Evolution of pre-main-sequence binary star systems

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Evolution of pre-main-sequence binary star systems
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West, Jon Kenneth, 1947-
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Thesis--University of Florida.
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Includes bibliographical references (leaves 210-211).
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Typescript.
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Vita.
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by Jon Kenneth West.

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EVOLUTION OF PRE-MAIN-SEQUENCE BINARY STAR SYSTEMS


By

JON KENNETH WEST











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











UNIVERSITY OF FLORIDA


1979















ACKNOWLEDGMENTS


There is no way I can give proper credit and acknow-

ledgment to all the people who have helped me in this en-

deavor. Time nor space allow me to fully express myself,

but my heartfelt thanks goes particularly to: Dr. Kwan-Yu

Chen, my advisor and chairman of the dissertation committee.

His genuine interest in my project and inspiring advice

have been a real encouragement to me. Dr. F. B. Wood

provided excellent literature for study and was instrumen-

tal in introducing me to astrophysics. I also thank Dr.

John P. Oliver and Dr. T. D. Carr and Dr. R. T. Schneider

for their efforts and time as members of my committee.

Especially I would like to thank Dr. Mario Levio for his

help in defining this problem.

I would like to express my gratitude to Ken LeDuc,

Butch Gould and Jim Madden for their continued support and

help wherever needed. And I wish to thank the Advanced

Degree Program of the General Electric Company Employee

Continuing Educational Program for encouragement and funds

needed to complete this degree.

The funds for the computing time were made available

by the Battery Business Department of General Electric

ii










Company. The computer was a Honeywell H6080 system oper-

ated by the Information Systems Business Department of

General Electric Company.

My appreciation also goes to Ginny McKann for her

willingness to edit the various drafts and smooth out the

rough edges of this dissertation.

And a very special thanks to Carl George for bringing

my attention to this area of study.

Finally, I wish to thank my lovely wife, Donna, for

her support through the many years of "after work" study.

She graciously typed the many drafts and really made this

work possible. Her love and understanding is a gift from

God.


iii
















TABLE OF CONTENTS


ACKNOWLEDGMENTS . .

ABSTRACT . . .


CHAPTER I.

CHAPTER II:


























CHAPTER III:


INTRODUCTION . .

PRE-MAIN-SEQUENCE BINARY MODEL

A. Kelvin-Helmholtz Time Scale .

B. Pre-Main-Sequence Mass
Transfer . .

C. Basic Assumptions for the
Binary Model . .

D. Equations of Stellar
Structure . .

E. The Hayashi Track .

F. The Gravitational Contraction .

G. Nuclear Energy Sources .

H. Initial Conditions .

I. Description of the Model
Program . .

RESULTS OF THE NUMBERICAL CALCULATIONS

A. TV Cassiopeia . .

B. IM Aurigae . .

C. U Cephei . .

D. AI Draconis . .

E. 6 Librae . .


Page

i

v










CONTENTS continued .

F. B Persei . .. 42

G. U Sagittae . 42

H. V505 Sagittarii .. .45

I. X Trianguli . .. 45

J. TX Ursae Majoris .. .48

K. W Ursae Minoris . 48

CHAPTER IV: SU-PARY AND CONCLUSIONS .. 52

APPENDIX A: PROGRAM LISTING . .. .58

APPENDIX B: TEST MODEL . .. 66

APPENDIX C: BINARY SYSTEM MODELS. .. 79

REFERENCES . .. 210

BIOGRAPHICAL SKETCH . .. .212
















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


EVOLUTION OF PRE-MAIN-SEQUENCE BINARY STAR SYSTEMS

By

Jon Kenneth West

June 1979


Chairman: Kwan-Yu Chen
Major Department: Astronomy


This paper develops a quasistatic polytropic model.

It represents the pre-main-sequence contraction phase. The

model was applied to both components of close binary star

systems. The initial conditions for the models were

determined from the Roche Lobe. The period and masses for

the systems determine the maximum stable radii for both

stars. Both stars were allowed to contract from the ini-

tial models. In the early stages, it may be necessary for

a secondary to evolve along the Hayshi Track. The time

for each contraction had to be calculated by an iterative

procedure. The time steps were corrected for nuclear en-

ergy sources between iterations when needed.

Models were calculated for eleven observed semi-

detached close binaries published in the literature. The

vi










combined mass of each system's components is between 2.5

and 6 solar masses. First, the results of the model were

compared with the observed systems, then a determination

was made concerning their pre-main-sequence nature and

age. The names and ages of the three that are likely to

be pre-main-sequence are: U Cephei (6.1 X 105 years),

X Trianguli (4.6 X 106 years), and W Ursae Minoris (1.87

X 106 years). The following four systems were found not

to fit the pre-main-sequence model in this investigation:

TV Cassiopeia, AI Draconis, B Persei and TX Ursae Majoris.

Finally, the next four systems could be pre-main-sequence

with modifications to the model: IM Aurigae (3.5 X 105

years), 6 Librae (8.2 X 105 years), U Sagittae (3.78 X

105 years), and V505 Sagittarii (1.89 X 106 years).


vii















CHAPTER I

INTRODUCTION


The problem of the formation of a close binary star

system has not been solved. No detailed understanding

exists for the physical process involved. The processes

for the formation of a single protostar are still in-

complete. At least three factors make the formation of

a selfgravitating protostar very difficult: conservation

of angular momentum, magnetic fields, and interstellar gas

dispersion.

Conventional studies of binary evolution usually do

not consider any mechanism of formation. Usually model

calculations, e. g. Pacznski (1971), are started with both

stars on the main sequence. The star that is originally

the more massive is called the primary. In most cases

only the evolution of this component is followed in detail.

Mullen (1974), however, followed the detailed evolution of

both stars in the case of a W Ursae Majoris type binary

model. Nevertheless initial conditions were that both

stars were on the main sequence.

The evolution of binary stars from the main sequence

is identical to that of single stars until the more mas-

sive star expands to fill its Roche Lobe. At this point,

1







2

Crawford (1955) proposed that a large fraction of the mass

is transferred to the secondary. The roles of the primary

and secondary may actually reverse. Sahada (1962) asked

whether such a star, originally a certain spectral type,

after acquiring mass from the other star could become a

more massive object that nevertheless would show a "normal"

spectrum of an earlier type.

The mass transfer, during the reversal, goes through

a rapid thermal phase. Then the star has a tendency to

expand slowly on a nuclear time scale because of hydrogen

burning in the core. The nuclear transfer phase is the

most likely to be observed.

Assuming that a binary can be formed, Thomas (1977)

has stated that the prototype of close binary evolution

with mass transfer has confronted theoreticians with an

increasing number of problems. He considers the post-

main- sequence evolution to be understood in principle.

The problems arise as soon as we investigate individual

systems.

Recently, three investigations indicate the need to

study individual systems of the pre-main-sequence type.

Some observed systems are shown to be pre-main-sequence

(Field; 1969), some observed systems violate the require-

ment that a zero-age contact system cannot exist (Leung

and-Wilson; 1976), and some show mass transfer rates that

are on a thermal time scale (Hall and Neff; 1976). These

cases are reviewed below.







3
Field (1969) found that some close binary systems

appeared to be pre-main-sequence. Field used data for

the contraction of single pre-main-sequence stars published

by Iben (1965) to find the age of the stars. Both com-

ponents of the binary were assumed to be formed simulta-

neously as single stars on their respective Hayashi Tracks

(see Chapter II). Four systems were determined to be pre-

main-sequence. They are KO Aquilae, TV Cassiopeia, WW

Cygni, and Z Herculis. Their total orbital angular momen-

tum are similar to that predicted by Roxburgh (1967) for

stellar fission.

Leung and Wilson (1976) have presented evidence that

zero-age contact binary systems exist. Results of their

photometric investigation show both components of V701

Scorpii and those of V1010 Ophinchi are in contact. Their

locations on the Hertzsprung-Russell diagram and their

radii both suggest that the systems are zero-age. However,

this result presents a problem. This is in contradiction

to the post-main-sequence model for the origin of contact

systems presented by Lucy (1968). This model requires the

components of a zero-age contact system with a common en-

velope should have equal masses. However, many detached

binaries with a mass ratio of unity are not observed. Con-

tact binaries have been observed to be on the zero-age

main-sequence, but none have a mass ratio of unit. In con-

nection with this, Whelan (1970) showed the possibility

that a mass ratio of unity was unstable for pre-main-sequence

binaries. An alternative possibility is that the adiabatic









constants of the stellar envelopes are not equal. The later

possibility implies short lifetimes for the individual sys-

tems. Bierman and Thomas (1972) allowed for different adia-

batic constants by requiring a luminosity exchange to main-

tain contact of the system. Their model did not solve the

problem of the light curves for W Ursae Majoris systems,

nor did it propose a mechanism for the required amount of

luminosity transfer. Therefore, their origin remains a

theoretical problem.

Hall and Neff (1976) evaluated the period changes of

24 observed semidetached binary systems. Their model as-

sumes mass transfer from the cooler to the primary compo-

nent. The mass loss results in a period change because of

conservation of angular momentum. Using the observed

period changes they calculated the mass transfer rate for

each of the 24 systems. All systems showed mass transfer

on a thermal time scale. This time scale differs from that

of the conventional post-main-sequence models for the forma-

tion of a contact or a semidetached binary system. For in-

stance, Mullen (1974) found that each component of a post-

main-sequence binary expands away from the main-sequence

on a nuclear time scale. Thus, the problem of the pre-

or post-main-sequence nature of a binary has not been solved.

Each system must be evaluated on an individual basis.

The problem that the writer wishes to investigate is

the evaluation of individual systems with a pre-main-sequence

model. The main purpose of this investigation is to find







5

possible pre-main-sequence binary systems. This will be

accomplished by comparing the positions of the observed

binary on the Hertzsprung-Russell diagram with the evolu-

tionary tracks generated by the pre-main-sequence model.

If both components appear to have the same age and fall

near the tracks of the model, then the system is likely

to be pre-main-sequence. The investigation will include

the evaluation of eleven binaries each with a total mass

between 2.5 and 6 solar masses. In all cases the secondary

star is over-luminous. These systems are classified as

semidetached.

The model being considered is independent of the ori-

gin of a binary star system. The origin may be either con-

densation of both components as described by Wood (1962)

or fission of a rapidly rotating pre-main-sequence core

as described by Whelan (1970). Yamasaki (1971) studied

the case immediately after fission where mass transfer may

occur. As the stars contract, they must eventually detach

themselves from their Roche Lobes. Yamasaki ends his in-

vestigation when mass transfer stops or a single star is

formed because of excessive mass loss. The writer assumes

the first case as the initial configuration of the pre-

main-sequence model. At this stage the individual stars

fill their Roche Lobes. Each star is allowed to contract

gravitationally and independently toward the main-sequence.

As a comment on the study of origins, I would like to

quote Whitcomb and Morris (1961) page 213.










After all, any real knowledge of origins
or of earth history anticedent to human histor-
ical records can only be obtained through divine
revelation. Since historical geology [or stellar
origin], unlike other sciences, cannot deal with
currently observable and reproductible events,
it is manifestly impossible ever really to prove,
by the scientific methods, any hypothesis relat-
ing to prehuman history.

This idea applies directly to the problem of binary star

evolution and origin. Experiments cannot be set up and

conducted that will prove or disprove any case of binary

origin. Only models of specific sets of conditions can

be made and compared with observed binary stems. The

"best" model which can be found has the fewest number of

ad-hoc assumptions and predicts the greatest number of

observed phenomena.
















CHAPTER II

PRE-MAIN-SEQUENCE BINARY MODEL


The masses and periods for eleven observed close

binary systems were selected from data tabulated by Gian-

none and Giannuzzi (1974). The stellar structure is rep-

resented by a polytrope of index n=3 :for "he radiative

solution. A polytcope of index n,: 5 ijs used for the Hayashi

solution. The Hayashi solution is 'eprc:r:ntec by tL'e line

A-B on the Hertzsprung-Russell diagram shown in Figure 1.

The radiative solution is line B-C. Both solutions are

calculated for each model. The Hayashi solution is selected

as long as it is above the radiative solution. After the

solution is determined, the time step is calculated by an

iterative procedure described in a later section. The mod-

el calculations are stopped when the primary reached the

main-sequence.

The variables and symbols used throughout this chapter

are listed below:

T Kelvin-Holmholtz contraction time (years)
AL Time step between Inolels
AR Radial contraction between mod'is
R Stellar radius
M Stellar mass
r Radial distance from Stellar C:ntLer
p Density
P Pressure
T Temperature
a Radiation constant






















/
/FOCRBIDDEN
REGION

C\
B


MAIN-
SEQUENCE-


< -LOG Teff


FIGURE 1.


Schematic diagram of the ev-lr,' ona;r trSrks
on the Hertzsprung-Russell :iiaT'.anm or a ciLV"--
mass. Hayashi solution line / 8, 'he rdi,-
tive solution line B C.


A


0
-J


0
-_j


_ I__~~









c Velocity of light
T(ff Effective temperature
y Ratio of specific heats
n Polytropic index
5 Polytropic radius
l Polytropic radius at surface of star
K Constant in polytropic relationship
K Opacity
a Stefan-Boltzman Constant
H Mass of hydrogen atom
L Total Stellar Luminosity
Ln Luminosity due to nuclear sources
La Luminosity due to gravi national contraction
L Luninosity due to the Hayashi Track
A Separation between the centers of the binary
components
G Gravitational Constant
C Total energy gener !;.io1 ial C
En Energy generation rate J',r nuclear sources
g9 Energy generation rr'te foyr gravitational contrac-
tion
R Gas constant


A. Kelvin-iHelmholtz Tijm Scale


The Kelvin-Helmholtz contraction is known as the

total pre-main-sequence phase of stellar evolution. The

source of energy is derived totally from the gravitational

potential. The time for this phase can be estimated by

the following relationship:

S= ro (M/M0)


Where To = 2.75 X 107 years and No is the solar mass, Lo

is the solar luminosity.

A fraction of this contr;ii- oin tojim,-e w is used as the

fi-rstl estimate for time step r .hPe modr calculations.

The first estimated time step was found from the following:

At T (AR) 3.17 X 107 seconds.
R
















B. Pre-Main-Seauence Mass Transfer


The study of pre-main-sequence mass transfer shows

that a binary will evolve rapidly from an over-contact

configuration to a stage where both stars will fill their

Roche Lobes. Mass transfer during condensation has been

proposed, e. g. Wood (1962). He describe; formation

mechanism for a close binary in which nuc '.i are poor in

hydrogen compared to the mixtur- from which they are con-

densing. Whenever two condensing nuclei of comparable

size are present, the formation of a double star, rather

than a single star or planets may occur. The difference

between the formation of a double star and a single star

is that the double star is constrained by the inner critical

equipotential surface known as the Roche Lobe. Wood argues

that the more massive component would continue to attract

the most gas.

Hayashi and Nakano (1965) and Nakano, Okyama and

Hayashi (1968) proposed a method for the formation of a

single star that could lead to the formation of a close

binary. Theyhey propose tht the .-Fre-fall ola [ n

opaque proLostIr is stopped by ine bounce 2n ,ntr-;l

core. For a protostar of one solar mass this takes ap-

proximately ton years. The bounce causes a shock wave.







11

The subsequent propagation of the shock wave toward the

stellar surface causes a sudden flare-up. The star finally

settles down to a quasihydrostatic equilibrium state and

rapidly moves toward the Hayashi Track. If the protostar

condenses with sufficient rotational angular momentum, the

above theory will predict the formation of a rapidly rotat-

ing core. When the bounce shock occurs it could be non-

spherical. The result may be the formation of a pro-main-

sequence binary system by fission.

Yamasaki (1971) studied the case of a binary system

immediately after fission of a collapsing protostar, He

suggested that a substantial fraction of the matter would

be crowded out of the critical lobes and would drift

around the components. The total orbital angular momentum

of the confined mass must be above the critical value to

cause rotational instability. He assumed that the balance

of the mass outside of the Roche Lobes would be lost from

the outer Lagrangean point. The loss rate is a parameter

of the problem. If only one component overfilled its Roche

Lobe, then the whole excess mass was assumed to fall on

the other component. It does not escape from the system

provided that the other does not fill its Roche Lobe. The

model calculations were based on a polytropic star with

n = 1.5. This represents fully convectiv LP stars -)ontrac t

ing on the Hayashi Track. The initial mass for all cil--

culations was held constant at 13.5 solar masses. In all

cases of stable mass transfer rates and mass loss rates,







12

the systems detached within several hundred years. For a

critical value of mass exchange rate the system would not

recover from the instability and form a single star.

The final configurations of these models still had separa-

tions from 2 to 4 A. U. Although these separations are

a minimum, they do not represent observable close binary

systems.

The writer evaluated the case of pre-main-sequence

mass transfer from the secondary to the primary. Table 1

shows the initial and final configuration after mass

transfer.


TABLE .

INITIAL AND FINAL CONFIGURATrION OF 1A3SS TRANSFER MODEL

INITIAL FINAL

MI/Mo 3.0 3.128
M2/M0 1.5 1.372
Period (days) 3.0 3.459
Separation (A. U.) 0.067 0.074


The mass transfer rate is chosen to be 1 X 10-7 solar

mass per year. This is representative of observed rates

in semidetached binary systems, (e. g., Hall and Neff;

1976). Table 2 shows the calculated models of the secondary

during mass transfer. Column 1 is the model number. Column

2 is the age of the model in second!. Colu in 3 is th

logarithm of the luminosity in solar units. Column 4 is

the logarithm of the effective temperature. Column 5 is

the ratio of the radius to the Roche Lobe. Finally, column

6 is the radius in solar units. The mass transfer stops







13

when the secondary is smaller than its Roche Lobe. The

initial radius of the secondary is arbitrarily chosen to

be 20% greater than the Roche Lobe. Conservation of

angular momentum and of mass requires the binary system

to increase in separation during mass exchange. Therefore,

the Roche Lobes of each star increase as the mass ratio

increases. The mass transfer stops within 28 years for

this case. This is similar to the results found by

Yamaski (1971). The initial configuration chosen by the

writer is immediately after the binary components form a

minimum contact binary. Both components will fill their

Roche Lobes initially.


TABLE 2

MODELS OF THE SECONDARY DURING MASS

MODEL TIME LOG(L/LSUN) LOG(TEFF)

1 2.506E+07 4.8421 3.2855
3 6.452E+07 4.7893 3.2888
5 1.044E+08 4.7364 3.2920
7 1.446E+08 4.6834 3.2953
9 1.855E+08 4.6304 3.2985
11 2.310E+08 4.5772 3.3018
13 2.830E+08 4.5239 3.3051
15 3.423E+08 4.4705 3.3083
17 4.101E+08 4.4170 3.3116
19 4.876E+08 4.3634 3.3149
21 5.761E+08 4.3097 3.3181
23 6.773E+08 4.2558 3.3214
25 7.930E+08 4.2019 3.3247
27 9.254E+08 4.1478 3.3279


TRANSFER

ROCHE ER

1.19
1.17
1.16
1.14
1.12
1.11
1.09
1.08
1.06
1.04
1.03
1.01
1.00
'' Q


R/RSUN

5.50
5.45
5.39
5.34
5.29
5.23
5.18
5.13
5.08
5.03
4.98
4.93
4.88
4,83















C. Basic Assumptions for the Binary Model


The proposed model represents a preamain-sequence

binary star system. The initial configuration introduced

is constrained by the Roche Lobes. The basic assumptions

which must be made are as follows:

1. The first law of thermodynamics requires mass
and energy to be conserved both for the in-
dividual stars and the binary system as a whole.

2. The second law of thermodynamics requires that
the entropy for any closed system be greater
than or equal to zero. This means that there
is no external source (or sink) of energy
available to do work. The system dynamics
are determined from the initial conditions.

3. The binary system obeys conservation of angu-
lar momentum. Circular, Keplarian orbits were
assumed.

4. The stars are spherically symmetric. Thus, we
are assuming that the stars are not affected
by rotation during the model calculations.

5. The stars are in quasihydrostatic equilibrium.
This assumption is required to calculate each
model. A contracting star is not, by defini-
tion, in hydrostatic equilibrium. Yamasaki
(1971) showed each contraction occurs over a
time step which is long compared to the dyna-
mical time scale of readjustment. Thus, the
approximation of hydrostatic equilibrium is
good.
















D. Equations of Stellar Structure


There are four basic equations of stell;


dP = GM(r) hvdrosta
dr r2

dM(r)
dr = 4Tpr2 continue

dL(r) 2
dr) = 4per2 thermal
dr

and either

dT 3 KP L(r) for radii
dr 4ac 3 4crr eauilibr

or

1 dT Y 1 1 dP for conv
T dr y P dr equilibr


M(r) and L(r) are the mass and luminos


ar structure:


tic equilibrium



ty of mass


equilibrium


active
ium


active
ium


ity a


t radial


distance r.

Along with these four differential equations

have the following relations:

P = P(p, T, chemical composition)



< = (r, T, chemical :omcosi on)

c = E(p, T, chemical comrositL-on)


we must



the equa-
tion of
state

I ;e Op' & c i.ty

the energy
Generation
rate Der
unit mass


Where







16

The boundary conditions for this problem are as fol-

lows:

At r = O, M(r) = O

L(r) = 0

At r = R, T = Teff

P = 0

M(r) = M


A polytropic star is one which obeys an equation of

the following form:

11
P = Kp + n


Since the pressure is an explicit function of density only,

polytropic stars are determined by the equation of hydro-

static equilibrium and the equation of continuity of mass.

By combining these two equations and using dimensionless

variables, one differential equation defines the structure

of a polytropic star:


1 d ( 2 d0) n
S2 dE dE

This is known as the Lane-Emden equation. The vari-

ables are defined as follows with the subscript c indicat-

ing the core valves.


On = p
Pc

0 = T
Tc

S_ r
r









Where rn is known as the Emden unit of length:



rn [(n + 1) P -
4lTGpc2


In addition, we have

R = 1 r


The polytropic model provides a fair approximation to the

structure of certain types of real stars. This idealized

model is often useful in qualitative and in rough quantitative

discussions and aid considerable in gaining an overall] in-

sight into the structure of stars,(e. g, Cox and Guili; 1968)


E. The Hayashi Track


Hayashi (1961) proposed a phase of gravitational con-

traction of a convective star known as the Havashi Track.

This is shown in Figure 1 as the nearly vertical line, A-B,

on the Hertzsprung-Russell diagram. The convection may

result from the large opacities caused by ionizing hydro-

gen. The adiabatic relation between the pressure and

temperature is a good approximation for most of the mass of

completely convective stars.


P = KT2.5


Here, the ratio of specific boats, y, is constant and equal

to 5/3. Even including the photosphere, this adiabatic

relationship i- a good approximation.






18

For stars in convective equilibrium, the adiabatic

temperature-pressure relation can be rewritten in terms

of Schwarzschild dimensionless variables as

p = Et2.5



where p = {(47r/G) R4/M2) P and t = {(R/G)R/uLM}T


and E = 47K(p/R)2.5 G1.5 M0.5 R1.5

If the convective envelope extends all the way to the cen-

ter of the star, then a limiting -.alue of i foi comnolete]y

convective stars can be found to be:


E = 45.48


Hayashi described the "Forbidden Region" for a star

of a given mass to be to the right of the line A-B. Line

A-B represents the surface solution for the luminosity

based on the limiting value of E. A star finding itself

in this forbidden region would not be in hydrostatic

equilibrium and would have to adjust itself to another

structure on roughly a "free-fall" time scale.

The luminosity for fully convective gravitationally

contracting stars, can be found by solving for radiative

transfer in the photosphere. Cox and Giuli (1968) found

a solution for the luminosity for stars /:. h


L > L


to have a strong dependence on the effective temperature in

the following form:









L cBlpB2MB3 B4
LIP M Teff


By assuming complete ionization of hydrogen at the base of

the photosphere, the equation for the luminosity has been

shown to be:



L a C-9 0.9M-1.56 T-16.9
eff


Schatzman (1963) found the solution for the luminosity in

terms of solar type population I chemical composition and

mass to be:



Lh = 10-87 M Teff15.3 Equation 1)


Where M, is the stellar mass in solar units.


F. The Gravitational Contraction


For a pre-main sequence star, the only significant

sources of energy come from the gravitational potential.

The relationship between the gravitational contraction and

luminosity can be determined from the equation of thermal

equilibrium:


dL
d- 4Tapr2


By using the ideal gas law, allowing the r.:.io of specific

heats, y, to be equal to 5/3, and defini:-9 c= n + Cg, the

equation for thermal equilibrium becomes










dL
d 4,r2p(n 3 2/3 d (P


By using the equation for hydrostatic equilibrium the fol-

lowing relationship holds:


3 2/3 d P 3 P L dR
a dt ( 5/3) 2 P P at


Substituting, en = 0 into the equation of thermal equilib-

rium we get the following:


dL. P dR
S- 6r2 P
dr R dt


Now this relation is rearranged in the form of polytropic

variables:


dL 6rrn2 p n+l 1 dR
d = 6rn P c6 R dt


Finally, the dR being rewritten in different form and by
dt
noting that R = 51 Ar,, the relation becomes


dL
= 6TrrnPc E28n+l AR
51 At

Where AR is the radial contraction and At is the time step

for the contraction, the total contributio- to luminosity

from a gravitational contraction -oL a pcl '-upe is found by

integration:












L = 6iPcrn2 Arn f r2Gn+1 da (Equation 2)
At 0


where El = 3.65375 for n = 1.5 and E = 6.89685 for n = 3.0.


This relation for the luminosity has the term Arn. The
At
Arn represents the polytropic radial contraction while At

is the corresponding time step. Arn is selected simply to

optimize computing time. The time step, on the other hand,

must be determined from the Virial Theorem and conservation

of energy.

The Virial Theorem simply states that half the energy

made available by a contraction causes an increase in the

stellar temperature, while the other half is radiated away.

The potential energy 2 is given by:

R
p = + f VdMr



For a polytrope the potential energy is given by:


= ( 3) GM2
5-n R


If only half the potential energy is available for the

luminosity, then the time step, At, is determined by


^t -



where is the average value of the luminosity for corres-

ponding time step. Therefore, we have












At = ti+1 ti = ( 3 ) GM2 (1 1
5-n Ri+1 -

(Equation 3)

and = (Li+1 + Li)


where the index i corresponds to the ith model.

The problem then is to determine, by an iterative

procedure, selfconsistent values for the new value of the

luminosity Li+l and the time step ti+1 t-. An initial

guess for the time step was obtained from the Kelvin-

Helmholtz time scale for contraction onto .he main-osequence.

The estimate was obtained by proportioning the time scale

for the radial contraction step. An estimate for the lumin-

osity, L', is calculated for the polytrope. An estimate

for the time step, At', is calculated from L'. Then using

At' a new value of the luminosity, L", is calculated and

compared to L'. If L' is greater than L", then a new in-

creased time step, At", is estimated. Likewise, if L' is

less than L", then the new estimate for the time step is de-

creased. This can be written as follows:



At = At" +jAt' At" l


where + was used if L"
The primed values are the old estimates and the double prime

indicates the new estimate. After L" and L' approach each







23

other to the desired accuracy then L" is set equal to Li+I.

The time step, At", is stored for that model as At and be-

comes the first estimate for the time step for the next

model.


G. Nuclear Energy Sources


If the assumption is that no contraction takes place

then the equation for thermal equilibrium becomes


dLn
dr = 4Er2p-n


By using the appropriate polytropic var:iabLes and the nuclear

energy generation rate cn for the proton-proton reaction,

the luminosity is



Ln = 4Trrn3 P2Tc4 oX2 f 22n+4d (Equation 4)


where Eo 10 X 10-30 ergs/sec/gm and X is the hydrogen

fraction of stellar material.

The nuclear luminosity is calculated and compared to

the luminosity of the "zero-age" main-sequence star of

corresponding mass. The model calculations ended when Ln

approached the main-sequence value.

The time that a particular componernt of the binary

remained on the rmain-sequence is considered to evaluate

the systems. Generally the primary should reach the main-

sequence before the secondary. It is of interest whether







24

the primary has enough time to evolve off of the main-se-

quence before the secondary can reach the main-sequence.

A logarithmic regression is performed on data published by

Iben (1965), and Strothers (1963) (1964) (1966). The fol-

lowing Table 3 represents mean published values for the time

on the main-sequence tIg, Novotny (1973).
TABLE 3
TIME ON THE MAIN-SEQUENCE

M/Mo log M/Mo log tMS

60 1.7782 6.4914
45 1.6532 6.5563
30 1.4771 6.6721
15 1.1761 7.0043
9 .9542 7.3242
5 .6990 7.8088
3 .4771 8.3443
2.25 .3522 8.7259
1.5 .1761 9.2966
1.25 .0969 9.6053
1.0 0 10.0086
0.5 -.3010 10.4771
0.25 -.6021 10.8541


The following relationship can be used to represent the

above logarithmic values: (The maximum error is less

than 5% over the entire range of masses.)


log tMS = (-2.00998) log (M ) + 9.62409
SM

In all cases evaluated, the secondary has sufficient time

to contract onto the main-sequence before the primary

evolves away.















H. Initial Conditions


With the masses of each component and their period

being free parameters, the Roche Lobe for each star was

determined by the program. The following relation, given

by Paczynski (1971) for the Roche radius, Rr, was used:


Rr = A (0.38 + 0.2 log (M1/2) )


The Roche radius was determined for each ftar and was

taken to be the initial stellar radius for the first model

of each star. The program then calculated the sequence of

contraction models.

The chemical composition is chosen to be of popula-

tion I type and assumed to be the same for each star. The

hydrogen fraction, X, is selected to be 0.750, the helium

fraction, Y, selected to be 0.224, and the heavy metal

fraction, Z, is selected to be 0.026. This selection is

arbitrary but is chosen to allow comparison with the work

of Iben (1965).

The total mass for each model is between 2.5 and 6

solar masse-. Eleven close bi --ry systemO are chosen from

Giannone and Giannuzzi (1974). The parame-ters are listed

in Table 4. Column 1 is the name of the System. Column 2

is the period in days Column 3 is the mass in solar units







26

for both the primary and the secondary. Column 4 is the

logarithm of their luminosities in solar units. Finally,

column 5 is the logarithm of their effective temperatures.


TABLE 4

OBSERVED SEMIDETACHED BINARY SYSTEMS


SYSTEM PERIOD M/Mo LOG (L/Lo) LOG (TEFF)


TV Gas 1.243 3.10 2.070 4.029
1.39 1.028 3.787
IN Aur 1.245 2.97 1.999 4.029
0.89 0.203 3.679
U Cep 2.493 3.19 2.066 4.079
1.53 0.875 3.678
AI Dra 1.199 2.18 1.467 3.982
1.03 0.650 3.756
6 Lib 2.327 2.96 1.976 3.982
1.31 0.770 3.675
8 Per 2.867 3.15 2.082 4.079
0.74 0.797 3.696
U Sge 3.381 4.27 2.593 4.134
1.60 0.573 3.567
V505 Sag 1.183 2.22 1.501 3.958
1.18 0.414 3.693
X Tri 0.972 1.72 1.044 3.947
1.00 0.000 3.642
TX U Ma 3.063 3.13 1.997 4.079
0.90 1.088 3.746
W U Mi 1.701 2.68 1.824 3.947
1.19 0.541 3.681



I. Description of the Model Program


The program calculates both stellar model sequences

simultaneously. This allows comparison of time steps and

structure of each component when possible. The initial

mass of each component and the period of the binary are

parameters of each model sequence. Each observable system

can be evaluated in turn.







27

Figure 2 represents a flow chart for the pre-main-se-

quence model program. By using the Kelvin-Helmholtz con-

traction time, an estimate for the time step, At, is made

for the initial radial contraction AR. The initial value

of the luminosity, L', is arbitrarily chosen from the mass-

luminosity relation for main-sequence stars:



log (L ) = 4.1 log (L) 0.1
Lo Mo

The radial contraction, Arn, wps chosen in all cases to be



(Arp)i1 = 0.005 Ri



A polytropic model is calculated for both n = 1.5 and n = 3.

By using Equation 2, the first value for Lg is calculated.

The effective temperature is calculated by using the follow-

ing relation:


4 L
Teff 4' R2


By using Equation 1, the Hayashi Track luminosity, Lh, is

found and compared to the radiative luminosity, Lg. If

Lg
the time step can be calculated fj-rm th(? Viral Theorem

using Equation 3. If the old falue fo r the urnmi nosity

L' is equal to L" within sufficient accuracy, then the

pro -'dure is stopped; if not, the time step is adjusted








ESTIMATE FOR INITIAL TIME
STEP AND LUMINOSITY


CONSERVATION
OF ANGULAR
MOMENTUM
---N -L


FIGURE 2. Flow chart of the program for the pre-main-sequence
model






29

as described previously and the procedure starts over

again. After the error between L" and L' is less than 1

part in 105, the values for R, Teff, L, and At are stored

for that model. If the program limits have been met, Ln

is large enough; the program stops; if not a radial con-

traction is made and the procedure is restarted.

Appendix A lists the main program, all subroutines

and polytropic constants. The language is Honeywell's

Mark III version of General Electric Company's BASIC.

A test model is compared with the evolutionary tracks

published by Iben (1965). The primary for the test model

is 3 solar masses, the secondary is 1.5 solar masses, an6

the period is 3 days. Initially both stars are filling

their Roche Lobes and are on the Hayashi Track. Figure 3

shows the plots of the test model and Iben's pre-main-

sequence tracks for single stars on the Hertzsprung-Russell

diagram. The initial point for Iben's models is labeled

"O". The points 1 through 5 can be used for age compari-

sons with the test model. Table 5 shows the ages at these

points labeled on the evolutionary tracks in Figure 3 for

each star. The same numbering is used for the test model.

The test model tabulation for both the primary and the

secondary can be found in Appendix B. Each table has 6

columns. Column 1 is the model number after the initial

model. Column 2 is the accumulated time ir seconds after


the initial model.











2.5 o
0

2.0- \.


1.5- 6 \ /
1 3 A
\ /o
S1.0- 5 ---
0o< 1o



o0. 3 0



-0.5 -


-1.0 -



4.2 4.0 3.8 3.6 3.4
LOG Teff

FIGURE 3, Evolutionary tracks of a pre-main-sequence test
model for a binary (S = 3Mo, 0 1.5 Mo) and the
evolutionary tracks for single stars published
by Iben (1965).









TABLE 5

COMPARISON OF AGE (106 YEARS) FOR THE TEST MODEL


PRIMARY = 3 Mo SECONDARY = 1.5 Mo

AGE POINT IBEN TEST MODEL IBEN TEST MODEL

1 0.034 --- 0.23 0.205
2 0.208 0.06 2.36 1.391
3 0.763 0.747 5.80 3.483
4 1.135 1.202 7.58 4.962
5 1.250 1.351 8.62 5.949
6 1.465 1.432 10.43 --
7 1.741 -- 13.39
8 2.514 --- 18.21


Column 3 is the logarithm of the luminosity in solar units.

Column 4 is the logarithm of the effective i-emperati-ue.

Column 5 is the ratio of the model radius to that of its

Roche Lobe. Finally, column 6 is the radius of the model

in solar units.

These ages and evolutionary tracks of the test model

show reasonable agreement with Iben's results. Up to

point 5, the age of a component of the test model compares

with the age of Iben's models. Suggestions for future im-

provements and changes are discussed in Chapter IV. The

major differences in ages occur near the main-sequence.

These differences are caused by the limitations of a poly-

tropic model to adjust for the core's changing chemical

composition. In particular, Iben followed 'he depletion

of C12 in detail. Normally, C12 is maintEined in equilib-

rium for the CN reaction chain. However, during the pre-

main-sequence contraction the CN reaction has not started.







32

Only very near the main-sequence is there sufficient ther-

mal energy in the core to initiate this reaction. C12

must be depleted to form the equilibrium concentrations

of the "secondary" elements in the CN reaction chain.

After equilibrium is reached, the concentration of C12

remains constant. This resulted in slowing down the con-

traction near the main-sequence as C12 was depleted in

the core. The test model is polytropic and cannot follow

detail changes in stellar structure. It is intended as

a survey tool to determine whether or not a binary system

is pre-main-sequence.

















CHAPTER III

RESULTS OF THE NUMERICAL CALCULATIONS


The age computed by the writer's model is the time

after both components filled their Roche Lobes. It may

not be the same as would be calculated for a single star

contracting onto the main-sequence.

Appendix C lists the numerical values for each of

the pre-main-sequence models calculated as described in

the previous chapter. The Appendix C contains 22 tables.

Each table is identified by the name of the binary system.

Each table has 6 columns. Column 1 is the model number

after the initial model. Column 2 is the accumulated time

in seconds after the initial model. Column 3 is the

logarithm of the luminosity in solar units. Column 4 is

the logarithm of the effective temperature. Column 5 is

the ratio of the model radius to that of its Roche Lobe.

Finally, column 6 is the radius of the model in solar units.

The computed binary model and an observed binary sys-

tem are compared by age and position on the Hertzsprung-

Russell diagram. The ages of each component aust match and

the observed binary must fall near tne evolutionary tracks







34

of the binary model for the systems to be pre-main-sequence.

Individual systems are discussed in the following sections.


A. TV Cassioneia


It appears unlikely that TV Cassiopeia is a pre-main-

sequence binary. The secondary is either over-luminous or

it has too great an effective temperature. This is in con-

tradiction to the results published by Field (1969) for

TV Cassiopeia. Field determined the pre-main-sequence

nature of the system by estimating the radii and age of

each star. The estimate is made by interpolation of pub-

lished models of a single star.

Figure 4 shows the plotted contraction of both com-

ponents of a pre-main-sequence model for TV Cassiopeia on

the Hertzsprung-Russell diagram. The observed luminosity

and effective temperature for both components are plotted

also. The problem of the over-luminosity of the secondary

is clearly shown.

If the age of the primary can be determined from the

model, it is approximately 6.3 X 105 years since the in-

itial model. To estimate the age of the secondary, either

the radiative solution or the Hayashi solution must assume

to be correct. Comparing the luminosity with that of the

model secondary on the Hayashi Track yields an age of 1.5

X 105 years. If the radiative solution is assumed, the

age is 3.1 X 105 years. This pre-main-sequence model does

not fit the observed system for either case.

















F M %


2.5


2.0


1.5


`\O0o'
-I f


6.3 x 105


4.0


YEARS


3.8
LOG Teff


FIGURE 4. TV CASSIOPEIA (A = 3.1 Mo, A = 1.39 Mo). The
evolutionary tracks of the pre-main-sequence
binary model is represented by the circles
(primary 9, secondary o).


0

8
o


\o9.


0
0
0
O
0


1.0


0.5


0.0


-0.5


.O-


4.2


3.6


3.4


__
















B. IM Aurigae


Figure 5 shows the plotted contraction of a pre-

main-sequence model for IM Aurigae on the Hertzsprung-

Russell diagram. The observed stellar luminosity and

effective temperature for both components also are plotted.

The observed system falls close to the contraction paths

of the pre-main-sequence model. However, the "ages" do

not match.

The age of the primary seems to be 3.1 X 105 years.

The age of the secondary appears to be between 9 X 105

and 3.1 X 106 years according to this mo-:l.


C. U Cephei


Figure 6 shows the plotted contraction of a pre-main-

sequence model of U Cephei on the Hertzsprung-Russell dia-

gram. The observed luminosity and effective temperature

for both components also are plotted. This system appears

to be pre-main-sequence.

First of all, the observed system falls close to the

contraction path of the model. This is true for both com-

ponents. Secondly, the ages match within 20% at 3.15 X

105 years if the secondary is on the Hayashi Track and has

a slightly different chemical composition. Iben (1965)

















\ ,CO


2.5


2.0


1.5


0
0


3.5x105 YEARS


-1. 01


4.2


4.0


3.8
LOG Teff


3.6


3.4


FIGURE 5. IM AURIGAE (A = 2.97 A =z 0.9 Mo). The
evolutionary tracks of the pre-main-sequence
binary model is represented by the circles
(primary a, secondary o).


0
0


1.0


0.5-


0.0 -


-0.5


I


















99


000 A
\ 000 0
N 0


6.1 x 105 YEARS







I


4.0


LOG Teff


FIGURE 6.


U CEPHEI (A = 3.19 Mo, A = 1.53 Mo). The evolu-
tionary tracks of the pre-main-sequence binary
model is represented by the circles (primary *,
secondary o).


2.5


2.0


1.5


S0-


0.5


0.0


-0.5


1.0-


4.2


3.6


3.4


|


I







39

showed a slight decrease in the metals composition would

move the Hayashi Track toward higher effective temperatures.

This would match ages and positions on the Hertzsprung-

Russell diagram. This model predicts that U Cephei is a

pre-main-sequence binary star system with population type

I chemical composition for both components.


D. AI Draconis


Figure 7 shows both the path of the pre-main-sequence

contracting model and the observed binary system AI

Draconis plotted on the Hert.spr ng-Russell diagram. The

system does not appear to be a pre-mainr-sequence object

according to this model. This problem seems to be the

same as TV Cassiopeia. The ages do not match. The secon-

dary component of the observable system is over-luminous;

or the effective temperature is too high for it to be con-

sidered a pre-main-sequence star for this particular mass.


E. 6 Librae


Figure 8 shows the pre-main-sequence contraction model

for 6 Librae plotted on the Hertzsprung-Russell diagram.

The observed binary system may possibly be pre-main-

sequence if it is assumed that the two stars have different

chemical compositions. This would follow the description

by Wood (1962) of the formation of a binary with different

compositions. As stated before, Ibn (1965) has shown that

such a change in composition could increase the effective











2.5- o
0
\ o
2.0 \o
0
I .5 \
\ o
SI .0 1 o


o 0.5 -
-J
0
0.0 /


-0.5 \
2.6 x 106 YEARS
-1.0 \



4.2 4.0 3.8 3.6 3.4
LOG Teff

FIGURE 7. AI DRACONIS (A = 2.1.8 Mo, A = 1.03 Mo). The
evolutionary tracks of the pre-main-sequence
binary model is represented by the circles
(primary *, secondary o).










0




0
0


2.5


2.0


1.5


8.2 x 105 YEARS


4.2


4.0


3.8


3.6


3.4


LOG Teff


FIGURE 8. 6 LIBRAE (A = 2.96 IMN, A = 1.31 Mo) he evolu-
tionary tracks of the pre-mai a-sequence binary
model is represented by the circles (primary o,
secondary o).


0
0
0
0
0
O
oo
0


1.0


0.5


0.0


-0.5


-1.0


'~OleCshq
as~







42

temperature of a star on the Hayashi Track. With this in

mind, the ages of both components are in agreement. The

age after both components filled their Roche Lobes is

approximately 8.2 X 105 years.


F. 8 Persei


Figure 9 shows the observed binary 8 Persei plotted

along with the path of a pre-maini-sequence contracting

model. This system does not fit the model for a pre-main-

sequence binary star system. It suffers LLom the same

problems as TV Cassiopeia and AI Draconis. The secondary

is not represented by this pre-Lmain-sequence model. The

age is also a problem. If the chemical compositions are

different enough to fit the model, the age for the secondary

would be younger than the primary by a large factor. This

is caused by the decrease in contraction time as the lumin-

osity increases. The luminosity must be much higher for a

given effective temperature for the secondary to fall on

the Hayashi Track.


G. U Sagittae


Figure 10 shows the plot of the binary system U

Sagittae on the Hertzsprung-Russell diagram. Comparing

the evolutionary tracks of the model with tile obser ved

system, the binary could be pre-main--equence. The ages

of both stars appear to match at approximately 3.78 X 105

























,0
\ o




-1.14xl 6 YEARS -


I I I


4.0


3.6


LOG Teff


FIGURE 9.


3 PERSET (A = 3.15 Mo, A = u. Mo) The evolu-
tionary tracks of the pre-mara-- 4 quence binary
model is represented by the circles (primary e,
secondary o).


2.5


2.0


1.5


1.0 -


0.5-


0.0-


-0.5 -


-1.0-


4.2


3.4


C$99,











2.5 s A 0
\ ** o

2.0- \ o
o
\ o
1.5 \
\ o
a \ o
oo 0


0 0. 5 -\
_J

0.0 -


-0.5- 3.75x 05 YEARS


-1.0 -


I I I I
4.2 4.0 3.8 3.6 3.4
LOG Teff

FIGURE 10. U SAGITTAE (A = 4.27 1M, A = 1.60 I1)). The
evolutionary tracks of tIhe pre--main-sequence
binary model is represented by the circles
(primary o, secondary o).







45

years. An increase in metals content of the secondary

would move the Hayashi Track toward lower effective tem-

perature. U Sagittae is probably a pre-main-sequence ob-

ject. Therefore, another alternative is that U Sagittae

may not have a fully convective pre-main-sequence secondary.

The secondary falls in the forbidden region described

by Hayashi. If this is actually the case, then the star

is not in hydrostatic equilibrium, and its structure would

be changing rapidly. Larson (1969) described pre-main-

seauence contractions that may not be on the Hayashi Track.

These evolutionary tracks had rpiidly varying luminosities

as a function of effective temperature, caused by different

depths of the convective envelope.


H. V505 Sagittarii


Figure 11 shows the paths of a contracting pre-main-

sequence model plotted on the Hertzsprung-Russell diagram.

Both components of the observable binary system V505 Sagit-

tarii are plotted along with the model. By using the same

argument that the components could have slightly different

chemical compositions, the system could be pre-main-sequence.

The age of the system would be approximately 1.89 X 106

years.


I. X Trianguli


Figure 12 shows the paths of the contracting binary

model and both components of the binary X Trianguli plotted













0
0
0
0


**



0
0
0
0


00




1.89 x 106 YEARS





I I


0O-


4.2


4.0


3.8


3.6


3.4


LOG Teff

FIGURE 11. V505 SAGITTARII (A = 2.22 t,, A = 1.18 M,).
The evolutionary tracks of the pre-nrain-
sequence binary model is represented by the
circles (primary e, secondary 0o)


2.5


2.0


I.5


1.0


0.5


0.0


-0.5































4.6 x 106


4.0


e o
0
0
0


YEARS


3.8


3.6


3.4


LOG Teff

FIGURE 12. X TRIANGUILI (A = 1.72 Mo, A = ].00 M). The
evoltuionary tracks of the pre-main-sequence
binary model is represented by the circles
(primary a, secondary o).


2.5


2.0


5


1.0


0.5


0.0


-1 .0-


4.2


\A


0








48
on the Hertzsprung-Russell diagram. The observed binary

system falls close to the pre-main-sequence path of the

model. The ages, however, do not quite match. The dif-

ference in the ages is a little less than a factor of two.

This could possibly be caused by a difference in chemical

composition. If we allow for a slight increase in effec-

tive temperature, caused by a decrease in heavy metals,

then the time of contraction would decrease. This is re-

quired by conservation of energy and the V--rial Theorem.

With these assumptions the binary X Trianguli could pos-

sibly be a pre-main-sequence object.


J. TX Ursae Majoris


Figure 13 shows the observed binary TX Ursae Majoris

plotted on the Hertzsprung-Russell diagram. The contract-

ing pre-main-sequence model is plotted also. The system

does not appear to fit the model. The age would have to

be _9.5 X 105 years as determined by the primary. This

yields the same problems that were found with 8 Persei and

others. The secondary is over-luminous for a pre-main-

sequence star with this particular mass.


K. W Ursae Minoris


Figure 14 shows the observed contact binary system

W Ursae Minoris plotted on the Hertzsprung-Russell diagram.

The path of the contracting pre-main-sequence model also














$ eve
\^A *
as9 9 e*


9.5x 105 YEARS


o

\ o

_ _


4.2


4.0


3.8
LOG Teff


3.6


3.4


FIGURE 13. TX URSAE MAJORS (A = 3.13 Mo, A = 0.90 Mo)
The evolutionary tracks of the pre-main-
sequence binary model is represented by the
circles (primary o, secondary o).


2.5


2.0


1.5


0
0
0
o


1.0


0.5


0.0


-0.5


.0-


L I_







0
0
0

o
\ o
- \o


2.5


2.0


1.5


4.0


3.8


3.6


3.4


LOG Teff

FIGURE 14. W, URSAE MINOEIS (A = 2.68 Mo, A = 1.19 M0) -
The evolutionary tracks of the pre-main-
sequence binary model is represented by
the circles (primary e, secondary o).


\\o

A o
0O

\ 0



!.87 x 06 YEARS--


1.0


0.5


0.0


-0.5


1.0-


4.2








51
is plotted. Both components of the observable system are

relatively close to the path of the model. The exciting

part of this analysis is that the ages for the pre-main-

sequence model match at about 1.87 X 106 years for a sys-

tem slightly different from the observed system. Iben

(1965) showed that an increase in metals could account for

the increased luminosity of the primary on the radiative

path. He also showed that a decrease in metals could in-

crease the effective temperature for the secondary on the

Hayashi Track. The results of this model cannot be dis-

counted concerning the possible pre-main-seque'ce nature

of W Ursae Minoris.
















CHAPTER IV

CONCLUSION AND DISCUSSION


This binary model allows discovery of pre-main-

sequence systems. Initially both stars must be forced

into their inner equipotential surfaces rapidly. Evolu-

tionary tracks calculated from -his initial configuraLion

define a new means for evaluating actual binary systems.

The major conclusions cf tnis evaluation are as

follows:

1. Pre-main-sequence close binary star systems can

be found by comparing observed binaries with this

model. Some observed closed binary systems are

found to fit this pre-main-sequence model.

2. The chemical composition of the two components

of the pre-main sequence binaries appeared to be

slightly different. Even though the actual dif-

ferences in composition were not quantified,

both stars could still be considered population

type I.

3. There does not seem to be a correlation between

the orbital angular momentum, ages, or mass ratios

for the pre-main-sequence binary systems.

52









Table 6 summarizes the evaluation of the eleven

binaries. The distribution falls fairly evenly into three

groups, those binaries that are likely to be pre-main-

sequence, those that are possibly, and those that are un-

likely to be. Improvements in the model could allow better

determination of the pre-main-sequence nature of a binary

system. A more realistic model, instead of assuming a

polytropic structure would guarantee more accurate struc-

ture and age determination. It would allow for changing

the chemical composition of both stars. The close binary

systems under consideration most likely are in synchronous

rotation. The periods are approximately a few days. This

rotation has been neglected in the calculations, but it

surely would slow the gravitational contraction to some

extent.

Roxburgh (1967) evaluated the effects of rotation and

magnetic fields on pre-main-sequence evolution of binary

systems. Rotation of an individual star could be uniform

or differential. Uniform rotation requires a large vis-

cosity of the stellar material to couple the core to the

envelope. If the star is in convective equilibrium or has

a sufficiently large magnetic field (>1 Gauss), then the

viscosity of the stellar material should he great enough

to -llow uniform rotation. In this case, Roxburgh predicts

loss of material at the stellar equator. Thus, rotating

pre-main-sequence stars in convective equilibrium will fol-

low a more vertical Havashi Track.
















o 1- W -i I n W L mo CO r Lo
4 4moNN 44N


in L* N t ) 11 IN 01 o CO mm
":iT In CN ma OcNiIn r- In M
"IT IN nr>N "rr- -q Cl


x X


X X X x


x x


in
I Ln r-I4 N m f, Lnr
Lo )o c D r-4co o
ooocsor-oiO


td
tM -: r; 10
to 3 a o *
U <; w C -r ( trin n

> I Hi Ln
E-i H D < a > X


rl


x3
X


U ~U ~ tI


z En
HE-
H

DD
Z 2


00
X En


cc2



MOH
0 HI




PI



I
Ci
(./i
o:







55

Stars in radiative equilibrium do not have the "con-

vective viscosity" mechanism. They are more likely to

approach the critical angular momentum valve and fission

may result. He concludes that contact binary systems could

be formed by fission of a single pre-main-sequence star

on the radiative track. For binary systems that are formed

before they reach the radiative track, the rotational

angular momentum should not reach the critical value.

Both components should be in synchronous rotation and most

of the angular momentum would be orbital.

Bierman and Hall (1976) and Oliver (1978) evaluated

the possibility that RS Canum Venaticorum systems could be

pre-main-sequence. Their arguments were based on space

densities and probabilities assigned by evaluating ages.

Because of the large number of these systems, they concluded

that they must be post-main-sequence. This conclusion is

valid as long as the universe is the accepted cosmological

age. However, there is no direct observational evidence

that these systems must be post-main-sequence. The writer

would like to apply an appropriate pre-main-sequence model

to RS Canum Venaticorum type systems. Bierman and Hall

concluded that RS Canum Venaticorum system resulted from

the fission of a post-main-sequence star. It would be in-

teresting to model a thermal contraction f hydrogen de-

pleted secondary. This inay represent a bi.ary immediately

after fission.









In studies of pre-main-sequence and post-main-sequence

evolution, the origin of a forming system is assumed. The

writer is assuming the origin of the binary system.

Clayton (1978) and Larson (1978) and others have presented

theories for stellar formation. They admit that unresolved

problems remain. These include gas dispersion, viscosities,

and turbulences within a collapsing gas cloud. The prob-

lems are more severe for the formation of a binary star

system. Many scientists have assumed the origin when mathe-

matical modeling beyond a particular point is difficult or

impossible. Whitcomb and Morris (1961.) imv-e proposed

special creation beyond the known laws of physical science.

Special creation states that God created the universe

(Genesis 1:1). He created each object with a specific

purpose ( I Corinthians 15:40, 41)

There are also celestial bodies and bodies
terrestrial: but the glory of the celestial is
one, and the glory of the terrestrial is another.
There is one glory of the sun, and another
glory of the moon, and another glory of the stars:
for one star differeth from another star in glory.






































APPENDIX A

PRE-MAIN--SEQUENCE BINARY MODEL PROGRAM LISTING













17n "T" c(?,50)

1 o7 "T T (
2io nr) -T (?IT'((
23 n0T)' "(T )
21.4 n TU 'T(2)




2r0 fTP' ,(2)
5.4 nr r P., ( r)

27n ~)" C(2)
260 I) T 'I1(2,2)

3r)n nT'T P(2)
31 0 nTI T(2.,95 )
120 0T'. V(2.25')
330 nT7' 4(?.2T)
14n nTi" 1(2,250)
S42 D -,NTIP .I 0f M A' -?"
3.42 T!PUT AP
350 PP TPT'l"i.,. r T M )r T T'T MS T "
3-j60 TIPTlJr '1I

39O THPUT "( 1 .(2)
17. D0 OP T r '"f',': ( r- PQ.' f rV 40 q r A, 7'3\)

s02 DRT.JTI(' Qon P T- T ',r) DRACTT)'7T (01 ,02 --?"
C4 TTMP!IT 01 ,0T
"0n ppT'"TTVIDP Trrr Tr "-'~vc=-
302 Tr'P'IT PI
304 pD.7fiTln Aq<(: TPA.ryg-cp ,ATC (,n/VCAD) --?"
30o/ TI'PUT TI
4 ; r)7:1, p (r)nv`irQOA TT)'l c'rT'-AqTVT Tq t
405 n rfI n ,
AIn C01". 00
420 r (1 )=''( (1 .=ooT 01
43)0 S(2)-'=i(2 -'-1 .o c1 +4-
440 0( ) -=A :! 1 .5 +1- 1+fL tn -n n o3-f ('((1i ) /*(( )) /2.3 r250)
450 O(3)=^ I-1 .F+ + --( 1.' -0- n.o "-nl r o( ('(3)/'( 1 )) /2.30250o)
455 ?PF IM!TNT TAT pT r rnCrlTIn' '
460 (i ,1)=(-0 )+ 1
47n P(2,1)--0()102
rn I) rp7f CAr C '(EAn? "rIT CfIT A 'T T.
51 Y r- 75r
3? ")

A 0' i i .)'(?"'* ^ v / --'7 ?














5 ) Dr?) '. f*2)-p)'T V 'T n T TL 'nF Y
57 'r?3 -
0on rlf)Qntc 5V1)n
A 'Q(n nYlq
*2n cop T=I T-) 5

,4Fn "JPYT I
aSrSo C0!0 T-1 r" 1
,QLn crop T=1 T'- 9
.62 l=n (T. ')
S^4 r)nl=n(T,2)
A,5 no cp 'fn~in
477 I TJ? TIT-:! 70n

so0n r on;i 7A ,"
700 'Th;!V9 22)2
7no -oqin ,? )
710n -r)qjI 2^-q
751 ) niD '=1 Tr) In

730 T r(,T.p)- P T-'4U 75
740 T ^ A^ P (( 2-_(- T. ) I/"( I,')) ". n"o l TIrl'l T qon
75n 0r) ll;[ i -Q
754 TF 02>(T,.2) T'"~ 70;
756 n( T.2)= ( T. 2)+( T A' ( ( T.2 1-n ) 4) r T2.
* 75) fl'Trn 770
7,V n( T )=n(T. )-(h ( n(T,? 1-"2 )) *'* .
77P C( T79) -
700 NIMyT v
8on pcr, CmUiTT'l!irU
ql n T I,1 )-nli
9q ( I. 1 ),--)lI
qAin irn;[qlq Airn
q5 T Tl -. ". A q59, T1=T1-."
S60 Tc TI >-( T, T) TI-cT I non
o00n ar!iq 49r00
00 V(T. T)--v( I. T-1 )+n(1.2)
ol0n M(F T T
0r 0 I n lX r"rT T


orrn rqrlq q 7Aq q

n 70r T 7r -
,)7n trr -T T













1 qo ~C Drn'TIfTp' T'1 T" nc 3 r) !Tn TTr

1 )o)oo q-rn)
S)n QiH 'flO' .iTTT c To) 7qT. r)--rT TA
?2 l n T.(_ )--1 <"(4. !-t nrO(u(T ) /2 .' 0250-. 1 )
2 rn ( 1 T((.7 7 7c] 4'( T =) ( .l-7 ) /.(r)
?qcn n( T.?)-- (I )
)2 0 r( T,2)= ( T,l )+ l

2 npn pRpTTQM
3'?n ",re iD0o0i!lT I! E Tre) n CATO pn TVTPnpTr ,nnfrr
2230 l'( T. )= (T, )/C
723n V'( T .? )-=P( 2)/
32250 C(T = c )/((1 ., ?447 )- "''(I ,)+r3+n)
33 t( T)-= (Q. TOR709-,.--7)4i-".f' ( .4. / ( ('" 41 )5 '( T) t-( 1 /'T?--1 ))
p?7" D(T)--'((T)w- (T)*+( -1+ /t? )
c2)', T(T T ), I -I (1 .1 1] A _o- or ( ::r (
?920 DPFTrITH
7400 pF'T RIt D')TTh! T',- T' 7 .T' Mp r: A;, T N'f f",' T '":
210 T1-,=lO'-q
A220 T1 =?2T n(C( T ))/2.3n2r)O
24 0 1 2 T ( 2 37,- TO-'n-l,'i*Y*- 2 ) /. ?1350
S2440 T3=3+T n('( T,2)) /2.0)25
94? n TAl-.4-T+ f (( T, T))/p.nt25o
4x T )( T ( )=1 n--+( Tl I-T 4 T7 -T )


-^)"1 r T -="'( T. I )-'-'( r )
262Q T 0 <0 Tr-c" 2720
),4,,o T 3=t.r: ( l *'. ,405-r<^-^ TQP (t ) /n T. ? ) ) /l> .? ^^ ^
2* 5 T.3=2*T ( "'(T,2)) /2.3 025?
944A ir 12T_-Cinq TruM i n)n
37T n= 1---~-I +-(T 4-1T )
2720 PFT!lwT
2?ro0 PF qrIJflQ''r)lTT TO') AfT M C, '! C) I TA-n con '- E ) DAV. t ri'(.--
910)A n Y=0l-r000 TLIyM Pon
2R30r Ti= 2*T or((T ))/.?n2o
?Qan) T2=T n( 1 /P(I ,2)- /D( I,1 1) /2.no) o
2_9 O 3=T. =n ((T2"( T,! ))/2)/?.R, To,
2QR6 TAl=--4. o)O7P-T.(n(5-'M2)/2'.025O0
357, n 2=1 n-*- ( T 4-T2-TI +-T-+ )
9 non DFTTM

q-rq qr2( ) ( 'r( 1 )) (?
-Z 11 DC T-'^














''2' [' D!')' T' p Tl=r -) AT9 T r c
crp T--I To 2

CAo D V- o 30
pP IT'IT "
MPYT
(n9'J" 59RO




Tri aJ

Tc TI crO9 'f T' 1 A


'41A00




44]74
46 A6
4640
4.650
4if55
4.560
;7 0I
4609
.4600
47 00
4709;
471 0
4720
-7 0r)
4740
. 4751
47 0 r
4770
4790
4? no0

4,q2 r)
491 0
4920
4930
4940

? nr)
5200
5210
tj ?7 r)
52q0
5200
5300
5,11 n
5210
5?20
5 .9 0
5340
53"0




561n
5 ,#, ( O


P ITI" "

p;py n cncp 1; e no(' r_'.) 1 r'(-TF c
DP .IT" "
II =-0
PPI'T VIT',Tq .4795. T.V(, T), .A(T, ). 1) r ( T I (T, T)
NMYT T
'JF T I
p:)( q'nlri)TITTM:C Qrp DAn TAT 0" MT '-rCTTT
T l= (I ,)
TrF Y )or000) Ttr:~r 4"0on
P( T,2)-D( T.2), C1
D( T, l ) -T 1
PFTIJR'1
I Qf I'WT(TTI'c q TrO qFT PrITVTPr-pTC C')'PTT'TRS"
TP r12=1 .5 THME 5320
C-=^.ROCR

(3=0. 90 4~46
N3=. 1 015A
PPTIPO,
C=* .45275





Dr- :.' IO'pIT T f- T') CATlI C E A p TT')A' ronA nOInTAT fT)CD rI'




A 1 T j -. ?
I -- ( ( I I )f.1 ( ) ) -,'+ )*' ( (* )


rno (T /, q:'j ) T .r (T'-r :)






- I J 1 4 1 Al{ ,
s ~


p


r)


I


"YH ~rY t~-'ih"~n JL 111/,1.













5-r DoIT q'IIrD!)'',I TT C T. ITE T i-r Ti- aT!"'I-
5RI 0 PPIIT" ". A$
5?';l pDT'Tr I" 'I
5q20 pADTrT ,I
.oj p-)? T.T-n' u qq 1 '( T ) D nT A )r", pcAt
r:;p D' pfTTI' ZDARATT V'!-", .) "A. "
1)'~ 1D) T'rT): T) D -D T 1 r A V/
iq pr T? ('T or;C.? Tnrr I ? l,,",A q "
595i TC TI T1IcM 9q-71
59r4n P T'TT"'" A T ,AM -'FD DAT -- )") Alq AF DA,.T.^ ( ^ql pCD VFAPiM
9Q7n (n9 V=1 TrI A
5972 PPT"iT" "
5974 MFYT v
5qqn rCT'lq,
Onr' DCPti tqIrlTir PlI Tr (A C T I MAVAqT TI I TDYV T ''I TMf PqTTV
6010 TV (T. T-1)>l.6 TW4FM' 615n
0 n5r I-l1 1 .Q -9(T, T-1 )


6'1 ? 1 C) T <'A (T 1 -1 ) T'Tcy! Ir 1
61 30 A(T. 1-1 T --
'l n ourTtl
e p ( Qn nqTT T \TI Tr) CAT r I r--;(T) -& / ('i' .)
4/410 TI ='(T)*-( T.2)
4s A n A(I, T )I=7t.y(T1 ) /2.? 3 25 o0-13. 5777
/4)A? R( Tr) T)='P ":('<. ?"1 +( TlI / ( T. 2)**? )**".2 ) /( T,-, o
41 Ar) CT, Q
A ion prr if CI'Rr3)ITTVep Tr) rFTrqcuTmD e p'rTVTr)ope
1,n IV 'n.((TT..O TI-CMT 6A..70



6Qf Tr'Jil P 'jC
6-3) n c T (T, I> ?, 9 T' F4 1 7n






5 fl' PF'. 1qTlreIlTTP!r T') t'Q'TF gq TPA Tqc DT
6P) TM=T1l( 9) /1.





4 qq V- '(2, 1-1 ),. fOO THc'! ~64$SC
, !. I -7 r)-'()
rion PPT' J R


69W O T2=TI(2.2 /2.17C47
69 V) T3=Tl+n(r( 7)/.-7F+7
'77 jr c(f.)-l)

4 '0, :(l )='(i )
5'-,70 ,r:(2. i)-"(;)

<.o.r [?CT'P-!'












7 -'n-) D c nIT '- c I- T -.T r.A ,T A T'T 7- r y'3-7 -rncn
7T 10 0(1 )= I .nu++- ")=A( +n -,-t ',( I ( 1 / (rr )); /I 3-n o)
70320 0(2)= 1+1 5+11 7 rn.'n o-,-q .- r 7 er,"( /"(I )) "2. ?25 O)
7n 0 c(l. ) )- (1 ) / (1 )
70 0 A: c(2. r )--'( )/O(C )
7 3r) n TrT'T
7 r 1 qQ1( fIlD,)I TTMp" TO' CATO ,rcm' q o DADATTr','
7' 1 T I =T -- 1
7220 Pr)T?='(( ) /(' ) ( )
7230 13=(1-T2)+*+2
72 ^A A -1 l /((q(1 )+'(2 ))* <((1 )-*+2) -T

S74mn nK q nllOrFITe Trc ,f'AT' PFPT-O qcr' coD D)TTiqT
741 n O (A 4- /(u( 1 )4'(2 )) + .5
7 l, Pl= -, -'4 .%9
7420 D T'I = 'I
* 7,00 RPF SID)'VUTTT T' yn r)Dn'rT '!C-lu -T'Fr Tif !n)T T1i- UAVA^gST Dpu4Sqp
'7 /10 1) -(T T. -i ) > .45' T."/-, 7720
7 2 I -I -=2- n (c (q( )) /?), ? o )
7 3 T2. 2= 00 (1 /P (T,2)- I /1D ( T i )) /2 02C -o
7 A n IT =( I n+4-.\ ( T- I ) -L1 -n4-- A-( T T )-- )/
76^A2 A T3<445n0) TU r 765-q
7 76I14 -1344500
745y l4--T I 4n(f*-T-.2Q+-3) /2. 0250
7 6Ar Tr=-4. o0e qo-T nr ( R-q -)T / V 05o0
74%7 TA-=I )-Y (- T 4[.1 -T.- T F))
7r 0 V( ,. T-1- )- T(T,.T-1 )_-n( T.34-TA
7720 rFT'IjD'T
')000 P"M








































APPENDIX B




































TABLE B-1

PRE-MAIN-SEQUENCE BINARY TEST MODEL FOR THE PRIMARY

MASS = 3 M0

PERIOD = 3 days










U) !o
1 .3 1 -+11 1 1. 0 .
2 3.' 47c+ 1 .7on7 63.955 ."9 0 o
1 5.T47n 4- l .12p ?. ?7 .on 4.25
4 7. 04m4.1 1. I O 4.o0o .oq 4 32
5 0. l7+4- 1 .1972 3.6004 .07 -. O
6 1 .04o +1 2 1 o00. zO1 .07 A.
7 1 7 A7 C+ 1 1. 1 .-1 0)7 .0 .
q 1.' ,0+ 9 1.1117 Q .<192 .06 4.
0 1. 1l+ 3 1.150- 3.5070 o5 i -.
10 1 .923+12 1.3101 2. o 6.0?,o5
11 ?.006+-12 1I.902 3.700 .04 'S
1 2 2. 1 00 12 1 .192 ^ .710 1 4 07
13 .-70F +12 1. ^^ .7n5 .0) 5 .0
1 2.551E+12 1 .32P 3.7051 .O 5.01
15 2.7159--19 1 175' ,70'7 09 5.go
1 6 970 7r+ 2 I :,??1 3.7 0'7 '*-, 04
17 3.1 00+12 1 .1 31 1 71 n0 5, 3
I A 55 '7 + 1 ? 1 .: 7r9 Io Q
19 2 0 7 5::- 1 w. nr 711 r 7,,

10 3 .462 F+ 2 1. -77 1 7 1 1 O 77
0 1 -,47C+- 1 1 .2- nn 3.71.'7 oA,1 5 7 -7
?1 3.920F+12 1".1,20 1.71Q PO 5 ,71
22 4. n19~+12 1. ^ 3.7192 .po .
? A. loF+12 1.34'(4 .710n .Q9 5.q5
24 ^.37,F+12 1 .1?A 3.7214 .1 .6,
?5 .5,50+2 l .400 3.7?91 .o,' +,
?6 A.7A1 F-19 1.150 .747 .07 5.57
97 4.o? c+12 1.3551 3.7 2. .97 5.5
29 5.106 F+1 1 .157 7.7pQO .oP 5.51
20 5. 99q1+12 1.05 3.7206 .94 5. 40
30 5.470F+12 .1 617 1.7112 .0o 5.44
31 5. 59 ,.C 1 1.319q 3.7320 .0C 5.4
32 5.935E+ 2 1 .3660 3.745 .45 5.40
33 ^.017+1) 2 .1502 3.7361 .94 5.39
31 6.20p0+12 1.3704 1.7?79 .94 5.35
35 6.3P21'1+l 1.1725 3.7304 ,s3 5.''
3A 6.54c:+12 1.3747 3.7A 0 .I9 5.30
37 6.747F+1- 1 .170 1.7A27 .93 5.27
39 .OPF+ 12 1.3701 3.7A43 .92 5.2
30 7. I1 1 12 1.017 A7450 ..2 5.22
40 7.?042+17 1.303 3.7476 .91 5.10
41 7. 476+12 .1.35^ 1.7402 .01 5.17
42 7.595 +1 9 1.170 3.7509 .91 5.l
44 l1 o. 2-4-11 ) ] -, o 1" 7:5 5 1 .5 o
7.7 A- I7I I

o o0 F:4-+ 1,1 .1 0 7 -r 7 70 5.
4/, 'o + 1 /, 7 c7 1 7' .
47 oa .570-+? ) 1-7 75'0 .70 5.
4-: 752-:+ 1 ) 4'0 ro 7a7n .79 4.0
4O .09q+ C 1. 4"11 1 .742 70 7 1.04








67

' iCTr TT T O(f T /T C:',M ) T q T T n(-rr ) 9 ,ri ro /r )"'!'

5 0. 1 17 .!1) 1.A ?7 1.7,<3O .77 4.04
51 0.'O00 +1 -) 127A 2,.7' 5 "77 4. 0
52 o0.49 C?+ 10 )0 74A7- .77 4.90
5 .A4~c+1 1.4 3.7A19 .74 4.A77
54 0. 3 4 6 1 4] A i.n A: ^77 7A 4
S ^-6+l ? .1 ^1^^ r.77n_ .75 4.9
1. +12 1 4?i 3A.7721 .7/) 1 q
5S 1. 121 +13 .41 3 77-7 .-75 .70

5 1.7~5c+i "?27 .777" 7. 4A.74
50 17 A7F+l- 1. 49A 3.77R/ .71 4.79
ne 1.O04c+lE? 1.4270 3.7Pr? .741 A.7)
61 1.112t +l? 1.450 1 7010 .71 4
? 1 .11 +13 1.431 A 4 i.7 .71 7.5
3P 1. 1 +1 1 7.7 5 .73 A.,5
A 1.1S7c 1.*435 3.7-,9 .79 4. l
55 1.1 5+13 3.370 3 7004 79 4 r1
S. 1 t 7 7 T- V o ..1 A C;
47 ). 222-+ 1 1. ^423 7 "' 7 71 !
62 1.9 r+ -l? I 1 ^c 3tO 3 4.51
o0 .25-n4l l 1 ^,7 .701 /" 4 0
703 .7 +1 .A4PP0 3. 701 .71) 4A4.(
71 .205r+13 .451n 3.700? .70 .441
72 1.313F~1 1. 452 7.7 00 .040 .42
7 31rl+ 1 1.455^ 3.901r5 .*0 4.40
74 l F+l ? .4579 ?. 1 .60 A4.)3?
75 .* 1. p-$.l 1.4507 3.?3047 .Axo 4.S
76 1.3PA^ +1? 1 1'SO 3.49 .I? 4.I 4
77 l.r F+ 1 .4^A 3.1000 .'i 4. 31
70 q..i? +? 1.4 3 .q07 .-7 4.t4
70 1. Anc+- 1. 4Z 3. 1 91 .7 4.97
q9 1.4505'+1- 1.47-7 3.120? .67 4.96
91 .477P+1 1 .A729 1l .4/ .A 4..21
92 1405qc+ 3 .14 750 3.9162 4.?l
3 1.5 13 F+ 3 .4772 3.21 7Q A. 10
4 1.5' ? -7 '- .4704 3.1 05 .55 4.1
95 1.59501 3 1.4316 ).9211 .^5 4.14
96 1.5AO+1 1 .4Aqp 3.9227 .65 4.1)
07 1.56q +13 1 .4950 3.3 92 .44 4.11
9 .16t05.1V 1.4+90 q.qp9a .6A 4.09
90 ~62?+1 "3 1 .4003 .9276 444.
on .64-lF-+13 1 .405 3.9o03 .63 4.04
o1 1.45O 4- 1 .4047 3.93QO .*S 4.-0
0 1 .-77c+ 13 1 4A0 ".90?3? 5 .I'3 4,0 3
0 o ,' .I 8. I ', .< ) "i -'-,7
-77
05 .7 1' 1 ?7 3 '7z 7 7 '" .
0 1 7 7 l 1 "5 -, ,] -, 4- .-n
07 1 7,<0.": ) 1 .C T o. 940- ^1 ". *,'T
0,: 1.7o -7 1'- 13 .51 ,r, 3-., 4 3L1 ,.Ao
0') '^O- 4 .^ 1 3) 3 9440' .Al '7'










S T T T 1( /T C-VT) T fn '(T t ) TDnT rc CD T /) '1


1I 1. 3~ 4I + 1 13. 4 .1 ,3 q- )*.0
1\ 2 1 ^ ":+1 1.517 C3.-4-0 ?Fl n
1| 3 1. 7' +1 -? 1 .5* 00 .- r-, .50 ..70
10. .oo)AU 1 1 .5231 3. 92 .'O .77
105 + 3 .53c, 1.q351) .o 1.75
1 1.01--+1 .1~75 3.55 4 .51 5.77
107 l.0o51c+) 1.5207 2 .971 ."0 1.71
109 .QOQ c+- 1 .510 'Q; 7 -, ,0)
l0 o 1.OP7 +13 1.54940 -.q,-. .0 o.67
110 2. '0".+1 1 4. 7*5 3.9+2q .57 .
111 29.924.+1 1 .5 '^ 3.9~ 4' .57 S.',
112 2. '^2c+ .54 1.9^ 2 .57 *.: f,2
11 2.nanc+13 1 .5423 3.9F50 ) ?..^
4 2.r-7o-+ 1 R545 t.5 .55 3.9

115 2.7o07c+1 1.5 72 7)7 1 % ", .^,
11 2. 1 15 + 1 1.5104 2.q ? .ri .5
1 17 9. 1 "3 4: -, 5 17 'i 7 "{ -! 5 "-', R 5
11 2 151F+1 + 1 .5537 2 7% 5.51
I 1 2.1 1 + .9:_ S75 T7 r) 4 "
10 2.1E 01 1 .C55) 1.70 -5. AQ
121 2.9 5, +.12 1.5 ', 3.pnn .54 ). -
122 2.22A +1 1.5525 3.9 1 .54 -.44
123 2.2A34+1 1.57 .q9P32 .5A 3.42
124 2. ST31-+13 1.5460 -3. 9qo .5, 5.41
125 2.270+1 1 .50oI1 2~.Q s .53 3.30
1 ^ 2.2 07c+- 1 .5713 3. -O4q .1. .1.37
127 2.316E+13 1.5725 3.990o .53 3.35
1 2 2. 316l?:+ 1 1 757 3.301 .52 2.
120 2. 52 + 13 1 .5770 .90ol .52 3.32
130 2.170 71? 1 .5T1 3.80A-7 .52 2.21
131 2.38qr+ 3 1.5923 2?.-o? .52 3.20
112 2.407-+1 1 .5945 2.9oq9 .51 2.27
172 2.425:+ 3 1.597 3.qoo0 .51 3.25
1?3 2.443+- 2 1 .59QO .0on13 .51 3.24
135 2.4,1V+13 1.501 1 3.002( .51 ?.?
13'5 2.48lP +1 3 1 .5'o) .on/l .50 3.21
1 7 2.40,oc- 1 .5055 3.o062 .950 .10
1 3p 2.51 6+ 3+ 1.5077 -.;on7 .50 3.1
t30 7 .574+ 5 1 .5000 3.n005 .5 r). 0
4n4 2.,535+11 1.6021 r.0 1 F.40 3.14
141 2.5715+1 1 I. 04 3.0127 .40 1.1-
I 2.5"'0o 1+ 1.6065 .o' ~ .Ao 3. 11
1.,t 2 27 4- l .. -;7 v 2 ..0"


* o. .4 4 + 15. 3.02')0 .- "5
147 2 + 74 .0 p .3 4) 3 1. 1
140 2.60 -+ 1 .41o0 3.2.42 .47 2. 1
1 "-o 2.7 17 +1 .6'2n 0)250 .47 .)01







69

U( r)t 0 T /T ") T U T/) (TT r/t- ) DP, '-) C D/D'I'IT

15n p.71,r71+ 1. ?A2 3.0275 .47 9.00
151 2.751+1 3 .' .00| .A7 2.09
15" ?.771 1~ 1. 97 0390 9.0
151 2.70"nc+-1 .61"3n0 .029 2.05
154 2 .q c+ 4 *'91 3.Oqal . 1 55 2. 92:+ 1 1 .4353 A1. 5 7 A.02
24 p. 944 +1- .. 7 S.0374 .^A4 7.0"
157 2.722+17 .430P 3.0o0o .4S 39.01
150 2. P -+ 11 A 2 1.0497 -. P2.7
150 2.Q o 17 3 C.4+4 2 3 .042 .45 P.R4
H1$ 2.017u+1 3 1.44~ 5 na0n .4A57 2.94
1 1 2.015F+1- 1 .6547 3.0L45' ,44 2.9q
152 2.0544+13 1.45o0 .0A72 .44 2.92
61 2.07.c+1 1.45' 2 2.04o0 .A4 .o,0
1 /4 2.0o0F+1 3 1 .4554 ".05? 2.70
1 5 ?. 2nc-R-1 1 57S 3.0)22 4 9.77
1 4 3.'277 +13 1. o *-. "r 2.7
1 7 3.45F--12 1. 421 '555 41- 2.75
1 .. ? 'i ^o^-c+ 1 3 .c4. '. '' ', 41 ? .3 7
1 7o r) ? I r) A' 7os y 1s 3.7?
3o 2. OR ) -12 1. ,A qo .-"-} 2.72
170 .000 I <,QO 3.A0 A /.4) 2.71
171 3. 11 9 +1 3 1.6 17 1 1 0 21 2.
172 2.134S+1i 1 .4734 3.037 .42 2.40
173 1 54E+3 1.67RF 0^ 54 .42 2.67
174 3.172-*-12 1.770 3.0670 .42 2.45
175 .l01 +!1 1 .6 01 .~0, 7 .41 2.6'1
176 2 *)on tl- 1 ^o A 3A.703 .Al 9.63
177 3.2 27F2+13 1.69A7 3.0720 .41 2.61
179 3. 745 -1 1 0- 4.07 37 ]1 2.64
170 .?44c+1 I .9o2 3. 07c5 .41 2.50
190 3.22, 2c1 14015 2 .0770 .40 2.57
11 3.3Fn-+13 1.6030 3.0796 .40 9.56
1 2 3.311F+11 33 1?,' 3 .o ?q .40 2.55
193 2. ",+1 1 .0839 3.0o10 .40 p,54
19^ 3.355F+13 1 .7o04 2. 09o .40 2.52
195 1. 37-+11 ) .70? 3.o003 .30 2.51
196 3.3o01+1 1.7)52 .n0040 .0o 2.50
197 .4 0OC +4- 1.7075 3.o036 .30 .q49
9 2.429 F+ 13 1.7009 2 ,00o .20 2.47
190 3.g 44'+1 1.7171 1.0010 .30 2.4,
100 1-i644+1 1.71 4 23.o0036 2.4a
101 3. 42 +)13 .71^7 .0059 .39 2.41
10) .fY nc r 1 .7100 3. 00,0 q.9 49






1 00 p + .71351 4. 6, .77 2.34










1nnrI TT'r n ( / y)) T ( c ) OnI /Dgly

2r T /T. ,1 1 .7177 .L Ci r? .r I7 J 1:

202 3.~)T+ 1 1.749A 4. "13 .3. 2.''
00 1.7 i 1 ] .7 4 -.1'1 .7; ?P.'
271 3.71) n- 1 1 .7471 A .'171 9 .29
p5 1~.7 7c 4- 1 .705 A 1 7 ? .77
q204 R.7cn+1 ) .75" ~ gq ," 2.4
727 1 77AC11 1.75-3 4^."21 .3 92.*9
2n29 1.702?r 1 3- 1 .75A7 4. '? 7 .5 9,.2
2nQ0 .Q(,c-1? .750'1 .r95A .15 .2,?
1 ". q 2.?OC+1 ? 1.7615 4.0971 .'-5 2.21
211 1.Q47 4-1 .7'1)0 4.7) 0 o .q1 97. 0
212 2. 9365:+ 1 1 .7 663 A.n .0, 2.1'
211 3. R F+1P4 1 .7s O A.0?22 .3*1 2.11
214 A .on?+2E- 1.7712 4.330o .4 "'.17
215 3.20:-+12 1.7771 4. "'', 3 2.
91 S ".o?0t+1 .7761 A.137-1 -' 2. 1
1 7 1.T3 ^ +l1 .770, ,l ,'on ,1 14
210 2 07.-7A+1 .-7Q10 A .A27 3') ?,1
O 0 '* 0 '')7 1 n 7it 4 I l) ') I 1
21" ) *. 7r)r-7T4 o)A) 7

?21 A.220'+13 1.7' A. mA o qq 2. A1
222 4.,027+-l2 1.7010 4.247- .33 .-00
22. 4.S.6+o13 1.7035 A.1407 2.nl7
2?4 l n 4)+A 1 1 .70 1 .051 .2 2.
225 4.1 02+ -13 1.7029 A.'T '7 ,'9 9.r5
2?6 -. 120+1 1 .IlA I 4.0545 .32 2.04
277 A.1 3?' +12 1 1n37 A.O5'2 .3? 2.0,
2?) 4.157+r41 3 o? A ,70 .1) 2.06
2?,0 17 + 1 3 .1.0Po A.,O50 90 6.01
2130 .l. I0 + 3 ." 1 1 R.I^ 1. .41 .2.O)A

232 4.23n0+13 l.,167 4.,040 .2! 1.0o
233 4.2 49n+-1 1.9I10q A.0q6~ .31 1.07
p14 4.2?^c +3 1. 920 A4.non .71 .O0
235 4.92 sI+1 : 1. '44 A.07r) 11 05
23, 4.3 1 +1 1.2R73 A.0710 .0n .04
237 4.3212P+1 1.n? I 4.0734 .330 1.03
239q A,. Oc+ r 11 .P327 A .754 ., .A10
230 4.357L:+ 1 .0A54 4.0777 .r)- 071
24 4a.,75~-+1 1 .'321 4.790 .2'0 .01
241 d.30AF-+ 13 1 .14400 4.0n07 .Q .0"0
242 n. .1 ~ +1 1 .Q947 ..2?55 .30 1.90
S94 4. .')7 4+1 + 1 RI40, A 7 i ) )

2 .5s 4. .." '. I ; 9 1 *. -, '70 4' n
245 I n. r4- 1 .7 o 2 41 7 .. 1 A ',
14 i-. r,+ l 7 .. t -7 q

047 ,.5 ?r+ 1 .057" /. 13 .-O
240 4. 21I+1 .04 "< 4.00 .po *.0q
?1 ?5~c~? ,?~j 49





































TABLE B-2

PRE-MAIN-SEQUENCE BINARY TEST MODEL FOR THE SECONDARY

MASS = 1.5 Mo

PERIOD = 3 DAYS









72

n TT' "" T rN ( /T q I I) T T r (TtNrc ) orn-r! ) ? QI pIv

1 1 .4,E0 + Q 4.1532 3."3 40 .00 ).5
2 2.?224- +0 .4.14 Ql A'. 1 l-5 .o00 "4.
.2 ?1. -n+09 4.116 3.343 1 .o0 4,54
4 3.A/A7--+0q .,004 9409 .0q 4.5
5 7* 2 [:c.Q A0774 3 1.3514 ,07 4.40
4 ,37EC+0 q A.5055 ;,151 .07 4.47
7 6.500E40Q 4 00OA 1.1547 .0 A. 45
Q 7. o0'-+q 4.onA 3." .0) ? A,4
0 ..6 6oc0nq 4. 017 91,57 .o a 4,41
1 O.70p-0q 3.0C27 3.3506 ,o5 4.?A
I I.205r5+0 W09 3. 5 1F .0') r.)
1 1 .21 p4 -no A.0440 3. 629 .04 34
1?3 .4727+oo 3.0250 A. A45 o? 41.3
1 4 1 .* ARpt-no -1 ,0070 3.3461 01 A4n
!5 1.424F+O0 3. P99'0 r ,77 ,,o .2?
1 6 4 77 F400 3. Q.'i .') 4. 5?

17 1 n .070+o .q2 n 3
1 0 2. 4OA ..-+00 .01 3 O ?. ,-75 .0) 4.9j

20 2n.447+no 4 .70' 7 o ,o 7 .17
21 2. -7F4+no 3.7744 3.3775 .on 4.15
22 2. 37E+0 0 1.75 C 3.170 o 4.1n
23 .0A[a:+no 0 .7365 3.316 .o0 4.1l
24 3.2 Ac+nO 3.717F ". gp4 .o 4.no
25 .407+00o ,00)A 3.3n41 4.07
24 3.73no+00 1.6707 Q c5,7 .7 A.05
27 004n "+o A +nq 3.3973 .R7 4.
2- 4.2.61 E+0 3.6410 .P o. on .Q, 4.01
20 A.5AlF+00 ro .320 5.30o6 .96 3.00
'0 49.35-+00 3..630 3.3022 .p5 1.07
31 5.145E+no0 3.5s50 .-0ogo .5 3.05
2 5.46+nO+o 3.5661 3.305 .95 0
33 5.l910 +00 .5471 2.5071 .94 3.01
14 6.1 7c+00 1.522 3.30Q9 .Q4 R,90
35 4.543F+00 3.5002 .4r0n4 .03 3.97
1 6.7F0437 o0 n. 4or 3.4020 .93 1.s9
17 7. 3501+00 3.4714 .401 7 Q 3.83
3q 7.7R95r+n0 3.4524 3.4053 .92 5.q1
?0 q.24 F+0 3.4n135 1. n0o .02 3.70
4 97' )7pFq+00o I.41 5 5 4n6 .1 3.77
41 0.)222?+no 3.3056 3.410 .l1 1.75
49 0 7S50 no '7Av 7 1110 .1Vl .73

1.44 1 2.o 1 4p 141 ) 4 ,7
44 1 I 5 O' + 1 3 1 O ". C,1 1 7 -, 7 7 /
5 1.! -4- 0 ".3102 ".4167 /r 0,
, .2 1 I) 3 0' ". 1 < 7 :. ).
47 1 -0ir:+ -. qp, "S 00 -70 )
S 1 5 t+10 3.2' 30 1~21 .7 .6 72
40 1 42 l n 4, 1 ". 2 3 A. ,l








73

W)rcT T T "' rnr(r /T qI'-T) T f ('T'r: -) r n 'r7 7'0 D /' 'TT'T

5" 1 :.50 +1n .p9 -40 .77 1. o
51 1 .5 0 +1 0 .?2"q1 3.4 65 .77 '.57
5 1 .671 + 0 .I V2 .. 4q1I .I7 .55
53 1 .7 1A n 7. 1 ,0 0. lo .7" 3.5I
54 .04+10 r .140A Q.) .7 .52
55 1.057=+2. .10 2w..-1. .72* '.52


o0 2. o2,4 4-s '5 .* 71': 3. "4270 .74 4.c,
50 2. 404+I .)05A7 .4204 .74 ). 4"
60 2.510 C~+ 1 ,0 A ,7 3.4 /1 .74 .41
41 2. 4, '+10 )3." 4 q .4 72 1.O
62 .5qm l++ 2.I070 .7 3. /445 .73 ?. *
A42 2.050F+10 2.07100 !. 44 7' 3.4
6A 3.1 +1- ?.204 o. 1477 .7? 1
2a5 7'25-IC-10 ').0410 2.A0zo -oL 3
655 3. 43A'C 1 '. 2 .o)1 2,45 .7!. )31
7 1. 19 0in ?.0 ')1 1. IT52 7)1
64 32.70n -+l' n p.oep ? 2, 154 71 .2 ?
7o 4. 01+1 .-153 ?.'155. ,27. 4.7
70 A.2ni +JO p.R4/-q q. ^4- 5 .770 1.25
71 4.417F+] ~ 2 .077 3-. 402 .70 3 2
72 A. /^4F+10 2. ROPI' 7.4^'4 40 ".21
73 4. E2-'1 2.7905 3.4i24 .60 4.I
74 5.132l -+1 2.770, 1.44) .40 3.i0
75 5.30F+lon 2.751^ .4657 .49 3. T
74 5.471c:+10 2.7327 3.4472 .40 .15
77 5.060o9 +.n 2.71 7 2.44o0 .A 3.13
79 A.S ^4-e+1 2.604,' 3.4706 .47 q.
70 6.9P3E+10 2.^750 .A47 ? .67 3.10
9q 4.01p+I)n 2.I560 3.4730 .A7 no
91 7.260F+ 10 2.61 30 r .47F5- .1 3.07
92 7.42 E+10 2. 6100 3 .4771 .6
93 J.025E+10 2.6o01 3.47qq .qq 3.04
A .4494-Q n 2.5 911 3.4P04 .rv5 3n,
95 IR.n50+10 2.5622 2.4q20 .45 3.01
96 .3 n7EE+n ?2.54?' 7.4R37 .65 2.00
q7 0.770t+10 2.5243 3.4r52 .^4 2.oq
qR 1.027 +11 2.50,5n4 .440 .44 2,07
qo0 .770r:+ 2.4,4 3.49')6 .44 ?, 0
o0 1.134F +1 1 2.44,75 .40n .63 2.o0
01 1.101+11 2.,i4R0 3.1019 .- 2.02
o 1 .25 C+ 1 2.40?oA ?!0 4 5 2 .01i
o0 | 1 4r7AT: 1 4 7 1 0,0 ', p 9 O
04 I.9O+11 2,.307 1. 1404 2.39
S 1. 40-.-n+T1 ",17)0 2 4' /- 92 o
06 1.5?02- 2 I .3I5 20 1a') .91 2. 5
07 1.5200! +11 3.n 4 .5 ., 2.'
On q 70 4- 7 "il -0 3.5'1O? 1 9
On .7,^c6 +i- .0070 .q 40l .';1 2.Q1










If)ncT T iF T n (T /T qcT) T ( r M-) r: r:u 2 ?/ D 'I -

' on i 3 o^+ n 3 o 0-701 3 50S .'SO >. 7-:'
1 4 +- o -7

1'o 2. 0'-^-i 1 ).3o? .5no0 .^9 2.7/
10 2 2.2:12 3. 11 .0 2 ,75
,104 2.29"50t+ i 0 ." ? .51! I 0 2p.7".
1 9.15 tce 1 .19 .5 4I A70 2.7
106 2.4 04-c+1 2.1 4 2.51 Q .r) 2.71
1 7 2.,no II 9.1 45 3.5190 .5 7." 3
1OQ 9.730+11] 2.1265 9.v104 .cR 2o.
100 ,.774c4l 1 ). 117 'T 212 .*5 .47
1n 02QP+l 2.O?" *p.c) o .57 2.'5
Ill 3.171+11 2.j407 3.52^5 .57 9.44
11 9 3.30Y+ r 2.0507 3.52)1^ .I 2.63
S1 3 3.407F+11 2.010 4. 579 .P5 +.62
11 2.6720+1 1 2.0120 3.o204 .9 *.'S
1 5 2.355.5 + 1 1 2 .9o i r.5o
1I 4.'4 11 1.0750 3.5~27 ,5 57
117 .2 ?51 + 1 1. 0r ,- ', % F5 2. 55
SI q .lr 1c+ 1 1.017 .0 F 2. 5'

120 4.02p 11 1 .0002 3.5302 .'
121 5. 16F+11 1. 01 2 -) l.5 ,o .A' 2..51
122 5.4?2- c+1 1 1 1 3.52'-?5 .54 ?.5
12? 5.6o5E+ 1 1. 424 3.541. .54 3 .4
124 5.0809-1-1 .QP924 1.5457 .53 2.49
125 ^.970+11 1 .Q0/5 3.57-74 .5 9.,
S. .50,+41 1.7_ 400 .53 2. 5
1-7 4.0721 11 1.7' 3.5506 2. 4
10 7.267+ 11 1.7474 2. 55 .52 2.
190 7.63V1+11 1.79-7 3.5530 .593 .4!
1 0 8.O11E 1 1 .7007 2..5c5 .p 2.40
1 1 .41 c+11 1.4on 3.5572 .52 30
132 P. 30E+11 1.671 R 2.5 O .51 2.33
S33 0.271 +1 1 .4520 3.504 .51 9.7
1 4 0.741c+11 1.6130 3,5 21 .51 7.35
1 35 1.022E- 12 1.4150 2.5537 .51 2.31
1' 1.n7:0+12 1.5060 3.5651 .51 2.23
137 1.127E+12 1.5771 1.5A70 .50 2.32
179 1 .l 0+1? 1.55'1 3.5496 .5n ?. I
130 1.29429+1t 1..50'2 1.5702 .51 2.30
Ain 1.,n 41 9 1..59 2 3.5710 .0o 7.99
141 1 .34F+12 1.5013 1.5715 .4, 2.27
1 4 1.470+4 1 ,407 3.5751 .9O 29
1 ") 1 .M'" r- I 4 "%7 *- -.
, I 2 I 4-" I -% o A : '
t ic, 1 O-' 4-. V- 1 I "-,. 5 c q o,-" ,-. 9 o -


) o 1 : 1 1 7 .4 -7 110
1. 0 ? ,o r:. +1 1 7 3.50R 'i. 47 1











7T T"F r ( / CT ) t I,-T(I:'- T 7-,,' C / T-pr

,'- 2.1.?1P ) +! "7 7 0oo0 47 .,17
1 l?. 997c+17 1. 11P I. 0o 47 71.
152 2.2"130 -l12 1.20pO .?.0 o ./5 ).15
151 ?2., e-1 1 .97r .O l a5.1 3.11
154 2.~77c+1 ..230 -)? 5 7 .4 .12
155 9.7r4c-1 .9)2,V 2.50o4 .4's 2.19
1A P2.1^1-+12 1.2170 0Q OA 9.11
157 2. 9 -3+' 1.10ca .5n) .A5 *. )1
150 1 71' +12 .1701 1. "1 .1 ". 0
150 3.2977+10 1. 1 ) -. po .,A 2.Qg
4 4I -lrc+1? 1 .14 ? 1.in.5' ,- 2."7
I 61 3. )2 C-+12 1 .1 9 2 ) .,4 tP.
1 7 0.1 4 -+l 2 .l103 3.;,07 .44 4p.n
1V1 .004F+12 1.02A ., V'0!A .A 2.04
1'S4 4.10 +12 1 0? '53 2. 111 ? ".
1 '5 4023+12 ? 1.044^ 7.^1 07 4 "

1 ,7 4.A 2r4c+1 9 1 .0'9)4 A 0 )
^q 5.qo t0 1 ?A .r : 7, 07>r
1lO 5. ,I:1) 07n3 261(Y 0 I
70 5.615 1 P .o5t -c. ,pn',0 ..? 1 .07
171 5.o05c+l2 .0'22 3.?225 1, 4
172 6.1 90-+2 .o*0 6^9A .2? 1.05
172 4.407c4+1 .)047 3.?'259 .04
174 6.?21E+12 .q757 6.2774 .42 .0
1 75 7. I 'i D 5e/ .75)7 "1. 20 l .41 .o)
17A 7.51 I 2 .R 7,P n7. -n 7 .A 1 .01
177 7.0. 4-c p .qA00 4. 632 .1 1 .00
17 R. P29 +12 .7009 ,*F42 .A1 1.90
170 q.7 rl+1- .79nq ".63c; l 1 .90
qr) .0.1?5 +12 .7 10 3.65 7 .7* 1.7
1l o.51Fi+)1 .740 21,19 .AA 1.96
17 1.0n7F+-!2 .7?20 3.0A05 .40 1.05
1 3 1.0957E+1 1 .7040 1.'A421 .40 1.9
194 1 11 -+ .46An 3.64^ .4^ .'-
1 3 1.165E+11 .6670 2.g54 .30 1.92
19S 1.22Te+l .s403O -.1^70 .30o 1.1
197 1.2?n45+ 11 .?600 n.7AQ7 .90 1. 1
1 0 1.441 "+1 .07? 3.6r50 .A0 1 .0
190 I.5072+1 2 .n7A7 7. 450 .10 1.70
IOO I .7 r+.'- .07! 0 r.6 .' .77
1)01 1.0 o +1 .70n 3:55o .> 1.77
"*-' i "Q -*, 'q 0 -1 ,




0)7 2.nAp + .O). 2' 7 I .71
1 o '' 0.4- 1 .OO, A/ ',47 7 1.71
10r) + ; 0 ? ./''? T-7 1 .-7r













n7n -7 C7 + 7 rr0
20) 4.4l 17-l .1 01-5 .71 .41 ,
2 7, c+i .T ?? ,. /,740 P. 4 57

0^ 7t.040+ 1 .1074 ". '7W4 .-x4 1.


507 A. 41 1 +* .1 0 ] 1 4 .0 .7 7 .
9'0o -.qA c+ i .116^ ?.6I'1 ^^ 1 ^"
200 4.721 >F+1 .1 1 ,* o. 7 3 A, 7
P1 4 q,9 lc+1 .l?'"1 3.60 ^p 1
211 5.136F+1 .1? r 1 -1 ,oo i .V- 1.6
212 5.o41."-1 .1252 3.4704 .3^ 1.50
P1 5, ? 407o + 1 127A ?.,o l ...i | .51
214 .5* so. 1 .I ? ro 0 .31 1 o30
21 5. l ^r PT ?D 1? .734 ) .5
21 7 .. 07. + 2. .1 3 ./ ,'7 .3. 1 .55
212 1 .2 r:+2 1 5o0 2 7 0' 1' i.
210 ^.27-+i .1 220 ) 4 + 1)-1 1 4 0 3 .7nge .5
221 7.a5nn4+l i? 151 3.7 ,143 .'? .5)
222 6.75 6+1 1 2.771 5 733 .51.
21 ,.01P2+1 ? 1 404 3.7077 ?2 1.51
22 7. C +1, .15 l 7 .7 ,.3 1. n
235 7. 93]5'- ? .150 1, 71 10 .?9 1.
22 7. ?T" 1 :+1 ? .15.2 7.7) 3 2 1 -A
227 7. 91nT+1 .150 32.71^ .12 1 .4
220 7.' 04--+1 1 n7 1 .71 0 o .47
270 7. q,1c4- 490 3.7176 .2 .A^
21 9. n07r+ 1 .1451I .710? .1 1.4-
231 9. 1 6 ? + 1 74 I.7 0 .31 1 .4,
2'12 9.1l0 +17 o1 0^ 3.7225 1 1 .44
233 9.476 -+1 .1710 2.7241 .31 1.43
234 0Q'S52c9 1 .17.1 3.7250 .1 1 .45
235 R.7P.-T+ 1 .175 2 .7274 .31 1.42
212; q.o905c+1 75 3.71201 2 .At
237 O.lnl+1 : .I1'nn 3.7q1n7 .13) .40
230 O.957f+4- .1V-(0 3.7 24 .1' 1.4A
27? 0.413+ 12 .153 2.7' 4 .7 7 1.30
240 0.57+ 21 .1775 3.7357 .30 1 .1
?41 0.72 r.+1 .1 30r ).7172 .q? 1 .3
2 92 q c-)r- I 2,' 7 ''00 1 0.7
2)1'4 I 0 ~- 1 -.? 7 U -<, 5 ) 1 .3'


21' 0.51rJ
2 t I 2'' +' 7 ]7,.)- .0) 32
S212 l i .r +l hI' 2.7( '0 0 17 .34
? 7 ] ?-..s. i < 7,{-!9 -09 1. 4

2 10 1 ,n -, '- I ? "' 7 o ? O 7 0 1 "








































APPENDIX C




































TABLE C-1

PRE-MAIN-SEQUENCE BINARY MODEL FOR TV CASSIOPEIA

PRIMARY MASS = 3.10 Mo

PERIOD = 1.243 DAYS











(r) r)7T T 'P T r (' /T q'I! ) r /** ( c ,'-:) 7" :' r5/n f''
unacy Tync rr T T It g y ( 7 n /1 7

1 1 A447 7+ i 1 4.50 1 07 ..<,
2 ". ?_O 4 .+11 1 .51 0,44 .00 A.5
- A.OACs4.11 1*.1 9 3.n0QO o 4 .Z
4 F6.5* 00 l j 1.51l 4 0 .I0 4.54
5 Q. )? .t 51QA 3.91i .07 4.F1
o 0.q?-+ 1 1.5100 1. 1o) .07 4.40
7 1)15-4c12 1.5I12 1.14 5 ."0 4.47
9 l.3I -+iV .59?1^ 3.31 .4) A
0 1.Pl, -- .-52t4 '.*l "7q *O' -.4?
1" 47-+12 1.5277 3.9104 .0 4.
11 1.712P+12 1.5900 '.q) o4 ^. 3
9 1 .077 +1 2 i21 3. 2?7 ,04 ? A ?.1

14 2.r4p+1 2 1.924 3. 240 .0' 4.31
15 2.471V+12 .536 -?7/4 o-> 4 ;20
1) 2.42^F+1 '.-4I0 3,70 .0> >.7
17 p.qnm-+ 3 .5I4 0 i c n. Io0 ^-,2:
1 2.06 2P+1- 1.5492 0.<3 .0; 92
10 c-+ l 1.5474 1 '< 1 4
20 3.2o04+1 2 1 .9409 o.' T ')on 4.10
)1 1.450 4-1Y 1.5517 .c9374 4, 17
22 3.62 F+12 1 .5990 .')01 .qO 4.15
2? 7.79OZ+1 .55 1 3. A07 .0 4.1
2 3.0531+412 1.5593 .q? 4.10
25 94. IP4-1) 1.59 5 3.PA4nn .A. 4.
9 4.2Pc+12 I.56246 Q.q^4^ .07 4.00
27 4. 447-' 1 .5Q40 3. A72 .07 4.04
9p 4. 12?c+12 1.5567 3.4qo0 --, 4.09
20 4.777F+19 1.5op02 '. n5 .-c 4.06
o A.oA2/I +12 1.57 41 '. 521 .0 ?,.op
31 5.10(6-+12 1.579 '^ -RQ o0 .Os
13 5.271+12 1.5759 3.955 .05 99
33 5.436cE+12 1 .770 9c7o 94 ,1-02
34 5. 4r0 +1 I .5901 3.9597 .. .on
15 5.765E+12 1.5923 9, q. 4g .99
36 5.0of, + 1 .5q45 3.961o .03 q,,
37 6.00oP+12 1.5967 A?.A2 .03 3.95
39 6, o50.+1 1 5.o0 3.9652 .92 3.92
-o0 424ac+l? 1.5011 ]3.94 .q 7. 1
40 6 Q,590rC+1I .r502p .g5 1.70
41 ^.79 +12 1.505 3.9701 .01 .77
2 6.01 +12 1 '9717 .01 3.75
-744 7? 7 --<- I 4 ? n 0 7'.--, n 0 ) 3 7
A 7 i "i -i 7 7,'1) i O-/~ 4 ~7 -In '> q
4. / 3 7'- 7 A ) *
4 ^, 7 T 7h:-7 .I O'- 1 ') '0 3, -;

47 7. 74')0- 1 F4 o/, 970 0 .70 < ,
I4 7.onCi+ 1 .100 on 70 3.64
40 .o 71 +17 l .1o .2 o7 3. -)








80

) rP rT T( T 'n ( /T /I ) T nO( T 'C : ) : r, 't r: C D D /0C'

C ~34c+12 1.'15 1 Q 0 40 77 -,.6
5t rni c+1 1 1713 0 ",365 .77 3
5? 9Q5 Ac+j5 1.C107 1.+pq6 .77 1.57
53 Q.7307 1- 1.4-17 2q.907 .76 3.55
5 4 Q.o0+r ? 1 .^~30 q,01 4 .74 .51-
5. 0.o50+-12 1.675 1 .0. .76 3.51
5' 0. ?2nr+l? .6?2 2.90A 7 .75 ,50
57 0.) 2 .I:+ 1 1 Cr 0 .75 -.4t
5 0.55"7c+1? 1 .427 0o70 .74 2.44
50 0.71E+2 l.60 21 .9004 .74 1.4
690 O.o+90+ 1 .1371 3.00r12 .74 2. 4
61 1* .Q05F+13 1 .603 1 0n o .7 3.41
12 n.214+1 1.415 3.00c5 .7 3.?0
4? 1 .0q? +1 1 .6437 .onS6 .71 ', 9
4 1.05 1[-I-2 1.4C50 ~,0n79 .72 3.36
,5 1.071 +12 1.6491 3.oCon r7 2 94
,4 1971+ 1 I 2 3
477 1. 71 1 1 !. x5 5 3 .0' 7 Q 1 V11


7 1.530+13 oi: 1 4 7 .2 60
71 1 .1 0" +.l .1 ",1 2. l .102 ,70 2*. 4
7 l.1 64t+13 1.535 ?.opnO .0o 3.21
73 1 .?2?p+-1 1.4~57 9.0225 .0 3.91
74 1.?)lo-+1 1 .470 3.0241 .Ao 10
75 1.2359-+1 1.47n1 1,p025 .69 3.1
74 1.g53c+1) 1 .6723 3.0274 .,Q .1
77 1.?26F-+1I 1.4745 ..ool .9 3.15
70 1. R5F--1 1.73.7 3..0r7 67 .
70 1.301 +1 1 .7P0 2,0903 .^7 3.1
n 1 19Pc+ -?) 11 3.0340 .67 2.1
91 1.133 9+?1 1.6223 2.0)256 3.09
92 1.151F+1 1 .655 3.0371 .66 .07
92 1.167 +13 1.6377 3,.090-o ;A .05
84 1.394 -+ 1P 1.60+n -,.OA)5 .65 3.A
95 1 A.4 +13 1.4022 3.042? .?, *1 .
96 1.417E+13 1.4044 r2)043 .65 3.01
- 7 1.42 +.1 1 1.6066 3.0Q 55 .64 ?.
3S I. .404F+1 1.600 1.071 .64 2.09
90 1. tC+-1 1.7010 3.'13^ .64 5.06
0o 1.4.92+1 1 .7033 .o0524 .6? 2.05
01 1.4o00 -1 1 .7= 3.0521 .43 2.0o
0 1.515.+ 1 1.7077 .,0517 .6 2).0
S7.' -4 ] 7 ) -) 7" 0

0, 7--O I 71 .71A -/ 1 Q.
07 I.5 "0 + 1 .71 o0 Q.0 1| .11 2.95
OO 14 -+ 12 1.7211 1.*OA5 .61 2.'q
00 1.i1 2+j .7722 25,2 .61 5, 'R










I -rr: T(P r /T 1'' ) C (T o (Te ) ro Df, rN : F7 '

S" 1. 7 -< 1 7 1 .7 0 0 0 o I onr)
1)01 1.7* A *1 Q) 9 Qo7 0 7

11 1 .7, ~4+ 1 .791. 9.0710 .0Q 9.71
1'- 1.r7)00 01 .717 r -.)07 o 2.76

S3 1 .7o0I + 1 1 .7360 .771 .rO 2.73


17Y 1.77"0T .1.719 1 r )-7 .5) -.57
106 1.7054FI? 3 .71 45 3.0917 .95 -,77
I n I .lp21+I ? ? I ,774- 07o A- .57 2.67
11 1.09 14-1~ 1 .7503 3.0"5) ,7 ,95
112 l .a45+1 1.7532^ ')n07 .57
1 1 1 .91 +1- 1 .75 Qpq? .59) p.
I 1 .979 +1 1 .7571 .00n0n .oo 2.61
Q 15 0.,Va +7 1 .750 I ...- 7 / -
11C^ l.01 c+ 1 .7?516 10.' .BA 2.50
117 1. Q'9 c+i 9 1 7/i^ 0 5O 2 Cz 0


12 1 .07/,+7 1 1.7707 3 )000 .V, 2,5a
121 1 .o03o c+ 1 1.77 n a. 'n l 4 2, p5
1 2 2. noc-- 1.7753 A .OOr .04 2.51
123 2.026,F+1 1 .777A 4. ,nnAo .54 2.5
1A 2.42M P7+1 1 .7700 4.nn0 .5 9. 240
195 2.150+ 11 1 .7922 4. *0' .51 2.47
194 2." 7C-4- .I.7Q4 A.9000 .5f, 2.
197 2.0Q027 +1 1.796 ^4.0ll .51 2. 5
12- 2.1 + 1 1 .7 01 4.n012 .52 2. 4
120 2. 125-+ 1 1 .7015 4.n 10 .52 93.4
130 2. l^ I ~ ? t .7029 4.O .52 2.11
11 2.15 -+1 1.7061 A.01? .59 2.4 N
132 2.174E+13 1.7094 4. 0100 .51 2.30
21 o.1o 1C+ I r00 4.n216 .51 2.39
1 4 2.207E+ 1 3 1.R'3l 4.0 I? .51 2.36
1 35 2. ? c-4+ I 1 .055 4. 2 40 .51 q35
196 2.2A^0F+13 1 .R07 4.n066 .50 2.31
137 2.25'S+1I 1 .0102 4.0293 .50 9.*3
13" 2.273'+11 1.R125 4.0300o .5 2.32
130 2.2P FC+-1 .1 A0 4.0316 .50' 2.31
140 2.3R Ac+1 I 173 .'99 ~ 7 .A40 .20
1 11 2.322? +1 1.9106 4.0r95 .'o0 2.29




S4- ? ) A .0 -
14A 7 4 -' 1- i C ) /A "7 o
14 .^ 4 ro- 1 3 j .o A.7 9 7.0 ?5 47 0.'1
4 *0 9. -RAN+l-1R I .q'{PO A P]4q- 7 9). I 0







82

'4q rTr T T fr T nr(T /T q I-I) fT r' (' 't- ) D)O'r" :- /I;T;r

150 9..71 +l 1.04-1 .4-7 ,. ,


l5A 2 0* 74-2 1.%1 4,ai .17 /15.1
154 2 5 4 9 ? 4 "7 4 7 1

155 2.71qc+-1 1.7064 4 7) ,A 2.11
15 2 ..5F0-+1 1.795'1 R.47h5 ..1
157 2.7^4 +1 1 .o51 ,4. ^'7 ..-, 2,1 1

S15 2.61 p+I ) .0 63 .. .- 1 ,"
150 -2. 1 .1 o 4. 473 .45 ,.0-
7 1: I 1" 9 / P7 'o 4 e) A0 r) r)7
11 2 .6523-0+ 1 .91P 7 -4.non ./A ?.07
1 62 2.5^')^0 1'7 1 .^ 717 n .0709 .^ :4 2.'S5
714 2.RQ72+12 1.P77 4A 47|2 .4 ..7 1

I7 7 7, n I F97 AA A r)O7 A 4 .A 4
S74 2.7015+ 1 3 .7A 9.4**074, .44 2.03



7 1 0 2. r0 1 41 1 07 1 r, A 717
171 9.7 p .007 /.^ no )7 n0
17.3 2.o'4Q+127 A0 t-) 000 4A 1- -
A)7 7)7 C-)7 r7 411 1 OqA? -7 4 41

) jO 3,7) ?C417 -o r) '


174 2.9,' +13 1 .r 021 4.'00 7 .4 .0
171 2. R1^2+ 1.005r) 7 Ano5 4 i .0


172 2.03l +-1 1.0,03 4.00?5 .41 .05
173 2.94 (I+4 Iana In onn .A2
1 74 2,q ET+ 1 3 1 .0or 4.007 .4| 1.04
175 2, 92 + 1.a d ?" ? 5 1 0 -9
1 7^ A.9 q 1 .cQ A 00i .-4 ..r)
1 77 2.Q15 + 13 1 ()I I )r )Q
170 2.0o95+ 1 1 .01 37 J.Y)Cq .41 1 oo
170 2.0-^5F+1 1 .0165 .1 ) 1 90
C0l 2 R.o5c +l? 1.o021 A. ~In An 1,9
19| 2.09 1 +1 1.o?21 .1 0. O 1].97
1q9 2.007+411 1.0o5 -. 1060 .40 1.s4
9 .IA-I+1I 1 .027P 4. 1079 .4 1.
194 3.03,05+-1 1.0 7 7 4. 100 .40 1.4
195 .47-+13 .031~ 4. 1 14 0
196 31 .6' r+ 13 1.01 5 4. 1 12 .3O 1.
197 .7'0 I-+1 1.0305 4.1150 .10 1.01
199 3.06E+12 .0424 4. 110 1 .3 .90
190 2. 11r + -l 1.0454 A. 1 17 .30 1.70
100 3.120c+41 I .044 4. 1 5 .79 1.70
10 i 4 4^c+- 1 .0514 .n1224 .9 1.7
10? 3.1A2F+1 1.0545 ,4. 149 .90 1.77

S4 ". 1 '05-'-1 2 1 7 75
1 5 4.-2i 1 1. 0 4o ) 0 '7 1,"7
S') 7 7)
S)7 2. 7 ,:4. 1 7 -76 41 1 '7I *1
00 9 7-7C 1 .077,4 A. 1? 7 1,7
2 03 1*'7-0* 12 T A6 X 7'! 77 1,71










r)r)T TT E' T OO(T /T Cl ) T nr (T r C -) )ira : -/Dg'

S' 7 Ar) --t ) .' 700 A.) ") 7 7
20) 3. 1 CnC 3 no ? 1 *10 l 1 4

20 3, 1 000 4 R -A 1.67
2)04 1~ *2 c- ." 44. 7. .
p 3 .37-7 i- 1 00 At1 1go 1 .67
206 ?.3 +4 9.'7. i .1500 7. .

2?7 400no+1 2 q.137 .1501 .- 1.6^
20- 3.42' 1 1 7 a. ; 4 ,+ 5,
190) 2 7 ++1 2 .01. 1 0 f 1- .

210 3 .450 7 3. 1 1 .7-
211 2. 474 1 2.0~11 .A140 1.q1
212 3. 01~7 +13 2.n7 ~ A.17 < l.il
31 2.315o01 ". 4.0155619 .34 1.5'
214 1.524 + ? 2.0203 ] A.17O 1.5?
215 .snr c+19 +,94 I I0.
21 3?.557r + 1 2.037 0. .1 71 .?4 .57
217 3.574-E- Q 2.04) 0 4 .1730 i .'4 .
21 s ? on- ".r '1 + 1 ) .2 44 1 5
210 3.*677412 2 o.40P, 4.1772 53 .47
21n ?. 7-~23 I? .o5 P '. 170 7 .5 4
221 3. A' r:t+1 2 .05 0 r4.101 3 1 .5
272 3.2^6E+11 2.0l11 .19?35 q 1.5
223 3.673+1 3 2. 5 A. 1 "5- 7 1 1.51
-24 1. p0F-1? 2 .nI05 n.179j .' 1.51
225 3.706F+13 2.0710 l.1000 .32 1.5.
232 .7p' 2+1 .0702 ^.1027 .2. 1.0
27 3.73"p+l 2.0122 a.)0o4 .? .40
2? .755C l'12 n .7 AI 10. 2 1.4
230 3.771 0+1 2.F016 A 1000 1 .47
230 3.7oq0n 12 2.1451 4.p2ol .-'I 1 .4
231 2.3n'c+n1 2.1000 4.n) .*1 1.45
232 3.8 21 F+1 2.1055 .30 .1 1 .4
23 23.937'-+I 2.1 i!2 A. 7? .Al
231 3.954. -+1? 3 11! 4. 101 .31 1.42
25 3.70 +: -1 2.1100 Q.2124 .C1 1 .471
236 3.9R7 +13 2.194 0 4.e 147 .30 1.4
217 1.o n c--*l 2.1 700 .2171 .3 .41

2370 .. q2! :r i 7. in2 4 .* 21 o'0 .40
240 3.07E41 3 2.1 54 4.-??47 .30 1.30
241 3.0~--l ? 2.15n7 4.?226 .30 1.2

A 19 0.



4 2 + 1 2 1-i ,4 1 5 ) r)
? 4n4|0 + 2.1 ;7 45 A4 < A .l



































TABLE C-2

PRE-MAIN-SEQUENCE BINARY MODEL FOR TV CASSIOPEIA

SECONDARY MASS = 1.39 MO

PERIOD = 1.243 DAYS










S^ T f (' /T (J ) Tr n'( TWC( ) !n' rTo 0 /T I r -T

7 ) ni(oo 1 7! 77 n7 1, to
S. o o *." 3 1 s?7 .0) ,0 .1
0 4.3F5c00 2.0476 0.42 54 *.15
4 5.560)+o 1 ,-oo7 0. 370 .0-)
r5 c+00o -. 0007 1. ?9 .07 .10
n -.1) y n L ona -). An -) 07 .1
7 O. n 55- + Or 2.0719 3..'1r) "r
S 1.102+101 2.0520 1*.115 ,0 3.37
C)0 .o c- i _~. 3Q.9020 .4 52 Q.5
14 1.416 +19 ..015 4.-, .05- 32.1
S11 1 .5 l+ o .l ni .4S4 2 04 ?0
12 1.7, I +10 2. 771 440I .0 m.01
13 0.o 54+" 2.Q952 3.4 1 7 .' 9 .00
14 2.1An+1, 2.9102 1344 .02 2.00
15 2.7-151 2.9207 .^^50 .0F 2.06
S 2.5-67 + r-) p.70,1 3. ^o .0- .0A5
17 2.70 -- 1 9*7 o1A ", A99 ,o* ,
10 1. 02 -rcl2 2.765 ". '400 I 2q 00
10 3.21F r+]10 2.7 ^AF, 4%51 2.' n
9n 2, 54":-4] ?9.7710 -, .4 O 9 -0
21 3.91 l '0 2.73/66 .447 .0Q 2.q"
22 4.1r 1 ^7 +10 7 2 77 ~L4^ .0') ".
91 4.411- l7+ 2.'69P ..4500 .00 2.
24 4.720rn .,oo ?..a0 ,o 0
25 5.~61' + I 2.'00 3.412ll .0 2.9?
26 ).41 4A^1+ 2.61 10. 1420 9.7 2.5"
27 5.70?c+13 2.5301 ?.0. .07 9 .7
20 6.160+10 2.5741 2. '!4 ) .0o 2.70
20 4. 57 xc 2.55 1 P.467 .0< '0 7<
30 7.0 2 -+1 2.5342 .4 604 20 .75
?1 7. 40onA 2.5172 3.4711 .0g 9.7
32 7.010+10 2.4003 1 .47 7 .05 2.72
33 q.4 n1 2.,47021 ?.4743 .04 0.71
34 030,3+1 2.46ln4 1. 76 0 .94 2.60)
35 0.474A 0 2n 14414 3.4776 .R, 2.^
I6 1. 04-+ i 2. 4225 3.4 709 .01) 0.67
37 1 .064E+l) P 2.4^' 1.4on0o .q3 2.65
?2 1.127+ 11 2.346 3.2425 .09 2.43
30 1.103 +11 2. 5 7 ?.40o .,2 2.63
.9 A .0 ~,1 2.3467 3.4tR .0) 9.61
41 1 3355+1 1 2. 3170 41074 .ql 2.6r
47 1. ) ." o 3.400o .01 9.50
41 I .40o?+1 2.2700 .r4007 0 2.5q
... 7. ^ 7 '7 .5
,- 70-7
S: .7 "- 1 -' 0 7Q .55
. i .!3 7. ^ 7';, 3.5
4a I 0 7n 1 1o "00. 0oo .7 o 2.51
40 7. -'=A + + 1 2. 1762 R.'n5 00. 7.7 5










86



5" 2. 17- 1ii .1 7 2.To2 .77 2.70
51 ?20E+: 2.137 ~C 3.5 17 .-7 47
52 2, 120r:' 1 I 9. 1 ] 5. 54 .77 p,1S
2 .5 1"-<-1 1 2j 2'1 2 -' A
o" P.o 0c^- ". "'^ /.5n7r .75 2.4.3

S 29.0 2-51+1 2. ". ? 2.5-110 .75 2.41
5'; 2 .o". FI lt l P. ''. 01 1 C .7), 4. 41
57 2.141Tc+11 2.272^7 3.515 .75 ?. 4
9 3.n17 F+ 11 2.0157 7 251kr .74 2330
50 3.n27C+-ll OQ ?-.51)9 .74 23.o
2n 0.. 44,c+ 70 0.51 "4 .74 9 .
~31 3.q55+ I .0'o 1,.5201 .73 2.35
~2 n4. n-+l 1 .o0 0 3.5217 .72 9. 4
53 4.?72+ 11 1 .011 1 ,577 .71 2 33
44 *.705-i7 -o ?1n fln':` 7) 2 1
5 .7? 05+ 1 1I 7 2.5 .7' 11
A, 4.074F+- 1 1 1 .? .71 0
77 572 3r:17+ 1 1I .? )500' i 4 72
0^ 5. .5 .+.11 1 1 3 1 : 9 '7
SO 5 7-,7 +11 1.7073 7 5 i 70 2,.
70 6,. P5E-L 1 1 77 0 q A7. A7 2 7
71 ,.00o +] 1 .750 4 35 4 .7 .
72 6.7271-11 ] .74024 1.5Q0 .o 2.21
71 7.07'9r 4 1 1.7?15 3 .507 O .9'9
74 7.435C+11 1.7pc 2.5 42 .Z 2.9?
75 7. 95cq ll 1.4,) i.5400 ,o Q 1 0
7/6 ,.2^ P+ l ^ -., 5 0 I. 1 0
77 O 4 n 4- 1~ 1,4 .47 3.SA ? .6, 9.17
70) 0.074E+l 1 .6 267 2.5470 7 2.16
70 0o.tcC11 A170 3.~5o0 .77 2.15
n 1 .0ln2F+12 1.59~ 2.5511 .7 2.14
91 1.052+l ? 1. .5 0 .5 527 .^6 2.12
2? 1.1n7t+12 1.5500 3.5544 .l1
.3 1.163IE+1 1.522 1. 553 n 4 2.11
94 1. 973F+19 1.510 3.5576 .,5 .1 n
'5 1.2AR :+12 I ,o7 3. 95560o .5 2.0o
2s 1.^'0-q+12 1.4751 3.5600 .5, 2.02
Q7 1 17F+12 1.^5 l 61 ., .2 4 2.07
2 1, 2l+12 1,4417 3.5642 14 79 4
o .5 3+1?2 1.412 .^ .4 2.04
0o I .^ c4-1 1 00 2.5675 .' p.0
0o 1.7?25+12 1.302 2.501o .n ? 2.,o
0 ~ .7" 1-I '1 ;707 .
0) 1 7n, 00 !-'1 .7 57- :1 ,

2/ ) .1 --:4 -'F 1 *7-7 3 /57 -* I '
-0 2 l .2r -4-I 2" 5772 1.07

7 Co.) I 1 .7V 3 .14l 1 .O
00 -) 22 .: I.











r *T]"!^ ] ( T /TR q '( ) q "1( T7- ) /1r '- rnsver

|00 92.,5'44c-+1) 2 ) q -'i 1.04
1r1 2. 1nt0r +1 1.1007 1.5P54 .A .1 '
lIW 2 I.0601-4 1.1717 107 1C7)
1, -7.1 1 7 1.1537 n .057 l oT
1 'q -l. i ,0 O l 1 ? 1 7 0.^oq7 -30 Al0
11 3.2754+12 1. 120 0..nn .50 1.01
105 K.4?20+~1 1 11A 3.5020 .51 .20
1"A nlc ?+ .noeC 3.5-04 .57 1.i
107 3.701i F+i) 1 .n74o0 ,505 .1 1.7
1) .0771 + 2 1.570 3.o50o .S 1 .54
100 ,.170F+12 ,-oo 1..nOQ .5 1 .*5
11, 4. 270o4.1 1l .2Ioo 3 53. .7 1. 4
1 4.50195 + 7 .,"I .57 1.91

113 5.071P+12 .0,430 2.5l0 .5 .1
1 I / 5.)12^: +1 .OF: '.nA .95 1.7o
115 5.501 1+ ? .0251 ?.", .50 1.90
11 5.971 +12 .o'7 2.4 00 .5S 1 .70
1 7 4. 1 ,59 +19 .O971 3. 11, 5 1.7
119 6.471E+1 .45 QI '. 1 ? .55 1.,77
AO 0.707T+ 12 .q940 3.61 AI401 1.72
1?0 7.117E+12 .o0102 .41^5 .55 1.75
121 7.40. 5+1 112 3.651 9 .54 1.7-
122 7.970-+12 .7022 3. 1o .,4 1 .77
1 31 .,'A -+l1 ? .7712 2.421 4 .5 1 .76
194 1.477E+12 .752 5.??11 ,51 1 .79
125 0o. i +1 .7-52 3.'247 .5 1 .71
17 o.566F+12 .7!52 2.A24 .5~ 1.70
127 1 I0-+ .407. 1. n1 ,e- 537
1? 1 .055c ~~ .67R2 3.690 .I5 .,
120 1 .07+1 .4502 1.~ 1 .52 1.57
l 1 .1 A +l 1 640n 2 3.6320 .5 .0,7
1 1 1 .221E+13 .^-1 2 .N 45 .52 1.
11- 1.205E+ 1 .6022 3.6342 .51 1. 5
1 33 1.3* ^E+1- .5'2 1. 1 7 .51 1.64
i.4 1.4 >^1R4+2 1 .5^3 2 3.5304 .51 .1 ,l
145 12.6 1+1 3 .5452 2.6 11) 1 1.62
136 1 .550 +l1 .5 ?2 3. 527 .50 1.
1 7 1 .6 6F+ I? .5071 .AA44 .5 1.61
!< 1.710+1 .40I ?.6n ,6 .51 1.6
1 0 I .n0.A+1 .Asol 3.6746 .60 1.50
14, I .90^4 -1 0, 1 40' .a 1 .53
1 2.00 1 +1 -.010 3.4500 .4 1 .5
? 4 I -) n.00 +I 17 -+ I7 I A. 1,) 5"0 1 .57
f .. 4 P. C r-i+ 2i -.O?.7 542; .' 1 75
144 12.9Q C +11 -.0 nl70 /Co .4," 1 .5

14 4 .0 .:+l1 t -.1R27 7. r01 4 1 .54
1407 .'Q C+I -.' ^o? ^ ?. <^OI' .- 1.57
14 0. 47or:+ --.1"1 4/,^0 47 1.54
1 "-0 4."%7 -- 1) -. > | >. 440) 47 1-7









88

1(o i "'T'n O (T /Ti CTPI) T T- T 7C) pI! T r- I- 7 r/)qr

15 !.3 -74 +1 -.. l on A ,7 .,1 5)
151 4 .' r7 +1 -'.0~1^7 ?. ^73 47 I.5'
155 4.27 +11 -.0145 3.i60on .44 1 .
153 4.-Qc+4. 1 2> 7 -* ,i
154 4-+1- -.0nlil .* 723 .46 1.
155 964+ l? -.0070 ".7 0 .4 I 1,47

157 5.?260+r) -.rni4 ".,77 E5 1 .4
1 5.2c 2'5i -.n n ? 3. 779 .45 4
150 5.'656fE+1 .,01r0 on 5 1. 44
160 5B.5) q c+ 7; .1 3.1 21 .45 1.4-
41 ^.051 F+ 1 .0055 m, ., 1. -3?
162 6.?4AOE-+1 177 2,50 .5 4 1 .
16- A. 147 :+1 .f0 3.5 71 .44 1 .4


1)6 75+ .'0144 1 0-1 .^1 A, 4
1 7 7.2304 +1 3 .11 q 7 1> ^ /42. 1. 3
A )7 7. 437^+ I ? ; :. 4" 1 7-4


170 7. .'!? +1", .o->57 r),. 7 F 16
171 q.'91c +1 .0270 3. 700' .49 1.4
172 .220oE+13 + 2 .7010 .42 1.35
171 R.427F+11 .*022) 3.70-3 .*A 1.'1
174 9.625 + 1 .5347 3 7r'53 .42 1.34
175 9.*Q?2-+1 n390 3.7060 .4n1 )
176 0.021e+1 .n03 2 70P5 .41 .
177 Q. 10T ^ 1 415 .71? .41 1 -Z
17Q 0. 17 +1 3 .0437 3.7110 .4 1.31
170 0.415P+1 .040n 1.71 5 .41 .30
qn O.ql c+ 12 .0493 3.7152 .0A 1. ^
I 1 1.0 + .nlF+l 5 ) .71 0 .40 1.20
192 1.21P+1 4- .052? 3.t 5 ..40 .
193 1.041 E+ .'n551 1.72 1l .40 1.29
I4 .06Cr+1A .0574 3.721 .4 1,.27
1s5 1.OPRE+1 n507 2.7294 .?0 1.25
196 l.1)ql + .n020 3.7251 .30 1.2
q17 1.120)+14 .0543 2.72 4 .30 1.25
q 9 1 .1401 4 .04AA 3.723A .10 1 .9
190 f. 5o-+; .n40o 3.7?nf .o 1 .24
100 1. 17o0 + i ."71 2 2.72 10 .39 1 .
101 1 00F+1 ..n70 3.771 131
)?2 +. In- 'I* 1) o 2 7 1 .'1 ?
') ) Or 4_) -7 F72 7 ) 7q !7 .
") 4 1. 5 30nq 4 7,7 ,'! .431 .4!
()5 1 070 77 + 7
1 05 1.07" +! 4' ." 2'7 0-' 1 )0
1'0 1.209P 4 .O" 5'1 7 A1 .27 t )
1 7 5. o )7A 7 1 4 7 1 .1

I 4 a + n 7 0 7 r ,77 1r.101








89
I)o T f -T'- T -(r /T C'T) Tr r T C" ) DTOTr)r /n ';

?" 1 77F+14 .00 4,A 2.7.40 -7 1.17
291 1 7-.1 4 .r A'o 3.7501 1 7
' .117 .+1 .-002 3.7r, 1.1
:03- 1 .437+ -?.7.? 1 .I -
2?"A 1 2cc+ 1 .n *o < 755. 1.1
2 .1476'P+1 .1 1.75..< .'5 1.1
,I2 1."~r1 I -197 3.755 .2'5 1.11
207 r.-16 ,-14 .7111 6 7 0 5 0.11
900 1.'?-c-In .11 2 1,7610 *3 1.12
290 1.555+-14 .1159 2.7^1' *25 1.19
210 1 .575c9+ l10 3.7,5" 1.1 i
211 1.5 c4- 1 1 7 ".7.7q .15 1.11
212 I.615,lC+14 191 3f.7.7 .2^ 1 .1
21nq l.s5c+1 12 51 3.77 .1^ .1
21 1. 5FI ^ E+ .1 9- 2.7721 .* 4 1,r)
?\\ I i +7 1 A .l .?n7 7 3 -?A n 1.1

21 1 .60A -+ 4 .1 g 2. 37 .3.1 )9
217 1.71 )4-,4 .13 l?~ 77?7 .4 no
21 1 ,7%d[-+ 1 1 A, 1.77 lr) -7 1P .)7
21 1. 73.t:+1A .1279 2 772q .1> 1 .7
210 1 .75' + -1 .1 409 7 3 07
2? 1 .773 4- .1 1 127 2. 70)2 22 1 .
221 1 .7o +1 .4 .1 42 3+.7Q4-0 3 1 .A
292 1.91 31 +1 1A7 4 .70 7 .7q ,07
223 1.92FE+-A .l (n3 3.7975 .33) .04
294 .q9524+ 4 .152R 3.7000 .04
22? 1 .972+ 4 .1 55 2.7000 .-22 1.03
9^ 1.qorc+l 4 .15 70 .4 702' .1 q
227 1.012F+ d .1 6 4 7.70 1 .32 1.2
)2o 1.02lC+4 4 120 70 .61 .1 1. 0
220 1.o51Q+14 .i ^ >.707? .?2 .01
9'10 1.071-+1 4 .1 4Q 3.7006 .31 1.1
231 1.001iF+ 1 70P i.Qan ? 1 1 0
232 9.011 r +14 .1724 3.0n30 .31 1.00
212 2.03n)E+ 170 1.q290 .1l .00
234 9.050+ r 4 .1707 q3.0465 ,31 .00
231 2.370-T+14 .1911 3.203 IAi .o0
236 2.0on+1 .14^, 2.91 00 .30 .O0
237 2.11 +14 .1967 3.9119 .30 .07
239 2.12o +14 ..104 1ol4 .30 .07
2o 2.140t+14 .10 21 3.q1151 .0 .0
240 2.!60o+14 .1040 .171T .30 .o0
241 2. or l- / .107 3.91 90 .3r .o0
249 2.2 00 C4-4 .2A0/1? .r7 .? .O5
-.3 ?. ? "-+11 9 .)9> .-o ,
4 1 2 .2, 1 .1; -0 ,9 .* O .01
9?) 0,-:. C..', I "' 9>n.1 ?i 0" ) 1 O-
2 *- 1 21- ) / 3
24'7 ?. "*O141 .2l 4I 2 03 .
2 9. 7- 4-' 1 .9I7" 3. 3 4 .20 .09
947. 2. '7''-: I 0 ',9 ".' 29 20 T99



































TABLE C-3

PRE-MAIN-SEQUENCE BINARY MODEL FOR IM AURIGAE

PRIMARY MASS = 2.91 MO

PERIOD = 1.247 DAYS










91

-rc T T T ( T ( T ( "r c~ ) 1r,, t/ 'f r -/

1 1 .Q 1 ^ I + 1 1.5112 1 1. 4 o2 r)
2 3.767C11l 1 511 ?.9560 .00 ')^
- 5.IcQ 17 1.51xn 3.I.7 /nI
4 7.5 C5 4- 1 1 1 77 q.2 -o .on ,7
5 o,4,nccl 1-5I ) .,1400 o 7 1.41
6 1 .120p+o 1 .59 1l o, s 07 1.o
7 1. 17 4-1 1.5- Q. A 42 .0. 1 -,7
I .5Q05 + 2 .)65 2 .*o 0 .of 1,55
0 1.40 41 ) 1.5 07 Q474 ,3A ?,
10 l.981V+12 1.5300 9.qAso .0, 3.52
11 9.) r)+ ) .1 -a 9 7 ,I q7? C4 ) ,5
? 1F9.+7) /3 1,5r)C+ q 1 1 4
12 2.?274-+1? 1.5153 2.7979)' .04.
13 2.445 -+12 1.5'7L ?.97.0 .0 3.4<
-4 .4~4,c+ 1 .51307 3.97rf 6? 0 ,45
15 2. ?22F- 4I '., 77 ')2 .
I )1 .f10 n) 1 5 17 0 70 P 941
1 7 3. 0 9 -+l .- ', l7 2 U '1 .

10 3.574 +12 ? 1 .55 s, q:n2 o 0o o, ?.
20 "i.7 ') l'V) 1 .952 3.Q054 o00 2 4
21 3.0o50- + 12 1.5550 -2 ,7 .1 (O. 3.
22 4.125 .19 7 ?. 1n 5 3 q97 .90 ".11
23 A.327F+12 1.5o50 ?.qoo .0o A .2o
24 4.5155412 t1. 51 1.2020 0 9,.
95 4.701C+12 .5630 ,.o?2 oq 2.
26 4.901 +1-- 1.54 4n 0051 .97 7.24
27 5. 7or+1 1 .56 2 1.9060 .07 1 .9
2P 5.267E+12 1.? 7 7.nqo ,/ / .
20 .455 -+17 1 .726 3.0009 .)p o,
30 5. 4A3E+12 .574AP 7,.oni .s 3.1q
31 5.q*q1l+ )2 .5770 3.0035 .5 2.0
37 6.022n+ 12 1.570?2 2.o'l .05 .15
33 :.2 00E+12 1 q15 3.on60 .94 2.*1
34 ~.1306+12 1.5937 2-. O~ .n4 1.19
35 6.58459+12 1 .5950 2.0 00 .( 3.10
3 .7721+12 1.59 1 3.0 117 .Rn .,o
37 6.060',-+12 1 .50n.Ol .01 .93 3.07
o" 7.1 4P:+l- 1.5025 3.0150 .Q l.
30 7. 336E+12 1 .5047 Q2.01 .02 3.1.4
40 7.52A'+1' 1.5060 3.0192 .1 -. '
41 7.713E+12 1.500) .01 00 .q1 .0l
.1 C 7 71 1)7 1 C oo I r n, .9

0 07 '1 :0' 7 7r- 0L I I o1 q.-')


q' O 20: C4,- 1 1 7 n. 0. 7q 2.0I
'1 r.) o 7 : 1 l /?5 .,"0)r07 .-0 7 0os

40 0 -7 r 7. 1 070 Q0














5) .n5 *19 0 1. 1 AI 7 .77 .oo

52 0.7495+12 1.C 0 15.0e1) .77 ?."
-5 0.504' +1 1 .7 7 1?.07'!i .7- 9.)
5') 0.07"-+i I 5 ".0 .77< .o
54 ).014 6 +1 1 .4S0" .O/I .7" 9 .Q
55 .n!r+l! 1 ,,q r! !.0ox?0 .7', 3.Q1
5 .Y3r+1 1 .142 .3 0420 .75 2.7)
97 1.O73c+ 1')7 T:3) ,74 3 .73
57 1.7I7+ 2 1.Ao 2.0A62 .7/ 91.7
5q !,o )+ l 1.470 9. 79Q .71 2.76
50 1.1 n+ 1 .6419 .0405 .74 2.75
, 1 ?.o1 c+ 1 1 15 3. 5 1 .7. 2.7
61 1. l .71, -1 1 .641?7 I. A5,o .71 ?.79

63 l.l 5c+3 1.6o9 2 2,RA) .71 9.70
,A 13.mc 41 1 .65^ '. 577 .790 7
65 1. 23+1 1 i .'^97 7 0R0 .79 2.67

67 1.2 ^O--+1 2 .6572 ", r'497 71 2.64

6 9")1 4 1. 61' < 7" 0 ?
70 1. 7317?-+1 .<. 0^77 .7'") .
71 '1,4' +1 1 .6663 3.,.02 .70 .50
72 1. 35) 1 1 .A6 2.o-0 1 .4o 2.51
71 17~'+ 1 1.670! .O74? .7 9P,54
7^ 1. 0 E+11 1.7 7 1 '.)7n l .Ao 2.5
75 ?1.4l1+4 1I.675^ 2.07 9n ,x .
76 1.410c+ 1 .6777 -2.0-77 .o 2.57
77 1 c--w 1 .o rnn 70 .4 .
70 1.46.7 c- 1 .6 23 .Cono) .A7 2.70
70 7 749 -+11 1.?.4 ."2 7 2.A0
c9 1.505n+13 1.696q .0042 .67 2.4
l 1.52 F-+12 1.01 3O.0eP50 .%6 2.4
02 1.5-49 -+1 1.6015 3.07- -).4

q4 1 50=+13 1.6041 3.0000 .oo5 ,42
25 1.5o00+12 1 .60ooA .Oo0 .,5 2.41
6 1 .6,10P+1 2 .7007 3.0043 .5 9 ,A4
07 1.6537 +1 1.70r .0050 4 2. 0
2 1 .6.5r4-1, ,+7054" 3.0076 .64 910
qo 1.674F+1 1.7077 ?.000o .,1 2.17
-n 1 .60.+1 1 71 n 4,.00 .-) 9,r5
01 1.7 1?c+ 1.712^ 4. ,o .9 2.3 4
02 .17'r:+ .71T Al 1 0"'3 2 .2
Q? 1.7',o + -1 1.7171 /. An .. -7
07 1 o7 -7 A, 1 7r.7 1 ">, "'1

DA 1. 2i50+1 1 709.9 4,0 <0 9 g
37 1 *)5o 1 .79 .7 ,;1 .72

S .1?=17 1.7312 4.911 .1 C.)5











r) 7T TT r) l 4 T ) o r / i'r7

'1 .o, -4-11 1. 7 7 7 ")1 77 ) 4

12 1.'010 +1 I1.7" 4.r ? 11 .*S 0.2 )


S .0I-71 Q 0! .-7a-o 0. 2.^4 .-7o q 7.1Q
4 1 7 0/.-+ 1 7 -o 4. 70 1 ? 17
S17 9 C,+I ] .75 4 ^. 20 .1, 71.1 0
1Io 2. 31'E+1 1.7571 ".r'7 1? .5 2.15
100 )7r+17 i -75r5 A.0n30 .571 2.)1A-
1i 2.QBp+ r3 1 .7500 4/.017 .57 2.1
I 2 .9 0r+) 1 .t ,2 7<,;4 .57 ?2 A1
117 2. In7c:+ 3 1 .7T 0 4.A P | .57 7. 11
1 1 2.124 *I l I .7 r, 4. .130 oq .A ,A 2.1
11 4 7.1 3 4l. + 1 S r. .-10 q. -o
1Ic 2. 1 "7+1? 1 7" ."'

I 7 2.2 -1 -. 7.1/,7 ., c
11 j2.29 + 1 7701 4 4 r)5 )

2 )57+ 1 .7q 4. 1 0 r9 "+'D
t 2.99 7 4 1r.7 I I
121 2.2746 +1 1.72,5 *.0~3- ,4 2.) 2
122 2.05a+-1- 1,79P4 A~'1554 .54 9.
1 2.31^ /-+12 1.7010 4.r571 .54 2.01
124 2. 339+ 1 1 .70>^ 4..035, .%3 1.Oc
125 3.151 I+l 1.7"0') 4.'n/S .c) .0
124 2.370+11 1.7000 4.0 1 .51 1.07
1 7 2.00-+| 1.915 4.0,41 1.c. )0o
129 2.4f09-+l 1 1 ..07 40,n",4 .52 1 .0
1 33 2 r 4 q i t 4 ", q 1 ,17-

190 2.42m 4-l A1 .U0 4.0)674 .A5 1.04
130 2. 445F+ ? .00q5 4.A004 .93 1.03
11 2.4A4 A ?1 1.21 1 '.0711 .52 1 .')
1?2 2. 4 '9, + l 1.l 15n A. '7 0 .51 1.01
33 2.Sn52F+ 3 1 .9177 4.0747 .51 1 .0O
1 4 ?.25 .l 1 1-1'7 A .0764 .51 1 ,00
1?5 2.530E+13 1 .9233 4.0797q .51 1.93
136 2.559n +-4- 1 .2 ?" 4.AP)o .50 1.97
127 2.577+1' 1.9200 4.ql0 q .5 1.Q9
13 9 2.50^ +-1 1 .0o17 4.091 .50 1K O5
120 7.615r+13 1.9^, 5 n.054 .5 1.4,
140 2.63' +- V .1')74 4.0'~07 .40 1 .2
I 41 2. 2 4pr l 2 I .9403 4.noon .40 1 .0
1 '! 947, "-: 4 n?4 C ') 'on

9. 7 00i -7 13 1. 1 A 'n0'% I 001
\ ,I. 1, ? .7. 4 :+ 2 I .q- 4.0 .4 4. I 7''
1 2 .. 47 0 .---7 .'7
I," 7! i I 1 7

1 -..- 1
1 40 '> f' ^ '-i7 1 '; .47 1 ~I