3NIAL J
Li F 1 C 1 LHC
Or
62 5 0
0 «
0 2 0
0 . 0
, 2 80 7 8 6 1
, 0 3 0 2 8 9 2 3
.1 1 7 0 0 1 6 6
o
*»
6 25 0
0 «
0 40
0 r 0
,26162587
r 1 01 • \ ■: > 4 3 6
. 4 0 B 6 0 6 0 0
o.
6 25 0
O c
06 0
0.0
« 2(..4 3 50 2 0
r 1 6 4 57 03 4
• 6 2 2 84 (.,<_> 8
Or
62 50
Or
080
U r. 0
< 2 6 6 7 6 8 4 V
< 1 9 2 0 3 0 5 73
. 7 1 9 <28026
0 *
6 2’jO
0 .
1 0 0
0 . 0
« 2 6 5 8.6 7 1 4
« 1 6 3 7 3 i_ i 6
r 7 2 85ó6 39
Or
0 2 b 0
0 •
C 2 0
~ ) 6 . 0
c 2 ; 0 9 tr 1 7 4
.03371 tic 3
. 1 2 9210 05
0 c
62 5 0
Ü r
04 0
- 15.0
> 2 t / o .3 7 1
. 1 0 42 8 1 63
. 3 9 573c-96
0 .
6 2 b 0
0 r
0 6 0
15. 0
« 2 (.5 04 81- 1
< 1 7 ! 4 28 6 8
. 6 3J 6 6 8 7 3
Or
6 2 5 0
Or
C 60
-1 5 . 0
. 2 (: ;' o ' â–  1 1 0
. 1 93 6 8 3 31
.728712(4
0 <
ó 2 5 0
0 «
1 00
â–  ] 6 t 0
. 2 c. 5 7«- ob 3
. 2 2:02050 1
. 82 8'. 99 0 2
0.
6 2 6 0
0 r
0 2 0
— 3 6 <• 0
• 2 6 2 6 2/ 6 ü 5
. 0 2 6 73054 5
. 10 2 157 o 2
0 <•
62 5 0
Or
0 4 J
•- 5 0 . 0
r 2l4967 5 8
. 1 1 2 3307 7
« 4 2 8 8 7 3 6 7
0 .
6 25 0
0 r
UoO
- 3 0 .0
.¿654567 v
. 1 7 4 3o6-> 4
r 65 6 574 , 3.
0.
6 25 0
0 .
0 6 ^
- 2 0,0
. 2 (â– â– < 3 2 6 4 :i
. 2' 0 0 7 :4 4 2 7
• 7 6 9 5 6 5 6 2:
0 .
6 2 5 0
0 r
i 0 0
~ 3 0 c 0
r A'f.C-i 6 ( ' 1.
.223434 36
• f• 6 2 0 6 9 . .. 6
0.
6 2 5 0
0 ,
0 2 0
5 rO
. 2 t. J‘. ÍJ 1 C. ‘ i Ci
» 0 1 9 3-.y c.5 0
.0/3 4 c> 6 3:
0 .
62 5 0
0 .
04 0
- 4 5 <. 0
. 2. t 2 ; 4 1 3 5
r 0 3. J 5 6 2
r 3 c i 1 3 31' 6 9
0 „
6 25 0
Ü «
0 0 0
- 4 6 . 0
. 6 6 ' 7’ 2 G .c
«• 1 60 613 96 7
. 60334 4 7 7
0,
62 5 0
Ü ,
G i.-» o
“4 6 r 0
2 6 4 i 7 6:1 9
. ?. 1 7? 7 -i 0 c. 0
* 8 032016 5
0 ,
6 2 5 0
U c
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.< 81 0 5 5 1 o 2:
. 3 ..s2 2535 5
.85139272
i" • 0

TABLE 8-l---Continaod
5 S k A T I C)
AL
PI) A
1 ML
1 A
ÃœKB! 1 AL J
ROT
AT I ONAL J
L 5 r ICILNC
0.555 6
0 .
0 20
0
. 0
.23065507
. 0
4 3 20 5 4 7
. 17 8 5 44 74
Go 555 6
0.
GAO
0
0
*233 2 5.654
. 1
29)6107
«. 5.5 3 7 ft' 9 7 C)
0 « 5 5 5 G
0.
06 0
0
r 0
* 2 3- ü 3 2 6 /
. 1
5 7 394 5 6
. 6 7 0 1 4 4 0 2
0.5556
0 .
OoO
0
f 0
, 2 3 7 ->• 1 6 5 5
, 1
8 2 0 8 3 7 2
« 7 o 5 3 2 90
0* 5 56 6
Oc
1 00
0
. 0
* 2 4 0 ] 6 0 5 0
v l-
0 6 7 7 '1 2 6
* 8 5 6 8 2 0 s ft.
C) c 5 5 5 6
0 c
0 20
~ ) :
. 0
e 2 j i t i 6» 5 0 6'
. 0
... 6 22 3 5. 5
. 297601 4 7
0-655 6
e
U 4 0
-1 5
« 0
ft 2 2 6 2 ft 6
« 3
2 3 23 civ3
. 56700 31 o
0 « 555 6
0 .
0 60
- 1 o
c 0
* 2 3 o 5) 8 6- 6 6
* 1
t; 2 (. 4 2 1 7
.7719760 2
0 < 5 6 6 6
G «
0 60
- J 6
« u
* 2 3 4 14 4 65
* i.
0 2 5 3 4 5 8
. 8 6 4 2’ 56 4 3
0,5566
0 e
1 0 0
--1 5
< 0
« 23723 c2 3
• ».
1 9 2 6 8 5 2
.924221 4 7/
0 <5 5 6 6
0 .
0 20
- 3 0
f 0
.234 00 •: .5 6
t 0
4 7 6 6 3 2 4
« 2 0 3 c 6 4 4 ft.
0 v 56 5 (j
G c
0 4 0
- 3 0
f 0
. 2 3 c 1 C .' :>
e 3
4 0 0 f. 7d
. 592 2 0623
Of 5556
G *
060
-3 0
< 0
* 2 3ft. 6 2.0 3.2
c 1
8807077
« 7 9 3 v 1 3 4
0,555o
G t
0 t 5 0
- 56
. 0
* 2 3 5 i. 7 1 ó (
'-i
t ¿5
0 â– . 1 7 5 0
. i874?6005
0» 5 5 5 6
0*
3 GU
— 3 0
f 0
. ft ’ 5 j 6. 7 6 2
r r.
1 9 6 3 2 6. 3
« 93360 821
0 « 5 5 '5 6
G «
0 20
- 4 5
f 0
. ¿33 6 r'o:-y
1. u
6 7 9 : 6 3 8
<. 24 6 063 5 1
0 * 5 5 5 6
0 «
ÍJ w/
— 4 L i
< 0
«26 5 v 2 5- 1 ;
. 1
2.9 J 6 2 7
* 3 2 4 9 6 0 ó c
0 * 5 5 5 6
0 <
06 0
— 4 6
0 0
.. 23 7 7 255 5.1
ft 1
7 7 37 7 61
. 7 4 o 9 9 c* o 3
0.5556
c <•
0 60
„ A ft
. 0
. 2355 2 392
ft. 3.
977316 2
« C 3 9 y 4 0 <5 9
0,5556
U f
1 0 0
„„ /. c.
. 0
« 2 3.2 0 5 1 0 :i
.2
1 2 9 i 6 c¡
. 9 1 7 5 3 9 1 6
f=0

208
T A B L E 8 - 1 - ~ C o n t i n u e d
BASS BATIO
ALPHA
THAT A
CKi.i
1 TAI. J
1 ’ l! 7
AT 1 UB'AL J
L.f 1 J C i !. B'CY
0 . 3 0 D 0
0 „ 0 2 0
0.0
. 2 I
026432
. 0
8 0 10! 8 6
. 3 81 3 3 8 59
C. 5000
0.04 0
0.0
.2 1
24/72T
. 1
3 .7 9 ó 8 3
6.3 9 1 1 2 3 9
0.5000
0.0 o 0
0, 0
0 c‘ }
4 7 2 2 A 3
« ]
o 1 2 04 3 6
«.73! 03 5 0 1
0.5000
0.0 ; 0
Ü . 0
.â–  2 )
00 7 0 0 J
. 1
y i . /;7 3
o 8 3 o 5 8 3 o 7
C. 500 0
0 , I c c.
0 « 0
.. 2 )
9 0 7 /â– . 7
. 1
so 9 3/aa
. 9 1 0 8 0 9 9 2
0.5000
o.o o
-15.0
< 2 1
) TOO AT
0
J 7i'9 JO
. 3 8 3525A 8
Cc 5000
0.0‘í 0
-15.0
„ 2 )
A 1 3 HI -3
( ¡
5 5 9 L. 9 8 0
.65 3 35 6 2 3
C c 5 0 0 0
0 . o.'.., 0
- } 5 « 0
. 2 1
5 0 A 7 i 0
. .1
5 Co 1 A 9 ?
.8375 6 0 09
0. 5 C 0 0
o. or, o
-15.0
„ 2 1
81575u
. 1
9.. ! i.O ! 5>
. 9 0 7 5 0 Í? 9 0
c.aooo
C „ ! 0 C
- 1 5«0
. 2 !
6 l 6 9 I
t c_
0 7 7 7 0 A A
. 9 o 1 1 6 6 3 3
0.
6 0 0 0
0 .
0 7 0
V "j ,’ A
« 7 1 3
A 5 7 4 5
r
0 8 7 0 0 3 A 7
. A 0 7 5 8 2 2 6
c.
5 0 G U
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TABLE 8-1--Con ti míe cl
MASS ¡MAT ! 0
At PI i A
ini ra
mtBlUL J
KOI AO 10WAL 5
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0.3333
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TABU
7T7
TABLE
5/
.1--Con ti
nued
f'A28 NA 1 J 0
Al. PH A
T i- í 1 A
OK
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nor AT i ÃœNAL.
0.2000
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C.2000
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1633184
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212
TABLE 8~l~--Cont inuod
f/A5iS fVATIC
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214
TABLE 8-1 â– --Continucd
36 RAT i C.
AL fJt lA
T Hu 3 A
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kCJTAT I ON A L J
Lf K J. C 1 L NC V
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7TT7
TABLE 8-1- - Continued
A5 0> H A T i ú
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.22u 3 <’■ o. Or b
.3 0 16 3 0 1
, 9 9 4 6 3 0 3 ft
0 r b b b 6
Or UÚ 0
vb 0
. 0
.¿13bo7bo
. . ’ 3 o1 6 8 2 o 0
» 5 Or 3 5 4 Or 4 Or
0.5b5C
Or . 1 0' 0
“ o U
. 0
r . i7 i râ–  l 7
e- C- v..- i i < o L1 i
, 1 b 7 b-(bo 3
. 9 4 0681 71
0. bbbb
u. 0/ 2 0
~ Ú \j
r 0
. 2 3 4 1 Oob7
« 2 .? 2 9 4 4 6 3
* 9 b O' 3 3? 3 1 3
U . b b 5 6
0. r 0 4 0
~ b
. 0
O - • b (.■ 3^
c 2 2 2 0 7 B b OS
. 9767 9202
0 « o* b o ó1
0- 1 Go 0
~ * j b
, 0-'
c b l v o b u v U
. 2 1 7 0.19 b - 7
.9 92 4 63 7 9
0 « bbb6
01 . G Or G
— /,
.0
« /b } 4- J U f.' - >
«2 0 7 0.4 71.6
» 9 7 03 5 0 7 ü 01
0.55b6
0.10 0
— /, >j
. 0
C ¿ U 9 4 / ¿3 o 3-
. 1 9 0 0 0 b Or b
. So 4 7 8 0 2 0) b
f=0.1

216
T ABLE 8-1--Continncd
v,\ sr.
HA 1 I 0
Al. (At A
1
1LI A
0 5'
l ? ; 1 A!. J
IIOIATIUNM 5
rrr r c. i i:MC
c.
5 0 0 0
c. o2 c
0« 0
«
] 5 4 5 5 2 4
. 7 0 4 0 4 0 0
< 93595 3 2 O
c.
3 0 G 0
0.040
(â– â–  . 0
« Í.
OP42592
. 2 0 7 2.3 2 7 1
. 0 0 3 0 3 4 1 0
c.
5 0 0 0
0. Co C
0. 0
«- /
0 5 4 2 5 ó 7
. 2 0 4 5 t 1 5 7
. 9 9 3 3 4 0 9 7
0.
50 0 0
0.0 \¡ 0
G . 0
«* 2-
G 2 7 • 0 5 (>
. 2 00 190 91
.93737106
c.
5 0 0 0
0 .10 0
0 . 0
*• 1
* i
0 0 .‘5 0 2A
,194910 73
.07065319
f
L L
c. f) ('â–  p
c . c :: o
-
15.0
. 2
1 3 ? 4 5 € V
. 2 0 04 5 3 0 0
.9 32 21 5 21
c.
5 0 0 0
0.0 4 0
-
) 5,0
’ >
0 7 9 7 7 ¡
, 2 0 ó 7 5939.
. 9 3 3 3 2 6 3 3
c.
C 0 0
C. CO C
-
1 5 . 0
« r
0.235575
, 7 0 93 5-. 3 M
.9 9 03 9 9 3 6
0.
50 0 0
0.0 A, 0
-
¡5.0
. i
9 7
, 1 0 14 3 0 31
♦ 5 ó i > y 2 1 3 6
c.
5 0 0 0
0.1 0 0
--
1 5 . 0
. i
9 3; >34 5, 7
. i 3 <. > O 1 ! 0 6
, 96 <4 7 79 6
c.
50-5
0,0 2 0
-
3 0» C
c
1 :â– ,? C 0 6 7
. 7 O 9 6 5 . 9 9 3 Q 4 2 0
c.
5 00i-
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-
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c.
t, p [1 (j
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-
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C C ! 5 / 4 9
. 1 07 2 30 ! 7
. 9 -5 3 3 0 7 6 6
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0.0{; 0
-
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. !
4 7 (. 5 >; •. 5
, 1 A 4 .? 2 4 ó 1
. 9 4 c; 3 1 7 3 0
c.
5 0 0 0
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-
3 0.0
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. 9 J9 1 1 7.7 0
c.
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4 5 . 0
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5 3 224 7
, 306741.2 0
, 9 67 90 / o /
c.
5 0 0 0
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-
4 5 . 0
« c!
: b 0 *'l 1 ] ' i
. 3 0 ó 57 1 7
. 9 6 35937 .i
c«
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-
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553327u
, 1 9 4 ‘3 9 3 5) 7
. 9 73 61 3 63
c.
5 0 0 0
0«3 A 0
-
4 5.0
.)
5 2 5 0 3 .3
.13 1 J 0 3 3 0
. 94 9 64C b 9
C»
o
o
0 . 1 0 0
-
â– l 5.0
i
2 4 j! 0 3 7
.1 r033195
. ° 2 3 •') 1 1 5 4
f = 0 . 1

217
TABLE 8-1
Con tinned
MAbS CM I 6
Al f‘h-\
1 Hi; T A
ÜKiJllÁl. J
COT M I ON AL J
U‘F IC I ONLY
0,3 3 3 3
0 < 03 0
0,0
e. 1 3 o 6 i ft / 0
. 1 54 42 09 7
. 990 6 4 3 J3
0.3333
0. GaO
G . C
. ! 31 o a 1 0' 0
. 1 A 4 9 -J 2 0 1
. 95 61 4 7 4 1
0.3333
0 . OgO
G . 0
. 1 A 3 333 03
. i 3 3 9 3 0 0 2
, 9)4694 2 4
C t 3 3 3 3
0 c 06 0
0 .0
. 1 A 3131 31
.12915330
.683703u3
0 « 3 3 3 3
0. 1 00
0 . G
4t 1 A A .> Vi g 0
, 12 13 2 A A 4
, 84 02 05 7 0
0 , 3 3 3 3
0 4 0 2 0
- ! 3 . 0
. 1 3 0 31, 3 1 7
.154)1174
*992136 o6
C * 3 3 3
0.0 0 0
~ 1 3 . 0
.130 (>V03 4
. 1 4 2 7’ 6 3 < '*
.9611 7 9 0> 9
C . 3 3 3 3
' 0 . 3 o 0
— 13. C
. 1 A 3 3 31, : <3
.13191023
c 9 0 6 0 6 04 4
0 . 3 3 3 3
0 . 0g0
-13. C.
4. i A 1 723 1 3
. 1 1 7637 2 4
. 6 3 0 2 4 933
0.333 3
0 . 1 0 0
- 1 3 „ 0
1 3 / 3 o 31 /
. 1 12 9200 7
. ft 1 9 04 31 9
0.333 3
0.0.. 0
-3C , C
. 1 33 3 1 7 7 9
. 3 3 337 94 1
, 9 3 9431 A 7
0.3333
0< 04 0
”30 . C
. 1 A y 2 v ) 7 4
.14137795
. 94 6 99 J 3 1
0.3 3 33
0 ( 060
-30.0
. 14 j 6 ? C V 4
, 1 20 71 79 2
,83230934
0 4 3.3 3 3
0 c O' C
“ .3 0 4 0
4 1 A13 0 3 '3 1 2
.1132 01V3
. ft 3 4 A 9 5 3 3
C . 3 3 3 3
G 4. I 00
- 3 0 . 0
. 132 GS3C1
4 10 2 1 3 2 63
.70973753
0.3 3 3 3
G < 0 20
— A 3 . C
. 1 3 3 v C (9. ?
.15297317
. 9 3 1 1 ft 3 1 1
0. 3.': 3 3
0.0 A O'
— L- .0
1 9 3 V 3 J 9
4 1 3 8 5" ! -1 0
. 9 2 6 4 2 4 3 0
0.33ó3
0 . 0 ' j 0
- A 3 . C
. 1 A 2 9 7 2. 3 J
. 1 2 g(37 39 0
.30740986
0,3 3.3 3
0.0 3 0
-4ij . 0
4> 1 3 u 3 9 3 J 3
1 5 2 9 j Í 3 A
. 02 7 6 05 9 B
f-0.1

218
TABLE 8 ~ 1 ~ - C o n t i. n u e cl
ASS RATIO
ALPHA
r he
Í A
OR
E> 1 i A ~ J
< or
ATI 0\’ A .. J
5 r r 1 2 1 E N O Y
o
ro
o
o
0.020
0.
0
c )
3 3 6 2 4 8 6
6 l
315 3 01 0
.93474707
0.2500
0,040
0.
0
. J
2 £> 5 7 6 7 )
. 1
1 9 591 4 4
.93011745
0» 2 50 0
0.050
1-0
0
.J
2 5 0 3 1 Í) 2
. 1
07/3143
.85)49573
0.2500
0.080
0 .
0
. 1
2 8 209;. 9
. 1
0344767
t84647350
0.2 5 0 0
C . 1 0 0
0 .
0
. 1
2043292
* 0
98)9979
. 7 6 5 Ci 6 9 6 0
0.2500
0. 02 0
5 5 <
00
< 1
3 4 1 3 5 5 6
. 1
3 3 2 5 4 1 4
, 99 3 3 4 7 2 1
0*2500
0 e 0 A 0
1 5.
0 0
c 1
2 9 8085 0
e 1
2535737
. 9 5 5 2 525 2
0.2500
0.0 j 0
1 5 .
0 0
c 1
2 75 7 3 96
. 1
1720740
.9!8740/2
0 «25 0 0
C . 0 8 0
1 5 .
oc
. 1
2 5 81 3 j 3
< 1
1367990
« 90 0 5 3 5 93
0.2500
0.100
1 5 <
00
<)
2 0 0 77 59
. 1
0 6 58 50 6
. b 4 5 3 8 9 9 4
0.2500
0.080
3 0 <
0 0
c 1
3 4 3 3 2 1. 6
. 1
3449 3 0 5
. 9 9 7 1 3 7 5 3
0.2 5 0 0
0.0 4 0
2> 0 .
0 0
. 1
3 1 54 7 7 0
c 1
2 9 0 1 5 0 5
< 9 3 07 4 7 2 4
O
o
ro
CT
o
0.0 5 0
3 0.
CO
. J
2 9 8 7 C 9 9
.)
2525945
. 95 4 491 4 7
o
o
â– J)
1
o
Oc- 08 0
3 0.
0 0
. 1
2 9 5 6!. 1 3
. 1
2 3 0 0 0 :> 0
. 349 3 0 ;• 4 3
0»2 5 0 0
0 v 1 0 0
30 ,
00
. 1
2! 0 3 2 5 6 8
. 1
2 Í 6 9 5 0 1
.93377514
0.2500
0.020
â– i Z> c
0 0
1
c i
3 5 9 6 1 0 9
c 1
355309i
,99633303
0.2500
G . 0 4 0
4 5 c
00
. .1
3 2 6664 b
. 1
3274939
, 9 9 9 1 1 5 7 7
0.250 0
0 : 0 5 0
4 5 r
0 0
. 1
3 ! 3 0 i 4 9
. 1
3 1 0 9 8 4 1
, 9 9 8 4 5 3 3 4
0,2500
0. 08 0
4 D c
0 0
. 1
5 1 2 6 0 6 0
„ 1
30 4 0 1 3 2
« 9 9 3 o 0 3 0 7
0.2500
0 . 10 0
4 5 c
0 0
, !
3 x 9 3 7 2 4
o !
3136508
, 9 9 5 0Í 4 9 3
i>o. i

2)9
ThBLE 8-1 — Continued
M A S S K A T I Ü
0* 200 0
0.2000
0.2000
0.2 0 0 C
0.2000
0. 2.000
0.2000
0.2 0 0 0
0:2000
0.2000
C. 2 0 0 0
0.2000
0.2000
0.2000
0:20 00
0: 2 00 0
0.2000
0.2000
A L P H A
T M r. T 4
0 <• 0^ j
0 . C
si*
o
o
0 < 0
0.06 0
0.0
0.0 3 0
o. e
0:500
0.0
0. 02 0
lb: 0 J
0.0 4 0
15.0 0
C. 0.1) 0
15.00
0 - 080
5 5:0 0
0 a 1 0 0
15:00
O
;\j
o
o
30. 0 0
0 a 0 4 0
3 0, 00
C a 0 3 0
3 0.00
0.06 0
3 0 . o f;
0 « 1 0 0
30 , CO
0:020
4 6., 0 C
0.0 4 0
4b, 00
0 i 05 0
4 5.00
GR3JVA- J
. i. I 4 9 5 1 4 4
. 1 0 9 9 9 Ü 2 1
«50b b :> ?. 3 8
. 5 0 4 2 02 80
c J I) 2 b7 5 3 8
» 5 lb 0 8 !> 3 3
, 5 1 1 Ob 5 6 4
«10934371
«. 1 0 b 4 0 8 7 2
, 1 0 8 10 5 3d
: 1 1 6 3 s 3 5 6
1131 7 O' 3 6
.55177053
.1116b489
. ) 1 ? b 0 4 3 4
,117b8010
.11490099
<11368809
2 014 T I J V \ _
« 3 0 7 38 I 3 9
< 0 9 68310 1
.084 38 1 3 4
« 0 76662 5 0
.0649181 8
. 5 09 6 ? 5 0 :>
« 0 9 9 8 6 2 7 3
.09 35 5 3 8 8
. 0 8 6 4 4 2 v 6
. 0 0 8 1 6 0 2 1
.. 1 1 3 4 9 6 S 0
.5 0 66 37 9 1
. 1 0 2 6 7 5 82
. 0901 0 8b 8
< 0 9 6 3 6 O 12
.115 0 7 6 3 6
.11157955
1 C 9 7 4 1 1 6
5 ~r1 C15 4 6 7
< 93 4 06440
. 86 04 7 7 9 0
. 79632 3o4
7 2 5 5 0 7 4 1
. 63 26 GO 0 4
.94719812
v 8 8 904 5 4 6
. B5 5 1 43 2 6
. 7 9 7 4 13 d9
. 76 91 51 7 8
.97511241
o 9 4 2. 5 7 7 9 4
. 9 1 8 6 3 3 3 7
.877105/3
«- B 4 5 4 2 7 3 5
.93721093
« 97 10 9 30 4
. 95520 2 8 0
f=0.1

220
T A ULE 8 -1 - - C o n t i n u u d
ASS I .‘AT 1 i.
ALPHA
Tilt
Í A
UPblTAl. J
PUT AT i'JNAL J
U F F ¡ C I l’NC X
0, 0.133
0 * bb Ü
0
. 0
«238A8828
.04242941
. 56800039
o. 3
0 < t u
0
. 0
.341 o 8 3 ’3 2
C 2 4 00 3 7 A 0
.70365962
0 . a 3 3 3
Or GOO
0
v 0
. 3 6 ! V 8 a 7
. 3 8 8880 3s
.4225190o
0.0333
Or 000
0
. 0
.3 70S 082 J
.3719 92 3 8
. 4 5 4 2 961 4
0,03 3 3
0,H; 0
c
.0
, 39 1 0 8 1 8.6
< 1 2 0 6 v 4 I Ü
. 32 32 971 4
0. 0 3 3 3
0 . Go 0
- 1 s
f ,
t
.2 GO 18A 0 2
< 0 3 0» 3 7 3 3. 0
, 1 7 (9 8 03 2 1
0. 03 3 3
0 . G 4 0
•- 1 3
<â–  o
, 3 A 7 73 8 9 3
» 2 0 s 3 8 2 u 0
,59477s73
G,0333
0.0 s 3
- 1 s
t 0
. 3 b 7 1 0 3 4 3
<â–  1 l 9 0 1. 6 7 2
.460 30 3 4 5
0, 03 33
Or CcG
" ] t)
. c
.3 01 5 A 7 0 4
,10011381
c 4 0 6 3 3 7 5 7
0.0 3 3
0 , 10 0
“ 1 3
* o
, 391 ijC 2C 3
r 18! A /• 7 0 7
, 31 02 0 74 6
0. 03 3 3
G < C2 0
-5 c
, c
« 8 0 8 7 A1 0 9 a
. 0 3 9 7 6 4 0 0
. 19 2 3 3 7 7 3
0.0333
s t, 0 4 0
**â–  j 0
i G
, 3 00 89 7‘3A
r (l. ) t ' i .4 b 0 b
<0160 Ó!0 4 4
C . 6 3 3 3
0 <• 0 L > b
Jj L
.0
. 3 0 d 2 7 7 t
c 14 908-907
„ 4 0 55 0 0 8,7
0,8333
C - CsC
-- 3 0
. c
t J bC b O b
.10101003
< 26 52 2 891
0 , 0 8 3 3
0 . 1 G 0
- 3 C
i. • A
«.88 0 0 A (32 A-
0 70 S 3 0 9 0
. 1 817 7 902
C« 08 3 3
0 . 08 0
—4 b
< «.:
. -< 2 4 3 A 7 1 0
. 0 A 3 07 1 2 3
, 10 2 0 7 «4 0
C« 08 3 3
/ r. / r
b r %.> ' i L-:
4 b>
. 0
c b b <■_ « 2 A 0 211 I 3 0
< 6 9 6 4 6 8 9 8
Or 3 3
0 <000
~ ^ b
,0
. 3 6 o 0 £: 7 8 9
. I 34 0 0833
,420 62 0 7o
0« 03 33
0. 080
- A 3
(•
» 3 7 5 3 9 s 3 4
. ) (> 1 O 2 s 3 9
,42713030
0 . i 3 8
0 .. 1 0 0
— ¿i b-
t ü
, 3 fc. 3 7 0 J C 3
.09604199
» 25 03 0404
f- -Ü . 1

TA BLE 8 -1- Co n t i n i i e d
S3 ('AT 10
AL
t ’HA
T ) ií; 1 A
OfUíi TAL J
r-:m ai ! onal j
f.í ¡ I C IíINC.
0.7)4 3
0 .
0 20
0 . 0
4- $ V 4- -'i- V 4
. 0 0 0 0 0 5 4 5
. 00 0 00 1V3
C . 7 i A 3
0.
o. o
:*. 4 Vi. y(. / i; jB
,0)02 2 0 9
c 0 0 6 3 0 3 4 4
0.7} 4 3
0 «
0 0 0
0 .0
s 0 8 3 3 7 1 0 1
. 0 06313 47
. 074063 49
0.7)4 3
l> c
03 0
0.0
.30 7 S3 108
, 0 ó 1 60 12 2
.22999974
0.71 A 3
0 .
! CO
0.0
, ¿fio 5) 7 38
.. 0 36 0 4 3 4 5
. 12 3 G 71 3 1 O
0.71 A 3
0.
0 ¿ 0
“ 1 S . 0
v v 8 -v .t- c
. 0 0 0 0 0 ! 4 3
. 0 0 0 0 0 1 2. 1
0.7 ) A 3
0.
0 4 0
- 3 3 * 0
r r. >; v- >:â– . \
. 0 0 C 0 o 5 1 0
« 0 0o o 732 3
o. 71 a
0 .
0_ 0
— 15.0
. 24 0 c;4 1 So
.C 0 9 32298
, 0 3 0 7 0 9 9 5
0.7) 4o
0 .
0;. 0
-13.0
. 3 7 7 7 4 ¿14 í.,
. 0 4 7 2 0 0 ) 4
.15993645
0.71 A 3
0 t
10 0
-13,0
.. 5 84 93 4 3 2
. 0 3 5 o o 0 1 6
.12090662
0.71 A 3
0 «4
Ü .7 C
- 3 0 . 0
/ i,». >;t A --a ... ?;i
. 0 0 0 0 J o 3 7
© 0 0 0 0 0 2 o o
0.71 4 3
0 <
C 4 0
-50.0
>' V * >’/ 'A 4' V v 4
.0 14 12 3 o2
< 0 0 6 7 9 0 7
0.71 4 .3
0 .
0 o 0
-30 .0
, 1 1 5 o 0 7 3 ‘3
c C 5 2 33 4 3 4
. 09 51 002 9
0. 714 3
0 .
0 t V1
—3 0 . 0
< 0 ' i¿ 4' 8 o v
.097 1 3!06
.34415263
0 c 714 3
0 .
1 00
-ó 0 l yj
. 5 9c. So 0-j 9
«03981442
. 2 01 4 2 20 9
0.7 1 4 3
0 .
030
5 v V- y >,s. V: )â–  ;[ :.r â– *
.00000539
. 0 0000449
0.7143
0 c
0 4 0
~4 5 c 0
>: â–  7 >3 4' *V V V 5 -V
t. 0 2 2 36 37
. 0) 5 2 9c14
0.72 A 3
0 .
0 0 0
— 4 L> c C
>s 4- # y 4 y y 7 i,.
.00304604
. 0 0 3 0 972 3
C. 714 3
0 .
k'üü
-4Í3 cC
• ¿ó*
.13 4 4 3 7 2 4
c 4 7 o 1 1 4- 0 1
C . 7 ] 4 3
0 .
1 0 0
~ 4 0 c 0
, Z9ük'i 93 6
. 0 3 9 6 1 4 L 3
i 3 3 7 2 1 2 1
£ ■ 0 .]

222
TABLE 8-J ---Coni: i nuod
MASS RATJu
AL
PMA
V i 2. T a
ü ¡i b I VAL
J
ROVATiUNAL J
Li i 1 C 1 LUC
0 «• C> ci D 0
U í
0 20
0.0
^ j , j ; jf. >. r.
v
* 0 0 0 0V0 40
. 0 0 0 0 0 0 31
0 » L> k. 1) 0
0.
V M 0
0,0
>;< v *. r
-
.00223110
.OU 1 1 9 o 4 9
0,6 23 0
6 *
OoO
0 c u
. 03} t;7 3t
1
» 0 1 2 o J 2 3 A
<14062363
Oi oc’SO
o .
OoO
0 < 0
<22? » V 3 4
6
«031 17 32 4
c 1 3 o j 3 c* 3 0
o,or so
o.
J UO
0 * 0
,2 4oJJ77
•>
< ü 3 6 2 1 4 3 3
< 2 2 S 2: 196 0
0.6 2 3 0
u <
0 A. G
- 1 3 < O’
« 0 G «-• 0.) 0 '.V 8
< OOOOOOol
C « 6 2 3 C
0 *
o-O
- 1 3 0
3 0, -v>,
. 0 0 4 3 1 6 3 3
» 0 0 2 2 0 ] 4 9
0 . 6 23 0
G *
u'o 0
- 1 j ( 0
.23 ] i)9 2 C
3
. 0 1 ó 7 9 9 3 3
. 037 3 332 4
0 . 0 2 3 0
U r.
G0
- 1 3.0
* £3 J> G 3- Gy ,3 O
7
• 0 4 7 3 3 3 4 4
e 3 0 Ü o ¿< ó 6 3
0. 62 0 0
O t
] GO
- 1 3 c 0
e 3 5.:, C; G G Í.
. 0 9 Li 1 2 2 9 1
« 3 7 9 0233 0
0.6?00
U e
0 GO
~ oO « 0
v ~ ^ * 4 v
. U 0 0 0 0 0 l -V
« OüO G 0 ü4-
0. 62 3 0
L- c
u 4 G
- j-u «0
3>: =. =:
V
» 0 J 3 1 2 r 9 7
« (' 0 i 6 3 o 4 1
0.6260
G «■
ÜOo’
-30,0
» o O / u .
G
ff u 1 0 v? j o G 3
, 02 9 4 1 1 iAt
0.6260
U *
03 0
6 G t G
c 2 3 .. 2; - tí 7
;>
i 0 4 a. 7 3 4 .1 1
. 1 70.3 3 3 3 0
0. 0260
0.
J ■•.'(/
- 3 u < 0
* P 6 G 3 Ó - 7 4
O
» 0 6 2 7 .-> 4 2 3
.32 4 290u0
0.6 ; 3 0
C <
0 ✓ r J
~ 4 „ G
4 JN ^ V V v V -v
.000 0 0! 7 9
,0000Olio
0 . C 2 3 i
G c
o *i o
4 G 4- 0
3 '••• -v -■ v v-
, 0 0 7 3 7 2 9 7
» C 3 4 4 0 0 2 4
0 » o 2 3 0
G «.
G L? 0
- - 3 . 0
. 0 7c 76 9 6
G
. 0 ! 9 u 1211
. 2 3 3 4 6 u 2 ¿V
0 c 0 P. 6 0
• 0 .
0 o 0
- 3 3.0
. 2 «, 2. 6 0 7 3
-v
« 0 (s 3 u ••;» 3 0 6
« 2 7 9 4 6 9 3 4
o. o; 3 o
g *.
J C' o
- 3 3.0
. 2 3 ‘ < 72 7 3
7
. 0 6 6 i 3 2 3
< 3 3 6 2 1 2 9 4
0 . J

223
TABLE 8
• 1--Cont:¡ nued
A b A
R A 1 1 u
AL-RI-íA
lli: 1 A
ÜRlíI tal j k
CAI A '1 ; dim AL J
[ t F I C 1 t'UC
0.
b b b G
0 c 0 rl J
0 i 0
v -v v * v >.•*
« 00 000)06
.U0000007
0.
b b 9 6
O « U 4 U
0.0
-.vv>. VG V-
. 0 0 4 Go5 1 4
*0021 5 15 0
0 .
55b c.
o . G G 0
G » C
«• 3 b c 7 7 y 9 ó
. (; 2 0 0 30 14
.09332049
e
O
5 5 o o
0 . Goü
0 , 0
« 3 o ó o 4 ! o ó
< 0 4 b 9 2 6 o 2
« 2 2 ó 1 92 32
0 *
55 5 0
0 « 1 C J
o *. 0
o ¿1 J 5 o ir i 5 / a'
« 0 ó 1 J o 1 b 1
.A ) 32 01U 6
0 .
5 5 9 6
L « o 25
“ 1 ‘ « 0
/' y ,¡ y y: v .y.
« Ü 00 A ! 23 G
« 0 0 0 2 1 /4 A
0 €
5 55 6
0 * 0 ^ 0
1 Oí 0
y y a- ^ i - >g
.00 7 92bó3
« 0 0 3 b 3 A t. a
0 ,
b S 9 o
O » O o j
- ¡ ó . 0
. 2 2 b 1 2 1 y A
« ) Go ¡ 30 bü
« 74 4 7 9b! 1
0.
b 5 b 0
0 «. 0 c 0
-Ib. 0
«2 1! i,kW
« 0 6 7 o 4 1 3 9
. 3 ! 9 2 9 0 G i
0
5 5 5o
0 « 1 Gu
- 1 :â–  < G
o ¿L ir. ■. 1 Oil
e J 0 5 ó / 5 O 4
c a b 97 62 9b
0 .
bbbó
0 t O £_ O/
-3 4. C* 0 0 5 i 1 5 5
.. 0 3 0 2 9 A 4 /
0,
5 Roo
u < 0 < u
- 0 < 0
7 * v ■■• * b v ñ-
« 0 0 o o 0 3 b 3
« 00296907
0 .
5 5 5 ó
0 r 0 o o
- 3 G . 0
,2 3o ÃœvV1 7
*215201 i 5
. 9! 9 7 74 Y 0
0.
5 5 ó
G * G O 0
— 5 0 < C
«. 2 1 o o 7 7 o 4
« 0 O D O ) O 1
3 02 1 7 94 3
0 «
5 5 5 ó
G i. i 0 0
- Ó 0 « ü
C ¿1. C . f ( O i V
.117'; j 0 0 b
« ¿ ) 3 ü A ; i 3
0 .
5 5 5 o
0 < ^ A.G
- A O , U
V -V- V A -.'-o -
. DOGO TOGA
«. 0 7 0 0 0 o 4 7
0 *
bbbo
Ãœi o A G
4 f ü
^ b- o
< 0 0 .3 / 0 ' J o 1
» G G 1 O 5 O 5 O
0 «
5 5 o o
ü « 0 o 0
— 4 b «. o
«Ib 2 o b b G 0
r 0 y C *J o 3 o 5
. 13 7 3 4 29 0
0,
995 6
o . Vt'Ú
b . ü
«21o 9 2b2j
« 0 . 9 !vj 0 o 0
« 2 / •_ 5 4 2 2 D 1
0 *
b:9b(:
0 « ’ 0 0
— 4 b * 0
* 2 5 7 7 lj ¿ o
« 1 ! 3 7 3 G 7 6
« b 0 Ob 4 29-;.
í — 0.
1
X

224
TABLE 8-1— Comv ¡ nuc'd
s 3 l < A T I u
ALOHA
lit: TA
U A b J 1 A J
ALTAI ION A L J
a Fr 1 C I F.NC
0. LOGO
0 t- 0 2* 0
0.0
.23 5 41] ! 1
• 0 0 5 0 ! 4 9 0
.0 3 4 59 036
0 . 3 0 0 0
u c 0^0
V . 0
« 1 6 u 2 0 u I 7
. 0 3 7 6 1 2 9 4
. 3 1 9 7 1 u 5 3
0.500 0
0 * 00 0
0 .0
. 1 9 0 7 8; 1
.0980 A 9 3 6
. 3 0 5 3 2 510
0. S 0 0 ü
0 C 0 L> U
U . 0
.. 8 01. 4 t. 5 0 0
. 1 2002 4 8 7
« 3 3 1 3 2 J 5 1
0.5 0 0 0
Ur ] 00
0 = 0
. 2. 1 0 O 5 7 3 '3
.13317034
.71010572
O
c
o
0 < 0 r.'..j
3. 5.0
, 130 37 731
. 0 0 8 5 0 0 0
.06536150
Co 5 0 0 0
U < 0 4 Ü
-15.0
. 1 8 4 58 U 0.2
.0 6 7 4 5 8 9 0
. 364 9 4361
C . 5 0 0 0
U s> 0 Li 0
- 1 5 « 0
.19b74344
. 1 0 -i vJ J 4 3
« 8.2 8 34 94 4
0.5000
0 c 0 c < 0
- 15.0
* 2 0 55 53/0
. 1 5 1 7 1 9 3- 9
« 7 2 8 (12 6 3 6
0 . 5 0 0 0
0 < 3 ü 0
-15.0
.81 7 3 Ü 0:5 1
. 1 0 3 a o 4 3 9
. 7 3 7 04 2 8 7
0.5 0 0 0
u ,00
-50 .0
. 17 0 7 4 8 1 0
. C 0 7 1 0 3 S
«041 3 1 0 0 8
0.5000
U * 0 4 0
-.50.0
.13 7 4 4 4 Í-.3
« U 5 9 0 •> 5 _ J
.5130t U 9 6
0.5000
O c 0 ó U
- 3 0 . 0
. 8 0 0 b 9 5/ 3 4
.110 6 8 5 7 6
. 5 3 17 83 3 7>
0.5 (.i 0 0
ü < 0 i .< 0
5 0 .1'
. 8 1 0 3 4 3 3
. 1 3 3378 9 0
.74 8.3 4 9 3 9
0.5 0 0 0
U r. ) u U
- 3 u . 0
.81303722
.171094I3
. 78 3 4 8 5 1 -
0.5 0 0 0
0 - 0 o J
-43. U
<■ 2.7 0 <•_ 3 < t 8 0
. 0 0 3 8 0 7 0
« 0 1 1 0 7 3 7 9
0.5000
0 . U 4 0
— 3 . U
. 1 3L-. 73 5 1 7
. '0 5 0 9 5 8 U
. ) c- 5 7 8 2 8 7
0.5 0 0 O'
0 o 0 vj 0
- 4 5.0
. 3Í 0 u 41 j 0 9 5
.10097917
.50373480
3»
O
o
U c (2 1Ó U
— 4 5 .0
. 2 0 3 4.- 1 ’ . 3
. 1 4 04 3 93 7
, (-â–  7 3 2 0 5 5 5
0. 50 0 0
0 « 100
-45.0
. 8 3 50 833 7
. 1 t 0 (54 4 9 1
. 7 9 3 1215
f=-0.1

2 / o
TABLE
M A S 3 H A 1 J u A L M i A Til;. 7 A
0 • 2> .3 3 3 0.02 0 0 » 0
0.33 3 3 0. 040 C.O
0.3333 0 < UOO LI . ()
0 « 3 3 2 3 0.0 3 0 0 . 0
0.3333 0.100 0.0
0.333 3 u <• 021 -lb .0
0. 3 3 3 3 0 . 0 A 0 -- 1. 3 . u
0.33 3 3 0.0 uG -Id. 0
0. 33 3 3 d . U30 — 1 u f 0
0 » J j b b 0.1 O {i — J 3 . u
U i 1 b u . 0 2 0 — O « C
0.33 3 0' . Os- 0 — o Í. ' • 0
0.3 33 3 O.0O0 -30 ,0
Os b J j Ü . G o 0 b u <0
0.33 33 0. 100 - o 0.0
0.3733 0.020 -a 3.0
0 . 3 3 3 3 0 « 0‘* 0 —4 3.0
0.3333 0.000 — A o . 0
0.33 3 3 0 0 0 0 -43 0
0 3 3 3 3 U , 1 00 - A 3.0
1 —Continuo'd
lit’.L'! TAL J
[-01 AT IUNAl J
Li f ic i i
. 2 O' y u 1 0 0 o
e 0 0 7 6 4 4 6 3
. 0 7 6 12 3 72
. 1 0- o 7 1. 3
. 0 2. 7 09 1 2 1
. 4 9> 0 0 7 7 9 3
. 1 3 O 03 6(07
.. 0 3 6 04 3 , A
. t.900.9239
. 1 ! i U ; â–  v 3 1 o
. 1 2 4 2 3 D b
. b ¿7 7 1 7 73
. 1 ; 3 3 7 4 .1 6
. 1 33 24 9a Ci
. 33 007 ) 0 3
. i 7 6' üb 1 3 1
c o i o; -1 3 u u
• 0 3 31: 3 oo9
. 1 JO 3290 J
* 6 7 1 2 0 U
. 2) 4 4 3 "7 4 o 0
. ) 4 2 u 4 ? u 3
.1076394)
.73 1 D9v 1
. 1 3 3 0 3 0 4 3
« 1 J 11) 7 0 3 7
. B.9 694 7 1 7
. 1 (. , 0 7 01 > 2 d
. 2 4 2 1 7 4 3 2
. 97 3.0 2 1 93
.20427 j a 3
.0032 V 47 0
. 0 4 39 0 1 -L
1 3 1 0 2 t > 2
» 0 o ii 97 0.1 9
. 3 0 2t '.lie, 3
. 1 4 0’ 3» 0 .. O' 7
. 1 13 4 9 2.2 4
. 71.7331 0
. 1 3 b. o. 7 o 3 3
,12363730
.37109302
c 2 3 : . 7 b 7 7
. 14 402303
9.' 3 . 0' 31 '
« 27 7 i U ? 2, (j o
.00 o 3 3 4 . - .0
. 0 329'6 4 34
. 1 3 0- 0 4 6 J 3
. 0 634492 3
. 4 6 9 ? 2 6 4 1
. 2 432 44 7 0
.10099027
. 7-t 7o 93 3 b
. 1 • í . -, . 0.' • - i
. 1 2.4 1 â– . fcc 2'
. 7 7 32 3.3 3
. 1 3(3 7 1 9-
«• 1 4 v> 1 3 1 - > 6
. 9 2 b. 1 1 9d9
£~ - 0 . I

TABLE
226
• 1--Conti miad
â– '.AS 5
(â– â– AT 1 U
ALPHA
1 HL 1 /..
ORBITA!. J
P Ü1 A 1 í ANAL J
E
rricilnc
0.
2 b 0 0
0 c 02 0
0 . 0
V- A -d A' G- v y y
.000 0961 6
í
00003493
0,
2 LOO
0 . 0 0
O c 0
« o- 4 b o c. 4 i, 7
.011 5935 1
c
0 3 7 96 36 o
0 .
2b 0 0
0*0 O 0
O . L'
.â–  1 2o- b 0 /
e C/b 1 9 0 51 7
-
4 24 5 131 ti
0 ,
2 b 0 0
0< 0;:.0
0 , 0
1 0 9 1 6 b 0
<â–  0 8 0 0 9 3 4
«
73374 é>90
0 .
2 'o 0 0
U » 1 u 0
0 »0
* 1 2 1 9 o 4 7 8
. 0 9 91 3 1 3 3
V
t â–  1 2 5 5 b * 14
0.
2. b 0 0
0 < (¡2 0
- ] D . 0
;;; /<. A . y, y i,
.000 0 6 1 7
<-
0 0 0 0 2 2 1 3
0.
2 b ó 0
U 4 0 ‘{ L?
- ib. 0
< 4 b 1 '5 i el
» 0 1 3 7 o ü 9 2
<â– 
0 29o 991 b
0.
2b 0 0
u L- 0 o 0
- 1 * 0
.106 2.0; 0
. 04 0 53092
«■
3 8 1 ... b 2 2 0
ü<
2 5 0 0
u « 08 0
- I 5 * 0
.11bti¿922
. 0 5 1 0 1 7.) b
*
7 8 5 /ñt.Lj 8
0 .
2 S 0 0
0 ..â–  i o J
-15.0
C 1 2 . - O' o b 6
.11027239
»•
87 2‘i GubG
ü.
2 5 0 0
0 s Or.O
--5 0 <â–  0
>: >. v â– ,
c 0 0 0 .O T â–  1
<•
00002081
0.
25 0 0
01 U 4 0
- 3 o t 0
« 5 0 4 j V’ 1 c. J
, 0 ! 4 ‘j 9 7 ü 3
fc
0 2 8 74 4 0 0
0 .
25- 0 0
0 c G í 0
- 5 0 < 0
, 1 0 5 5 ) 7 o
0 o 4' ■. ’ 2 íj f 5
*
3 fj 2 3 1 b 2 3
0 c
2 b 0 0
0 < 0 i > 0
- 3v . 0
1 1 f,2bcbo
. 0 5 4 1 â–  3 i; 3
c
/ 9 6 b i 3 6 o
ü <
2 5 0 v)
0 t J 0 0
- 5 G < 0
. 1 i 0 â– : . 5
e ) 1 bb P. 2
'.3 1 9 9 84 o
0.
2 5 0 0
G <â–  0 2.1
— A O . 0
•’ '■ •'■■ '■ •• '■■ 7
o 0 0 0 5 2 2 8 9
*
0 0 0 1 3 3 4 4
Ü.
? b 0 0
U. U 'tú
- 4 5-, u
,78 9 94 290
. 0 1 2 V 0 4 0 6
t
01 7 4 39 8 7
0 ,
2 5 0 0
0*0' o 0
- 4 o <â–  0
. 1 B7o.9i.52
. 1 2.1 6 94 1 o
c
9 5 0 9 2 0 / 4
c.
2 b 0 0
(j t 0 i j 0
*4 € t >
. 1 1 o >7 7 0 4'
. 0 ¿ '■ b 3 o 2 4 3
e-
72298020
0 r
2 b 0 0
u . 3 0 0-
“•4:> c U
.i2b5245b
. 1 1 o 55 91 8
t
ü80774 2 9
f=-0.1

TABLE 8-1—Con 1 i míoel
S S í < A1 i L
AL PmA
1 1-tL
1 A
c¡
í V tí I ¡AL. J
r- ÜT Al ¡ ONAL J
tFt IClcNC
0, 2ÃœC0
O e 0 0
0
eO
e-
3 A 4 0 7 1 tí .3
,00003770
.00 0400 4 7
0 , 2 0 ü 0
0 , U 4 0
c
t 0
tf
12/11 30. 3
, 0 1 0 3 3 0 b b
.08126294
0 c 2 0 0 c
0,060
0
e- 0
&
1 LHíáooVü
« 0 G) 3 4 3 3 1 O
. 34 1 5 7 4 31
0, 20 0 0
G< 00 0
0
. 0
i
0 033oh 33
« 0 / 1 L. 4 4 0
. 763 [7o o3
0.2 0 0 0
0 , H/0
0
. 0
<
1 OH. 7.3 b 2, A
0 b 2 0 7 4 4 o
, 2.6 2tí 3 33o
C . 2 0 0 0
ó . 0. 0
< ti
c
3 0 7 6 4 37 9
.000 0 77A o
« U 0 0 2 o 2 3 7
0 . 2 0 C 0
G . G A 0
-- ¡ b
< 0
♦
0 5 0 9o o oü
, 0 1 í* Yj í ■ í 7 2
. o 2 7 9 7 7 3
o. 2 o o o
O , 0 . :>
- 1 b
c 0
«.
05GbG7bO
.C44 71b b3
e 4 9 3 9 3-1 6
Oí 2 0 0 0
O» 0¿>U
— i O
< 0
»
1 Ü ü Gol O Ol
. 0 '7 4 '3'., 0 o 9
. 2. 3 ’3 0 4 9 o 3
0,200 0
G . 1 O ü
- 3 2>
, 01
f
i 1 1 04 35 2
C 1 0 3 2 3 â– '! i 7
. 0 2 9 7 3 0 9
0 . 2 0 0 0
o«o,' o
-2 0
t. O
<â– 
2 ) 2 o 7 4 2 o
< 0 0 o G 3 1 c 0
. UOO3 83o9
0« 2 0 00
0 s 0 4 0
- 3
r. ü
•
0! 0 0 0 3;' ó
, 0 lo 324 79
. C 7 9 boo A 3
o, o o o
O « G i;0
vi' U
«u
V
OboOG i 3 i
« 0 4 o 4 7 4 h o
, 49 b 3 4 1 7 4
0. 2 00 0
u. o.-o
-3 o
c U
<
102oo037
.0Mb23 0 0 o
« ii 3 5 3 9 i 0 ó
0 * 2 0 v G
0 , 1 ü 0
— 0
c 0
•
i 1 U Oi'O j J
í. 1 0 b o 2 7 7'
. 9 2 3 3 A>7:
O
v\¡
o
G í 0 í 0
* 0
•
4 ü 1 0 1 o 0 7
, G 0 U 1 o- 0 >\ 4
. J0033355
V , 2 0 C 0
O « 0 4 0
- 4 3
* 0
V
2 í 4 4 O 3 3 3
c U 3 3' } 3 ' ) b ,2
. 057 0 j 2 '1 0
() o 2 0 0 0
G . 0 o U
~ '* -
« 0
f.
1 1 ,0 7 2 7 3 =7
. 0 73 5 i 03 3
. ó 5 2 0 4 9 2 1
C« 200G
u . 02 ü
-4 3
* u
*
i C - 3 4 7 b 3
t 0 'o /•' 3 02
. 6 0 ó 8 ‘o; 13 b
0,2 0 0 0
C1 , loo
- 4 3
e 0
,
I 0 v 0 4 o 4 “'i
«■ 3 0 J 1 3 o 3 b
, 92 --+0 o , 7-,
f=-ü.1

228
TABLE 8-2
HON-SYNCH I
(CHOUS ROTATION PARAMETERS
(Secondary Com.] ¡one nts)
FOR DLSB
System
V”»>
VHU
v? (kiv./sec) vsyn
(km/sec)
v / V
7 syn
Í Phoe
6 - 3.
3.0
7 6
60
1.26 6
AG Per
54
4 . 6
73
65
1.123
V Pup
ICO
10.0
18 0
184
0.978
CV Vel
5.6
54
. 8 5
4 0
2.12 5
a Vir
10.9
6.8
4 4
4 7
0.93 6
U Oph
5.30
4. 6 5
7 5
93
0.806
6 8 Her
7.9
2.8
91
106
0.858
RX Her
2.7
2.3
6 8
57
1.19 2
356 Sgr
1?.. 1
4 . 7
90
62
1.4 51
y Aql
6.9
5.5
80
88
0.909
Y Cyg
17.4
17.4
14 4
10 0
1.4 4 0
AH Cep
16.5
14 . 2
9 3
17 4
0.534
CW Cep
10.0
9.8
13 8
81
1.7 03

CHAPTER NINE
LIMITING VALUES OF clM and dJ
9.1. Limiting dJ
It would bo desirable if an upper limit could be cal¬
culated for the maximum possible period change which would
result due to non-conservation of orbital angular momentum. An
examination of the terms in Equation 9.1 J will show the relative
importance of non-conservation of J with respect to that of
mass transfer in calculating dP/dt. Wilson and Stothers
(1974) have done this for the usual case of conservation of
total mass and orbital angular momentum. We have
dP^r T
l\ L-U
Ci I ■ ‘ i:
i'll
M-. M2
J(M1 -
M:.)
d.J
dM2
Wilson and Stothers allow
(9.1.1)
J = k(2GM«R„)
max u 2 x
(9.1.2)
where k is
strilling th
result can
efficient e
a factor which depends on latitude for material
e star tangentially fell ling from infinity. This
be extended by inc.luci.Lng the efficiency co-
, as calculated in this study. We now have
^ ma x -- f-
JO
229

230
and hence
dJ/dM2 = -ek. (2GM2R2) "% (9.1.4)
Substituting as before, we eventually obtain
dPNCJ
'32,:21
1/6
1/3 1/2
(1+q) q
1/2
R ¥
dPMT
G
kc
1 - q
1/61/3
1
where
q - VM!
Here is given by
p — r fM /m l n / o i r 'i
k2 K©'M2/no' (-/ . l. u
where n is the exponent of the mass luminosity law. If we
adopt n-0.55 for non-rotating zero age main sequence stars,
we obtain
dPNCJ
dpMT
0.69kc
(H-q) "q
)l/3q0â– 7 7
(1 ~ q)
(Mi
/M
0.11
(9.1.7)
with P in days.
One might ask why it is the
star, rather than that of the Roche
tutad for the value of R0. The reas
it
from ORBIT, which shows that the eje.
cases under consideration in this st
: radius of the accreting
lobe which is subs Li¬
on for this is apparent
-:tea particle, in all
.udy, strikes the primary
component
.rec ci
■!• h
striking

2 31
tangentially as put forth by Wilson ¿ind Stothers gives a
generous upper limit, as for the most part, material does
not s t r i ke an ywh ere near t a agen t i a 11 y .
However, for purposes of calculating an upper
limit of the effect of dPv,-,T/dPf we will allow material
to strike tangentially giving
dpNCJ
dPf^T
= 0.69 k
(l+q)l/3q°*77
(1 - g)
(M3/M_)0
11
(9.1.8)
Furthermore, integration of Equation 9.1.8 over the entire
accretion process yields
2k , , t0
“tot = —3 (2GW)V
M2 (n + 3)/2
M
(n+3) /2
M
(9.1.9)
Wilson and Stothers liave shov.m that tlie ratio (dJ,.^, /dJ , ) ,
rot' orb mux
is such that the largest observed values of this ratio
based on the underluminosity of binary components assume
reasonable values, three to four times smaller than allowed..
Another possible way to lira!
observe binary systems where one or
non- sy r> a h r on ou s r o t a t i o n . H e r e w e a
transfer time scale is sufficiently
t the quanity dJ is to
both components exhibit
asume that the mass-
short so that an

232
orbital-rotational momentum interaction is possible which
is not obliterated with respect to the tidal time scale.
The tidal influence will attempt to resynchronize the
p
stars on a time scale of 10 years.
Basically, the total amount of orbital angular
momentum which is converted to rotational angular momentum
can be calculated in three different ways.
1. The theoretical limit obtainable over the
entire accretion process can be expressed as in Equation
9.1.9. Naturally, this amount is subject to the average
efficiency of conversion of orbital to rotational angular
momenta. Realistic values of AJ ^ can be obtained by
multiplying the value of AJt ^ in Equation 9.1.9 by the
M
2i*
a
.ency
coe
.r- f ;
i. I. i.
cient.
The
.e 9-1
f o
1C 3
ever
al
no ir-
calcu
lat
ion
of
the
ave:
ilv be
done
i n a
c;
r i c t
L1T1 ÍS
ava
1 la
ble
(Me
thod
.â–  cl 0 G -C
to
” Pr
oduc
e"
an a:
:f f ici
ent
f o
r th
is
case
mere]
y a
ver
agin
9 t
h e e
w e a
s s u
that
v;h
a to v<
¡non o
f n
on-
O! \r y-,
h r o
nous
cal e
(at.
le
,S; O
a s
shor
. cab:
u].a
te
the
obs
evvci
tover process is leading
'rot
of the

233
system, and compare this value to the rotational angular
momentum of the star at some previous time when it was
rotating synchronously. Thus we have
(9.1.10)
where M = mass
a := semi-major axis
ano.
and • w = angular velocity
These values are also located in Table 9.1.
3. Finally, we may start with a value of J^, and
increment it for an evolutionary time step value according
to the appropriate value of the efficiency coefficient at
this mass ratio from ORBIT simultaneously with the mass
flow rate of dM from the evolution model. Hence, the next
evolutionary time step will depend on both of these factors.
Eventually, after a suitable interval of time, we arrive
at a value of This method has not been carried out by
this study owing, again, to the huge amount of computer
time necessary to do this star by star. Nevertheless,
it is certainly a much belter method of attack than either
of the other methods, and will be pursued in future in¬
vestigations of individual systems.
9.2 Ca .1 c n 1 a t:i on o f dM
Each System
The mass flew rato dM/dt
conservatxve ease,
in the first column of

2 34
TABLE 9-3.
NON-SYNCHRONOU3LY ROTATING COMPONENTS
System
Theoretical Maximum AJ
Measured AJ
R CM a
9.956xl050
4 9
9.08x10
TV Cas
5.946x1051
9.225x1050
U Cep
4.738x1052
1.508x1052
XX Cep
9.5 7 8 x 10 01
1.142xl051
TW Dr a
1.377x1052
4.497xl050
S Equ
2.54 6xlC)S2
1.160x10s1
RX Her
6.220x10s1
7.758xl050
U Oph
1.520x10s2
1.138x1051
W Ori
7.870x10s2
7.244x1051
356 Sgr
2.582x10s3
1.160x10s3
505 Sgr
3.172x10s2
1.395xl051
Z Vul
5.6 67x3.052
7.2566x1051
RS Vul
5.013x1052
3.947x10s1
(Units
3 , .
g-cra /sec)

235
Table 9-2 for each system. The value of dMNCJ for each
system is to be found in the next column, while the value
of the maximum mass flow dM»,ñV is found in the last column,
9.3_ Limits on _d_M
If we assume that all of the gravitational potential
energy of an element of mass is converted to thermal energy
during the accretion process, the limiting luminosity L
can be. expressed as
L
GM dM
r dr
(9.3.1)
due to Huang (1956). The conversion efficiency of potential
to thermal energy is considerably less than 100 percent for
most objects. If we make the additional assumption that
only the radial component of velocity for an impinging
partido contributes to such a. conversion, we arrive at
e' = (1 - e)k
(9.3.2)
where e' = the efficiency of the conversion of
gravitational to thermal energy
and e = the previously calculated coefficient
of angular momentum conversion from
orbifa1 to roLahiona3 momenturn
The facto
constan':, depending upon the medium v
â–  7> ,
:e
impact
place.
Since we would like to use the value

236
of dM/dt from Equation 9.3.1 to set an upper limit on the
mass flow rate, and as no previous work exists which enables
us to deduce a value for the constant k, we will arbitrarily
set k equal to unity. This corresponds to perfect conver¬
sion of potential to thermal energy for the radial component.
Thus, Equation 9.3.1 becomes
L
CM dM
r dt
(9.3.3)
Expressed in solar units
L
3.3x107
M
R
dM
dt r'
(9.3.4)
Solving for dM/dt yields
dM
dt
3x10"8
(9.3.5)
This equation, gives an upper limit to the mass flow
rate dM/dt,, for several reasons.
NCO
1. Only the radial component of the accreting mass
was used to calculate the efficiency of conversion from
potential to thermal energy. In principle, the tangential
component will alsc contribute to this process for synchron¬
ously retí.ting stars.
2. We are assuming that the entire luminous
output of the star is due to mass accretion.

2 37
TABLE 9-2
LOG(MASS FLOW RATES)
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TABLE 9-2—Coniinucd
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N A !â– ' E
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TABLE 9-2---ConV inued
DM/CT CC \ '• Df’/DT NCJ
DM/DT MAX
¡ h
li (:1

CHAPTER TEH
CORRELATIONS OF COMPUTED AND ABSOLUTE PARAMETERS
3. 0 «I Perro cl Co r re I a Cions
It has been the aim of this study to investigate
the possible correlations of different parameters for a
number of close binary systems, which were chosen primarily
for their rapid period variations. A technique has been
proposed which will account for non-conservative effects
with regard to total orbital momentum, whenever this effect
is present. The question arises, does the inclusion of
non-conservation of orbital angular momentum improve the
correlations of period change and/or mass flow for a sample
of close binary systems? In this chapeer we will also
deal with some other correlations or anti-correlations of
various parameters.
One of the most obvious parameters of an individual
system is its period. Although our group of systems
displays a somev?hat homogeneous sample with regard to
this quantity, our group also represents the most meticu¬
lously observed set of systems. Hence, other systemic
parameters are relatively well determined. Figures 1C-1
and 10-2 show Log (P) plotted «gains Log (M-;) and Log (M^) ,
respectively. It can be noticed that the mass of the loss
24 0

241
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Figure 10-1


massive component is relatively constant compared to the
mass of the greater component. Also, there is a general
systems to the left relative to detached systems, while the
masses of the greater components do not show this effect.
Thus, we may surmise that not only has the less massive
component undergone mass loss, but that loss is arrested at
some minimum value, regardless of the initial configurations.
Log (P) - 0.198 + 0.18 7 LogUO
.045 .111
(10.1.1)
Log (P) - -0.0/5 + 0.477 Log(M )
.062 .125 1
(10.1.2)
Figure 10-3 contains a plot of Log(P) against
Log(total mass). Using Kepler's Third Law, we expect that
on the average for all binaries longer period systems will
be generally less massive. However, for our sample of close
binaries, the longer peric-d systems are in fact more massive
than the shorter period system
(10.1., 3)
Figure 10-1 displays no relation between Log(P) and the
mass ratio M.,/.

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246

24 7
for contact and semi-detached systems, although no over¬
all-systems relation exists. Figure 10-6 shows the relative
error in dP/P for each system, as well as a line indicating
at what
Log(P) - 1.357 + 0.125 (dP/P)
.608 .06 5
(10.1.4)
point the error is equal to the quantity dP/P. As most of
the systems lie below this line, there is strong evidence
to explain the overa]1 period variation in terms of a
parabola. Systems plotted above the line either cannot
be explained in this fashion or contain relatively few
data points. A number of these systems display periodical/
parabolic behavior in which the error of the parabolic
term is less than its magnitude. Hence, there is, in
general, a small correction to the quantity dP/P for these
systems using the periodic/parabolic representation. How¬
ever, this effect in no cases exceeds a factor of three.
Log (error dP/'P) â– â– = -3.037 + 0.697 (dP/P)
.476 .052
(10.1.5)
Figure 10-7 shows the correlations of Log (P) with
quantity is the actual change in period with time. We would
of dP/dt than short
j/ o n o
if only for the reason that

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249
2 . O O C <â– 
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250
the period cannot decrease below zero days. In fact, there
appears to be a qualitative change of behavior around
Log (P)=0.280.
Log(P) = 1.838 + 0.184(dP/dt)
.263 .029
(10.1.6)
Figure 10-8 is analogous to Figure 10-5, but it
plots Log(P) versus Log (dP^-.j/P) . This represents a "recti¬
fied" period change rate, due to the inclusion of non¬
conservation of orbital momentum terms. At first glance
it can be seen that the inclusion of NCJ effects improves
this relation. While all systems undergo some displace¬
ment, it is the contact systems which undergo the largest
systematic shift between these two diagrams. The use of
NCJ allows us to include both contact and semi-detached
systems in the same relation.
Log (P) = 2.357 4- 0.221 (dP )
.728 .028
(10.1.7)
10.2 d P/P Corre1a tio ns
Figures 10-9 and 3.0-10 display the variation in th
and the primary is basically irrelevant.
dP/p
. 0 8 7
2 J 0
(10.2.1)

251
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Fiijure 10-10

254
Again, dP/P is understood to be Log(dP/P). Figure 10-11
shows run of mass ratio against dP/P. Here y=M2/(M-j_+M2) .
]i = - 0.559 - 0.115 dP/P
.358 .038
(10.2.2)
Figure 10-2 is included to indicate the general
run of dP/P with the size of the larger component as a
fraction of its Roche lobe size.
Figure 10-13 shows a plot of Log(dMC) versus dP/P.
Here Log(dMC)-Log(Delta Mass Conversion Case) . This diagram
indicates the relationship of the fractional change in
period with calculated mass flow rates for the conservative
case.
Log(dMC) = -1.277 + 0.833 dP/P
.730 .077
(10.2.3)
10.3 dP/dt Corre led: ions
Figures 10-14, 10-15, and 10-16 illustrate the
anti-correlation of the period change rate with time dP/dt,
with M2 .• Mj / and • Figure 1.0-17 shows that the criti¬
cal parameter is the mass ratio M^/Hg. Finally, Figure
10-18 serves to indicate the relative period change rates,

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in-'!.:'
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2 64
as a function of size of the larger component compared to
the size of its Roche lobe. We see that there is a
greater intrinsic dispersion of semi-detached and detach ed
systems in this diagram than in Figure 10-11, which
indicates the fractional change in period.
10.4 dPppj Correlations
Figures 10-19, 10-20, and 10-21 show the anti-
correlation of dPpcj with M2, Mp, and respectively,
while 10-22 shows the mass ratio versus dPNCj• Under the
assumption of NCJ, this relation becomes
QU = 0.813 + 0.020 dPNCJ
.267 .027
(10.4.1)
Which is somewhat more satisfactory than Equation 10.2.2,
again illustrating Die advantages of an NCJ approach.
Figure 10-2 3 illustrates the run of cIPjjcj with
Log (Delta Mass Non -conservation of Orbital Momentum) , dMKCi7 .
dM--
NCJ
1.371 + 1.110 dPppj
.723 .074
(10.4.2)
This r e 1 a t i. o r.
correlation of
by Figure 10-1
between the co
s also superior numerically to a similar
mass and frac
Liona1 pe
r iod
change
illustrated
Figures 10-
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n ~ o "
U /-
display
relations
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DELTA P/ DELTA T
Figure 10-26

272
cases, the contact systems are the most heavily influenced
by NCJ considerations. Here dMC is short for Log(dMC) .
Log(dP/P) = -3.043 + 0.641 dPNCJ (10.4.3)
.869 .088
dMC
-3.287 + 0.605 dM
.588 .061
NCJ
(10.4.4)
10.5 Summary Tables
Results of our study are summarized briefly in this
section. Table 10-1 lists a variety of effects which may
lead to period variation. Also listed are
1. The expected type of mathematical
representation for each effect,
subject to (2)
2. Conditions within each group leading
to different possible causes of the
effect under consideration
3. Possible causes of the effect; in
general, several causes will be
possible for each effect
Table 10-2 summarizes which of these effects is
most probable for a given system geometry and "best fit"
in the least squares sense. The table is used in the
following fashion. After data on a given system have been
analyzed, a "best fit" is selected. If the system geometry
is known (contact, semi-detached, detached), reference is
made to the appropriate column. If the system geometry

273
PERIOD CHANGE
TABLE 10-1
EFFECTS/EXPECTED
RE P RE S ENTATIONS
i-i feet
Representation
Conditions
r ot= si ble Cao re
1}
Apsidal ration
Periodic
Sccond a r y 1 ?. 5 ° cut c f
phase vita i-.: ary
Mecen'. :ic orbit# dc.jioo cf central
condensa*ion
2)
Light-tire effect
Periodic
Secondary i r. r he.se
with prir-ary
Distant third body
3)
Constant macs transfer
a)
Upward parabola
«2 < M*
NCJ < r.T# J-j r.isr transfer
b)
Pow;vvurd parabola
K? > D
NCJ < K,\ mss transfer; r.^ss
less
e)
Upward parabola
fj > ka
NCJ > X?, Lj naso transfer
d)
De» wr. vs rd pa r a bo 1 o
«2 < >!X
NCT > MR, l, mss transfer; rnsr.
loss
c)
Linear
r.2 < Mj
Mass transfer - NCJ temporarily
f)
Multiple upward parabola
•D < Ki
Mi ir.ann-Nal 1 r. ue
<)
Increased rass transfer
a)
Cubic ( + +)
>;2 > <‘l
Sever. NCJ ir.crcasiro.
b)
Cubic (+ *1
«2 < K,
Koti-tr. of NCJ to orbit and linear
rv.sa transfer; rass transfer
i r.srcasi a;
c)
Cubic (" *)
M2 > Mi
Mass ler.s/transier deer suing;
return NCJ and linear mass less
d)
cubic r ’)
n-, < M,
NCJ df»c;c..:. ir.g; mass lc.:s decreis-
ir.g
5)
Decreasing r-iss transfer
a)
Cubic {â–º ")
< Hj
Mass transfer slowing; NCJ
deer uniting
b>
Cubic C1 ")
M2 ' !:1
Distortion wave; nsyeretry per¬
turb cion; severe NCJ .c*cr* auir.g
c)
Cubic ~)
>‘2 ' b.
NCJ inerecsii.g; r.iss lots
increasing
d)
Cubic ( )
k2 » X1
Mass loss :-c.ri asir.97 i;cj
decreesin*
6)
r.cj
&)
Change in s? tioi*. or cccc 1 vr 2 t; on
of variation
Mass transfer
Storage of orbit norre r.i tun ►ro¬
tational raTinon.
b)
Change in sign c f varia¬
tion or «cede rati cm
of variation
Mas*. transfer
Circula!.o:: of r.iss and momentum
i:i contact system
D
Periodic instabilities
Several lin ar i.:
disccntirntit- -o i.n -*.y
other rep:esontation
Hail ir.stnbi) i ty; occasional
epoch- 0: cro.it inisit flew
caunri by other types cf :n-
stibi 1 i ti 0:i
e?
Systemic rass ]fss
Arv Piv.r.-/ard rey ‘e'-t.-nta¬
il on
—
dear s * Mode
9)
Pi s tortJon v.'óve
Pseu-Jv-j eriodic Ap4>:ar-
ar.ee or cubic
—
Spotted it a:-; asyuttiotr 1c
eclipses
10)
111
Ko r.«»s loss
Asymmetric eclipsen
Linear
Cuasi-poriodic
r£" *'» - b'-
CJ ’• 2; .OCHu-
Spu’.te.; . tars, disk Lriyhtr.cus
dir rib tió; ^ istion; ;
i;t r« ..: ; :.ij uaiin.j to cnissio:.
and ahsorptier*
id
Tidal braking
Any upward 2 opreccii*. .* t : i-n
>’>: .rr.dd .i..i distri¬
bution
Tidal forces

274
TABLE 10-2
SYSTEMS GEOMETRY/DOMINANT EFFECT/ASSIGNMENT CONFIDENCE
2
0
0
'
1
l’eriodic/parabol ic /sue out of ¡ r.ase
1-Aj
3c-L
1-A;
3a-r.; 12-::
1-A;
3.W«; 12-E
3-?.
sr-c
Pcriodic/par^bolic /--.••c out
: or' pha&e
i-A,
e-O;
3 b-C
i-A;
3*1— !>; C-E
1-A;
3d-:»; G-C
C-E
1
1
0
1
ivriodic/r.on-parabolic/sec
out of phase
1-A
1-A;
2e-b
1-A;
3i?-b; Gb-D; 12-E
¿C-C
C-E
3
0
2
3
Porioii.c/parabolic+/&ec i»-
phase
?-A;
3c-D
2-A;
3a-ü; 12-3
2-A;
3.1-F; 12-3
D-C
SD-E
Period:c/parabol*c /sec in
phase
2-A;
8-D;
3b-C
2-A;
3d-3; P-F.
2-A;
3J-D; t-r.
c-c
0
0
0
0
D-3
rprií^íic/r.or.-par aLol: c/sc r
in phase
2-A;
10-A
2-A;
3.:-L*
2-A;
3o-B; 6b-D
SC-3
C-E
0
9
4
13
1-C;
3-C-D
1-D;
3a-bj 13-F
i-D;
3a-B; Cb-D; 52-3
Period.c/?uralo 1 io'*/r.o sce
2-C;
3w- D
2-C;
3-»-:.; li-i:
2-C;
3.-Í-P; tL-D; 12-3
2-C;
S-D;
3b-D
1-L;
8-3; 2d-I.
1-D;
e-F; C; Tb-D
Pericdie/paraboiic“/r.o c
2-C;
8-D;
3 b-D
2-C;
¿-E; 3J-3
2-C;
S-3; 3d-f>; tb-D
C-D
2
0
C
2
Per iceic/non-paralo 1 ic/r.o see
1-D
2-£V
10-A
1-D
2-D;
30-3
1-D.
3o-F>; 6b-D
S C-E
C-3
0
0
0
0
Cubic 4 4
4a-D;
9-P
4 b-D;
9-E; 12-3
4h-D;
Cb-E; 12-3
D-C
SC-3
1
1
0
3
Cubic 4 -
5 b-Dr
0-b
Sa—B;
9 !.; 12-t
5a-B;
CbC; 12-3
3-3
s:-d
C-E
I
0
2
2
Cubic " +
4c- D;
9-0;
e-c
4 d-C;
8-E; 9-E
4d-C;
B-E; CL-C
3-D
SI -E
C-D
0
0
1
0
r-E
Cubic
5d-C;
9-D;
fc-C
5c-C;
S-3; 9-2
5c-C;
6-K; 6b-C
sc-s
c-C
5
9
9
21
Farii-olic 4
3c-C;
9-D
3a-b;
12-H
3a-B;
C-b-C; 12-E
D-D
SI-r
c-c
2
8
4
10
Parabolic
3b~D;
8-C
3d-fc;
8-3; 3b-E
34-!i;
£b-C; 8-E
-
i_-C
c-:>
4
3
0
17
Linear
10-A
3e-B;
t « 3a-E
3c-A
SD-C
4
2
8
12
21 Xo«5«lar cha;u;•?s
7-A
7-x;
9-E; 3-A
7-1»; <
Cb-3
sc-r
c-:»
C
4
0
c
Multiple upward parabola*
—
3&-A;
;2-;;
—
D-r.
5>.\
c-c
•best tit-
DL7AC1ÍLD
f. l:m-
DE/ACih-D
CONTACT
C
Cor.:
!• —- r>:: t
c--hi: í
t—sor.a
I — ii-W

275
is unknown (other), an indication is presented for each
type of best fit as to which system geometry is most
likely, based upon the distribution of stars under survey.
Once the appropriate column has been selected, the various
possible effects are listed along with a rough indication
to their possibility, which is given below:
"Confidence" Level Designation
all A
most B
half C
some D
few E
For example, a semi-detached system has a "best fit" given
by an upward parabola. Effect 3a is seen to be mass
transfer with NCJ "B," or most of these systems will have their period varia¬
tions explained by this effect. Effect 12 (Tidal Braking)
is also found in this column, but it will be the dominant
effect in only a few, if any, systems (E rating). Also
provided in this column is the number of systems (9) which
fit this designation out of our total number of systems.
10.6 Statistical Trends
Out of a total sample of IBS close binary systems,
four systems displayed definite evidence of apsidal motion.
Eight systems displayed pronounced light-time effects. In

276
all, 48 systems could be represented by some periodic fit.
Similar statistics can be obtained for whatever groupings
the reader desires by an inspection of Table 10-2. We can
draw the following general conclusions:
1. Apsidal motion is a relatively rare,
but easy-to~distinguish cause of
period variation.
2. Periodic representations can be applied
to about one-fourth of the systems in
this study. There is some preference
for semi-detached systems.
3. The parabolic term is significant
(larger than its error) in about two-
thirds of the sample. Cubic
representations are significant for
32 systems. Of the semi-detached
systems, only two binaries have para¬
bolic terms which are not significant.
4. The concept of constant mass transfer
in semi-detached systems seems well
established.
5. Abrupt changes in period offer repre¬
sentations or alternate representations
for one-sixth of the systems. This
representation slightly favors contact
systems.

277
6. A number of "detached" systems which
do not display periodic behavior under¬
go some other type of period variation.
Isotropic mass loss could account for
all of the detached systems with de¬
creasing periods. Perhaps the observed
variation of the others is caused by
some sort of non-mass transfer process
such as a distortion wave of the light
curve.
These trends seem well established on the data set used in
this study. A continuing study of the times of minimum
light for close binary systems is planned to extend and
verify these results.

APPENDIX ONE
OBSERVATIONS
The observations for all of the systems used in
this study are listed system by system on the following
pages. For each system, the first column contains the
number of observations while the second column lists
either the time of primary or secondary minima in terms of
JD heliocentric. The type of observation is indicated in
the third column. Table 5-1 contains the key for this
column. The cycle count is given in column four for each
observation from a selected initial epoch as given for each
system in Section 5.6. Column five contains the observed
minus computed value from a linear ephem.eris which in most
cases is given by either the GCVS or GPC. This value has
the units of days. The final column contains the residual
of each observation in cycles. This residual is in all
cases from a parabolic fit, except for systems listed in
Section 6.2 which follow a periodic representation, in which
case it is the periodic fit residual. Cubic and linear
residuals are available upon request.
278

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