Group Title: formation of a contact binary star system
Title: The formation of a contact binary star system
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Title: The formation of a contact binary star system
Physical Description: viii, 144 leaves. : illus. ; 28 cm.
Language: English
Creator: Mullen, Elisabeth Fentress Ferratt, 1942-
Publication Date: 1974
Copyright Date: 1974
 Subjects
Subject: Double stars   ( lcsh )
Eclipsing binaries   ( lcsh )
Astronomy thesis Ph. D   ( lcsh )
Dissertations, Academic -- Astronomy -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 141-143.
General Note: Typescript.
General Note: Vita.
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Bibliographic ID: UF00098344
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000582495
oclc - 14116821
notis - ADB0870

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THE FORMATICN OF A L :,C' lIiNArA STAR SYSTEM


EIISABETI;I ENTlVESS FErA4T:'f IULT 3















A DIESL ETATICN I~RESLNTI TO T'H GIADU.nTE? COUIKCl
01' !i UTIVf;SIY OF Fi'ORI'/, IN PARO TA]. ILFMLLENT OF TE" R'ULMLEN'tS
FOE Tie DEC.TCRE OF Dt.i.:TO, O' EI!ILCSO]HY















UNiV.CRS] TY OF FLORIDA


1I97q










ACKNOWLEDGMENTS


There are many people without whom this dissertation could never

have been completed. To my advisor and Chairman, Dr. Kwan-Yu Chen,

goes my undying gratitude, not only for his support, advice, and

encouragement during my time as an astronomy graduate student, but

also for accepting me as a student and giving me the opportunity to

w"rk on this problem. Dr. F. B. Wood has been an inspiration and

a source of both wit and wisdom. I thank him for serving on my

committee and for providing me with the historical perspective one

gains in working on the Card Catalogue of Eclipsing Binaries. The

several efforts of the other members of my committee, Dr. A. G. Smith,

Dr. B. S. Thomas, and Dr. E. D. Adams are also acknowledged with due

appreciation. Dr. H. L. Cohen is also to be thanked, both for attending

my preliminary oral examination and for many enjoyable as well as

useful discussions.

It is also fitting to acknowledge my parents, Thomas Littelle

Ferratt and Elisabeth Fentress Ferratt, not so much for their fine

efforts in raising me, or even for their sacrifices toward my education,

but rather for whar they have given me in spirit, in character, and in

strength by their own examples. Not only have they been steadfast in

support and wise in counsel, but I truly believe that my mother, in

particular, is uniquely responsible for -ry interest in the sciences, my

enthusiasm for my work, and my basic philosophy of both life and science.

For all this, these poor words are paltry thanks.


i. _










Many friends and associates have also rendered invaluable aid;

John P. Oliver, in particular, has been most generous with his time.

I have reaped both profit and pleasure. from our discussions. It is

with deep gratitude that I acknowledge the assistance of Dr. J. Ziol-

kowski, who generously provided both program listing and advice which

were incorporated into the modification of the single star program

which was kindly provided by Dr. B. Paczynski. Fellow graduate students

are also to be thanked, not only for their daily insights and suggestions,

but also for making the astronomy department a pleasant place in which

to work; particularly, I would like to mention Richard B. Pomphrey,

Christopher A. Harvel, Dr. Raymond H. Bloomer, E. Whit Ludington,

Frank P. Maloney, and Andrew S. Endal. Dr. Thomas L. Bailey of the

physics department has also been a source of assistance, as well as

an excellent teacher from whom I learned a great deal.

Two friends outside this department have also been of immeasurable

assistance. For their willingness to educate an ignorant "user", for

their vast talents, both pedagogical and logical, and for their under-

standing and support as very special friends, my deepest thanks go to

Charles John Young, Jr. and John G. Schudel, III.

Computing funds through a grant from the Northeast Regional Data

Center are most gratefully acknowledged.

It is a pleasure to thank Mrs. Jeanne Kerrick for typing this

dissertation with such care, for her kindness and support during the

writing:, and also for her excellence and assistance during the last year

in her role as department secretary.










The figures in this dissertation were. drawn by ry husband, Joseph

Matthew Mullen, and it is with deep affection and thanks for performing

this arduous task that I dedicate this dissertation to him. It seems

far too insignificant a gesture to acknowledge all that he has done and

said and sacrificed on my behalf. He has truly been my advisor, my

friend, my strength in time of need; there are no words with which to

describe the completeness of his generosity or the depth of his under-

standing, no numbers with which to count the blessings of his daily

presence.


iv









TABLE OF CONTENTS


Page

ACKNOWLEDGMENTS . . . . . . * * ii

ABSTRACT . . . . . . * * * * * vii

CHAPTER I: INTRODUCTION . . . . ............ 1

CHAPTER II: THE I URSAE MAJORS TYPE ECLIPSING
BINARIES

A. Observational Background . . ........ 3

B. Theoretical Models for
W Ursae Majoris Systems . . . . .. 5

C. The Rotation Problem . . . . .. 14

CHAPTER III: THE PRE-CONTACT CONFICURATTON
AND SELECTION OF INITIAL PARAMETERS

A. The Suggested Formation Mechanism . . .. 19

B. Selection of I.Aitial Parameters ....... 21

CHAPTER IV: DESCRIPTION OF THE COMPUTER PROGRAM

A. Calculation of Evolutionary Sequences
of Stellar N odels ............ 24

B. Binary Star Modifications for the
Mass-Losing Star ......... . 29

C. Binary Star Modifications for the
Mass-Gainir g Star . . . . . . .. 34

CHAPTER V: THE MASS-GAINING STAR

A, Ionteral Structure of Mass-Gaining
St . . . . . . . . .. . 38

B. Relaxarion Calculations . . . . . 49

CHAPTER VI: THE FORMATION OF A CONTACT BINARY. . .. 60












CONTENTS continued .


APPENDIX A .


APPENDIX B ....


REFERENCES . . .


BIOGPCPHICAL SKETCH


SIGNATURE PACE


. . . . . . . . . . . .


. . . . . . .


. . . . . . . . .


. . . . . . . . . . . I


. . . . . . . . . . . .









Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the Requirements
for the Degree of Doctor of Philosophy



TIE FORMATION OF A CONTACT BINARY STAR SYSTEM

By

Elisabeth Fentress Ferratt Mullen

August, 1974



Chairman: Kwan-Yu Chen
Major Department: Astronomy


The process of forming a contact binary star system is investigated

in the light of current knowledge of the W Ursae Majoris type eclipsing

binaries and the current rotational braking theories for contracting stars.

A preliminary stage of mass transfer is proposed and studied through the

use of a computer program which calculates evolutionary model sequences.

The detailed development of both stars is followed in these calculations,

and findings regarding the internal structure of the star which is receiving

the ;aass ore presented. Relaxation of the mass-gaining star is also studied;

for these stars of low miss and essentially zero age, the star eventually

nettles to a state very similar to a zero-age main sequence star of the

new' mass.

A contact system was formed through these calculations; it exhibits

the general properties of a W Ursce Majoris system. The initial masses

selectcJ for the calculation were 1.29 M and 0.56 M An initial mass
D 0D










transfer rate of about 10-10 solar masses per year gradually increased

to about 10-8 solar masses per year. After about 2.5 X 107 years, the

less massive star filled its Roche lobe and an initial contact system

was obtained. The final masses were 1.01359 M1 and 0.83641 ~1 The

internal structure of the secondary component is considerably different

from that of a main sequence star of the sane mass.

















CHAPTER I

INTRODUCTION



This study was undertaken in an effort to determine the likelihood

that a theoretical model based on the contact assumption can explain

the observational anomalies of the W Ursae Majoris type eclipsing

binaries. In view of the fact that the existence of contact binaries

has been questioned in the past, it seemed appropriate to consider

first the theoretical possibility of forming such a system. It is to

this question which the present work speaks rather than to the problem

of matching individual systems.

In any theoretical pursuit it is desirable to have in mind some

object or class of objects, present and observable in the real world,

which may eventually give the investigator confidence in his work by

exhibiting certain properties which lie predicts. It is with this end

in mind that considerable effort is devoted to a discussion of the

observational properties of and previous theoretical efforts regarding

the U Ursae Majoris systems; indeed, it was the study of this material

which first inspired the work of this dissertation.

Other chapters dealing with background provide a description of

the assumptions and calculation techniques employed in the computer

code which produced the actual results, and a discussion of the problem









of rotational braking and stellar winds insofar as these topics pertain

to the proposed mechanism of contact binary star formation.

This proposed formation mechanism involves a preliminary stage

of mass transfer, and the investigation of this phase has required

stellar model calculations for the recipient as well as for the donor

of this mass. As such calculations have not been previously performed,

considerable discussion of the properties of this mass-gaining star

is included.

Finally, the resulting initial contact system is described and

suggestions for following its subsequent evolution are made. An

attempt is also made to place this work in the context of the obser-

vational material.


1














CHAPTER II
THE W URSAE MAJORS TYPE ECLIPSING BINARIES

A. OBSERVATIONAL BACKGROUND



For a variety of reasons which will be elucidated in this chapter

and the following ones, observational confirmation of the work to be

presented in this dissertation will involve eclipsing binaries of the

W Ursae Majoris type (hereafter referred to as W IUMa systems). It

therefore seems appropriate to review the basic properties of these

systems in some detail.

In a review paper on W UMa systems, Kraft (1.67) describes some.

of the basic observational features of these systems. He points out

that the two components 6f a W ULa system generally have nearly the

same spectral class, and the average difference in their magnitudes

is 0.25 mag. Mass ratios range from 4:1 to nearly 1:1 with a mean of

2:1. The range in masses of primary (more massive) components is

2 Moto 0.5 H Averages for masses are 1.2 M0 for the primaries and

0.6 M for the secondaries. Periods range from 2d0 to 0d25 with a

sharp cut-off at the lower limit. Eggen (1967) goes into some detail

in pointing out that this period cut-off is not due to observational

selection. Space motions of W UMa systems are typical of disk population

F and G dwarfs. According to Eggen, there are several of these systems

which are members of galactic clusters (e.g. TX Cnc is a member of

Praesepe). He also indicates that while most W UMa systems consist









of main sequence stars, there are some which are pre-main sequence and

some wtich are post-main sequence. It is also very important to note

that W UMa systems do not obey the main sequence mass-luminosity relation.

In particular, the more massive star is generally underluminous, while

the less massive star is generally overluminous. Eggen also says that

there is evidence that W UMa systems are less numerous among brighter

stars, although Kraft says that this effect will be less pronounced if

one takes into account the fact that the primaries are underluminous.

One important observational feature of W UMa systems is the period-

color relation which these systems appear to obey. Basically, this

relation simply implies that bluer systems tend to have longer periods.

Evidence for such a relation is presented by Eggen (1967); its existence

is one important reason for considering W UKa systems to be in contact.

The light curves of W UMa systems have many interesting features.

In addition to the well known continuous light variation, which is

due to distortion of the stars, and the approximately equal depths of

the minima, there are three unusual phenomena. One is the reddening

during eclipse which occurs in about two-thirds of these systems. This

means that the facing hemispheres of the stars are brighter than those

which are turned away. Attempts have been made to attribute this

phenomenon to the reflection effect or to gravity darkening, but Mauder

(1972) argues that neither effect is large enough. The second interesting

effect is that short term fluctuations are often found in W.UMa systems:

These occur within a single night, and thus the scatter in the light

curve is often considerable. In many systems there are also differences

in brightness at the two maxima. The third effect is the period changes













which occur over a single year in many of these systems. A partial

list of such events has been compiled by Herc=eg (1970). For example,

both ER Orionis and W Ursae Majoris itself have exhibited three rapid

changes during the last thirty years.

Next we turn to the question of whether or not W UMa systems

actually are contact binaries. This discussion is primarily due to

Mauder, who considered the effects of the different models used in

the rectification procedure on the resulting elements. Mauder notes

that models generally used are not for contact systems, and that the

elements depend strongly on the rectification procedure. For this

reason, he attempted to account for effects of distortion, which will

be much more pronounced in a contact system, using Fourier analysis

of the light curve. He found that this does indeed lead to values of k,

the ratio of the radii, which imply contact. The important feature of his

results, however, is that they show much better agreement with the spectro-

scopic observations. This lends support to the contact hypothesis,

although the question is certainly not settled observationally. Indeed,

Mauder seems to feel that a better theoretical model is needed before

the question can be definitely resolved.



B. THEORETICAL MODELS FOR W URSAE MAJORIS SYSTEMS



Most of the current models proposed for W UMa systems are based on

the nodel developed by Lucy (1968). This model was proposed in an attempt

to deal with a problem first noted by Kuiper in 1941. He showed that it











is not possible to have a zero-age main sequence contact system unless

the two stars have equal masses. In order to see that this is so, one

has only to note that both theory and observation indicate that for the

region of the Hertzsprung-Russell (hereafter denoted by H-R) diagram

occupied by W UMa systems, the mass-radius relation can be expressed as


0.7
R/R = (M/Mg )



while a contact system must obey the relation


0.46
R/R1 = (M2/ )



It is immediately clear that the only solution to these two relations

is that of equal masses and radii for the two components of the system.

Now this result is in complete disagreement with the observation; indeed,

W URi systems are striking in that they never have exactly equal masses.

Lucy proposed that this problem might be resolved if the two stars had a

common envelope. The difficulties, he found, remained if the envelope

were radiative, but for a convective envelope, Kuiper's argument was

found not to hold. This was due to the fact that such a common con-

vective envelope would require an energy exchange between the two stars.

This, in turn, would cause the radii to be considerably different from

their main sequence values. Thus Lucy was led to construct a model of

1W IMa systems, which consisted of two spherical stars of the same chemical

ccRamosition with convective envelopes having equal values of K, the

adiabatic constant. This required determining a new mass-radius relation

based on this assumption of equal adiabatic constants. The resulting











plots of mass vs. radius for various adiabatic constants reveal two

distinct branches of different slopes. This is due to the two different

dominant sources of nuclear energy. When the contact mass-jadius relation

is added to this, one sees that it is indeed possible to make suitable

choices of mass provided one star is on each of the two branches of the

curve. Thus, Lucy was forced to consider systems in which the more massive

star is dominated by the CN cycle while the less massive derives its

energy from the p-p chain.

In spite of its attractive features, this model has faced several

problems in comparison with observations. First of all, the original

model did not obey the period-color relation at all. In attempting to

improve this situation, Lucy found that by suitable choices of energy

generation rates, he could obtain a period-color relation, but it did

not span the observed range, and the agreement was not good. In these

models, however, Lucy used very early opacities, which are quite inaccurate

in the region of the H-R diagram which is populated by W UIa systems.

Another problem arose when Lucy calculated theoretical light curves based

on this model; namely, the deeper minimum was due to the eclipse of the

more massive star. Observations, however, often show the reverse to

be the case. In addition, it should be noted that the rxCquiremcnt that

one component have the CN cycle as its dominant energy soerr;t while the

other is dominated by the p-p chain presents considerable difficulties,

since there are a number of W UMa systems in which both l .far have masses

small enough to require the dominance of the p-p chain energy source.









Before discussing the attempts to treat these problems, perhaps

it would be worthwhile to mention another paper which relates to the

Kuiper problem. Whelan (1970) considered the pre-main sequence case

of an equal mass contact system. He found that such a system is unstable

to mass transfer, and that if one star transferred mass to the other, the

systs would tend to continue this transfer with the now less massive star

losing mass to the more massive star. Thus, the construction of a zero

age, unequal mass, contact model appears to be both relevant and necessary.

Perhaps the most obvious thing to try in attempting to improve the

Lucy model is the use of more accurate opacities. Moss and Whelan (1970)

did this and found that,using conventional values for energy generation

rates, they could not construct models with a normal population I chem-

ical composition. They did succeed by using an extreme population I

composition, but of course, this is not consistent with observations.

There are W UMa systems which are known to belong to older clusters and

could not have such compositions. These authors also tried alte:ring

the energy generation rates, but in order to obtain viable modeir, they

had to increase the CN cycle energy generation rates far beyond the

range of experimental uncertainty. An attempt was m rdn to caluate the

effects of altering mixing length theory, but a)ppscr tly nhiit did not

have any significant influence on the results. Better agraciient with

the period-color relation was achieved, but the models did not span

the observed ranges for periods, colors, or mass ratios.

Further work along this same line was reported by Whelan (1972),

who went into considerable detail in discussing the effects of various

parameters on the models. He treated chemical composition, mixing










lengths, various methods of simulating energy transfer in the convective

envelope, and energy generation rates. Ey considering the case of

superadiabatic energy transfer, he was able to predict that the secondary

might be hotter than the primary as is sometimes observed. The dif-

ferences in temperature depended on the mass ratio. The main result,

however, was much the same; an extreme population I composition was

still required in order to find a suitable model. Also, it remained

impossible to span the observed range of properties.

The models discussed so far have dealt only with zero-age W UMa

systems, Perhaps at this point it is appropriate to consider other

possibilities. H!azlehurst (1970) pointed out that any system whose

total mass is less than 3.5 M0 could not be a zero-age system if Lucy's

results were correct, since both stars would then derive their energy

from the p-p chain. He then checked both the less massive and the more

massive stars and found that while the secondaries appear to be of

zero age, the primaries of such systems have radii which are too large

and therefore might be evolved stars. In fact, lie found that the

larger the radii, the larger the mass difference for this special group

of W UMa systems. The trouble with this, of course, is that the first

stage of mass exchange which occurs after an evolving star expands

to fill its Roche lobe proceeds very rapidly. Thus, some of these

systems should, it would seem, show the less man;sive scar to be the

more evolved.

Post-main sequence evolutionary model sequences have been calculated








by Miss (1971). He used the Lucy model for his initial system, and

he selected his parameters carefully in order to obtain a contact system

rather than a semidetached system. For example, if the mass ratio were

chosen too large, a contact system could not be formed. If, however, it

was too small, the system would overflow its outer critical surface, and

such cases were not considered in this paper. In addition, only the

evolution of the initially more massive star was followed; that is, no

mass-gain models were attempted, although the possibility was suggested.

The resulting evolutionary tracks did lie within the observed region of

the period-color diagram and the H-R diagram. In addition, the observed

overluminous secondary with underluminous primary was obtained. The

major problem here is that in this same region of the H-R diagram are

sc e W UMa systems, which belong to clusters much too young for these

systems to have evolved off the main sequence. Also, W UMa systems are

much too numerous compared with the other types of binary systems for

a large fraction of them to be explained in this way. Surely, of course,

this explains some observed systems, but the large majority of W UMa

systems still remain to be dealt with.

At this point, therefore, we return to consideration of the Lucy

modal. Several of its problems have alre-edy been outlined, but others

remain to be discussed. Robinson (1972) calculated ratios of luminosities

and surface brightnesses as functions of maIc ratio using the Lucy model.

He then made a comparison with observations as listed in the Catalogue

of Kopal and Shapley (1956) which showed lo relation at all between

theory and these observations. Despite the variability in accuracy of

this compilation of data, Robinson's findings do seem to give a clear

indication of difficulties with the Lucy model.










Another serious observational difficulty is discussed by Lauder

(1972), who plotted pri-ary mass vs. mass ratio of 18 W UMa systems.

He then plotted curves corresponding to various adiabatic constants.

If Lucy's model is correct, then all. W UMa systems should be on or above

the curve corresponding to lC:g K =- -4. This is due to the fact that a

star has no outer convectiva zone below this value. One immediately

notices the observations fall uniformly belo; the limiting Lucy model

curve; the region in which the observations occur is populated by curves

for various pairs cf adiabatic constants. This appears to imply that the

two stars have different adiabatic constants.

The implication of this work is that models should be attempted

using different adiabatic constants. This has been tried by Bierman

and Thomas (1972). Allowing for different adiabatic constants required

a luminosity exchange; the amount was determined by requiring that the

contact condition be fulfilled. These models removed the previous

difficulties with chemical composition and nuclear generation rates. In

addition, they apparently fit the period-color relation fairly well, and

they can be forced to span the observed range of properties. T:o prolelms

remain, however. The light curves based on this maidc do inot give r.ninma

of roughly equal depth as is always the case for VW Uila systems, and the

eclipse of the Imor massive star is still the deeper. In otLher words,

the theoretical iLght curves have the same sort of problem as Lucy's,

only it is more pronounced in this case. In addition, no physical

mechanism has been proposed which could account for the required luminosity

transfer. This model, in essence, is a trial and error model built by









adjusting parameters.. It does rot appear r( have a physical basis. A

later attempt by Bierman and Thman C12973) limited the luminosity

exchange to the outer regions of the convective zone and then followed

the evolution of the system. The results chowed that the contact zone

moved deeper as the stars evolved and thus reduced to the usual Lucy

model.

Whelan (1972b) has made several remarks concerning the work described

above. Robinson's objections, he states, can be overcome by allowing the

secondaries to be hotter, while Mauder's objections apparently did not

include the entire framework of Lucy models. The problem associated with

the light curves predicted by the Bierman and Thomas models was also

mentioned. In this paper, several additional tests of Lucy models were

mentioned; these included the effects of opacity changes, which were

found to be very small, and the effects of including rotation, which can

produce an overall cooling.

Further work along this line was carried out by Moss and Whelan

(1973); this modification permitted energy transfer through the super-

adiabatic zone and permitted an entropy gradient between the stars, thus

allowing for different values of adiabatic constant for the two stars.

These models were somewhat more successful, although still not entirely

satisfactory.

Two papers have appeared recently in which post-main sequence mass

exchange was studied in relation to contact systems. The first of these

is by Hazlehurst and Meyer-Hofmeister (1973). Here the initial model

was a contact system with equal adiabatic constants. One of the major

results of this study is that the system evolves very rapidly toward an









equel mass configuration. Thus, the products of this process will not be

i Ula systems.

A paper by Vilhu (1973) also treated rmass transfer with the primary

losing mass until the secondary filled its Roche lobe. After that point,

mass and energy exchange between the components was determined by requiring

that the contact condition be fulfilled. Vilhu followed only the initially

more massive star in his calculations, assuming as is customary that the

secondary will remain a normal zero-age main sequence star. lie continued

his calculations until the secondary become hotter than the primary, and

remarked on the similarity of his findings to those of Whelan and Biermann

and Thomas regarding energy transfer between the two stars. In general,

however,these systems were, in their final state, far too old and too

evolved (in the sense of central hydrogen depletion) to provide a suitable

model, for most W UMa systems.

Some enlightenment regarding the Lucy model and W UMa systems can

also be obtained by considering the recent papers involving synthetic

light curves. A number of workers have developed computer codes to pro-

duce such light curves, and these have produced attempts to match various

W Uia systems. The first of these (excluding the original ones by Lucy)

were computed by Mochnacki and Doughty (1972a, 1972b). They noted that

for the systems which Binnendijk (1970) called A-type systems,a very

good fit was obtained using a Lucy-type model. On the other hand, the

W-type systems, which exhibit the various types of anomalous behavior

such as period changes, hotter secondaries, etc., could not be matched

at all. Lucy (1973) himself did similar, but less efficient, calculations

with similar results. lie found some indications that the cause of the

difficulties with the W-type systems might be that the common envelope








was quite shallow. Lucy suggests that the difficulties in fitting

W-type light curves might cause one to consider abandoning the common

convective envelope model if it were not for the mass-luminosity problem

which seems to be explained fairly well by the energy exchange which

results from the common envelope. Still a third synthetic light curve

analysis of these systems was reported by Wilson and Devinney (1973).

They also analyzed several A-type systems and found substantial agree-

ment with the other two groups.

Mochnacki and Whelan (1973) have since extended the technique of

Mochnacki and Doughty to permit temperature differences between the

components. This has made it possible to synthesize the symmetrical

features of W-type systems, although the asymmetries still remain to

be explained.



C. THE ROTATION PROBLEM



There is one anomalous observational feature of W UMa systems

which was not emphasized in section II.A.; namely, these stars are rapid

rotators. When one considers the normal main sequence distribution of

rotational velocities with spectral class, the most striking feature

of the distribution is the rapid drop at F5 and the very low velocities

for all later type stars. It is generally assumed that this rapid

rotation of W UMa systems is due to synchronization of rotational and

oribital periods due to tidal friction. Only recently (Huang, 1966),

however, has there been any attempt to relate the problems inherent in










the rotational velocity distribution to the binary star situation. It

is, of course, clear that a contracting, rotating star must increase its

rotational velocity as it contracts; this is required by conservation

of angular momentum. Observed velocities are, however, much too small

to fit this picture.

Before discussing the implications of this problem for binary stars,

it is necessary to summarize briefly the basic mechanism which will

account for this loss in angular momentum. Huang refers to the earlier

suggestion by Schatzman (1962), who considered the effects of matter

ejected from the surface of a star in the presence of a magnetic field.

This theory was later treated in much greater detail as a stellar wind

theory by Mestel (1968). Matter leaving the surface of a star will be

forced by the magnetic field to co-rotate with the star as long as the

ratio of magnetic to thermal energy density is large enough. Beyond this

critical point, the field is dragged along by the gas flow. Eventually,

a very small amount of matter is able to carry away a large amount of

angular momentum.

It is, of course, still necessary to explain why this theory should

be of special importance In relation to W'UMa systems; in order to do so,

it is necessary to consider how the theory will predict the shape of the

distribution. Schatzman, invoking Parker's (1955) theory of surface

activity, suggested that since the presence of an outer hydrogen convective

zone played a key role in this surface activity, the presence of such zones

at the surface might also mark the cases in which the braking mechanism









is effective. Thus, one might axpEct G, K, and M stars, which retain

thick outer convective zones after they reach the main sequence, to

rotate very slowly, while 0, B, and A stars, which lose their outer

convective zones very early during the pre-.-in sequence contraction

phase, would rotate more rapidly. Not only is this prediction borne

out by the observed rotational velocities, but additional confirmation

is provided by Wilson (1966). Here, the presence of the II and K lines

of Call in emission is stated to imply chromospheric activity due to

a convective zone.

This latter confirmation is particularly interesting, since H

and K line emission is a spectral feature of many W Ueta systems. Thus,

one might expect a considerable rotational braking effect for these stars.

Indeed, Mestel did consider the implications of his stellar wind theory

for binary protostars. His hypothesis was that fragmentation of gas

clouds into a cluster of protostars might produce some for which the

separation and radii are comparable. Since the stars will still contract

further, however, they cannot become close binaries unless they lose

orbital angular momentum. The contracting stars forming th; pair can,

of course, lose rotational angular momentum according to Mestel's theory.

Eventually, the rotational angular momentum will be much less than the

orbital angular momentum. Since they are coupled by a tidal force which

tends to produce a synchronous orbit, however, the rotation will be increased

at the expense of the orbital angular momentum, and the two stars will move

closer together. This suggested process not only provides a way of forming

a close binary system, but it also explains the rapid rotation of W UMa

systems.









There remains some question regarding the time scale for this process.

Mestel states that the time for synchronous rotation to be restored by

tidal friction is of the order of (AIR)6 years, where R is the stellar

radius and A is the separation. The time a star spends in the Hayashi

phase of its contraction, where the braking mechanism will be most effective

is on the order of 106 years. Thus, if A/R is initially much greater than

ten, this process will not be effective, which leaves some pairs of stars

to assume large separations. Huang, in discussing this point, suggests

an upper limit for the initial separation of about 100 RE He also takes

into account the fact that one can expect this braking to continue after

stars of the size and spectral type of W UMa systems have reached the

main sequence. In fact, he mentioned three stages; namely, pre-main

sequence with decreasing radii, main sequence with constant radii, and

the contact situation. The second phase would result in two main sequence

stars rhich move together until they reach stage three and form a contact

system. After that time, any further braking will require a new technique

for decreasing the angular momentum of the system. Huang then suggested

that the only possible way for this to occur is for the less massive star

to transfer mass to the more massive star.

In spite of the fact that this suggestion appeared in 1966, only

one paper dealing with W UMa systems has even mentioned the effects of

this process. Moss (1972) did allow the two stars to come into contact

and continue to move closer together. He did not, however, include any

mass transfer. Instead, he assumed that the two stars would continue to









move together, with their surfaces corresponding to a single equi-

potential surface. If this overcentact is large enough, the system

vill eventually lose mass through the outer Lagrangian point. This

modification did extend the range of viable, Lucy-type models very

slightly, but no attention is given to the actual process of forming

a contact system. It is also interesting to note that Moss and Huang

have actually made two entirely different assumptions regarding the

behavior of stars which have just become contact systems. One is immed-

iately inclined to wonder which is correct. More to the point, however,

is to inquire as to how this contact system actually forms in the first

place. Will two stars of arbitrary mass move together and form a contact

system without any sort of intermediate stage? This question will be

answered in the next chapter.















CHAPTER ITI.
THE PRE-CONTACT CONFIGURATION AND SELECTION OF INITIAL PARAMETERS

A. THE SUGGESTED FORMATION MECHANISM



As was mentioned in the previous chapter, it is of considerable

importance to determine the validity of assuming that two stars of arbitrary

mass can actually foim a contact system simply by moving together. Pre-

liminary calculations designed to answer this question were carried out

using the extremely simplified technique described below.

After selecting two masses and an arbitrary initial separation, the

radii were calculated by assuming that both stars obey the main sequence

mass-radius relation. The Roche critical radius of each star was obtained

from the usual approximate formula (Paczynski, 1966),




rc = A { 0.38 + 0.2 log (ML/M2) }



where A is the separation and M and 2,, are the masses of the stars.

The critical radii were then compared with the main sequence radii. If

neither star filled its Roche lobe, the separation was decreased by some

arbitrary small amount, and the new critical radii were compared in the

same fashion. This procedure was carried out for a wide range of masses

and mass ratios. When the initial mass ratio was exactly one, the result

was, as Huang assumed, a contact system. For all other mass ratios,










however, the more massive star invariably filled its Roche lobe first.

Since V UMa systems are observed to avoid a mass ratio of one, it appears

that the formation of a contact system by the rotational braking mechanism

will require the existence of a preliminary semi-detached system. Any

further braking or evolutionary expansion will then produce mass transfer

from the more massive to the less massive star.

In an attempt to predict suitable initial mass ratios for observed

systems, rough calculations, intended to follow this irass transfer.

qualitatively, were also performed. A small amount of mass (about 0.1.%

of the mass of the star) was transferred and new radii were calculated,

still assuming the applicability of the main sequence mass-radius relation.

Applying conservation of angular momentum, a new value of the separation

was calculated. New critical radii were then obtained and the results

compared as before. Two interesting points emerged from this. First,*

the amount by which the main sequence radius exceeded the Roche critical

radius increased with each iteration. This indicated that the mass transfer

should proceed on a time scale short enough to permit stellar model cal-

culations to follow the mass transfer without specific inclusion of the

details of any further angular momentum loss.

The second interesting point is that in these rather crude cal-

culations, the less massive star filled its critical radius before the

mass ratio reached unity. An assortment of total masses and initial mass

ratios were considered; these results were used in selecting the initial

mass ratio for the model calculations.









B. SELECTION OF INITIAL PARAMETERS



Chemical composition of the stellar material was the-first para-

meter to be selected. This would seem to be of considerable importance,

since rather small variations in the composition can make noticeable

differences in the parameters of the initial models. For example, early

test runs of the program (which will be described in the next chapter)

showed that for a 5 1b star, altering the hydrogen content, X, the helium

content, Y, and the heavy element content, Z, produced the following

differences in bolometric magnitude, '.ol, effective temperature, Te,

and radius, R/PI



X Y Z Mbo logTe R



0.70 0.27 0.03 -2.033 4.2234 2.707

0.75 0.223 0.027 -1.837 4.2081 2.653



On the other hand, the scatter in any observational plot involving the

above quantities is sufficiently large th-at it is not possible to dctc:r-

mine chemical composition by comparison of theoretical and observed curves.

Thus, one must find some other basis on which to make the selection.

Since the actual chemical composition cannot be observationally deter-

mined; it seemed reasonable to choose values which would facilitate compar-

ison with other theoretical calculations. A recent paper by Worden and










Whelan (1973) includes a thorough analysis of the observational material

available for W LTMa, the proto-type W Ilra system. In addition, it includes

a reasonably successful simulation of this particular system. The avail-

ability of such a combination of observational and theoretical material

made it seem worthwhile to select initial parameters which would facilitate

eventual comparison with this work. Hence, the adopted values of chem-

ical composition were 75% hydrogen, 22.3% helium, and 2.7% heavy elements.

Selection of the total mass was also based on the paper by Worden

and Whelan; they found a total mass for W UlMa of 1.85 Mg. This parti-

cular system is particularly suited for comparison, since it is very well

observed and exhibits all the anomalous types of behavior reported for

this type of eclipsing variable. Not only has it undergone several rapid period

changes in the last several decades, but a flare has also been observed

in conjunction with one of these period changes (Kuhi, 1964).

It seems reasonably clear, in view of the predicted phase of mass

transfer, that the initial mass values should not correspond to the

observed values. Hence, the preliminary calculations described in the

first section of this chapter were used to predict the initial mass

ratio which would correspond to the observed final mass ratio. Admittedly,

this will be a crude approximation, not only because of the assumptions

on which the calculations were based, but also because the end product

of this stage of mass transfer will, at best, be only an initial contact

system. It seems quite likely that if a contact system is, indeed, the

result of these calculations, as predicted by this theory, then mass




23




transfer need not stop with the formation of the contact system. Thus,

the initial contact system to be. calculated here would not necessarily

be expected to be identified with an observed system. Nevertheless,

for lack of any more accurate prediction method, the initial masses were

chosen, based on these preliminary calculations, to be 1.29 for the

tore massive component and 0.56 M for the less massive.

Having set up the problem to be solved and selected the initial

parameters, we now turn to a description of the computer code used in

the calculations to be described in the chapters which follow.















CHAPTER IV
DESCRIPTION OF THE COMPUTER PROGRAM

A. CALCULATION OF EVOLUTIONARY SEOTUENCES OF STELLAR MODELS



In attempting model calculations for binary -'tar systems, it is

both necessary and valid to begin with a basic, singic star trealta:ent.

Needless to say, the portions of the program which pertain to th'e outer

regions of the star require considerable modification. Yevsrtheless,

this first section, which considers only one star, pertains almost in

its entirity to the final version of the program.

Tha object of a stellar model calculation may be rendered in

mathematical form by stating that we wish to determine the run of the

physical parameters at every point within the star. The basic assumptions

which must be made are as follows:

1. The star is spherically symmetrical. This means, of course,

that we are treating a non-rotating star during the actual

model calculations; attempts to deal with angular momentum

must be inserted separately from the model calculations.

2. The star is in hydrostatic equilibrium. This assumption is

actually quite good in general; geological evidence plainly

indicates that the sun has been in hydrostatic equilibrium

for at least a billion years.









3. The st:r is in thermal equilibrium.

There are four basic equations of stellar structure. They are

given below along with the physical principle which they express.


dlP G y G(r)p
dr r


dM(r) = 4pr2
dr


dL(r = 4rTper2
dr
adT 3 K O Lr
and either d = r 2
dr 4ac T 4TTr


I dT = 1 dP
T dr y P dr


hydrostatic equilibrium




continuity of mass




thermal equilibrium



for radiative energy transport



for convective energy transport.


In addition to these four differential equations, we use three constitutive

relations; namely,


P = P(p,T, chemical composition) the equation of state,


K = K(p,T, chemical composition) the opacity,


and e = c(p,T, chemical composition) the energy generation rate

per unit mass.


or











All of the symbols have their usual meanings; r is the radial distance

at the point in question; P, M, L, T, and p are the pressure, mass,

luminosity, temperature, and density at the point; a, c, and G are the

radiation pressure constant, the velocity of light, and gravitational

constant; e is the energy generation rate per unit mass; K is the

Rosseland mean opacity.

The boundary conditions for this problem are as follows:


At r 0; M(r) = 0

L(r) = 0

At r = R, T = T


P = 0

M(r) = M


p =1012 g/cm3

where the surface temperature, To, is the temperature at the point for

which the optical depth is two-thirds; it is slightly lower than the

effective temperature. Thus, we have a well defined boundary value

problem involving a fourth order system of ordinary, non-linear, dif-

ferential equations.

To begin the solution of this rather complex problem, an initial

integration using the method described by Schwarzschild (1958) is performed.

Here, the star is divided into two zones and integration are performed

starting at both the center and the surface. When the match at the fitting

point has achieved sufficient accuracy, the resulting solution is used

as an initial guess for the more accurate and much faster procedure used









for the remaining models. Before describing this procedure, however,

it will be useful to make a few comments concerning the theoretical

study of stellar evolution.

It is not at present feasible to do model calculations which

include time explicitly. Hence, stellar evolution is studied by cal-

culating a sequence of static models with time steps between the

models chosen in such a way as to maintain the changes in the physical

variables within certain limits from one model to the next. In the

single star program, which was kindly made available by B. Paczynski,

the change in chemical composition due to nuclear processes is the

factor which produces evolutionary changes for main sequence stars.

Naturally, other factors will become important in this respect in

treating mass exchange; these will be discussed in the next section.

The technique used once a preliminary model is available, again

involves dividing the star into Lwo regions; the outer 10% is treated

by integrating a grid of static model envelopes in the luminosity -

surface temperature plane. This process is described in detail by

Paczynski (1969). The results of these integration are made available

to the main program as outer boundary conditions. The inner portion

of the star is then treated by the method of Henyey, Forbes and Gould

(1964). Basically, this method involves dividing the star into many

radial mass shells, the number and distribution of which are altered as

needed by the program. The equations of stellar structure are recast

in finite difference form and an error matrix is inverted to obtain ccr-

rection terms. Iterations continue until the desired accuracy for each










variable is achieved at each Tass point. The details of this process,

as implemented in the code used in this work, are described by Ziolkowski

(1970).

At the conclusion of each model, a new time step is determined in

such a way as to prevent drastic changes in any of the variables while

simultaneously endeavoring to optimize total computing time. A starting

point for the next model is then obtained by linear extrapolation based

on its two immediate predecessors.

Despite the fact that this material is fully discussed in the

references listed above, it seems prudent to document here the sources

of input physics employed in the particular version of the program used

in this work. In addition to the outer boundaries mentioned above,

there are several other auxiliary programs which calculate input data

used for interpolation by the main program.

The opacity calculations are based on radiative opacities due to

free-free, bound-free, and bound-bound transitions interpolated from

the tables of Cox and Stewart (1969). The radiative opacity due to

water vapor is based on the data of Auman (1967); the abundance of water

is calculated by the method of Mihalas (1967). Conductive opacities of

the electron gas are interpolated from the tables of Hubbard and Lampe

(1965) for densities below 1016 g/cm3; for higher densities, the tables

of Canuto (1969) are used. Energy losses due to neutrino emission are

accounted for by the work of Beaudct, Petrosian, and Salpeter (1967).

Energy generation rates due to hydrogen burning are calculated









according to the discussion by Reeves (1965). Rates are also available

for helium, carbon, and oxygen burning, but as none of these occur

during the period treated in this work, discussion of the sources of

this material is omitted.

One of the supplementary programs calculates all the necessary

thermodynamic functions and gradients needed by the main program. In

addition to the opacity data mentioned above, this program also requires

data for an electron gas; this information was obtained from the integrals

given by Chandrasckhar (1939) and made available with the program.



B. BINARY STAR MODIFICATIONS FOR THE MASS-LOSING STAR



As long as two stars in a binary system remain sufficiently

separated, their evolution will cuntinuc just as if they were single

stars. If, however, their separation is such that one of the stars

fillsits Roche lobe, then we assume that mass transfer between the two

stars must occur. This will, of course, affect the subsequent evolution

of both stars.

Post-main sequence evolution of binary star systems has been exten-

sively studied; in that case, the filling of the Roche lobe is due to

the increase in radius as the star leaves the main sequence. Here, the

mechanism is different, but computing procedures will be similar in

many ways. In particular, most of the basic assumptions are identical.

These assumptions are enumerated and their justifications discussed in

some detail by Paczynski (1969). This list is outlined below:

1. Both components are treated as spherically symmetric stars,

Even if the star fills its Roche lobe, this assumption seems

to be valid, since the resulting distortion occurs only in









the outermost region. Discussions by Paczynski and by

Jackson (1970) indicate that this assumption is no worse

for close binaries than for single stars.

2. The mean radius of the Roche lobe, which is the radius of

a sphere whose volume is equal to that of the lobe, is

calculated using Paczynski (1966) formula. The use of this

formula does assume circular orbits, but this should be

a reasonable assumption.

3. The radius of a ster, Ri, cannot be larger than its critical

radius. If an excess occurs in R then mass transfer

through the inner Lagrangian point will occur. For the

semi-detached phase treated in this dissertation, this

assumption should still be valid.

4. The star is in hydrostatic equilibrium except possibly

for a very small region near the surface. Sometimes the

star is also in thermal equilibrium; however, in post-

main sequence evolution it has been found that thermal

instability is the driving force for the mass exchange.

5. The total mass and the total angular momentum of the

system are constant. This assumption requires some care-

ful consideration for the proposed problem. At first

glance, it may appear that the constant total mass could

not possibly apply while incorporating the rotational

braking. In fact, this assumption seems to be a good one









for these pre-contact rass transfer calculations. There

are two reasons for this. The first is that one of the

original attractive features of this braking theory is

that very little mass loss is required. The second reason

involves the relative time scales of the braking and the

mass transfer. The actual calculations show that the average
4
time step is approximately 10 years. The total period

of time studied is about 107 years. Employing Mestel's

approximations for stars of approximately the mass of the

sun, we find that the synchronization time scale is much

smaller than the individual time steps, while the angular

momentum loss rate can reasonably be taken to be slow

enough to be neglected over the length of time required

for this pre-contact mass transfer to be completed.

In order to discuss the major alteration required in utilizing the

single star program for binary star calculations, it is necessary to consider

a few details of the operation of the program. In the single star version

there are actually two different programs which perform calculations for

the envelope and the core. This division.is primarily a matter of

convenience in computing and is a result of the different assumptions

pertaining to the two regions. Under this scheme, all nuclear reaction

considerations are restricted to the core. This means that the most

rapidly varying quantities will be treated with the most efficient code.

Conversely, it requires considerably more computing time to treat the

complexities of the outer layers, where simplifying assumptions, such









as complete ionization, are not necessarily valid. As long as one can

assume that the envelope is static, however, considerable savings can

be achieved by integrating a grid of these envelopes and interpolating

to fit the evolving core solutions. When mass transfer begins, however,

the envelope expands rapidly, and time dependent terms become important.

In particular, the entropy change with time is no longer negligible.

Several alterations are required to overcome this difficulty.

First, it is necessary to increase the proportion of the mass of the

star for which detailed calculations are performed for each time step.

Tnis is handled by dividing the star into three regions rather than two.

The interior portion, which uses the Henyey calculations, remains

essentially unaltered while the outer region is divided at the point

at which the temperature reaches 105?K. Although the exact number is

of no particular significance, the region beyond that point constitutes

a very small fraction of the total mass of the star. Calculations confirm

that since this point is so close to the surface, the structure of this

portion of the envelope depends almost entirely on the effective temper-

ature and the surface gravity; mass, luminosity, and radius have virtually

no effect. Thus, it is still possible to use a grid of envelopes for

this small outer portion of the envelope. In this case, the grid involves

effective temperature and surface gravity rather than luminosity. This

grid again provides outer boundary conditions for the main program.

The inner portion of the envelope remains to be discussed; this region

is -integrated starting at the point at which the temperature is 1050K

and moving inward to the core fitting point. For each new model this










fitting point is checked to be sti-e the inLerior contains 90% to 93%

of the total mass. The results of this intergrtion are then matched to

the core program; often several iterations are required to obtain a

convergent model.

It seems useful to mention a few of the details of calculation of

this new portion of the program. The integration itself is essentially

identical to that performed in computing the static grid of envelopes.

In fact, it became necessary to include the portions of that program

which allow for incomplete ionization.

Performing this type of integration is quite expensive; it is

therefore desirable to minimize the numbers of times it has to be done.

For this reason, the trial values of luminosity or mass loss rate and

effective temperatures are used to determine the center of a bo, .the

four corners of which are used as exterior starting points for an inward

integration in each of the four variables, temperature, density, radius,

and luminosity or mass loss rate. The corresponding four points for

each variable at the core fitting point are then used for interpolation.

An additional integration is performed at the end of the model calculation,

provided the complete list of variables for all mass shells is to be

printed. This procedure keeps the number of required integration to a

minimum. Accuracy of the fit across the boundary is affected by the

size of the box, as well as the accuracy factors from the single star

version of the program.

In the preceding discussion, mass loss rate and luminosity were

mentioned in a synonymous fashion. This is related to the fact that the










entropy change with time is significant in the case of mass transfer.

It turns out, in fact, that the entropy change with time can be expressed

conveniently in terms of the mass transfer rate (Paczynski, 1969). Thus,

when mass transfer i8 occurring the luminosity solution in the envelope

region is actually determined by the mass transfer rate. For convenience,

the program is designed to operate either with or without mass transfer.

Other additions to this mass transfer version include a subroutine

which calculates mass ratio, separation, period, Roche critical radius

etc., and another subroutine which calculates the radius excess due to

a given rate of mass transfer. The formula on which this calculation

is based was developed by E. Jedrzejec and reported by Paczynski and

Sienkliewicz (1972).




C. BINARY STAR MODIFICATIONS FOR TIE MASS-GAINING STAR



The modifications described in the previous section apply to only

one of the components of a binary star system. It has been customary

to assume that the star which receives the mass will, at the end of the

phase of mnss transfer, still be an essentially zero-age main sequence

star. This assumption is based on the fact that the initially less

massive star should evolve on a much longer time scale. Even Vilhu

(1973), who considered the possibility that the initially less massive

star might fill its critical radius before the initially more massive

star withdrew from its Roche lobe, did not make any attempt to follow










detailed models of the mass-gaining star.

In addition to assuming that the mass-gaining star does not require

detailed treatment, many discussions h:ive mentioned the'myriad problems

of angular momentum transfer. Specifically mentioned have been the

possibility of ring or disk formation, alteration of rotational angular

momentum, possible particle trajectories, accretion mechanisms, and loss

to the system of both mass and angular momentum. All of these are valid

points. In tbp case of mass loss to the system, for example, it seems

intuitively that it could hardly fail to occur. Nevertheless, it is in

the best. scientific tradition to begin a problem by making any simplifying

assumptions which cannot be demonstrated to produce gross inaccuracies

in the results. In this way, one acquires a first approximation which

can later be corrected to account for as many of the neglected quantities

as can be made mathematically tractable. With this in mind it was decided

to attempt calculations which would follow both stars. Thus, it seems

appropriate to reconsider the basic assumptions.

All of the previously enumerated assumptions are retained, but a

few additional remarks pertain to the mass-gaining star. In particular,

the following assumptions are made :

1. No rings or disks are formed around the mass-gaining

star; these would provide a storage location for both

mass and angular momentum in contradiction to earlier

assumptions.


2. Mass is absorbed by the receiving star through its envelope

from a uniformly distributed outer shell. No surface


interactions are considered.











3. Changes in separation are due strictly to redistribution

of mass from one star to the other; no spin interactions

are considered.

The first assumption requires a few comments with regard to its

validity; if a ring or disk is to be stable, there must be ample space

available between the star and its Roche critical radius. For systems

under consideration, then,this first assumption is probably valid,

since the star will be close to its critical radius. At worst, rattcr

might collect briefly beyond the surface of the star and introduce a

slight time delay in the accretion process. This, however, should not

alter the basic findings of the calculation.

The major alterations required to follow both stars involved

communication from one star to the other. The mass-losing star, of

course, determines the amount of mass to be transferred. Initially,

it was also the selector of the time step. This information was then

passed to the second star. With the obvious exception of appropriate

sign changes for the mass transfer rate and related quantities, the

calculations were essentially the same. Of course, the portions of the

envelope treatment which calculated radius excess were by-passed since

the mass-gaining star does not yet fill its critical radius.

After about 150 models had been calculated, it became apparent that

the mass-gaining star was sustaining sufficient perturbations that it should

be considered in the time step determination. The selection process is

designed to facilitate convergence; it is desirable that none of the

variables change by too large a factor. Here the calculation was carried





37



out at the end of the model calcC;lation and transmitted to the first

star to consider along with its ovn calculation.

It was also necessary to add a new criterion in the time step

determination; namely, that there be an upper limit to the amount of

mass transferred in any one time step. If necessary, of course, the

time step is shortened, so that this does not set the mass transfer

rate. It merely facilitates convergence.

In this version of the program, calculations were stopped when

the initially less massive star filled its critical radius.















CHAPTER V
TIE MASS-GAINING STAR

A. TNTERSAL STRUCTURE OF MASS-GAINING STARS



Model sequences were calculated in order to study the mass transfer

phase which is predicted to preceded the formation of a contact system;

these models followed the detailed development and changes in structure

of both stars. Although models of the mass-losing member of a binary

system have been studied extensively (Paczynski, 1971; Plavec, 1970),

no such previous calculations are available for the mass-gaining star.

For this reason, it is necessary to consider the features of the mass-

gaining star before discussing the formation of the contact system.

The internal structure of a star which is receiving mass in the

rmnner described in the previous chapter is altered quite significantly,

not oiLly from its initial state, but also from that of a zero-age star

of the same mass. In order to compare these models of a mass-gaining

star with other suitable systems, a series of masses were selected as

test models. These particular masses were selected so that they would

be spaced evenly over the relevant portion of the H-R diagram. A

practical criterion was that they be models for which restart capability

with the program was available. Table 1 shows a list of these model numbers

and the corresponding masses of both stars. M1 and M2 are the masses in

solar units of the mass-losing and mass-gaining stars, respectively.











MODE


11
13
19
23
21
28
3!
3:
34
4:
4'


TABLE 1. LIST OF TEST MO1D.S

iL M2/NSUN 1i1/NSiUN TI'mE (

60 0.58293 1.26700 1.'925!
10 0.63299 1.21702 1.8226(
30 0.652021.19799 1.9153!
90 0.70976 1. 14024 2.1258
30 0.74471 1.10529 2.2366
60 0.76334 1.08665 2.2924:
30 0.77353 1.07647 2.3226
00 0.78478 1.06522 2.3565!
30 0.79673 1.05327 2.3918
60 0.81232 1.03768 2.4379
20 0.83072 1.01923 2.14940
43 0.83641 1.01359 2.51114


YRS)

3D 07
0D 07
6D 07
1D 07
3D 07
3D 07
8D 07
5D 07
2D 07
1D 07
7D 07
9D 07


For each of the mass values listed for 1I1 in this tab)e, a zero-age
2

main sequence nodl was calculated. In order to be sure that age effects

were not the cause of the changes to be described for the mass-gaining

models, each of these zero-age models was then used as an initial model

in order to calculate a single evolved model using a time step exactly

equal to the age of the system as shown in Table 1. These suitably

aged raiin sequence models were found to be negligibly different from

the zero-age main sequence models. This is demonstrated in Table 2,

which lists the important parameters for the main sequence and aged

main sequence test models; namely, .the log of the effective temperature,

LTEF, boloret:ric magnitude, M BOL, radius, R/R SUN,log of the central

temperature, LTC, log of the central density, LRHOC, and central hydrogen

content, XC. Headings containing a number 1 refer to the zero-age models;

those containing a number 2 refer to the aged models.










TABLE 2. 'COMPARISON OF ZERO-AGE AND AGED MAIN SEQUENCE MODELS



NASS LTEF! L.TEF2 BOL1 M EOL2 Ri/RSON R2/RSUN


0.58293
0.63299
0.65202
0.70976
0.74471
0.76334
0.77353
0.78478
0.79673
0.81232
0.83072
0.83641


3.6126
3.6262
3.6315
3.6503
3.6632
3.6700
3.6739
3.6778
3.6817
3.6870
3.6931
3.6950


3.6127
3.6263
3.6316
3.6503
3.6631
3.6699
3.6736
3.5775
3.6815
3.6688
3.6930
3.6949


7.772
7.476
7.362
7.005
6.784
6.667
6.604
6.535
6.462
6.368
6.258
6.225


7.773
7.476
7.362
7.005
6.785
1.669
6.606
6.537
6. 464
6.369
6.259
6.225


0.50458
0.54323
0.55875
0.60389
0.62999
0.64427
0.65170
0.660"72
0.67097
0.68395
0.69928
0.70406


0.50424
0.54289
0.55841
0.60374
0.62986
0.64417
0.65203
0.66108
0.67114
0.68417
0.69952
0.70429


MASS LTCI LTC2 LRHOC1 LRHOC2


0.58293
0.63299
0.65202
0.70976
0.44471
0.76330
0.77353
0.78478
0.79673
0.81232
0.83072
0.83641


6.9623
6.9797
6.9864
7.0079
7.0216
7.0289
7.0330
7.0374
7.0420
7.0481
7.0552
7.0573


6.9623
6.9797
6.9864
7.0079
7.0216
7.0289
7.0329
7.0373
7.0420
7.0i 31
7.0552
7.0574


1.8986
1.90145
1.9070
1.9153
1.9202
1.9228
1.9241
1.9256
1.9271
1.9290
1.9313
1.9320


1.8986
1.9046
1.9071
1.9155
1.9205
1.9231
1.92A5
1.9259
1.9275
1'.9295
1.9318
1.9325


Perhaps the altered structure of the mass-gaining star is best demonstrated

by considering the luminosity as a function of mass shell for a representative


rcase.










Figure 1 shows the dependence for a normal main sequence star with a

mass of 0.74471 M0 and for gravitational, nuclear, and total luminosity

for model 230, wlich is the cocrespondiig mass-gaining star. Before

trying to explain the features of these curves, it is important to point

out that the value plotted for luminosity at a given point is not the

luminosity produced at that point, but rather the luminosity produced

interior to the mass shell. For the zero-age model, the trend of the

curve is exactly asexpected; the nuclear energy sources cause the total

luminosity to rise to a constant value at the point at which nuclear

energy generation is no longer effective. The total luminosity of the

mass-gaining star, on the other hand, behaves in a very different

manner. Not only does the central rise in the total luminosity reach

a higher level, but there is, in effect, an additional energy source

much closer the surface. This curve is actually much easier to under-

stand if one considers the two constituents of the luminosity; namely,

the luminosity due to nuclear energy generation, L and the gravit-
nuc
national luminosity, L gra, These quantities also appear in Figure 1.

The nuclear term for this model. actually turns out to be quite a bit

larger than its main sequence counterpart. This is due to the increased

central temperature and central density which can be observed in Table 3.

(Abbreviations used are the same as those for Table 2, with the addition

of the mass of the donvective core, MCC, and the luminosity, L/L SUN.)

It is evident, however, that the gravitational luminosity term is actually

responsible for the unusual shape of the total luminosity curve. Note

that in order to obtain the total luminosity, it is necessary to subtract


























60 H 4 1
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071 /













Table 3. Comparison of Zero-Age Main Sequence and
Mass-Gain Test Models.

Headings containing the number I refer to the zero-age
models; those containing the number 2 refer to the
corresponding mass-gain models.


M/MSUN LTEP1 LTEF2 M BOL1 M BOL2 R1/PSUN R2/RSUO


0.58293
0.63299
0.65202
0.70976
0.74471
0.763311
0.77353
0.78478
0.79673
0.81232
0.83072
0.83641


3.6126
3.6262
3.6315
3.6503
3.6632
3.6700
3.6739
3.6778
3.6817
3.6870
3.6931
3.6950


3.6098
3.6168
3.6198
3.6308
3.6410
3.6515
3.6603
3.6688
3.6784
3.6904
3.7009
3.7050


7.772
7.476
7.362
7.005
6.784
6.667
6.604
6.535
6.462
6.368
6.258
6.225


7.866
7.764
7.714
7.502
7.299
7.117
6.968
6.816
6.638
6.408
6.197
6.114


0.50458
0.54323
0.55875
0.60389
0.62999
0.64427
0.65170
0.66072
0.67097
0.68395
0.69928
0.70406


0.48960
0.49677
0.50157
0.52545
0.55062
0.57050
0.58691
0.60519
0.62826
0.66107
0.69414
0.70743


xC1

0.75000
0.75000
0.75000
0.75000
0.75000
0.75000
0.75000
0.75000
0.75000
0.75000
0.75000
0.75000


H/HSUN LTC1 LTC2 LRHOC1 LRHOC2


0.58293
0.63299
0.65202
0.70976
0.74471
0.7633 1
0.77353
0.78478
0.79673
0.81232
0.83072
0.836041


6.9623
6.9797
6.9864
7.0079
7.0216
7.0289
7.0330
7.03714
7. 0420
7.0481
7.0552
7.0573


5.9713
7.0046
7.0160
7.0159
7.0591
7.0645
7.0669
7.0699
7.0725
7.0763
7.0811
7.0825


1.8986
1.90115
1.9070
1.9153
1.9202
1.9228
1.92,41
1.9256
1.9271
1.9290
1.9313
1.9320


1.9165
1.9540
1.9649
1.9876
1.9932
1.9940
1.9936
1.9938
1.9932
1.9933
1.9934
1.9934


MCC1

.0.00100
0.00100
0.00100
0.00100
0.00100
0.00100
0.001S00
0.00100
0.00100
0.00100
0.00100
0.00100


FCC2 XC2

0.00100 0.74985
0.00100 0.74979
0.00100 0.74977
0.00100 0.74971
0.00556 0.74967
0.00759 0.74955
0.00754 0.74964
0.00884 0.74963
0.00937 0.74961
0.00952 0.74960
0.01050 0.74957
0.01067 0.74957










L from L This is due to the fact that the negative of the
grayv nuc
gravitational term was used as a convenience in plotting. Thus, a

positive slope corresponds to an energy sink while a negative slope

corresponds to a source. Normally, one thinks of this in terms of

contraction and expansion, with the former releasing energy and the

latter absorbing it. It is, however, important to remember that the

gravitational potential of a particle at a given radial distance from

the center can be altered not only by increasing or decreasing that

distance, but also by increasing or decreasing the amount of mass

Contained in the region interior to that radial distance. Thus, in

the inner regions, we find a gravitational term which absorbs some of

the nuclear energy. The same is true of the outermost regions. Between

these two zones, however, there is a zone which acts as a source of

energy and would therefore generally be associated with a contracting.

region of the star.

Perhaps an even more interesting result of these calculations is

the finding that in this region of the main sequence where stars have

radiative cores, mass-gaining stars develop convective cores. In this

particular model sequence, the convective instability first became

evident in model 197, when the mass of the star was only 0.71249 Mg.

At that point, the logarithm of the central temperature of the star

was 7.047. At the end of the sequence, it had only increased to 7.083.

Normally, a convective core is the result of energy generation via

the CN cycle, but that cannot be the case here, as the minimum central









temperature for a cycle time less than the age of the sun has a logarithm

of about 7.2 (Cox and Guili,196S). These authors also conclude, after

a survey of the literature, that the mass at which convective cores

disappear is about 1.1 M Thus, it is certainly unexpected to find

a convective core in a star of such low mass.

Still another rather surprising result of these calculations is

the fact that the radius does not increase rapidly during the mass transfer

process. In fact, the radii of mass-gain models were considerably

smaller than the corresponding main sequence radii during most of this

stage; only at a point very near the end of the sequence of models

did the radius of the mass-gaining star actually exceed that of its

main sequence counterpart. This is also readily observed in Table 3.

In view of the many unusual properties of these models, some comments

regarding qualitative explanations seem appropriate. Mass-gaining stars

have been assumed to be accreting matter uniformly and smoothly. The

time scale for this accretion has been determined by the program in such

a way as to prevent drastic changes from one model to the next. This

time scale is considerably longer than the dynamical time scale, but

considerably shorter than the thermal time scale; both of these time scales

have a role in this process. First of all, the addition of mass to the

star would immediately destroy its hydrostatic equilibrium. The subsequent

collapse and heating in the core will take place so rapidly as to be

negligible with respect to the time steps used. The star is, however,

not in thermal equilibrium; it must expand in order to remedy this

situation. In the meantime, more mass is added. It seems likely that

the competing processes of expansion to restore thermal equilibrium and

contraction to accommodate new matter are responsible for many of these



























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anomalous results; it would certainly explain the behavior of the

gravitational luminosity curve. In addition, the core and envelope are

effectively decoupled due to the fact that radiative energy transport is

much slower than convective transport. This indicates that heat will be

trapped temporarily in the core. As this heat accumulates, the temperature

gradient will eventually become unstable to convection in the center. The

star will be cooler, less luminous, and have a smaller radius than it should

for its mass until the trapped energy can, on a thermal time scale, work

its way outward. At that time, the star becomes brighter, hotter, and

bigger than its main sequence counterpart.

Actually, there is further evidence for this picture; it comes from

a consideration of the time development of the luminosity profile of the

rmss-gaining star. In Fig. 2, we see total luminosity plotted vs mass shell

for-several of the mass-gain test models. The effective energy source

described previously does indeed move outward. As it reaches the surface,

the star becomes hotter, brighter, and bigger.



B. RELAXATION CALCULATIONS



Another series of test calculatioiwas carried out in order to

investigate the effects of stopping the mass transfer and allowing the

star to relax. Each of the mass-gaining test models listed in Table 1 was

used as the starting point for one of these relaxation sequences. This

time, the mass transfer rate was set to zero and models were calculated.






























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The criterion for stopping these relaxation runs was that the gravita-

tional luminosity term in the envelope should have returned to the main

sequence value. A detailed list of relevant quantities for each of these

runs is to be found in Appendix A. In each case, the star did relax to

its corresponding main sequence position. This process is illustrated in

Fig. 3, which shows tracks of relaxing stars in the H-R diagram, in Figs. 4

and 5, which trace the changes with time of several quantities of interest,

and Fig. 6, which demonstrates the changes in total and gravitational

luminosity profiles as the star relaxes.

First, it is interesting to note that, in general, the relaxation

process compares favorably with the thermal time scale for these stars.

It is also of considerable interest to relate the disappearances of the

various anomalies to the tracks in the H-R diagram in Fig. 3. In Fig. 6,

the total luminosity and gravitational luminosity are plotted for a

series of relaxation times beginning with model 230. This process is

actually easier to appreciate, however, by considering the trend plots

in Figs. 4 and 5. The ordinate units are arbitrary in these two graphs;

this is to permit plotting all of the interesting quantities together so

that simultaneous events can be detected more readily. The numbers in

parentheses are the maxima and minima of the indicated quantities. Fig. 4

is for model 230, while Fig. 5 is for model 420. If one compares the

former with the H-R diagram track for model 260, which is very similar

to 230, one can see that the first turn in the track corresponds to several

events recorded in Fig. 4. At that time, the luminosity increases,

indicating that the effective source is reaching the surface, and the










convective core begins to disappear. The star then decreases its surface

temperature, moving to the right in the H-R diagram, while the convective

core disappears completely. The H-R diagram track then turns upward toward

its proper main sequence location while the central density and temperature

and the luminosity due to nuclear energy generation, decrease to their

normal values. During this time, the radius and the total luminosity

increase to their normal main sequence values. This track appears to

be representative of those for which the effective luminosity source,

which was discussed above, has not yet reached the surface. Model 420

in Fig. 5 and model 360 on the H-R diagram, on the other hand, exhibit

slightly different features as a result of the source starting to reach

the surface. The radius and luminosity increase steadily again until

the convective core begins to disappear. At that time, the track shows

a reversal of direction, with both effective temperature and luminosity

decreasing until the convective core is completely gone. During this

period, the core processes and parameters all return to their main sequence

state, while the surface parameters all decrease. Once the nuclear energy

generation rate has settled to its normal value, the radius and total

luminosity return to their respective main sequence values.

All of the tracks are the result of the same basic relaxation process;

model 60 (see Fig. 3) is almost a straight line because it has been perturbed

by only a small amount. Its envelope must expand, but its core is fairly

normal. The other models, however, all show evidence of core cooling;

with double reversal marking the disappearance of the anomalous convective

core.










CHAPTER VI.
THE FORMATION OF A CONTACT BINARY




The sequence of models calculated was terminated at model 443. At

that point, the less massive star had reached a radius such that it

filled its Roche lobe. The more massive star would, of course, have

continued to transfer mass had the calculations not been stopped. In

addition, the relaxation models show that the tendency is for the mass -

gaining star to expand. Thus, the two stars will continue to fill their

respective Roche lobes, and it is clear that this is, indeed, the initial

contact system which is predicted to occur through the preliminary mass

transfer phase suggested here.

In Appendix B the details of the mass transfer phase are presented;

the first few entries do not include every model because the changes were

very small during the early phases. The mass transfer rate was very low

during this time; it started at about 10- solar masses/year. Gradually,
-8
however, it increased until the rate at the end was about 2 X 10- solar

masses/year. Several test calculations have also been made for other

initial mass ratios, and there is some evidence that the initial mass

transfer rate may be affected by the mass ratio. Systems with mass ratios

very close to one transfer mass much more slowly than systems with initial

mass ratios further from unity.

The mass transfer process involved a much greater change in mass

ratio than was predicted by the preliminary calculations. This, however,

is not surprising, since there is no reason to expect that the main

sequence mass-radius relation would be obeyed during such a process.

The fact that the radius of the less massive star is found to be so









much smaller than normal explains the fact that much more mass was

transferred before the less massive star reached its critical radius.

The initial and final parameters of calculated model-sequence

are listed in Table 4. Although the total mass of the system was selected

in the hope of comparing the results with W UMa, the final mass ratio

immediately precludes any such endeavor. (The best available values of

the masses of the components of W UMa are given by Worden and Whelan

(1973); they find masses of 1.20 0.06 for the primary and 0.65 + 0.03

for the secondary.) This problem of being unable to select the initial

mass ratio which will produce a desired final mass ratio is one which

is encountered regularly in post-main sequence evolution. Since no

immediate solution to this problem presents itself, one must inquire

as to whether the calculated initial contact system will fit into the

class of H UMa type binary systems rather than whether it resembles a

particular observed system. For this purpose, one must refer to the

typical W UMa system discussed in the second chapter. The mass ratio

of the computed system is 0.8252 (see Table 4). This is somewhat higher

than average, but not outside the range of observed systems. Certainly

the masses themselves are acceptable. In this system, the difference

in magnitude is 0.3714 magnitude, while for a typical W UMa system,

the difference is 0.25 magnitude. The effective temperatures are close

enough that the two components might be said to be of the same spectral

class, probably about KO. On the other hand, the secondary is not

the hotter star, as is often observed to be the case.

In Fig. 7, the relevant portion of the H-R diagram appears; on it

are plotted both the zero-age models calculated for comparison and

the paths followed by the two stars during the mass transfer process.













TABLE 4

Initial and Final Parameters of Calculated Model Sequence


Masses, radii, luminosities, and separation are in solar units. The period
is in days; the age, in years. Effective temperatures are in degrees Kelvin.


Mass

Log Tef

Tef

"bol
Luminosity

Radius

Roche critical radius

Mass ratio

Period

Separation

Age

Mass of Convective Core

Log of Central Temperature

Log of Central Density

Central Hydrogen Content


Initial

Primary Secondary

1.29 0.56

3.7995 3.6067

6303 4043

4.001 7.905

2.06871 0.05677

1.21182 0.48773

1.21182 0.823

0.43411

0.373

2.676

0

0.01061 0

7.2067 6.9547

1.9712 1.8964

0.75 0.75


Final

Primary Secondary


1.01359 0.83641

3.7233 3.7050

5288 5070

5.743 6.114

0.41576 0.29531

0.77166 0.70743

0.77140 0.70650

0.8252

0.2311

1 9446

2.5 X 107

0 0.01067

7.1146 7.0825

1.9232 1.9934

0.74651 0.74957
























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E

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Comparison of the positions of models marked 443, which is the initial

contact system, shows that the primary is considerably underluminous

for its mass, while the secondary is overluminous. This agrees with

observations of typical W UMa systems.

One of the most encouraging results of this calculation is the age

of the resulting contact system. With an age of 2.5 X 107 years, it

is well within the age range for W UMa systems, such as TX Cancri,

which are members of clusters for which ages have been determined

(Sahade and Friebos, 1960). This is particularly interesting, since

previous efforts involving mass transfer without accounting for the

less massive star led to ages which are much too large for W UMa systems.

Probably the biggest problem in identifying this calculated system

as a W UMa type system, however, lies in its period, which is 0.2311

days. This value is, unfortunately, too short to agree with observations,

since W UMa systems have a sharp cut-off at 0.25 days. This particular

problem is discussed last because it indicates that another phase of

mass transfer will probably follow the formation of the contact system.

Mass transfer after contact was suggested by Huang, and it certainly

seems logical on the basis of these calculations. Since the relaxation

models have Cdemonstrated the expansion of the mass-gaining star, it is

quite likely that at some point the direction of mass transfer will

reverse. This would, of course, lengthen the period. Calculations

following this phase of mass transfer after contact is achieved would

be of great interest; as a major revision of the program would be

required, they are outside the scope of the present study. It does seem

worthwhile to note, however, that such continuing calculations might











also produce the case in which the secondary is the hotter star.

Certainly the relaxation models indicate that the secondary will

become hotter.

The major conclusions of this work are as follows:

1. A preliminary stage of mass transfer from the more

massive to the less massive star will, indeed, produce

a contact system with the general properties of a W IUa

system.

2. The structure of the star receiving the mass is

considerably altered by the process, although it will

relax to a normal main sequence state if mass transfer

is stopped and sufficient time is allowed to pass.

It this connection, however, it should be noted that

the star studied was of low mass and initially of

zero-age. Alteration of either of these conditions

might change the results.










APPENDIX A

TABLES OF RELAXATION MODELS



These tables contain the most important quantities for each model

calculated in the relaxation study described in Chapter V. The meanings

of the table headings are as follows:





TIME age of the model

M/M SUN mass in solar units

LTEF log of effective temperature

M BOL bolometric magnitude

R/R SUN radius in solar units

L/L SUN luminosity in solar units

MCC mass of convective core in solar units

XC Central hydrogen content

LTC log of central temperature

LEHOC log of central density

TEFF effective temperature in K.

LGRAV log of the surface gravity





The last entry in each table (model 2) is the corresponding aged

main sequence model. It is placed at the end for purposes of comparison.














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APPENDIX B

TABLE OF MASS EXCHANGE MODELS



These tables contain the most important quantities for

each model calculated in the evolutionary model sequence which produced

the initial contact system discussed in Chapter VI. Where they appear

in the headings, the numbers one and two refer to the mass-losing and

mass-gaining stars, respectively. Meanings of table headings are the

same as those in Appendix A; those headings which do not appear in

that appendix. are explained below:



DT time step for the model

DM/DT mass transfer rate

PER period in days

Q mass ratio, M1/M2

AA separation in solar units

RCR Roche critical radius

MOB mass at outer boundary of Henyey calculation

MCE mass of oonvective envelope








81













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