Rotation rates of algol-type binaries from absorption line profiles


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Rotation rates of algol-type binaries from absorption line profiles
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xiv, 158 leaves : ill. ; 29 cm.
Mukherjee, Jaydeep, 1961-
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Subjects / Keywords:
Double stars -- Rotation   ( lcsh )
Astronomy thesis Ph. D
Dissertations, Academic -- Astronomy -- UF
bibliography   ( marcgt )
non-fiction   ( marcgt )


Thesis (Ph. D.)--University of Florida, 1993.
Includes bibliographical references (leaves 151-157).
Statement of Responsibility:
by Jaydeep Mukherjee.
General Note:
General Note:

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University of Florida
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Full Text







This dissertation is dedicated to my parents and to Dr. Deepak Chatterjee.


A number of people have given me their time, support and guidance over the years

that I have been here. I would like to express my gratitude to them and thank them

for all their support.

First of all, I would like to thank the chairman of my committee, Dr. R.E. Wilson,

who though being my advisor treated me as a friend and not a slaving graduate student.

His patience and immense knowledge of binary stars made my thesis a very pleasant

one. Also, our numerous tennis matches helped improve my game.

I would like to thank Dr. Oliver for all his help and for writing all those reference

letters. I would also like to thank the other members of my committee, Dr. H. Smith,

Dr. Dermott and Dr. Ghosh, for reading the thesis and for their constructive criticism.

Special thanks go to Dr. Peters for making all the observations, which I have used

in this thesis.

I would like to express my sincere appreciation to my fellow graduate student David

Kaufmann, who time and again has helped me get out of a difficult spot. I thank him

for his patience and understanding and for the long discussions on every topic that one

can imagine, though just one victory in a tennis match would have been appreciated.

My special thanks go to the other graduate students especially, J.C. Liou, Damo

Nair, Billy Cooke, and Sandra Clements. I would like to thank my office-mates Jane

Morrison and Ricky Smart for putting up with me over the last three years.

Special thanks go as well to the department secretaries, Debra Hunter and Anne

Elton, for their help with grants and for sending all those reference letters.



LIST OF TABLES .............

LIST OF FIGURES .............

ABSTRACT .................


Overview of the Problem ....
Roche Lobes ...........

. . . iii

. . . . v

. . . vi

. . . vii

Algols .

Relation Between W Ser Systems and Algols

Rotational Modulation of Light . .
Rossiter Effect .......................
Light Curve Analysis ....................
Line Profile Analysis ...................
Profile Fitting and Line-Width Measurements ....

Fourier Analysis.

Other Methods .................. ........ ...

Formation of Absorption Lines in Stellar Atmospheres .....
Equation of Radiative Transfer ................ ....
The Milne-Eddington Model ....................
Center-to-Limb Variation ......................
Line Broadening Mechanisms ................. ..
Natural Line Broadening ...... ....... ........
Collisional Broadening .......................
Thermal Broadening ........................
Turbulence ..............................
Rotation .. .. .. .. ... .. .. .. .. ..
Combining Radiation, Collisions, and Thermal Motion .. ..
Instrumental Broadening ......................

4 DATA REDUCTION .........................
Observations ..............................





Corrections ....................... .. ... 31
Correction for Earth's Rotation . ... ... 31
Correction for Orbital Motion of the Earth . ... 32
Observed Algol-Type Systems ...... ............... 35

5 METHOD OF ANALYSIS ........................... 36
Introduction ..... ....................... ....... 36
Light Curve Code .... .............. .......... ..36
Line Profile Code ........ ............... ......... 38
Doppler Subroutine .............................. 41
Lineprof Subroutine .............................. 42
Hav Subroutine ......................... .. ... 46
Light Subroutine....................... ........... .. .....48
Instr Subroutine ............................... 52
Phase Smearing ......... .................. 53
Parameter Fitting ................... .......... 54
Simplex ........................... ......... 54
Differential Corrections ................ ...... ...... 56

6 RESULTS AND CONCLUSION ........................ 59
General Discussion............. .............. ..... 59
Discussion on Individual Systems . ..... ... 72
SCnc............................ ......... 72
RZ Cas ............. ......... ............... 73
TV Cas ................... .... ........ 74
U Cep ................. ................... 75
SW Cyg ................... ................. 76
SEqu ... ................................... 77
RY Gem .................................... 78
RW M on .................................... 79
TU Mon ..................... ........ ..... 79
DM Per .................................... 80
RW Per ......... ....... ......... ... .. ....... .. 80
RY Per .......................................... 82
/ Per . . . . 83
YPsc .................................... 84
USge ............. ......................... 85
RZ Set ...... ............ ................... 86
ZVul ........................................ 87
Rotation Statistics ..... ................. .... ......... 87
Conclusions .................................... 90
Future Work ......................... .......... 92

A LIGHT CURVES .............................. 115

.. .. .... .. 133


BIBLIOGRAPHY ................. ............ ....... 151
BIOGRAPHICAL SKETCH ............................... 158


1: Stars observed by G.J. Peters ............. ......... 35

2: List of parameters for the LC code ................... .37

3: Line Profile parameters ......... .............. 40

4: Other input quantities for the Line Profile code . .... 41

5: Basic system data .............................. 62

6: Spectroscopic data ............................... 64

7: Photometric elements......... ........ ........... 66

8: Line profile parameters (Differential Corrections method) with their
probable errors ................... .............. 68

9: Line profile parameters (Simplex method) . ..... 70

10: Rotation values from various methods . ..... 93

11: Rotation statistics of Algols from the literature . ..... 96


1: Equipotential diagram (mass ratio = 0.25) .......... ......... 3

2: Rossiter effect (Twigg, 1979) .......................... 8

3: Radial velocity curve of a model of RZ Sct, showing the Rossiter effect. 9

4: Figure showing light curves of RY Gem for both synchronous and
nonsynchronous rotation................ ............. 12

5: Flow chart of the Line Profile code. ... . 39

6: Variation in line profiles (pure absorption lines) due to limb darkening 43

7: Variation in the intrinsic line profile by changing, clockwise from lower
left, the effective temperature, number of absorbers, micro-turbulent
velocity and the damping constant. . .... 44

8: Variation in the intrinsic line profile by changing p and . 45

9: H(a,v) as a function of a for various values of v . ... 49

10: H(a,v) as a function of v for various values of a. . .. 50

11: Intrinsic line profiles for synchronous (below) and non synchronous
cases. The line profiles are choppy because they were calculated using a
very small grid to illustrate the effect of non synchronous rotation. 51

12: Line profile for the entire star for synchronous and non synchronous
cases .. ........... ....... .. ........... ...... 52

13: Instrumental profiles of various widths. . .... 53

14: Two dimensional simplex illustrating the four mechanisms of movement.
(Caceci and Cacheris, 1984) ......... ................ 55

15: An example of the simplex moving on the response surface's contour
plot (Caceci and Cacheris, 1984) ...................... 56


Histogram made from rotation measures in Table 10 and 11 showing the
numbers, N, of Algol primary stars with various rotation rates. Stars
with both kinds of determinations are represented by half an open box
and half a shaded box, so as to conserve the total number of stars. .

Fit to the observed line profile (dots) for S Cnc

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

Fit to the

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

observed line profile (dots)

.......... ... 98

for RZ Cas . .

for TV Cas ............

for U Cep ............

for SW Cyg ............

for S Equ .............

for RY Gem ... ........

for RW Mon ...........

for TU Mon ............

for DM Per ..... ......

for RW Per ............

for RY Per .............

for # Per ..............

for Y Psc .............

for U Sge ........ ....

for RZ Set .............

Fit to the observed line profile (dots) for Z Vul

















. 114

34: The system S Cnc at different orbital phases along with its light curves
and photometric elements .................... ... 116

35: The system RZ Cas at different orbital phases along with its light curves
and photometric elements............. .......... 117

36: The system TV Cas at different orbital phases along with its light curves
and photometric elements............... .. ..........118

37: The system U Cep at different orbital phases along with the light curves
and photometric elements..................... .. 119

38: The system SW Cyg at different orbital phases along with its light
curves and photometric elements . . 120

39: The system S Equ at different orbital phases along with its light curves
and photometric elements............................ 121

40: The system RY Gem at different orbital phases along with its light
curves and photometric elements . . 122

41: The system RW Mon at different orbital phases along with its light
curves and photometric elements . . ... 123

42: The system TU Mon at different orbital phases along with its light
curves and photometric elements . .... 124

43: The system DM Per at different orbital phases along with its light curves
and photometric elements....... ........... ......... 125

44: The system RW Per at different orbital phases along with its light curves
and photometric elements....... .......... .......... .. 126

45: The system RY Per at different orbital phases along with its light curves
and photometric elements .................... 127

46: The system 3 Per at different orbital phases along with its light curves
and photometric elements ........................ .. .. 128


47: The system Y Psc at different orbital phases along with its light curves
and photometric elements .......................... 129

48: The system U Sge at different orbital phases along with its light curves
and photometric elements .................. ......... 130

49: The system RZ Set at different orbital phases along with its light curves
and photometric elements............................... 131

50: The system Z Vul at different orbital phases along with its light curves
and photometric elements....................... .....132

51: Spectrometry of S Cnc over the wavelength range 4420 4530
Angstrom Units ............. ............. .......... 134

52: Spectrometry of RZ Cas over the wavelength range 4420 4530
Angstrom Units ................ .. ............. ..135

53: Spectrometry of TV Cas over the wavelength range 4420 4530
Angstrom Units ................................ 136

54: Spectrometry of U Cep over the wavelength range 4420 4530
Angstrom Units ... .... ............... ........ ..... 137

55: Spectrometry of SW Cyg over the wavelength range 4420 4530
Angstrom Units .. ... ........... ....... ......... 138

56: Spectrometry of S Equ over the wavelength range 4420 4530
Angstrom Units ............................... 139

57: Spectrometry of RY Gem over the wavelength range 4420 4530
Angstrom Units ..... ... ............ ....... ....... 140

58: Spectrometry of RW Mon over the wavelength range 4420 4530
Angstrom Units ................... ............ 141

59: Spectrometry of TU Mon over the wavelength range 4420 4530
Angstrom Units ................ ............. 142


60: Spectrometry of DM Per over the wavelength range 4420 4530
Angstrom Units ............................... 143

61: Spectrometry of RW Per over the wavelength range 4420 4530
Angstrom Units ................................ 144

62: Spectrometry of RY Per over the wavelength range 4420 4530
Angstrom Units ............................... 145

63: Spectrometry of f Per over the wavelength range 4420 4530 Angstrom
Units ......................... .......... 146

64: Spectrometry of Y Psc over the wavelength range 4420 4530
Angstrom Units ..................... ........... 147

65: Spectrometry of U Sge over the wavelength range 4420 4530
Angstrom Units ................................ 148

66: Spectrometry of RZ Set over the wavelength range 4420 4530
Angstrom Units ................................. .149

67: Spectrometry of Z Vul over the wavelength range 4420 4530
Angstrom Units ............. ................... 150

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



Jaydeep Mukherjee

December, 1993

Chairman: R.E. Wilson
Major Department: Astronomy

Rotational velocities of stars normally are determined from spectral line widths by

means of a calibration of width versus v sin i from suitable standards. Over the last

few years, it has been possible to extract rotation rates from light curves of binary

systems. The light curve model has been mainly used to compute theoretical light

curves, which is to say that it deals mainly with continuum radiation rather than line

radiation. However, the model can be used to predict the behavior of absorption lines.

The dissertation consists of modelling and fitting line profiles generated by a program

that includes the main local broadening mechanisms (damping, thermal, rotational and

micro-turbulent broadening, etc.) and binary star effects (gravity and limb darkening,

ellipsoidal variation, rotational distortion, etc.).

Previous estimates of rotation from line profiles assumed that all or most other

broadening effects are negligible compared to rotational broadening. This means that

the previous methods are at their best for fast rotators and at their worst for slow

rotators. However, it is for the slow and moderate rotators that information is most

needed, because light curves are beginning to provide rotation rates for the fastest

rotators, and the light curve method is best for fast rotation.

Knowledge of rotation rates of mass transferring binaries will be useful in gathering

information on mass transfer rates and the general mass transfer process and hence on

the nature of close binary star evolution. It will also help us ascertain how well rotational

velocities can be determined from light curves. Finally, it will help us establish the

statistical distributions of Algol rotations, including how many systems are close to,

or in, double contact (both components filling up their limiting lobes, one Roche and

the other rotational).


Overview of the Problem

Traditional methods for determining stellar rotation rates from absorption line

profiles neglect most or all other broadening mechanisms. They are subjective and

do not provide standard error estimates. Previous models also neglected effects such

as gravity darkening, limb darkening, reflection, tidal distortion, and eclipses. The

measurement of line profiles can provide much information about a stellar atmosphere. It

can tell about element abundances, surface gravity, and turbulence. Due to developments

in detectors such as CCDs it is possible to get high resolution spectra, which is very

important if one wants to analyze line profiles.

One of the most important fundamental parameters of a star is its rate of axial

rotation (surface rotation). This parameter affects the shape of the star and the

distribution of light over the surface. By analyzing eclipsing binary systems, both

photometrically and spectroscopically, and combining the results, one can get values of

fundamental astrophysical parameters such as mass, radius, and luminosity. To ensure

accurate determinations of these fundamental parameters one must have accurate light

curve solutions. This can be achieved by treating all the parameters correctly, including


One parameter that can be obtained from light curve solutions is the rate of

axial rotation. However, this determination may not be easy in practice because of

correlations of the rotation parameter with other parameters. It is difficult to discuss the

correlations, but previous work by various authors has shown that rotation rates can be

strongly determined for some fast rotators. For slow rotators and in-between cases one

has to rely on line profile analysis, which is the subject of this dissertation.

Knowledge of rotation rates yields important information on the evolutionary status

of a star. In particular, if there is a binary system in which one of the components

is rotating rapidly and there is evidence of a gas stream between components, which

implies a slow rate of mass transfer, one can infer that there was a recent large-scale

mass and angular momentum transfer. If there is a binary system where the components

are in synchronous rotation and there is evidence of circumstellar matter around one

of the components, as in the case of some Algol-type binaries, one can infer that the

component was once in rapid rotation but tidal braking has slowed the rotation to near

synchronism. There are cases of binary systems in which one of the components is

surrounded by a thick accretion disk. In this case, the system is in a rapid phase of

mass transfer and possibly the gainer of this mass is in rapid rotation. Thus statistics of

primary star rotation, especially of Algol types, will provide information on the mass

transfer process and therefore on close binary evolution.

Roche Lobes

For binary star systems the idea of level surfaces (equipotentials) is used to define

the shapes of the stars. In the case of synchronous rotation, the largest equipotential

surface that completely encloses one star or the other is the one that includes the

inner Lagrangian (L1) point (see Figure 1). At this point, the sum of the two forces

(gravitational and centrifugal) equals zero. The volume enclosed by the equipotential


surface which passes through the L1 point is called the Roche lobe. The whole system

is assumed to have uniform angular rotation about the center of mass.

If one considers a system in which star 1 is filling up its Roche lobe and star 2

is not (i.e., a semidetached system), then, if star 1 tries to expand (due to evolution)

there will be mass transfer onto the second star. This is because at the balance point

the material from the lobe-filling star faces a vacuum and will flow toward the second

star. Actually, the material in the whole photosphere faces a vacuum, but everywhere,

except at the balance point, the outward pressure is balanced by the inward effective

gravity and so the star does not expand (except slowly due to evolution). However,

the balance point is a null point of effective gravity, and so there is nothing to stop the

flow of matter from star 1 to star 2 in a small region around L1 (see Figure 1).

Figure 1: Equipotential diagram (mass ratio = 0.25)


Mass transfer will continue until there is pressure equilibrium across the balance

point, when there will be no more need for mass transfer and the stars will be able to

support a common envelope (overcontact system). If the system extends still further

out, it can reach outer contact. The equipotential, which includes the outer Lagrangian

point, L2, will limit the size of the system as a whole because L2 is a null point of

effective gravity and material can flow from the system at this point. If the two masses

are equal, one could get another outer contact surface on the left hand side (behind

the higher mass star), but if mi > m2 this configuration could never be reached since

mass escapes from L2. L2 is at a lower potential than L3 and is always behind the

lower mass star.

In defining these categories, certain assumptions have to be made. They are as


i. The stars are point masses. (Real stars are very centrally condensed, even on the

main sequence, and as they evolve they become more condensed.)

ii. Orbits are circular. In many close binaries, tidal effects will circularize the orbits,

under commonly found circumstances.

iii. The stars rotate synchronously, at least near their surfaces. This often is a valid

assumption, unless there is an active spin-up or spin-down mechanism such as mass

transfer. However, there are models that take asynchronism into account.

iv. Radiation pressure can be neglected. This effect can be added if necessary and

should not pose a problem unless the radiation comes from outside the star.

If the mass receiving star rotates faster than synchronously, the balance point moves

closer to the star, thus shrinking the limiting lobe. If one star has filled its Roche lobe


and the other has filled its rotational lobe, the binary is called a double contact system

(Wilson, 1979).


The primary star is a main sequence star in the hydrogen burning stage and is

chemically uniform or nearly so. The secondary is a subgiant and usually is the larger

star, with a helium enriched core. The mass ratio (m2/ml) ranges from about 0.1 to

0.35. In Algol type systems the primary (mi) is the higher mass star. Star 2 is more

evolved, usually a subgiant, except in very short period Algols. Originally, m2 was the

more massive star, filled up its Roche lobe, and transferred mass to the other star. Thus

mi became more massive, leading to the Algol stage where it is now stable against

mass transfer. The light curves of Algols have deep eclipses, which make them easy

to discover.

Relation Between W Ser Systems and Algols

When mass transfer (M.T.) occurs in a close binary system, the matter expelled by

the mass losing component can gain an appreciable amount of angular momentum along

its path to the other component, and in typical Algols it begins with considerable angular

momentum. The accreting part can cause the gainer to rotate faster than synchronously.

It is found that quite a large fraction of angular momentum can be converted into

rotational angular momentum (Wilson and Stothers, 1975 ). Calculations show that the

secondary has to accrete only 5 to 10 percent of its original mass in order to spin up to

its break-up rotational velocity (Packet 1981). The mass that is transferred early in the

rapid phase is quickly assimilated by the original secondary (present primary), which


is spun up at the surface and to some extent internally. When the surface rotation rate

reaches the limiting rate (i.e., the limiting velocity for which the surface gravity goes

to zero for at least one point at the surface ), no further accretion is possible and the

accreting material has to accumulate into a thick geometrical disk. Such a system is a

W Ser-type system (ex. RX Cas, SX Cas, 3 Lyr, W Ser). This is a class of binaries that

was introduced by Plavec (1980). These systems are transferring mass at a relatively

high rate, probably on the order of 10-610-4 solar masses per year. W Ser stars show

spectacular spectroscopic behavior due to large-scale ejection of matter from the system,

as well as rotation and eclipse effects by circumstellar rings or disks and flow of gas

from an evolved component into the ring or the disk. However, the thick and opaque

disks make observations of W Ser systems very difficult.

One can learn more by looking at systems in which the rapid phase of mass transfer

(R.P.M.T.) is just ending or has just ended. They can be recognized by the rapid rotation

of their more massive components. These are called Rapidly Rotating Algols (RRAs).

Some examples of RRAs are AQ Peg, RZ Set and RW Per. This term was introduced

by Wilson et al. (1985). It is presumed that after tidal braking has acted for a sufficient

interval, RRAs will become normal Algol systems. Thus it seems reasonable to assume

that the W Ser systems are in double contact and that some of the RRAs may be in

double contact.


The four basic methods for measuring axial rotation of stars are as follows:

1. Rotational modulation of light

2. Rossiter effect

3. Light curve analysis

4. Spectral line profile analysis

Rotational Modulation of Light

Apart from the sun and a few other stars (a Ori), none of the stars can be resolved as

stellar disks. However, one can determine the rotation period of a nonuniform surface

by studying the periodic changes in light that it produces. Slettebak (1985) divided

this work into two categories: (a) study of a variable continuous spectrum as star spots

appear and disappear on the rotating disk, and (b) measurements of the variation in

strength of emission lines (e.g., Ca II, H and K ) that arise in plages in a rotating


Rossiter Effect

As a star rotates, one of the limbs will be moving towards us and the other away from

us. As a result the spectrum lines are normally symmetrically broadened by the Doppler

effect. Assuming direct rotation, when one of the stars (star 1) is entering eclipse, one

of its limbs is gradually covered by the eclipsing star (star 2). Thus lines from star 1

are fully broadened on one side only (Figure 2). When these lines are measured for the

determination of radial velocity, the center of density of the line will be shifted towards

the broadened edge and away from the center of the symmetrical line. The measured

center of the line is thus displaced towards the region of longer wavelengths. The

radial velocities are then too large positively. When the star is emerging from eclipse,

the receding limb is covered and the approaching limb is visible. Thus the measured

center of the line is displaced towards the region of shorter wavelengths and the radial

velocities are too large negatively. This is called the Rossiter effect (Rossiter, 1924).



Figure 2: Rossiter effect (Twigg, 1979)

Twigg (1979) determined Veq. for 19 Algol systems by carefully analyzing the

Rossiter effect. An example of a binary system showing the Rossiter effect is shown in

Figure 3. He used Wilson and Devinney's (1971) light curve program for the analysis.


Twigg's procedure was iterative since a computer program for the simultaneous solution

of photometric and spectroscopic data was not available then. He first used the light

curve program to get a good fit with an assumed value F (see section on light curve

analysis for a definition of F). Then the theoretical radial velocity curve was plotted

against the observed one in order to check the amplitude of the Rossiter effect. If the

initial guess of F and hence the amplitude of the Rossiter effect was incorrect, then

light curve runs were carried out for another value of F, until the correct amplitude

was achieved. Twigg then carried out a light curve differential correction solution for

the new F.


-1 0


I I I I I I I i I I

RZ Set



- 2.

I I II i iI i

0 .2 .4

Figure 3: Radial velocity curve of a model of RZ


Sct, showing the Rossiter effect.


Light Curve Analysis

Determination of rotation rates from light curves was introduced by Wilson (1979)

and Wilson et al. (1985). Rotation affects the figure of a binary component and hence its

surface brightness distribution. These rotational effects influence light curves. Rotation

of the primary component will result in polar flattening, which in turn influences the

shapes and depths of the eclipses. The brightness ratio of the primary to the secondary

star also influences eclipse depths but is less obvious. If a component of a binary

system is in rapid rotation it is less bright, seen equator on, than a slow rotator with the

same mean effective temperature. As a result the tidally distorted secondary provides

an increased fraction of the system light. Thus there is an increase in the observable

ellipsoidal variation. Since the primary has polar flattening, the secondary intercepts

relatively little of the primary's emission. This reduces the reflection effect. All these

effects help in estimating the rotation rate, especially for fast rotation. Figure 4 shows

light curves of RY Gem for both synchronous and nonsynchronous rotation.

One can compute light curves from given parameters using a light curve program

(e.g., Wilson and Devinney, 1971, Wilson 1979). To compute the parameters from the

observations one applies the method of differential corrections (Wilson and Devinney,

1971), using the expression for the total differential of the light values,

Of Of Of
f(obs) f(comp.) = -- AP + P2 + ....... + APn (2.1)
OPi aP2 ap,

as the equation of condition for a linear least squares analysis. Here P1, P2,.....Pn

are adjustable parameters. The quantity f(obs) is a light value (i.e., relative flux) from


photometric observations. AP1, AP2, ...etc. are the parameter corrections which will

be determined by least squares, and f(comp.) is to be computed from a light curve

program. The derivatives, Of/OP, are to be found by

Of f (P + ) f(P )
P AP(2.2)

The increments in P must be within a reasonable range (neither too small nor too large).

An important parameter is F, the ratio of spin angular speed to orbital angular

speed. Although it is included in the list of parameters in the light curve model, one is

not sure about the reliability of an estimated value of F obtained from a solution. The

determination of F depends on how large the effects of rotation on the light curves

are, and on how serious the correlations between F and the other parameters are.

Wilson (1989b) has plotted rotational effects (polar flattening and equatorial dimming)

vs. F/F(critical). It was seen that departures from the synchronous case grow slowly at

first, reaching only about 18 percent of the full effect when F/F(critical) is 0.5. Previous

work has shown that, at least for fast rotators, solutions for F often converge well. For

known fast rotators the Differential Correction method usually will find a large value

of F even when starting with a small value. For known slow rotators it finds small

values even when starting with large values.

Rotation rates have been determined from light curves for RZ Set (Wilson et al.

1985), U Sge and RY Per (Van Hamme and Wilson, 1986b), AW Peg, AQ Peg, and SW

Cyg (Wilson and Mukherjee, 1988), RW Mon and RY Gem (Van Hamme and Wilson,

1990) and TT Hya (Van Hamme and Wilson, 1993). All the above systems, except

for U Sge, are fast rotators. In order to test the reliability of a value of F, one must


compare it with rotation rates obtained from other methods such as line profile analysis

and the Rossiter effect.

-.2 0 .2 .4 .6

RY Gem

F1 -14
F =1


Figure 4: Figure showing light curves of RY Gem
for both synchronous and nonsynchronous rotation.


- Oa




Line Profile Analysis

Traditionally, the method used to obtain Veq sin i, the projected equatorial rotational

velocity of a star, is to match the observed profile of a spectral line with the correspond-

ing profile of a standard star. The standard star is usually of the same spectral type

and luminosity class as the observed star. This method may work for single stars but

may have problems for binaries. The accuracy of Veq sin i is the same as the reciprocal

dispersion used in recording the spectrum. Thus to get accurate results one must have

high resolution spectra.

Profile Fitting and Line-Width Measurements

Shapley and Nicholson (Slettebak, 1985) showed that an undarkened rotating stellar

disk would broaden an infinitely sharp line into a semi-ellipse. Shajn and Struve (1929)

developed a graphical method of computing rotationally broadened line profiles. The

computed line profile is then compared with the observed line profile to give V sin i, the

component of the rotational velocity in the line of sight. Following the graphical method

of Shajn and Struve, Elvey (1930) determined the effect of axial rotation of a star on the

contour of Mg II 4481. He assumed no limb darkening. Instead of using line profiles in

the spectrum of the Moon as nonrotating lines, as had been done by Shajn and Struve,

Elvey chose profiles from sharp-lined stars of early type. The computed contours were

then compared with those observed in 59 stars of spectral classes O, B, A, and F.

Slettebak (Slettebak, 1985) and Slettebak and Howard (Slettebak, 1985) also used

Shajn and Struve's graphical method, but included limb darkening effects to investigate

stellar rotation in some 700 stars across the HR diagram. Collins (1974) calculated

rotationally broadened line profiles for 09-F8 main sequence stars using the ATLAS


model atmosphere (Kurucz, 1970), taking into account the effects of limb darkening,

gravity darkening and rotation.

Fourier Analysis

Fourier analysis of line profiles was introduced by Carroll (1933). The observed

profile is a product of the transforms of the nonrotating profile and the rotational

broadening function (Slettebak, 1985). Taking the Fourier transform of the observed

profile and comparing the zeroes of the transform with the zeroes of the observed profile

gives the value of v sin i. Gray (1976) has extended Carroll's method from a location

of zeroes to a fitting of the entire transform.

Other Methods

Rucinski (1979) developed a line profile synthesis model where the spectral feature

was assumed to have four lines (characterized by strength, di, and wavelength, Ai), all

broadened as Voigt profiles with the same damping constant 'a'. However he assumed

that the primary star was spherical. Rucinski (1992) describes a method for determining

the spectral-line broadening function based on simple linear mapping between sharp-

line and broadline spectra. Anderson and Shu (1979) computed bolometric light curves

and rotation broadening functions of contact binaries for a grid of values of mass ratio,

filled fraction, and orbital inclination.

Vogt et al. (1987) describe a technique -Doppler imaging-for obtaining resolved

images of certain rapidly rotating late-type spotted stars. There is a correspondence

between wavelength position across a rotationally broadened spectral line and spatial

position across the stellar disk. Cool spots on the surface of a rapidly rotating star

produce distortions in the star's spectral lines. If the shapes of the spectral lines are


dominated by rotational Doppler broadening, as is the case with rapidly rotating stars,

a high degree of correlation exists between the position of any distortion within a line

profile and the position of the corresponding spot on the stellar surface. The above

technique can therefore be modified to extract information on rotation.

Most of the previous modelling of line profiles assumed that all or most other

broadening effects are negligible compared to rotational broadening. Thus the previous

methods are at their best for fast rotators and at their worst for slow rotators. Since light

curve analysis is beginning to provide rotation rates for the fastest rotators, information

is most needed for the slow and moderate rotators.

Thus, one introduces a new method, which consists of modelling and fitting line

profiles generated by a program that includes the main local broadening mechanisms

(damping, thermal, rotational turbulent broadening, etc.) along with instrumental

broadening and phase smearing, as well as binary star effects (tidal and rotational

distortions, reflection effect, gravity darkening, limb darkening, etc.)


Formation of Absorption Lines in Stellar Atmospheres

In the frequency range of an absorption line the absorption coefficient is composed

of the coefficient of continuous absorption corresponding to bound-free and free-free

transitions, added to the absorption coefficient corresponding to the discrete transition in

question. Ionization results in bound-free transition and the acceleration of one charge as

it passes close to another results in a free-free transition. Bound-bound transition gives

rise to a spectral line. A point to note is that there is rarely a sharp transition between

where the line absorption dominates the continuum absorption and vice versa. However

for the development that follows it is assumed that a clear distinction does occur.

Equation of Radiative Transfer

Assume an axially symmetric radiation field that varies with 0 but not with azimuthal

angle 4. Energy crossing a differential area, dA, in time dt from the pencil of radiation

in solid angle dQ is given by

dE = I(v, x, 0) cos OdgdtdvdA (3.1)


I(v, x, 0) is defined as the specific monochromatic intensity of radiation (ergs/cm2/

sec/ Hz/ steradian)

v is the frequency

x is the distance of the differential area dA from the surface, and

0 is the angle between the direction of the beam and the normal to the surface.

The difference between the amount of energy that emerges from the volume element

and that incident must equal the amount created by emission from the material in the

volume minus the amount absorbed. This is expressed by the equation of transfer,

which is written as follows

9(I,/9r,) = I- S, (3.2)


p = cos0

7T = optical depth, and

Sy = source function = total emissivity/total opacity (ergs/cm2/ sec/ Hz/ steradian)

The source function can be written as (Mihalas, 1978b)

S, = {[(1 p) + e ]B + [p + (1 e)/]J,}/(1 + ,,) (3.3)


(l-e) refers to the fraction of photons absorbed that are scattered. Hence if e =1,

all emission is thermal (pure absorption) and if e = 0, then one has pure scattering .

p is defined as the ratio of continuum scattering coefficient to continuum opacity.

If p = 0 there is no scattering in the continuum.

/i is defined as the ratio of line opacity to continuum opacity and can also be

written as 3oH(a,v), H(a,v) being the Hjerting function (see Chap. 5).

/3o is defined as the ratio of line opacity to continuum opacity for a line with a

Voigt profile (see page 30 in Chap. 3).

By is the Planck's function
Jv = f I, v(p, 7v,)dpu and is known as the mean intensity.
Or, defining

=[(1 p) + eP,]/(1 + f,) (3.4)

the transfer equation becomes

l9(Iv/9T,) = I, A,Bv (1 A,)J,. (3.5)

The above equation is called the Milne-Eddington equation (Mihalas, 1978b). A point

to note is that one assumes that line scattering is coherent and local thermal equilibrium

holds. These are not accurate approximations.

The Milne-Eddington Model

Consider the Milne-Eddington equation under the assumptions that A,, e, and p

are all constant with depth and that the Planck function By is a linear function of the

continuum optical depth 7 (Mihalas, 1978c); i.e.,

B, = a + b = a + [brT/(1 + a,)] = a + pVr. (3.6)

Under these conditions an exact solution may be obtained (Chandrasekhar, 1947), but

this solution differs only slightly from the approximate solution derived below.

Taking the zero-order moment of equation (3.5) one finds

dHf,/dr, = J, (1 ,)J A,,B, = A,(J, B,) (3.7)
where Hr(7y) = f I,(rv,, 1)dy/ and is called the Eddington flux. The first-order
moment gives

0K,/7r, = HE


where K = J I,,(r, j)l2di and is called the radiation pressure. Using the
Eddington approximation K, = J, and substituting equation (3.8) for H, into equation

(3.7), one obtains (Mihalas, 1978a)
1 I1
1(02Jl/Or,) = A,(J, B,) = [2(J, B,)/Or2]. (3.9)
3 3

The solution to the above equation is (Mihalas, 1978c)

J, = a + pvrv + (p, v3 a)exp[- /3-A, r,,]/[ /3+ \-/3]. (3.10)

Assuming that on a mean optical depth scale (Mihalas, 1978d),

B,(r) = B,(To) + (aB1,/T)or = Bo + B17 (3.11)

and using the Eddington-approximation result for the grey temperature distribution,


T4 = T4(1 + 3/27) (3.12)

one can show that (Mihalas, 1978d)

B1 = -XoBo, (3.13)

Xo = uo/(1 e-u) (3.14)


uo = (hv/kTo). (3.15)

Thus the parameters in equation (3.6) are
a = Bo
3 (3.16)
b = -XoBo(R/K),


R = mean opacity, and K = monochromatic continuum opacity


p, = XoBo(R//)/(1 + 3,). (3.17)

Center-to-Limb Variation

The specific intensity at x=0 emergent at frequency v, at an angle 0 = cos-1 ,

is given by (Mihalas, 1978e)
I, (0,) = S,(rv)e-(r/J),l-ldr,
0 (3.18)

= [B, + (1 A,)(J, B)exp(-Tv/p)p-dT
where the form of S, has been taken from equation (3.5). Substituting from equation

(3.10) one gets

I,(O, ~) = (a + p,,p) + {[(p, v3a)(1 A,)]/[V3(1 + A,)(l + 3A )]}. (3.19)

In the continuum, /3, = 0, hence

Ic(O, p) = a + by + (- /3a)p](3.20)
[v3-(1l + V( ~p)()1 + V3(1- p)p)]
If there is no continuum scattering i.e., p = 0 then

Ic(0, ) = a+ bt (3.21)

The residual intensity is given by

r,(P) = I(O, l)/Ic(O, t) (3.22)

and the absorption depth

a() = 1 r() (3.23)

For a pure absorption line, e = 1, Av = 1 and the residual intensity is given by

r,(p) = [1 + (b/a)p/(1 + /l)]/[1 + (b/a)p] (3.24)

Line Broadening Mechanisms

Natural Line Broadening

If a photon passes very close to a bound electron, the electron will oscillate in

response to the photon's electric field. Classical broadening occurs because the bound

electron can oscillate in frequencies not exactly at resonance. Thus a radiating atom

emits a damped wave that gradually dies out, or it emits a wave of fixed frequency v

for a short time interval. If either kind of wave is Fourier analyzed, it is found that it

is not monochromatic but consists of a narrow range of frequencies (Aller, 1963).

Quantum mechanically, although each emitted quantum has a precise energy E and

frequency v, the atomic energy levels themselves are actually fuzzy. This is due to

Heisenberg's principle. According to that principle, AEAt ~ h where AE is the

energy width of the level and At is the lifetime of the energy level. The lifetime of

a ground level is very long, hence AE is very small and the level is quite sharp. For

an excited level, At is small, hence AE will be relatively large. Since the transition

can take place from any part of the broadened level, the observed spectral line will be

slightly widened (Aller, 1963).


Let the line absorption coefficient be denoted by a. Then from classical electro-

dynamics one gets

a = (e2/mc)[(7/4r)/{(Av2) + (7/4r)2}] (3.25)

The above equation is called the dispersion profile. 7 is called the damping constant.

From classical dipole emission (Carpenter et al., 1984),

Classical = 0.22 x 1016/A2 (se-1) ; A in A (3.26)

In the above equation, 7c is called the classical damping factor because it fixes the rate at

which oscillations are damped as a result of the radiation process. The classical damping

constant is usually smaller by an order of magnitude than inferred from observations

and a quantum mechanical interpretation must be made.

If the transitions take place between two excited levels (u and 1), the transitions

from those levels will result in the broadening of the line and the value of 7 will have

the form (Collins, 1989a)

r = Ai + Anj (3.27)

Al = transition probability from level 1 (lower level) to level i.

Aui = transition probability from level u (upper level) to level j.

One can also have broadening by electron collisions (ye). Then 7 = 7r + y, although

7e is always much smaller than -,.

Collisional Broadening

Interaction of neighboring particles with the absorbing atoms also can result in the

broadening of spectral lines. Usually the particles are charged and their potential will

interact with that of the atomic nucleus and thus perturb the energy levels of the atom

(Collins, 1989b). The net result of all these perturbations is to broaden the spectral line.

As in the case of radiation damping, one gets an absorption coefficient of the form

a, = (e2/r)c{ [(yc/4r)]/[(Av)2 + (7c/41r)2]} (3.28)

Here -c is the collisional damping constant.

If one takes into account simultaneous radiation and collisional damping, then

the form of the absorption coefficient remains the same, except one has to introduce

7 = 7 + 7r + Te. Thus,

a, = (e2/mc){[("/4Tr)]/[(Av)2 + (7/47r)2]} (3.29)

Here a, is the absorption coefficient per oscillator.

So far the absorption coefficient has been given for an oscillator. To find the

absorption coefficient per atom one has to find an expression that relates the number

of oscillators Fi to the number of atoms or ions n in the state from which the transition

begins. One introduces a quantity called the oscillator strength f (ii=fn). Thus the

oscillator strength gives the number of classical oscillators that reproduce the absorbing

action of one atom in the given line. Therefore the absorption cross section per atom

is given by

a. = (e2f/mc){[(y/47r)]/[(Av)2 + ('/47r)2]}



Radiation damping is of primary importance for strong lines in low density media.

However in most cases collisional damping is significant.

Thermal Broadening

Another process that leads to the broadening of lines is the Doppler effect, brought

about by the motion of the absorbing and emitting atoms. Even if damping by radiation

or collision were absent, the absorption line would still be broadened due to the velocities

of atoms or ions taking part in the absorption process (Ambartsumyan, 1958). The

absorption cross section per atom due to the thermal process can be written as

a, = (v7e2 f/mcAvD)[exp{-(Av)2/(AvD)2}] (3.31)


AVD = Doppler width = (v/c) 2kT/m

m = mass of the atom considered

k = Boltzmann's constant

T = temperature

If both damping and thermal broadening are significant, a spectral line will have a

core that is dominated by Doppler broadening while its wings are dominated by the

damping profile.


Introduction. Motions of photospheric gas on a scale large compared to atomic

dimensions but small compared to the size of the star are called turbulence. Turbulence

alters the line profiles in a stellar spectrum through Doppler displacements. If one

assumes that these turbulent velocities have a velocity distribution similar to the


distribution of thermal velocities (i.e., Gaussian), the resulting velocity distribution is

again given by a Gaussian with a reference velocity

S= thermal + turbulence (3.32)


AD = (I/c) (2kT/m) + ence (3.33)


a, = ( e2 f/mcA/vD) {exp [(Av)2/(AD)2]} (3.34)

Turbulence is characterized by the Reynold's number (Gray, 1988) and is defined

by the product of density, average flow velocity, and linear dimensions across the tube

containing the flow, divided by the viscosity. If this number exceeds 1000, turbulence is

expected. It is difficult to estimate the Reynold's number for a stellar atmosphere. Gray

(1988) made some numerical calculations and came up with a number around 3 x 1013.

Such numbers indicate that a stellar atmosphere is certainly turbulent. Beckers describes

turbulence as a word used "to describe motions which cause line broadening and

changes in line saturation (curve of growth effects) but which are otherwise intangible

as a specific kind of motion because of the absence of additional information such as

spatial resolution, characteristic line profiles, line shifts, etc." (1980, p. 85). According

to Beckers (1980), turbulence in astrophysics may have nothing or little to do with

hydrodynamic turbulence. In astrophysics, turbulence can include convection, waves,

stellar winds, large-scale flow patterns and even stellar rotation if the spectral resolution

is insufficient to identify the characteristics of any of these non-thermal motions. The


observed velocity fields are better understood from the point of view of macroturbulence

and microturbulence, which are the extremes of the cell-size spectrum.

Microturbulence. If the resolution of spectra is low, as in the early days, then one is

limited to measuring equivalent width. One needs high resolution spectra to study line

profiles. Equivalent width (W) is a measure of the total line strength and is defined as
0 (Fp-F_,)
W = J F where Fc and Fv denote the continuum fluxes (before any instrumental
distortion). W is the width of a perfectly black, rectangular absorption line having the

same integrated strength as the real line. Early work found that curves of growth did not

saturate as expected from theory. Struve and Elvey (1934) and others simply enhanced

the thermal broadening beyond that corresponding to the known temperature, and this

developed into the concept of microturbulence.

These velocities are much smaller than rotational velocities or turbulence dispersion.

Strong lines are desaturated due to microturbulence such that the equivalent width

increases. Weak lines reflect the Gaussian shape of the thermal and microturbulent

velocity distributions (Gray, 1988). For weak lines, the opacity is small and hence a

photon's mean free path is almost the same as the thickness of the line-forming region

in the photosphere. Each absorbed photon displays the Doppler shift for the turbulent

velocity of the material where it is absorbed (Gray, 1976). Microturbulence delays

saturation by Doppler shifting the absorption over a wider spectral band (i.e. reducing

,, at each v). In order to minimize thermal broadening, one should select lines from

the heaviest possible elements. In practice, however, one has to select unblended lines,

which is a more important criterion. Elements showing hyperfine structure also should

be avoided. If there are other factors affecting the saturation portion then it becomes


difficult to measure the microturbulent velocity (C). Some of these are listed by Gray


i. Non-LTE and T(r) effects

ii. Systematic f-value errors

iii. Errors in damping constants: quite frequently the damping constant is stronger than

expected according to classical theory.

iv. Zeeman splitting: magnetic broadening of the line absorption coefficient will delay


v. Hyperfine structure

Recent investigations by Gray (1988) show that C increases with luminosity, but

within any given luminosity class there is no strong change in C with effective temper-

ature. For main sequence stars, a value of 1 km/sec is supported by some solar studies.

However, others find higher values for of about 3 km/sec (I. Furenlid, 1976). For

dwarfs, a slight increase from 1.0 km/sec to 1.5 km/sec may occur with increasing

effective temperature. A much larger temperature dependence was found using the

older results reviewed by Gehren (1980).

Macroturbulence. If a cell of gas is significantly larger than the length corresponding

to unit optical depth, it produces a specific intensity spectrum of its own, and that

spectrum will be Doppler shifted according to the velocity of the cell. By assumption

there is no differential motion within or across this cell, hence no change in the line

absorption coefficient. This is the concept of macroturbulence (Gray, 1988).

In a review paper, Huang and Struve (1960) describe macroturbulent velocity as a

geometrical effect rather than a physical effect. Thus macroturbulence has no effect on


the equivalent width. In general, one velocity is used to describe both types although it

is very difficult to single out one characteristic velocity. The only way macroturbulence

can be detected is from profiles. However, even at relatively high resolution, rotational

broadening and macroturbulent broadening may not be separable.


Due to rotation, one half of the star moves away from us, while the other half

moves toward us. The component of motion in the direction of the line of sight is

v, = vr(O)sinf, where vr(O) is the rotational velocity for the colatitude 0. For rigid

body rotation vr(O) = vesinO and the velocity component vy along the line of sight is

Vy = vesindsinO, where ve is the equatorial rotational velocity and 4 is the longitude.

vy is constant for all points on the sphere for which sinqsinO is constant.

Combining Radiation, Collisions, and Thermal Motion

All the above broadening processes take place simultaneously. If one considers any

two of the absorption coefficients and imagine the first one to be synthesized by a series

of delta functions, then each of the delta functions experiences the broadening of the

second absorption coefficient. Thus the two distributions must be convolved together

to obtain a combined result. If one expands this to include all the processes, one gets

a multiple convolution,

total = natural acollisional thermal (3.35)

The first two coefficients have a dispersion profile. If two dispersion profiles are

convolved one gets a new dispersion profile with a damping constant, 7 = natural +

7collisional- To obtain the total absorption coefficient, one combines radiation, collisions

and thermal motion. Therefore,

total = 7re f/mc{(7/4r2)/[(AL )2 + (7/47)2]}
*{[1/v'lAv]exp [(Auv)/(AvD)2}.

The convolution of the dispersion and Gaussian profile is called the Voigt function.

V(Av, AvD,7) {(7/442)/[(Av)2 + (-/47r)2]}
*{(1/V'AvD)exp [(Av)2/(AD)2]} .

The Voigt function is normalized to unit area.

V(Av, Av,,7) = f {(-/4i2)/[(A A )2 + (-/4)]} (3.38)
x{(l/IVrAvD)exp [(Avi/AVD)2]}dAul.


v = Av/AVD and a = (7/47r)/AvD. (3.39)

The quantity a is different from the one described on page 19. Therefore,
V(a,v) = (1// AvD)(a/7r) (e-2)/[(v y)2 + a2]dy (3.40)

y = (v vo)/AvD. (3.41)

The Hjerting function H(a, v) (Hjerting, 1938) is closely related to V(a, v) as shown


V(a, v) = (1/vr AvD)H(a, v) .



ay = (v/'e2/mc)(f /AD)H(a, v) (3.43)


ao = ( Ve2/mc)(f/AVD) (3.44)

Thus, (Gray, 1976)

av = aoH(a, v) (3.45)

Instrumental Broadening

Assuming that the spectrometer is receiving strictly monochromatic radiation, the

profile recorded by scanning the spectrum will have a finite width due to the effects of

diffraction, slit width, and other causes such as aberrations. It is considered adequate

to adopt a Gaussian profile for instrumental broadening.

If the energy distribution of the source as a function of A is a triangular function

6(A Ao) centered on the wavelength Ao, the recorded spectrum will be a curve

representing a function O(A' A). Then by definition O(A') is the instrumental profile

of the spectrometer. The narrower the instrumental profile, the less the recorded curve

R(A') departs from the true spectral energy distribution R(A) of the source. Therefore

the recorded function is the convolution of the source function and the instrumental

profile (Bousquet, 1971),

R(A') = / R(A)(A' A)dA. (3.46)



The observations were made at the Kitt Peak National Observatory by Dr. Geraldine

J. Peters from 1989 to 1991. The Coude Feed telescope with Camera 5 and the TI3

CCD detector was used to observe Algol systems in the spectral range AA4400-4550.

In order to study depth effects in the rotation one needs lines formed under a range of

excitation energy. Two such lines are He I (A4471) and Mg II (A4481).


The observed wavelengths must be corrected for the orbital revolution and axial

rotation of the Earth. Since the observations were taken at a particular phase, the

primary star's velocity around the center of mass can be computed by the light curve

program as part of the synthesized velocity. Hence the observations need only to be

corrected for the Earth's rotation and revolution. To compute the sun-earth system

corrections one needs to know the time of observation (sidereal time), the co-ordinates

of the star, the latitude and longitude of the observatory and the longitude of the Sun,

which, is defined as the angle at the Earth between the direction of the Sun and the

direction of the Vernal Equinox.

Correction for Earth's Rotation

For an observer on the Earth's equator at some point x, the eastward velocity of x


is V( x) =circumference/seconds per sidereal day. If R is the radius of the earth, then

V(x) = 21rR/86164 = 0.47 km/sec (4.1)

The velocity is reduced by a factor cos 0, where 0 is the observer's latitude. Therefore,

V(x) = 0.47 cos km/sec (4.2)

A star's radial velocity will be affected by the earth's rotation. If V is the velocity

of a star towards the west due to the eastward velocity of the observer at latitude q, then

the component of V which is in the direction of the star is given by (Birney, 1991),

Vd = 0.47 cos cos 6 sin 7 km/sec (4.3)


6 = declination of the star, and

r = hour angle of the star

Correction for Orbital Motion of the Earth

The observed radial velocity will also be affected by the earth's motion around

the Sun. This is corrected by calculating the earth's orbital velocity and then finding

the magnitude of the component of the velocity in the direction of the star. First an

expression is found in terms of the star's ecliptic coordinates, celestial latitude, /, and

longitude, A. It is then transformed to an expression in terms of the star's equatorial

coordinates, declination and right ascension.

The heliocentric orbital velocity of the earth (Ve) is obtained from the FORTRAN

subroutine BARVEL (Stumpff, 1980). In terms of the star's ecliptic coordinates, the


component of the earth's orbital velocity (Vy) in the direction of the star S is given

by (Birney, 1991)

V = -Ve cos 3sin (O A) (4.4)


f and A are the ecliptic coordinates of a star at S, and

o = celestial longitude of the Sun = angle at the earth between the direction of the

Sun and the direction of the Vernal Equinox.

In this equation the negative sign has been included to maintain the convention that

the radial velocity is positive when the distance between the source and the observer

is increasing.

Corrections in Right Ascension and Declination

One now has to convert the star's ecliptic coordinates to equatorial coordinates.

The last term in eq. (4.4) can be written as

sin(O A) = sin O cos A cos 0 sin A (4.5)

After multiplying both sides by cos3 and rearranging one gets

cos 0 sin(O A) = cos 3 cos A sin 0 cos f sin A cos 0 (4.6)

Vy = Ve(cos f cos A sin O cos 3 sin A cos 0) (4.7)

Thus one has to find expressions for (cosf/cosA) and (cos/sinA).

The components of any vector in the ecliptic system (E) can be transformed to

the equator system (Q) by pre-multiplying with Ri(c), which is a matrix that rotates a

coordinate system by the angle e on the x-axis. Therefore,

E (A, [90 3]) = Ri(e)Q9(a, [90 6])


The present value of e is approximately 23027'

(sin(90 ) cos A 1
sin(90 -/3) sin = 0
cos(90-/ ) 0 -

(cos cos A
cos sin A
sin /3 I

- sin

0 sin(90 ) cos a
sin e sin(90 6) sin a
cos e cos(90 6)

cos 6 cos a
cos e cos 6 sin a + sin e sin 6
cos e sin 6 sin e cos 6 sin a



The inverse relation is as follows

iQ(a, [90 6]) = R1(-e)iE(A, [90 /3)


cos cos a 1
cos 6 sin 0
sin 6 0

cos C
sin e

- sin e
cos n/ (

From eq. (4.10)

cos 3 sin A = cos e cos 6 sin a + sin e sin 6



cos 0 cos A = cos 6 cos a

cos 6 cos a
cos 6 sin a
sin 6




Substituting eq. (4.13) and (4.14) in eq. (4.7) one gets, (Birney, 1991)

Vy = -Ve {cos a cos 6 sin O cos 0(sin 6 sin e + cos 6 cos ea)}
= V {cos 6 cos a sin cos G sin 6 sin e cos O cos 6 cos e sin a}

If VM is the measured radial velocity before any corrections have been applied, and

if VR is the radial velocity with respect to the sun, then VR=VM+Vd+Vy. Once this

velocity is known the shift in the wavelength can be calculated.

Observed Algol-Type Systems

Table 1 lists the binaries whose line profiles have been analyzed here. In this table

the term "noisy lines" refers to those for which the spectral distributions have large

scatter, whether due to observational noise or to large numbers of spectral lines. Any

blending is usually with lines of the same star because Algol-type secondary stars are

much less luminous than the primaries. The term "clean lines" refers to those which

are reasonably free of scatter and blending. The resolution of the CCD as it was used

is 0.15 Angstroms per pixel.

Table 1 : Stars observed by G.J. Peters

RZ Cas 3 Per S Cnc TU Mon
TV Cas Y Psc U Cep RW Per
S Equ U Sge SW Cyg RY Per
RW Mon Z Vul RY Gem RZ Sct
DM Per



The line profile program has been combined with the light curve program (Wilson

and Devinney, 1971), which normally generates light and velocity curves, but now is

used for its computation of surface quantities within a physical binary star system. The

line profile code generates an absorption line profile, given the damping constant, the

effective number of absorbers along the line of sight, the micro-turbulent velocity, and

the rotation rate.

Light Curve Code

This code computes light curves from given parameters (Wilson and Devinney,

1971), which are listed in Table 2. It computes light and radial velocities for each

component for a given phase or a range of phases. As a part of its calculation it also

computes the radial velocity and the intensity at angle 0 relative to the outward normal

direction (limb darkening law) of the elemental surface areas. Since the line profile (LP)

code requires cos 0 (0 = aspect angle) and the local flux, both of which are calculated in

the light curve (LC) code, it is very important to have a good photometric solution. The

light curve code is used as a subroutine in the line profile code, which is described in the

next section. The light curve code includes binary effects such as tidal and rotational

distortions, reflection effect, gravity darkening, and limb darkening.

Table 2: List of parameters for the LC code

Ratio of spin angular speed to mean orbital angular speed for
F1, F2
each component

i Orbital inclination in degrees

Phase shift : This parameter permits a phase shift of the
computed light curves by an arbitrary amount. It is the phase at
which primary conjunction would occur if the longitude of
periastron (w) were 900. It is not the actual phase of primary
conjunction unless the longitude of periastron equals 90 or the
eccentricity (e) equals 0.

e Orbital eccentricity

w Longitude of periastron (orbit of star 1) in degrees

Gravity darkening exponent for each component with value unity
gi, g2 corresponding to classical (Von Zeipel) darkening (i.e. for stars
with radiative envelopes)

SBolometric albedos of the components having values of 1.0 for
radiative envelopes and 0.5 for convective envelopes.

P Period in days

TI, T2 Mean surface temperatures of the components in Kelvins

12 Modified surface potential for each star, which specifies a
surface of constant potential energy


Table 2- (Continued)






X1, X2

L1, L2


Line Profile Code

The line profile code (LPDC) generates an absorption line profile as an array of

wavelength and residual flux, given the line profile parameters listed in Table 3 along

with the light curve parameters listed in Table 2. The other input quantities are listed

in Table 4. All the parameters and input quantities are identified according to their

FORTRAN names. The parameter fitting procedure is described later in this chapter.

A flow chart of the code is shown in Figure 5.

Mass ratio (m2/ml)

Semi-major axis of the relative orbit (al + a2) in solar radii

System velocity in km/sec

'Linear cosine law' limb darkening coefficients

Monochromatic luminosities of the stars at a particular
wavelength. It is a 47r steradian light output, and hence it will be
about 47r times as large as the computed (output) light of the
model star, which is per steradian

Amount of third light present at a specified wavelength in the
same unit as the computed light values.

Read input

Write o/p
and errors

Figure 5: Flow chart of the Line Profile code

Table 3: Line Profile parameters

(FORTRAN names)









Effective temperature

No. of absorbers along the line of sight

Damping constant radiativee + collisional)

Micro-turbulent velocity (km/sec)

Ratio of angular rotation speed to orbital angular speed for
the primary star

Ratio of continuum scattering coefficient to continuum
opacity (p=O: No continuum scattering )

Fraction of the line emission due to thermal processes.
(e = 0 : Pure scattering)
(e = 1 : Pure absorption)

Table 4: Other input quantities for the Line Profile code


LAMBDA Rest Wavelength in Angstrom Units

M Atomic Weight of Element Producing the Line

INSTBR Full width at half maximum of the Instrumental Profile

PHAS1 Phase at the beginning of observations

PHAS2 Phase at the midpoint of observations

PHAS3 Phase at the end of observations

Doppler Subroutine

This subroutine is called for each surface element by the LIGHT subroutine in the

LC code. The central wavelength and the value of cos 0 are inputs to this subroutine,

which in turn calls the LINEPROF subroutine for each observed wavelength in the

line profile. The central wavelength for each surface element will be different from

the rest wavelength and will differ according to the velocity of each element. At each

wavelength, LINEPROF calculates the residual intensity. Thus the output of DOPPLER

is an array of wavelength and residual intensity for each area of the star.

Lineprof Subroutine

The input to this subroutine consists of the central wavelength, effective temperature,

damping constant, micro-turbulent velocity, the number of absorbers along the line of

sight, the atomic weight of the element producing the line, ratio of continuum scattering

coefficient to continuum opacity, the fraction of line emission due to thermal processes,

and the wavelength at which the residual intensity is to be calculated. One of the other

inputs is the limb darkening factor, which is provided by the LIGHT subroutine. The

output is the residual intensity at the wavelength specified.

The decrease in continuum strength as one approaches the limb of the apparent

stellar disk is called limb darkening. For a spherical star, the outward normal direction

corresponds to 0 = 00, or cos 0 = 1. The ratio 1(0)/I(0), which gives the intensity at

angle 0 relative to the outward normal direction, is referred to as the limb darkening law.

A rough approximation to real limb darkening is given by 1(0) = I(0)(1 x + x cos 0).

The coefficient x (provided by the light curve data file) used in the linear limb darkening

equation is obtained from tables ( e.g. Al Naimiy, 1978). These values are wavelength

and temperature dependent. Limb darkening also affects line strengths. Figure 6 shows

how limb darkening changes the line profile (pure absorption line). For a pure absorption

line as the limb is approached, the line vanishes.

Figure 7 shows how the intrinsic line profile changes as the effective temperature,

number of absorbers, damping constant and the micro-turbulent velocity are changed.

Here one is assuming a pure absorption line. Figure 8 shows the variation of the intrinsic

line profile when the parameters p and e (see Table 3) are changed.


0 0 0
O l O _

I I I _

D "0


CD C- 0

MLISN31NI "ivnaissy



o bo

-I I

/ I -
3 0

> -
.. , 0

AIiSN3iNI 1vnaIS3I ,o I

o a ao
0 0 0 00

I I Io

a 2
r fII I f f i vI

.& '




- o 7 v1 o0 -7 o 0 3



o' .E


> .


C o




Hay Subroutine

This subroutine calculates the Voigt function H(a, v). The function alone shapes

the absorption line, and accurate values of H(a, v) are required to generate line profiles.

The function is given by the following expression (Aller, 1953)

a f exp(-y2)
H(a, v) = a / exp(-- y dy
SJ a2 + (v y)2dy (5.1)


a = and

F = damping constant (natural + collisional)

AVD = Doppler width

v = frequency departure from the center of the line in units of Doppler width


Much work on this problem has been done by Finn and Mugglestone (1965) and

Hummer(1965). This function has also been calculated by Hjerting (1938) who provided

a table of H(a, v) for a up to 0.5 and v up to 5. Harris (1948) gives a table which

contains the coefficients of the Taylor expansion of H(a, v) as a power series in a For

small values of v and a (v < 2.75 and a < 0.2) one uses the Taylor series (Harris, 1948)

H(a, v) = Ho(v) + aHi(v) + a2H2(v) + a3H3(v) + a4H4(v),


Ho(v) = e

Hi(v) [1 2vF(v)]

H2(v)= (1 2v2)e-v2 (5.2)
22 2
H3(v) = --[~-(1 v2) 2v(1 -v2)F(v)
1 2 4- v2
2 3
H4(v)=( -2v2 + v 4)e-V2

and F(v) = e-v2 / edt

If v > 2.75 (for all values of a, except a = 0)

H(a,v)= J 2 +-(vy)2dy (5.3)
f a2 + (v y)y
We have truncated the limits to [-7,7]. The above integral is evaluated using Simpson's


Finn and Mugglestone (1965) have shown that H(a, v) can be written as

H(a, v) = e-2 + [-1 + 2ve-2 e'2dx]

00 (5.4)
+ [e- 1 + ax]e-'/4 cos(vx)dx
If a = 0

H(O,v) = e-


If v = 0

2a 1OO
H(a, ) = 1 -2 + (e-ax 1 + ax)e-/4dx (5.6)

Thus the integration was divided into five parts according to the values of a and

v. These five divisions are as follows:

i. a = 0. (equation 5.5)

ii. v = 0. (equation 5.6)

iii. v 2 2.75 (equation 5.3)

iv. v < 2.75 and a > 0.2 (equation 5.3)

v. v < 2.75 and a 5 0.2 (equation 5.2)

Figures 9 and 10 show the broadening function H(a, v) as a function of a and v


Light Subroutine

The LIGHT subroutine is a part of the light curve program. This subroutine calls the

DOPPLER subroutine for each surface elemental area. In turn the DOPPLER subroutine

calls the LINEPROF subroutine. Thus for each area we have a line profile. The next

step is to weigh and sum them accordingly. This is all done in the LIGHT subroutine.

As mentioned earlier, since each element has an associated velocity due to the rotation

of the star, its central wavelength will be Doppler shifted. One must account for the

possibility that the star might rotate faster than synchronously. This is where the FI

parameter (in the case of the primary star) comes into effect. The velocity of each

element, with respect to the center of the star, is multiplied by F1.

a a a, a a a a a a





D o



Q ucaca

Cr j Cl)It Lo



O 4








Figure 11 shows intrinsic line profile shifts due to rotation for two cases, one for

asynchronous rotation and one for synchronous rotation (F1 = 1). Figure 12 shows

the line profile for the entire star, again one for the synchronous case and the other

for the non synchronous case. In all these cases instrumental broadening has not been

taken into account. The convolution of the line profile with the instrumental profile is

described in the next subsection.

z -
U = 4483.804 A (Ti II)
.2 F, = 4.0

AA (A)

Figure 11: Intrinsic line profiles for synchronous (below) and non synchronous
cases. The line profiles are choppy because they were calculated using a
very small grid to illustrate the effect of non synchronous rotation.

1- -


.i .6

.4 = 4483.804 A (Ti II)

I --- F = 1.0

.2 2 --- F = 4.0

-4 -3 -2 -1 0 1 2 3 4

Figure 12: Line profile for the entire star for synchronous and non synchronous cases.

Instr Subroutine

The instrumental profile is represented in the program by a Gaussian. It is given

in terms of full width at half maximum (FWHM). The instrumental profile depends on

the instrument. The resolution for a 2-pixel line width for the CCD setting used was

0.30 A. This means that an infinitely sharp line that strikes just one pixel will appear

0.15 A wide. Thus the FWHM in our case would be 0.15 A.

Figure 13 shows instrumental profiles of various widths. The idealized theoretical

line profile is to be convolved with the instrumental profile. If the line profile before

convolution is broad, then the resultant line profile after convolution is not significantly

different, so convolution with a broad line has little effect on the overall profile.


The INSTR subroutine convolves the idealized line profile with the instrumental

profile. The line profile is an array of residual flux as a function of AA. However this

array is not tabulated finely enough. Thus it is necessary to fit a smoothing function

to the line profile over some range of wavelength. The smoothing function used was

a polynomial of degree three. The coefficients of the polynomial are obtained by least


z 6 ---- FWHU 0.30 A
3a -- FM 0.105

0 L 3---w o.,o -

-.6 -.4 -.2 O .2 .4 .

Figure 13: Instrumental profiles of various widths

Phase Smearing

The last stage of profile generation is to account for phase smearing. This is done

by averaging computed profiles from the beginning and end of the observation interval.

However if the observing interval is very small, it suffices to calculate the line profile

at the phase corresponding to the middle of the interval. In all of our cases the interval

was very small and hence the profile was calculated at the phase corresponding to the

middle of the interval. The program can calculate the line profile either way.


Parameter Fitting

Two methods have been used to fit the theoretical line profile to the observed one.

The first one is the method of Differential Corrections (DC) and the second is the

Simplex method. Each method has its advantages and disadvantages. An advantage of

DC over Simplex is that it provides error estimates. The best procedure is to fit the

observed line profile by the Simplex method, and then use the DC method as the last

step to get error estimates.


The Simplex method was introduced by Nedler and Mead and is discussed by

Caceci and Cacheris (1984). It requires only function evaluations and no derivatives.

One of the advantages of the Simplex method is that divergence is impossible, while a

disadvantage is that one cannot get standard error estimates. This method also is slow.

A simplex is "a geometric figure that has one more vertex than the space in which

it is defined has dimensions" (Caceci and Cacheris, 1984). For example, a simplex on

a plane is a triangle. The basic idea in the Simplex method is to build a simplex in the

N+1 dimensional space described by the parameters one wants to fit. Thus if there are

two parameters 'a' and 'b' then one can consider them as axes in a plane on which one

creates a simplex ( a triangle). Each vertex is represented by three values: 'a', 'b', and

'R' where 'R' is the weighted sum of squares of the residuals.

To reach the lowest value of 'R', the simplex is moved "downhill", accelerating

and slowing down as needed. At a given stage, the program finds which vertex has

the largest R. It rejects that vertex and substitutes another. The program computes the

new vertex according to one of these mechanisms: reflection, expansion, contraction,


and shrinkage. Figure 14 illustrates these four mechanisms while Figure 15 shows an

example of a simplex moving on 'R's contour plot for a two parameter fit.

Although the Simplex method never diverges, some problems did arise. If the initial

guess is very bad or if the increments are very small there will be a failure to converge

and the program can end prematurely. For some cases the method was restarted at a

point where the program claimed to have found a minimum. In some cases where there

was suspicion that there was more than one solution the program was run again with

different starting guesses.

Figure 14: Two dimensional simplex illustrating the four
mechanisms of movement. (Caceci and Cacheris, 1984)



C B= Vertex
W w = worst vertex
R = reflected vertex
E = expanded vertex
C = contracted vertex
S = shrunken vertices

Figure 15: An example of the simplex moving on the
response surface's contour plot (Caceci and Cacheris, 1984)

Differential Corrections

The equation of condition for the least squares method is

Of Of af
O C = Api+ Ap2+ Apa +..... (5.7)
Opi 9p2 9p3

where, for the line profile problem

O C = observed computed flux

f = residual flux

p = a parameter

= partial derivatives

Ap = parameter corrections

Thus for each observation one has one such equation. The normal equations can be

written in matrix form as




X = nx 1 matrix of computed corrections to our estimates of the parameters

n = number of parameters

d = mx 1 matrix of residuals based on the previous estimate of the parameters

m = number of observations

A = mxn matrix of partial derivatives

W = mxm matrix whose diagonal elements consists of the weight assigned to each


Using matrix methods these equations are solved for the correction terms which will

then be added to the initial guesses for the parameters. The partial derivatives are

obtained numerically from the following equation

9f f(p + dpl,P2, -)- f(pi,P2,P", (5
8pl dpi

where dp's are the parameter increments.

The weighted least squares iterative differential correction solution is given by the

following equation

X = (ATWA) -ATWd (5.10)

The covariance matrix (ATWA)-' provides a means of estimating the errors of the

solution vector X. One assumes that the errors in X depend only on d, and that the

errors in d are independent and follow the normal frequency distribution. If c is the

error in X caused by the jth component of d then

E2 = 2(AT.W.A) (5.11)


where a is the mean error of unit weight. Thus,

wr2 = rT (5.12)

where r = residuals. To get probable errors, Ej is multiplied by 0.6745.


General Discussion

Seventeen systems have been analyzed and the rotation rates of the primary stars of

these systems have been determined by the Differential Correction method (DC) and the

Simplex method. The basic data on these systems are listed in Table 5. The photometric

and spectroscopic elements were obtained from the literature. Each system will be

discussed individually in the next section. Table 6 lists the spectroscopic elements for all

systems while Table 7 lists photometric elements. The basic data and the spectroscopic

data were obtained from the Eighth Catalogue of the Orbital Elements of Spectroscopic

Binary Systems (Batten, Fletcher and MacCarthy, 1989). In Table 6 some of the systems

have no K2 values, which means that a2 is not measured spectroscopically. In these

cases a is obtained from al and the mass ratio (m2/ml) listed in Table 7. Table 8 lists the

line profile parameters along with their errors from the Differential Correction method

and Table 9 lists the parameters obtained from the Simplex method.

Table 10 lists the equatorial velocities obtained from this work and other work. Fi

values obtained by analysis of the Rossiter effect are averages of those from Twigg

(1979) and from Wilson and Twigg (1980). The values in the fifth column, Fi(sp),

were taken from Van Hamme and Wilson (1990), who listed values of F1 obtained from

previous line profile work. The seventh column lists the limiting value of F, which

tells how fast a binary star component would rotate if it were centrifugally limited and

had the same size (as it does with its present rotation). The procedure to calculate


For is explained by Van Hamme and Wilson (1990). The equatorial velocities listed in

column eight were calculated from the values of semi-major axis, side radii (rl(side),

and the period.

Figures 17 to 33 show the theoretical fit to the observed data. Figures 34 to 50 (see

Appendix A) are computer generated pictures of these systems at various phases along

with their light curves and photometric elements, while Figures 51 to 67 (see Appendix

B) are plots of the observations.

It is important to choose suitable lines for analysis. A number of systems were not

analyzed either because the lines were very asymmetrical or seem blended. If the higher

excitation lines such as He I and He II are compared with H lines then it is seen that in

some cases the He lines are broader than the H lines. Hansen and McNamara (1959) for

RZ Sct and Hiltner (1946) for RY Per suggested two reasons for this. One possibility

is that the H lines arise from a more slowly rotating disk or envelope structure around

the primary, while the He lines arise in the true stellar photosphere. Another is that the

normal H lines are severely modified by absorption and/or emission.

Some of the lines seem to be blended. Near the bottom of primary eclipse the

light from the secondary star may equal or even exceed that of the primary star. In

this case some of the observed line profiles may be blended. This could hamper the

determination of the line center. However, for the case of Algol type binaries, light

curve solutions show that for most of the partial phases the light from the primary is

much greater than that of the secondary. Hence there should not be significant blending

of the spectral features of the two stars, except for a few weak lines (Twigg, 1979).

As mentioned earlier, two methods were used to fit the data. The Simplex method


usually required a large amount of computer time but there was no convergence problem.

The Differential Correction method found the minimum quickly, but, in some cases,

ran into a convergence problem. This is possible if the computation of derivatives

and residuals lack precision. It can be overcome by using a fine enough computing

grid, which usually increases the computing time. Another reason could be that

the derivatives and residuals contain systematic errors, which vary from iteration to

iteration (Wilson, 1988). Numerical derivatives will contain systematic errors if they

are computed asymmetrically. The Differential Correction method has a provision to

compute symmetrical derivatives. However this will double the computing time.

The parameters adjusted in both the DC and Simplex methods are the damping

constant (F), the number of absorbers along the line of sight (Nf), turbulent velocity

(Vturb), and the rotation rate of the primary star (F1). The value of the effective

temperature was taken from the photometric solution and was not adjusted. Two other

parameters (e and p) also were kept fixed. The value of p was always 0, which means

no continuum scattering. Continuum scattering becomes important for O-type stars.

Since all the stars in this sample are late B-type or later, continuum scattering is not

important. The other parameter, e, was kept fixed at 1, which means a pure absorption

line. e equal to 0 means a pure scattering line. Realistically e should be between 0

and 1. However it is difficult to obtain a theoretical value for e. It also is not well

determined by fitting line profiles. Solutions were made by fixing values of c at 0.3 and

0.7, but, the fits were not good, so e was kept fixed at 1.


Table 5: Basic system data


S Cnc

RZ Cas

TV Cas

U Cep

SW Cyg

S Equ

RY Gem

RW Mon

TU Mon

DM Per

RW Per

RY Per

3 Per


(in days)














(Component 1)
B9.5 V

A3 V

B9 V

B7 V


B8 V


B9 V

B2 V

B5 V


B5 V

B8 V


(Component 2)









Table 5- (Continued)


(in days) (Component 1)
3.7659 A3 V

3.3806 B7.5 V

15.190 B2 II

2.4549 B4


(Component 2)





Y Psc

U Sge

RZ Set

Z Vul


Table 6: Spectroscopic data

K2 km sec-1




S Cnc

RZ Cas

TV Cas

U Cep

SW Cyg

S Equ

RY Gem

RW Mon

TU Mon

DM Per

RW Per

RY Per

3 Per

a /(Rsun)














km sec'1














f(m) /Mun

V7 km sec-1












4.3 x 10-2

1.3 x 10-1

3.3 x 10-2

4.4 x 10-3

8.2 x 10-2

8.7 x 10-3

1.3 x 10-2






Table 6- (Continued)

a /(Rsun) K1 km sec' K2 km sec- Vy km sec-1

13.80 37 6


Y Psc

U Sge

RZ Sct

Z Vul












f(m) /Msun

1.9 x 10-2

7.7 x 10-2


o0 M ON
0 0

\o \o
00 c0

af o

o '0
0s '
0s 0

n -

C/)S o


00 en0 0 01
o d d d 6 6 d d d

0 0 N

-4 O O 1 0 '-4N

1-4 0) 00 0 W
0 0 00 00 0o
D It en o C)o c
00 cq- 00 o cq co

0- uO N No 0- 40
oC 0 0. 0. 0 0. 0

% 0 r- 00

O 0 c
s o 8 % ^ S

I O I 6 'O N~ O
-4 o

0o ;- d

;3 oo ~J)
u, d 0
H ~4 N


Table 8: Line


profile parameters (Differential Corrections method) with their probable errors


S Cnc

RZ Cas

TV Cas

U Cep

SW Cyg

S Equ

RY Gem

RW Mon

TU Mon

DM Per

RW Per

RY Per

3 Per

Y Psc


13.15 + 0.23

1.38 + 0.02

0.89 0.01

6.59 + 0.07

8.06 + 0.11

1.17 0.04

8.80 + 0.10

1.30 + 0.03

4.87 0.05

2.64 0.04

16.21 + 0.24

9.63 + 0.16

1.14 + 0.01

1.01 + 0.04


1.23 0.26

11.0 + 0.26

0.43 0.10

2.62 0.29

1.83 0.14

1.01 0.43

5.12 0.23

2.11 0.23

3.12 0.12

2.74 0.13

2.55 0.43

6.02 0.82

6.18 + 0.49

2.80 + 0.42


2.66 0.58

0.67 0.03

0.41 0.13

4.33 0.43

6.89 0.52

0.16 + 0.05

3.48 0.21

3.31 0.34

9.10 0.28

5.60 0.18

2.18 0.35

3.04 0.47

0.19 0.01

1.05 0.17

Vturb.(km sec-1)

2.66 0.20

2.98 0.01

8.84 0.11

3.36 0.13

2.75 0.15

8.33 0.47

2.51 0.22

1.26 0.08

2.77 0.08

2.72 0.42

6.27 0.36

2.32 0.30

6.86 0.05

7.14 0.26

Table 8- (Continued)


1.34 0.05

4.49 0.31

1.41 0.01


27.3 1.46

2.97 0.92

3.18 0.04


0.05 0.01

3.55 0.99

3.12 0.07

Vmrb.(km sec-1)

4.83 0.02

2.20 + 0.17

4.48 0.15

U Sge

RZ Sct

Z Vul


Table 9: Line profile parameters (Simplex method)

STAR F1 Ir [108] Nf [104] Vturb(km sec-1)

S Cnc 13.01 1.23 2.68 2.65

RZ Cas 1.38 11.38 0.66 2.93

TV Cas 0.88 0.42 0.40 8.60

U Cep 6.62 2.68 4.34 3.38

SW Cyg -

S Equ 1.15 1.12 0.15 8.41

RY Gem 8.99 5.14 3.48 2.53

RW Mon 1.29 2.17 3.25 1.24

TU Mon 4.89 3.12 9.10 2.76

DM Per 2.64 2.75 5.56 2.73

RW Per 16.11 2.53 1.99 6.58

RY Per 9.60 6.03 3.03 2.33

p Per 1.13 6.18 0.19 6.86

Table 9- (Continued)

F [108]




Nf [104]




Vturb(km sec-1)




Y Psc

U Sge

RZ Sct

Z Vul






Discussion on Individual Systems

S Cnc

The photometric and spectroscopic elements were taken from Van Hamme (private

communication). The primary star's rotation is asynchronous. However, there are no

signs of circumstellar matter in the form of emission lines, although the line profiles

are asymmetrical. Etzel and Olson's (1985) photometric and spectroscopic results

show that there might be an extended atmosphere surrounding the Roche lobe filling

component. Usually it is the primary of an Algol type binary which shows anomalous

photometric and spectroscopic behavior. Etzel and Olson (1985) mention that S Cnc

represents an unusual case among longer period Algol type systems, in that there is no

apparent disk about the hotter component despite the fact that it rotates much faster than

synchronously. Popper and Tomkin's (1984) spectroscopic observations and Etzel and

Olson's (1985) photometric and spectroscopic observations give a very small mass for

the Roche lobe filling component (0.18 solar masses). According to Popper and Tomkin

(1984) the Roche lobe-filling subgiant of S Cnc has the smallest known stellar mass

established directly from spectrographic observations. Absence of an accretion disk and

the small mass for the secondary may be indicative of the end of normal mass transfer.

Van Hamme (private communication) found an F1 of 13.0 from his photometric

analysis. This work gives a value of 13.15 from Differential Corrections and 13.01

from the Simplex method. The critical value for F1 is 28.5. Since the small mass of

the secondary is indicative of the end of normal mass transfer, the system is at a stage

in which the primary's rotation, which probably was very fast during the normal mass

transfer stage, has slowed down due to tidal braking.

RZ Cas

The photometric elements by Chambliss (1976) were used for the analysis. The

primary's temperature was obtained from the spectral type (A2 V). The light curves show

no gas streaming effects. However a period study by Hall (1976) shows sudden period

decreases. Other studies also have shown an overall period decrease. Conservative

mass transfer from the less massive component in a binary results in a period increase.

Since the period of RZ Cas is decreasing one can infer that the mass transfer is non-

conservative: some angular momentum must either be lost from the system or stored

temporarily as rotation. According to the Biermann and Hall (1973) period change

model a dynamical instability causes a sudden outflow of mass from the cooler star.

Sudden outflow of mass results in period decrease because angular momentum is taken

from the orbit and stored temporarily as rotation in or around the hotter star. This mass

transfer spins up the primary which means that F1 must be a little greater than 1.

The spectroscopic elements were taken from Duerbeck and Hanel (1979). They

obtained orbital solutions from a least squares fit of the hydrogen lines and metallic

lines. However the solutions differed in the 7-velocities and in the shape of the curve.

The adopted solution was one from the metallic lines. The deviations of the hydrogen

lines from the adopted curve show that outside eclipses the disturbances are strongest

around phases 0.2-0.3 and 0.8-0.9. Duerbeck and Hanel (1979) suggest that this result

is similar to the one described by Plavec (1967) where he mentions the presence of red

displaced absorption lines around phase 0.3 and 0.8 in Algol systems S Equ, U Cep, U

Sge and RW Tau, and ascribes them to the absorptional effects of circumstellar matter.

Previous line profile work gives a value of 1.16 for F1. Twigg (1979) finds a value


of 1.6 through an analysis of the Rossiter effect. Line profile analysis gives 1.38 from

both the DC and the Simplex methods. Probably the secondary is in the slow phase of

mass transfer and sudden bursts of mass exchange spin up the primary, with subsequent

tidal drag.

TV Cas

The photometric elements are from Wlodarczyk (1983). The primary temperature

was obtained from the spectral type (B9 V). The spectroscopic elements were taken

from the DAO catalog (Batten, Fletcher and MacCarthy, 1989). Period studies show

some interesting results. Tremko and Bakos (1976) show that the period is decreasing.

TV Cas is presumably semi-detached with its secondary filling its Roche lobe. Hence

there should be some mass transfer from the secondary to the primary. Because of this

mass transfer the orbital period should increase. If the period is decreasing, as in this

case, mass transfer could perhaps be taking place in the opposite direction. Tremko

and Bakos (1976) state that such mass transfer could be caused by a gentle stellar wind

from the primary component. This seems to be a very unlikely explanation. Another

explanation (Frieboes-Conde and Herczeg, 1973) could be the light-time effect of a

third body.

Previous line profile work gives a value of F1 as 0.73. (Van Hamme and Wilson,

1990). This work gives 0.88 for F1. Here we have a close binary system where one of

the stars seems to be rotating slower than synchronously, which is difficult to believe.

Van Hamme and Wilson (1990) suggested that one explanation could be the mixing

of a slowly rotating inner envelope with the observable surface layers. Twigg (1979)

found a value of 2.1 for F1 from the Rossiter effect.

U Cep

The photometric elements are from Terrell (private communication). The primary

temperature was taken from the spectral type. The spectroscopic solution is from Tomkin

(1981). The primary velocity curve has large scatter. This is because all the lines of

the primary have irregular, asymmetric and broad profiles. The primary line profiles

are distorted by the gas stream from the secondary to the primary.

Light curves of U Cep show many disturbances. Olson (1978) noticed large, rapid

changes in the light curves outside eclipse. The ultraviolet light curve showed a broad,

deep dip near orbital phase 0.6. Olson attributed this to a cool spot on the primary

star about 2500 K cooler than the normal photosphere. Compared with other classical

Algol-type eclipsing binaries, U Cep has unusually high circumstellar activity. Some

eclipsing binaries exhibit very strong emission lines of N V, C IV, Si IV, Fe III, and

other species in the far ultraviolet region (Plavec and Koch, 1978). Plavec (1980) called

this group of binaries W Serpentis stars. Observations during the primary eclipse by

Plavec (1983) in 1981 August and December showed a spectrum rich in emission lines

similar to those observed in the W Serpentis stars. According to Plavec (1983) the

ionization energy for these emissions comes from the accretion process since all the

binaries in this group show large scale mass transfer. The energy cannot be supplied

by the component stars, and it has been shown (Plavec et al., 1982) that it does not

originate in the chromospheres of the late type components.

It is reasonable to assume that all W Serpentis systems are in double contact (see

section 4 in chapter 1). Thus one might suspect U Cep to be a double contact system.

However light curve solutions give a value of 5 for F1, whereas F1 should be 7.44


for double contact. Published spectroscopic results give F1 as 5.24 (Van Hamme and

Wilson, 1990). This work finds Fi to be about 6.8. However Twigg (1979) found a value

of 8 by fitting the Rossiter effect. If the absorption line comes from the circumstellar

matter, then an anomalously low F1 may result from the line profile analysis.

SW Cyg

The photometric solution by Wilson and Mukherjee (1988) was used for this anal-

ysis. The spectroscopic parameters were taken from Struve (1946b). The photometric

solution corresponds to a double contact configuration. The rotation was found to be

11.66 times the synchronous rate. This corresponds to an equatorial velocity of about

300 km/sec. It is possible that, at such a velocity, all photospheric lines are broadened

beyond observability. Yet Struve (1946b) saw lines of Ca II, Mg II, and Fe II. Thus

either the lines are circumstellar or the photometric F1 is wrong. Published spectro-

scopic velocities (Olson, 1984) give a value of 5.6 for F1. Thus it is possible that the F1

determination from light curves is wrong. However it is worthwhile to see the effects

of circumstellar gas on Fptm and Fsp over a period of time.

This system is in a state of active mass transfer as shown by the frequent period

changes, emission lines, fast rotation, and distorted light and velocity curves. According

to Wilson and Mukherjee (1988), the presence of equatorial gas could make the primary

star to look like a fast rotator, photometrically. This is because the fitting procedure

makes use of the shape and surface-brightness distribution of the primary star. For

line profile determinations, equatorial darkening diminishes the influence of the fastest

rotating regions, and thus should lower the measured rotation rate. In the case of SW


Cyg that is the sense of the discrepancy, but, the difference is large. This makes the

photometric F1 value doubtful.

A semi-detached configuration was assumed for the line profile analysis and the size

of the primary star was kept fixed. This was done by changing Q1 every time F1 was

changed in the DC method, so that rl(side) was constant. This gave F1 as 8.06. Again

one is not sure whether the line used for the analysis was circumstellar or photospheric.

One way to get around this problem is to analyze more lines. Unfortunately there were

not many "clean" lines to analyze. It would be worthwhile to make more observations

through many spectral windows. The available data is limited to the spectral range of

4400 A to 4500 A. Due to the nature of this analysis (having a fixed size for the primary

star) the Simplex method was not used. Because Q1 has to be changed after every

iteration if Fi changes, the Simplex program is not applicable in its present version.

S Equ

The photometric parameters are from Cester et. al. (1979). Plavec's (1966) spec-

troscopic parameters were used. When Plavec arranged the radial velocities according

to photometric phase he found that they showed very little variation except before pri-

mary eclipse, where the velocity increased suddenly. At that phase he found that the

hydrogen lines from Hp to HE were all strongly asymmetrical, with the cores shifted

toward long wavelengths. This was the case with the MgII line and the K line. When

he tried to measure the line centers he found a systematic shift toward positive radial

velocities. All this suggests the presence of gaseous streams. Another important result

was that the hydrogen lines displayed the same rotational disturbance as the other lines.


According to Plavec (1966), this was the first binary with gaseous stream where this

fact was demonstrated.

The light curve of S Equ shows a slight brightening or weak emission just after

primary minimum, probably caused by the gas stream projected onto the sky (Piotrowski,

et al., 1974). Due to these gas streams the primary's rotation rate should be at least

somewhat above synchronous. In fact line profile analysis obtains a value of 1.17 for

F1 from the DC method and 1.15 from Simplex.

RY Gem

The photometric solution was taken from Van Hamme and Wilson (1990) who

analyzed the B,V light curves by Hall et al. (1982). In their analysis Hall, et al.

adopted a model with circumstellar rings around the primary. Van Hamme and Wilson

(1990) included the rotation effects, which Hall et al. did not take into account, but did

not include the disk effects. The results from both work are similar except for the FI

value, which was taken to be 1 by Hall et al. Spectroscopic parameters also were taken

from the simultaneous solution by Van Hamme and Wilson (1990) from radial velocity

curves by McKellar (1949) and Popper (1989).

Line profile analysis gives a value of 8.80 for FI from the DC method and 8.99 from

Simplex, while the photometric value is 14.1. Fig. 23 shows the fit to the observed

line profile which is asymmetric. Blending of lines could explain the asymmetry in

the observed line. Thus our value may not be accurate. Twigg (1979) gets a value of

4.5 from the Rossiter effect. It seems that one gets different values depending on the

type of observations. There are no references in the literature on this binary regarding

equatorial velocities obtained from line profiles. One explanation for the different values


could be that the analyzed line is not photospheric. Popper's (1989) radial velocity work

contains no observations during primary eclipse, so the Rossiter spike does not show.

McKellar's (1949) observations for the primary star show much scatter. However his

observations show the Rossiter effect. The Rossiter spike may be affected by both the

rotation of the primary and the circumstellar ring, and Twigg's value may not represent

the true rotation of the primary star .

RW Mon

The photometric and spectroscopic solution is from Van Hamme and Wilson (1990).

RW Mon is suspected to be a fast rotator because of the emission lines seen in totality by

Kaitchuck, Honeycutt, and Schlegel (1985). Presence of emission lines during totality

means that circumstellar gas is present around the primary, indicating mass transfer and

the possible spin-up of the primary star. Unfortunately no good radial velocity work has

been published for RW Mon. The only radial velocity work is by Heard and Newton

(1969), whose observations were published by Van Hamme and Wilson (1990).

Van Hamme and Wilson (1990) get a value of 4.99 for Fi both photometrically and

spectroscopically. There is no other published evidence of fast rotation for RW Mon.

Line profile analysis gives an Fi of 1.30 from the DC method and 1.29 from Simplex.

So the question of whether the line is photospheric or circumstellar arises again.

TU Mon

TU Mon is an ordinary semi-detached close binary system consisting of a bright

B5V primary and an A5 subgiant which fills its Roche lobe. The photometric elements

are from Cester et. al. (1977). The spectroscopic elements are from Deutsch (1945)

and Popper (1967). Both the DC and the Simplex method yield a value of 4.9 for F1.


The analysis of the light curve assumed synchronous rotation. Fig. 25 shows the fit to

the observed line profile. The line is asymmetric and is probably a blended line and

hence the value for F1 may not be correct.

DM Per

DM Per is an Algol type eclipsing binary consisting of a B5V primary and an

A5III secondary. It is a part of a triple system with the third component having a mass

of 3.6 solar masses (Hilditch et al., 1986). DM Per is an early type semi detached

system and, although it has a large total mass and a small difference in the spectral

types, it can be classified as a classical Algol system The first detailed photoelectric

study of this system was made by Colacevich (1950). He noticed that at the end of

the primary minimum the brightness of the system remained nearly constant during the

phase interval 0.08 to 0.1 and then gradually reached a maximum value. He explained

this as due to an obscuring gas stream which leaves the secondary component and flows

toward the primary component, turns around the star, and comes partly or entirely back

to the secondary component.

The photometric and spectroscopic elements are from Hilditch et al. (1986). The

primary's rotation is asynchronous. Line profile analysis gives a value of 2.64 for F1

from both the DC and Simplex methods. However the fit to the observed profile is not

good. The line seems to be blended and asymmetric which accounts for the poor fit.

RW Per

RW Per is a long period Algol type eclipsing binary system with a B9 primary

and a K2 secondary. The orbital period is 13.199 days. The photometric solution is

from Wilson and Plavec (1988) while the spectroscopic solution is from Struve (1945).


According to Wilson and Plavec (1988) the primary of RW Per is in rapid rotation.

From their light curve analysis they find that the primary is rotating about 28 times

faster than synchronous. Olson et. al. (1992) gets about the same F1 from independent

observations. Wilson and Plavec (1988) also find the primary to be nearly filling it's

rotational lobe. If it filled up its rotational lobe, the system would be a double contact

binary. Such fast rotation implies considerable mass transfer. Hence one can expect

an accretion disk around the primary. However neither spectroscopic nor photometric

results do not show any concrete evidence of an accretion disk.

Hall (1969) studied several old light curves and two new photoelectric light curves.

He found that there was a steady decrease in the duration of totality of the primary

eclipse and the eclipse was partial in 1967. He suggested that the primary star of

RW Per has more than doubled its size since 1900. Later Hall and Stuhlinger (1978)

suggested a growing disk around the primary to account for the changes. Young and

Snyder ( 1982) also tried to explain the change of eclipse from total to partial by an

accretion disk whose radius was increasing due to an increase in mass transfer. Olson et

al. (1992) suggested that the variations in the duration of totality are due to the slowly

changing polar radius of the primary star.

Wilson and Plavec (1988) looked at a number of light curves and came to the

conclusion that "there is no substantial basis for a long-term secular decrease in the

duration of totality." In fact, they analyzed light curves with two models, one having a

disk and the other none. However they did not find any evidence for a disk. If there

were a disk there should be the expected 'W-Ser type' emission lines. But Dobias and

Plavec (1987) did not find such lines (UV emission lines of Si IV, C IV etc.).


The present line profile analysis gives a value of about 16.1 for Fl. Since the

photometric solution corresponded to a value of 28 for Fi one had to re-analyze the

light curves using the F1 value obtained from a preliminary line profile analysis, and the

new photometric solution was then used for the profile analysis. This procedure was

continued till the DC method gave a very small correction for the Fi parameter.

Light curve analysis gives a value of 28 for the rotation parameter. If there is no

disk, the rotation parameter can be found directly as suggested by Wilson and Plavec

(1988). If there is a disk, then the photometric value for FI may be incorrect. A possible

explanation (Van Hamme and Wilson, 1992) for the differences in Fi (photometric vs.

spectroscopic) is that most of the matter from the mass losing star is falling on a small

zone on the equator of the mass gaining star thus spinning up that region. The higher

levels may or may not be spun up. Photometrically, if the equator is spun up, there will

be polar flattening and one will get a high value for F1. Spectroscopically it is possible

that a fast rotating equatorial zone is so narrow that the absorption line from the zone

makes little contribution to the overall profile (Van Hamme and Wilson, 1990). Also the

rotation could have changed between the photometric and spectroscopic observations.

RY Per

Van Hamme and Wilson's (1986b) photometric and spectroscopic solution was

used for this analysis. The primary star is classified as a Rapidly Rotating Algol

(RRA). RRA's are binaries whose primary stars are rotating faster than synchronously

(Wilson et al., 1985). The spectral type of the primary is B3V, which corresponds to

a temperature of 18,800 K. Van Hamme and Wilson (1986b) assumed a mean surface

temperature of 20,700 K since a rapidly rotating star seen equator on will appear cooler


than its true average temperature. The scaling procedure has been described by Van

Hamme and Wilson (1986a). The scaled up temperature of the primary was used for

the line profile analysis.

RY Per has been investigated spectrographically by Hiltner (1946) and Popper

(1982). Hiltner (1946) observed both stars' radial velocity curves. However, for the

primary star, the velocity curves for hydrogen and helium are not the same. The

most significant difference is in the rotational effect, with helium showing twice the

rotational velocity of hydrogen. The velocity curves for the two elements are different

in amplitude and there is a large difference in the -y velocities. Hiltner (1946) explained

this discrepancy by suggesting that a stream of hydrogen flows from the secondary to

the primary star. This explanation was further strengthened when spectrograms taken

just before first contact showed that hydrogen lines had a second component displaced to

the red presumably coming from the flow of material from the secondary to the primary.

The photometric F1 is 10.82. Previous line broadening work reviewed by Van

Hamme and Wilson (Van Hamme and Wilson, 1986b) gives a value of 9.11 for F1.

Twigg (1979) found a value of 10.0 from the Rossiter effect. Line profile analysis

produces a value of 9.63, which is very close to that of previous line broadening work.

)/ Per

The photometric elements are from Kim (1989) while the spectroscopic elements

are from Tomkin and Lambert (1978). Algol is a triple star system in which the brightest

member (Algol A) is eclipsed every 2.8673 days by the subgiant secondary star, Algol

B. Algol C orbits Algol A and B in 1.862 years. The secondary star (Algol B) fills its

Roche lobe. The primary star is in or close to synchronous rotation.


Algol is in the slow mass transfer stage, which is believed to follow the W Serpentis

stage. One theory (Wilson, 1989b) suggests that a W Ser star is in the rapid phase of

mass transfer. After that stage has ended, one has a Rapidly Rotating Algol (RRA).

After tidal braking has acted for a sufficient interval of time, an RRA will become

a system like Algol. This system does not show much evidence of mass transfer.

However, there has been observational evidence of mass ejection from the primary.

This was suggested by Kondo et al. (1977) to explain the shortward shift of the entire

set of Mg II lines in the UV spectrum of Algol. Even after Kondo et al. (1977) took

into account the primary star's orbital velocity, there still was a shortward shift. They

claimed that the Mg II lines originated from an optically thick expanding shell.

Light curve analysis of Algol (Kim, 1989) assumes synchronous rotation. The

present line profile analysis gives a value of 1.14 for F1. Previous spectroscopic work

found a value of 1.04. Rucinski (1979) has developed a line profile synthesis method

from which he finds that the rotation of the primary is very well synchronized with

the orbital motion. An unusual result from his model was the very high value for

microturbulent velocity (35 km/sec) compared to our value of 7.5 km/sec. He also

found that the primary's angular rotation is not very far from uniform.

Y Psc

The members of Y Psc are of spectral types A3 V and KO IV Spectroscopic data

by Struve (1946b) shows that the radial velocity curve is distorted by a gas stream. A

H/ violet-shifted emission feature seen during egress from primary eclipse reveals the

presence of the gas stream. The photometric solution was taken from Mezzetti et al.

(1980). Both the DC and the Simplex method give a value of about 1.01 for F1. Twigg


(1980) obtained a value of 1.65 by fitting the amplitude of the Rossiter effect.

U Sge

The components of U Sge are of spectral types B7.5 V and G IV The photometric

and spectroscopic solutions are from Van Hamme and Wilson (1986). Their simultane-

ous (photometric and radial velocity) solution was used for the line profile analysis. U

Sge seems to be a slightly asynchronous rotator. The light curves are not very erratic,

suggesting a small mass transfer rate. Dobias and Plavec (1985) have compared U Sge

with U Cep by means of IUE spectral scans. They mention the fact that although the

systems have similar characteristics, U Cep is more active than U Sge. Van Hamme and

Wilson (1986b) suggested that the difference is because of the very rapid rotation of U

Cep brought on by active recent mass transfer. Due to the rapid rotation the primary

may be unable to accept the transferred material.

Even though mass transfer occurs in this system, emission in the optical spectrum

was seen only by McNamara (1951). Dobias and Plavec (1985) state that UV emission

lines were present during some total eclipses. McNamara (1951) finds that the hydrogen

lines are strongly asymmetrical. The hydrogen lines give velocities which are different

from other line measures, especially between phases 0.14 and 0.3 and from 0.76 to 0.92.

This can be explained partly by absorption in gaseous streams. Also the H lines do not

show the rotational disturbances displayed by other lines. As explained before for S

Equ, this either could mean that the H lines are circumstellar or that there is a severe

blending problem in the cores of the H lines during eclipse. Plavec (1983) suggested

that all gainers in Algol-type systems are surrounded by a hot, turbulent region. All of

these effects suggest mass transfer and non-synchronous rotation.


Van Hamme and Wilson (1986b) list a table of published rotation rates for U Sge.

Most of the values are from the line broadening method. However the values are mostly

inconsistent. One interesting fact arises from the table. Olson (1984) gives two values

of F1 separated by 12 years. His 1970 observations yield a value of 1.12, while his

1982 observations give 1.65. This suggests that over this interval the primary (the mass

gaining star) may have spun up due to mass transfer. However this analysis gives a

value of 1.34. This could mean that the star's rotation has slowed down over 9 years,

probably due to tidal braking. Van Hamme and Wilson (1986b) adopted a value of

1.44 for FI for their light curve analysis of observations made by McNamara and Feltz

(1976) in 1976. The theoretical line profile does not fit at all at the shoulders, possibly

due to a blending problem.

RZ Sct

The photometric and spectrometric elements are from Wilson et al. (1985). RZ

Set may be a double contact binary in which both components fill their limiting lobes.

A fit to the light curve (Wilson et al., 1985) yields FI = 6.66. However a Differential

Corrections solution of the Hansen and McNamara (1959) single lined radial velocity

curve (including the Rossiter spikes) yielded F1 =9. This is about 35% larger than

the photometric value. A fit of the light curve with a fixed value of Fi(9.0) produced

poor results. Also, rotational spikes obtained by keeping FI fixed at 6.66 did not match

the observed ones. Hence, insofar as the primary rotation is concerned, the light and

radial velocity curves do not agree. Wilson et al. (1985) suggest that the orbital semi-

major axis is larger than previously estimated by about 35%. This not only brings

into agreement the photometric and spectroscopic rotation rate, but also improves the