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PARAMETERS OF THE'SPHERICAL. 7' :''::',_ ._:MODEL FOR EC1PSING BiNARiES

N72-10892 (NASA-TM-X-65746) CORRELATIONS AMON GTHEPARAMETERS OF THE SPHERICAL MODEL FORECLIPSING BINARIES S. Sobieski, et al

6nclas (NASA) Oct. 1971 13 p CSCL 03A;~ 08688 G3/30

STAN L-EY- S'OBE SKI.N7 JULIA WHITE

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https://ntrs.nasa.gov/search.jsp?R=19720003243 2018-09-16T20:41:06+00:00Z

CORRELATIONS AMONG THE PARAMETERS OF THE SPHERICAL

MODEL FOR ECLIPSING BINARIES

Stanley Sobieski*

and

Julia White**

Goddard Space Flight CenterNational Aeronautics and Space Administration

Greenbelt, Maryland

ABSTRACT

Correlation coefficients have been computed to

investigate the parameters used to describe the spherical

model of an eclipsing binary system. Regions in

parameter hyperspace have been identified where strong

correlations exist and, by implication, the solution

determinacy is low. The results are presented in

tabular form for a large number of system configurations.

INTRODUCTION

Russell and Merrill (1952) have pointed out that indeterminacy

in the elements of eclipsing binaries can persist even though

high quality observations are available and regardless of the

formal method employed to solve for the elements. Such a

situation exists for the case of shallow partial eclipses; large

changes in the ratio of the radii result in only minor changes

in the shapes of the eclipses. Another example is the

low determinacy of the limb-darkening coefficients for

partial eclipses. In Russell and Merrill's discussion,

generalization of the determinacy of a solution is based on the

size of the angle formed at the intersection of the shape and depth

relations. The shallower the intersection the

* Presented at the IAU Colloquium #16 on Analytical Procedures

for Eclipsing Binary Light Curves held at U. of Pa., Phila., Pa.

Sept. 8-11, 1971.

** Present address: Dept. of Astronomy, Univ. of Massachusetts,

Amherst, Mass.

lower the determinacy. A number of troublesome situations

are described by Russell and Merrill and these results will

not be repeated here. Kopal (1959) has discussed this problem

also and points out that the rate of convergence is related

to the determinacy of the elements and that a solution depends

upon the nature of the eclipses. Again, shallow partials

produce low determinacy.

It is difficult to express the determinacy in a closed

analytic form and for this reason a numerical approach is

used in the present work. Rather than considering the weight

or determinacy of the parameters directly, it was found more

convenient to employ the correlations among the parameters as

indicators of the reliability of the elements. The primary

objective of this work then is to identify those regions in

parameter hyperspace where the correlations are most serious and,

by implication, the determinacy the poorest. In practice one is

required to use data from both eclipses for partials due to the

ill-conditioning of the eclipse relations. To simulate this

practical aspect, the correlation coefficients have been

computed with model data from both eclipses. Preliminary

results indicated that the maximum geometric eclipse depth,

po is a suitable choice for the independent variable in

order to demonstrate the trends in the correlations in an

efficient and meaningful way. The matrix of correlation

coefficients is found for a range of geometric depths at grid

points representing a wide range of eclipsing configurations.

Only spherical stars with no radiative interactions are treated.

Method

A program for calculating the light and light loss during

eclipses of spherical stars was written by the authors for the

IBM 360 computer at GSFC. Although in its most general form

eccentric orbits and non-linear limb-darkening can be treated,

the eccentricity was set to zero and the non-linear terms for

the darkening were suppressed in this work. In this initial

treatment of the problem it was decided to express the model

in terms of the most basic set of parameters. These are given

below.

rg r - radii of the larger and smaller stars, respectively,rg s

in units of the semi-major axis.

i - inclination of the orbital plane relative to the

plane of the sky.

ug u - linear limb-darkening coefficients of the respective5

stars.

R - ratio of surface brightnesses, the cooler to the

hotter star.

The light of the system is given by the following:

1- IdO

where the limits of the integral refer to the eclipsed portion

of the eclipsed star, dO is the elemental surface area, L is

the sum of the light of the uneclipsed components, and

I = I0 (1 - u + uO) expresses the conventional linear limb-darkening

relation.

In an orthogonal system this equation can be integrated

analytically over one dummy variable. Tests are made within

the program to determine the limits for the first integration

as a function of the geometry of the eclipse appropriate for the

given mean anomaly. Then the resulting expression is evaluated

numerically by Gaussian quadratures using double precision

arithmetic throughout. A ten point Gaussian form is used.

The accuracy is such to reproduce the tabulated a values given

by Russell and Merrill. At each grid point, and maximum geometric

depth the light is calculated for 40 equidistant phase values

between minimum and external tangency for the ascending branch

of both eclipses. Of course J E 1 outside of eclipses.

The correlation coefficients, Pij of the six parameter

models are given in terms of the elements, Ei., of the momentmar1/(E E )1/2

matrix as Pij = Eij/(.iiEi). In turn, the moment matrix is

defined as twice the inverse of the matrix of the second partial

derivatives Rij where, in terms of the light, X, Rij =2 (61..6)/

)e a e. The e.'s are the six parameters defined above. In orderto compute the derivatives, a total of 73 distinct models mustto compute the derivatives, a total of 73 distinct models must

be computed for each grid point at each geometric depth; one

to define the run of the light curve for the parameters given

at the grid location and the other 72 for the light curve

variations induced by the cross term perturbation of the parameters.

Subsections of the subroutine STEPIT, written by J. Chandler (1965),

were used to organize these computations and perform the matrix

inversion. The results appear in Tables 1-5. Unless noted the

computations are for rg = 0.25 and equal limb-darkenings, ug = us = 0.6,

but Tables 4 and 5 show results for significant changes in these

base parameters. The explanation of the tabulated material is

as follows:

Column 1 ratio of the radii, occultations have k > 1.0.

2 geometric depth, po

3-7 correlation of rg with rs, i, ug, us, and R.

8-11 correlation of rs with i, ug U and R.

12-14 correlation of i with ug u and R.

15-16 correlation of u with u and R.g s

17 correlation of u with R.s

A row of zero values means that no computations were made for

that particular geometric depth. A negative sign means anti-

correlation.

Discussion

No attempt will be made to discuss the form or significance

of' all the correlation trends apparent in this material.or to

generalize them. The intention of this work is to provide a

framework within which the analysis of individual solutions can

be assessed. High correlations imply poor convergence and

certainly indicate anomalous error assignments for the solution

parameters. For perspective, the commonly accepted rule is

that p < 1.5 implies no correlation while p>10.81 implies high

correlation. The generally conceded increase in the indeterminacy

for shallow partial eclipses is evident and is quantified in

this work. The high determinacy for total eclipses is verified

except that a moderately strong correlation between us and R

appears to exis.. As R approaches zero, the main parameters

rg, rsj and i become strongly correlated. Troublesome

correlations also appear in partial occultations for internally

tangent configurations.

From a different point of view these results tend to reaffirm

the assertion made by Kopal that a solution is not complete unless

the errors of the elements are evaluated. As a future

investigation, it would appear desirable to study selected

combinations of parameters which would minimize the correlations

or remove the skewness of the probability error ellipse and

thereby improve the determinacy of the solution.

REFERENCES

Chandler, J. P. 1965,Quantum Chemistry Program Exchange,

Indiana University, Bloomington, Indiana.

Kopal, Z. 1959,Close Binary Systems, The International

Astrophysics Series, Vol. 5, J. Wiley and Sons, Inc., New

York.

Russell, H. N. and Merrill, J. E. 1952, Contr. Princeton Univ.

26.

CAPTIONS

- Correlation coefficients for a light ratio of 0.

- Correlation coefficients for a light ratio of 0.6.

- Correlation coefficients for a light ratio of 0.2

- Correlation coefficients for a range in the size c

the primary star but with k = 1.0.

- Correlation coefficients when the limb-darkenings

differ from 0.6.

Table

Table

Table

Table

1

2

3

4

Table 5

of

of

Table 1

CORRELATICNS FOF A LIGHT PATIC = 0.9

-C, 43-. 00 55- C044-0,61-0 350, 64

0C 930 0

-0 21-0 61-O S54-0.53-0. 79-0, 68-0 26

C.27

-0 11-0C30-C0 51- 20 75

-0Co91- 0 77-00.54-0. 38

-0. 57- 0.62-00 85-0083-0308

-0,44-0*45

- Ce 67

- e 75-0,e85-0 a 7- Co 388-Co 9l-0G 9;-Co 99

0C.0

0O 630. 56O 570. 78

0O 830. 7300 60O, 0

0O 700O 830. 74

0O 8200 90o. as

0a 82O 73

O, 32Oa 340. 480. 740G 91Oa 76

00 550, 47

Oa 38Os 3800 58O 7000 650. 13

-0. 050o 13

Ca 3_CO 380o 530G 510 320. 27

-6, 21- Co 47

C a' 7G7 C

0O 080 200 1 90.4500 82O 87Oo 74

0.0

-0. 01-OD O I

0.370D 3 70 6200 840 92

0O920*e30.83

0 56

-0 25

OD 1 50.150.5000770C 920 800 61

0O44

-00 390 110O580O 73Of@ 6 90 19-0 07-0.23

-0 38-0O30-0 34

0O200. 28

- C 02

-0o 3400 C

-0 010 100.050 11

0.4900 690 030.0

-0. 25-0. 21

0 13O 440 540.400.03

-0 37

-0*29-00 07

O 09

0o120*09R

0. 100 05

-0. 14

-0.43-0,23-00 16-0O 26-00 260.04

-0. 10-0*47

-0e 38-0e 52

-0 51-0o21-0 130,41

-0o 020 00

-0.77-0087-0.89-0.90-0 * 9 Al

-1.00

-00 9 900

-0.60-0o 6 3-0,63-0086-00 86-0073-0C.36-0050-0.75-Os, 7 I

-00,1 I-0.29-Oe 2 g-0.49-e. 4 S;-074-009 1-0* 7e-005 5

-C045

-Go 2 S

-Co6- I-0054-0.73-0 86-0.75-0*40

O 1 2

-0050-0.6.-005 5-0.67

-0o 5 200550 a5 20e92CoO

-0 35-0 29-0 0 350O210 0 19

-0 06-0 .33

000

-0 o370 .130 .570 .660O 51

-0.0c-0 * 28-0 o 34

-0 32

00030.039

0.720 090,

0 *740 *47

-0001, 0 0 3,0 . 340 .580.,83

0 e93o .es008?0 750 ,8 2

0.010

0 e29

0.250.45;0 o,40o67

0e7O

0 075O 02

0O250.370.360.250o14

-0 28-0.47

Oe 30005

0.53

-00 30

-0.0Ot

Oe 19

C0o 180,33

0.6600880.6900 39

0.29

0*640 780.680C 80

Oe9C00850o720,64

0.5600480.49

00710 760.73

0,0

See text for explanation of tabular material.

0. SC

0.50

0.50o . 5 0

0.50

0,500.e5c

OeP00.80

0eS0

O 8 C

c e,o00eoC

1008,005

1 C C0

1.0001.000

I . cc

leeCI

100C

1020

1.25

1.25

1 oe =

1o281o251.251.25

I * C 0

I aCC

I 2 5

.O25

2.00C

.2eC'

1 -2 '5

2,002000

1 *25

I2C0

I o 2 e

2 3' 0 C29C O

2aCC2*00

2 aC Z

I 5e1 * 52152

I . 4

10401 361o 321 28

I 521 48

I 401 * 3~6

1 o 321 28

1 o 561 52

1o481l44

1.401 * 381 32

1 * 28

1 E5610521 .481 441 401 a 36

1* 321 028

10 _61 E2

1e481 44

1o a 01 361 o2

1 28

C0o 35

Oc 370047

O It0o ee

_ 03

0o0

-0,070 120Go 17

Co 250,17

-Ce43-OeG3-Oe e9

-0037-0,61-0.42

0.15e 0 eeCo4G~

0.68

Oo 1'3-Ce 40

-0062

-0,000.13

0O 160O34C0 39

-Co 26-0.62

-Co 72

03?

005fCo4S

Ce 51

0, c3O C

0.250.360.370.120.02

-0.02-0 0020.0

Oo 360 16

0 0080.15

0.10-CO * 16

O0C5

004A0 .4k

0260.05

-O. 10

-0.13-0 0080.08

-0 10-0*04

0 15

0.250*22

-0.12

-054-0.31

0.04

e042

-0.03-0.21-0*15-0.16

-0 .e 67

0o0

00030*05

Co 14-0* 40-0 29

0. 040 39

0.0

-0* 23-0. 76-00 86

- 0*94-00 86-0. 50C022

-0 27-0. 68

-0. 89-0.97-O 99-00 91;-00a 98-Oe 1;2-0* 74

-0.48-0 59

-0. 79-O 92

-0.96- 0 0 9 1

-0 72-0. 29

-0043-00 58-0a57-0.45-C 5 I

0O 12

0o.0GOe

-00 31-0.42-0.42

-0.37-O0 32

0 240 50

00

-00 43-0 57-0. 74-0. e88-0.9;2

-0, 84'--0.55-0. 1 1

-O .28

-0065-0, 85-0*96-0. 99

0-e 97-0 90

-0 72

-0 26-0.76-0.86-0 94

-00 96-0.91-0.62

0 13

0 02

0 03

OeC9

-0e 40-O0 38-0 04

006

- 0.28-0.39-0040-0* 18-0.21

0.42-0,05

0.0

- 0.260*02

- 0.14-0.35-0.44

- 031- O 1 9

-0.01

f0 0O 010000.000

000. 00. 0

0.28

000. 170o4 10050

0.2900 1 90.47

Oa.& I

0.540.54

0. 280039

-0.310o13Oa O

Co 1 30. 3

-0009-O-C8-0e07

0e 36

0*7200 900. 90

00

-0 110 37

0*65

0e 87

00 cS400 c30. 86

0067

-0.37

0. 150. 650.910 98

O0 g50*87

0. 76

-0.16Oe 320.63

0.95O.92

0. e10059

-0.20

-0019- 0 19

0 22

O. 64

0,910.e1

00 1Oeg0 Sl0

- 021- 022

-O 27-Co 10

0 200. 33

-0 340O 0

-0 45- O0 35

- 00 090o 24Oe 37C 20

-0e 20- 0 54

- 0. 80-0.61

-0. 38- C 18- 00 08-0. 13- 022- 0 32

-00 91- C 72- 0. 67-0.66-0.64-0 e 46- Co 27

- Co 25

-0.99- 0 99- C0o 98-0. 91- C. 94

- Oa 67- 0 020O0

O0 99O. 98O0 97

O0 89O0 82

0O 71O0 090. 0

0O 92O0 73O0 690. 670. 66O 54O 32

O 24

O 840, 670, 45O 230. 11O. 180 290, 36

0, 520, 41O, 14

-0. 23-0 38-0, 07

0. 34O 62

O0 30O 300, 350a 19

-0, 13-0s 30

0, 390 O0

(1) (2) (3) (,A) (5) (61 (7) (8 ) (9) (10) (11) (12) (13) {14) (15) (16) (17)

Table 2

CORRELATIONS FOR A LIGHT RATIO = 0.6

0O 03

0o 0900 070.44

0. 79

0o 860 72

0. 0

0,0Oa046

Oe 670 83

00 900. 930.9109 70

-0 08

00330063

0 8200 906.830 740. 962

-0040

0. 0700580.82

00850.400o23

-0 08

-0. A 1-0O 25-Oe 26

00190.37O 06

-00 39

0. 0

-0 060 0

-0C 070 07

0, 450,70Oo 090, 0

-0,25-0020

0013

0035

0.410 30-0 . 08-0.53

-0. 34-O0 13

0 01e003

Oe01

00 01-O,07-00 24

-0.52-0 34-0 32-0 45-0.44-0e 13-0* 24-0,54

-0e42-0.55-00 54-Oe 30-0 29

0O 26-0013

0O0

-0 82-0o91Co 9 1

-00 93

-0097

-100-1.00

00

-0,65

-0.62

-008 7-0.85

--0 71-005 1-0o78-09 2

-0,09

-0C26-0o44

-0*70

-0.83-0,70

-0.56

-0059

-0.4 3-0*64

-C05 1-o-0. 7-o-0.90

-0C76-0.49-0.08

-Co6 3-0e77-0o71-0O78-0 700.28

0.86000

-0 36-0 42-0 a55

0019

0 o02-0 a 14

-0 O 340 00

-0 039Ol C o520 .470O 16

-0.41-0 067-0 *68

-0 045

-0 . 25011

005300730.480.10

-0O 15I

0 0070 o370 0480 0780O910. 77

0e480e10

0 190 04000370O530 08500860 770 0O

0. 170,290 29OD I 70.17

-0*04-0. 4 1-0.5 10.0

0. 150. 190,400 .37

-0044

-0O63-0O 4 1

-coose-0,05-0 00

00480 700.430,08

-0005

005200 68

0.570.7700 90

0.7300500o49

Oe480,450044006200670072

o0063000

0,180 3000340 06

-0009-0.58-0,09

000

0240a07

0O020,03

-0 009-0, 18

0o140,59

0,190.06

-0 06

-0.09-0.08

-0O030.070926

0.180 20

--006-0.39-0,46

-0C070.22

0.53

-0.10-0.30-0e25

-0 . 19-0 o48- 0060

00090O0

0 0 200 31

0.46

-0 32-O. 06

0. 160 39

0 .0

-0 08-0.s76

-Ce 83-0. 84-0079-0.53

0 160 63

-0o. 2-0 67-0. 88

-0 96-0, 98-0. 95-0o83--053

-00 55-0. 68

-Oe 8e-0 97

-0099-0*96-0 84

-00 42

-00,47

-00 62-0,61-0 5-0o7 1

-0 o 180045

000

-0.33 -0.23

(1) (2) (3) (4! (5} (6) (7) (81 (9) (101 ( II! (121 (13! (141 1151 (16} liT)

See text for explanation of tabular material.

-0O 22-O0 33

-O 33

-Oe 27

-0 050.410e 55

0,0

-0O22

- 0.33-0.37

- 00 1 I-0,02

0,56

0.05

0e 0

0.500,50

0 , 5 ^

0050

00500.500_5 C0.50

0800..8C

00803.eo0,800.08

1 .CO

0 e el

10089

1. 3 0.8c0080

1 c 0

1 001 lCO

1 00

I a 0

I a 2 5

1.00

1l251.25

1 .2'I 25I 25

I .251 25

200O

200

2.00

2.002.002O00

2 0020CO

I a ! _5e* E2

I 0 52

I .441 *4I 40I 1 361 32

o1028

1 E 61 a 521.481 441*,01, 36I1a32loJ21.2

I o =6

I o340

I 32l o 28

1 D E,

1 . c21 4 9

1 X 401 9 10

lo 361 321 28

1 .561 521 e.81 .441 401 l 36lo 321 e28

C.25

C I C

0. 33C0.100o40

-0o C2-0.77-0.9500 C

-0.17

C . I-3Co 19

C-O 19-0. 66

-0.84

- a 80

- a 29-0*69-0. 4

C , 3c0.0 C'.Co 39

-000",-0050-0,69

0o010.13

. 10

0o61-C, a C -C 0c,-0o56-0. 7

00590C520. 57

-Ce 5a-00o2

CoO

-C0 24

-C033

-0.1 C

-Oe %9

0O 770 9500

-O 01

-Co62

-0.55-0056-0.5 a

-00 30

Co 34

0G69

-00 1 I-0 30

-0. 52-C0 74

- O 84- Co68

-0.43-0 1 4

-0049

-0. 56

-0co5-0. 90

-Co 8-3-00 57-0042-0048

- Co 66-.0030

-0 81-C0o84

-0088- Co 95-0o 97

Co0O

O. 73

O. 62Oa 65C. 88

0. 89?O0 71

0 570, 0

00 73O0 91C. e5Oa 870, 91C. 940O 94O. 30

0, 61O. 66O. 730e 8408 4)00, 90Oe 830, 77O0 76

0, 4 ,O. so500. 750 85eO. 85'O. 32C, 370, 46

OD 420. 57O, 550O 46

0, 50-0, 08-0O 4 9

0, 0

-0.07- 0 12

- O0 120.410 740.870087O.eo0

-0.030.480.700,860.92

0 940. 92

0.77

-00 180 320o720. 910O 96O 940 880 79

-0e 120 360-670.910. 96

00 900e 760.53

-0 17- 0. 09-00 10

00 240, 60

00 91

oOeO0.900Q,0

-00 19-0 24

- 0. 30- 00 030, 28

0 34-0 23

00 0

- 0 36- 0 26

0e 00 240 29

0- 20-0. 17-0 58

- 0 67

- 044- 0 24-0. 13-00 10- O 11

-O0 16-0 27

-O. 86- 0 62- 0 60- Oe 62- 0 57- 0 25- 0 16- 0.24

- 0.98- 00 98- Oe 96- 0. 84-00 71-0Oe 58

00 09O0 0

0O 990 990o 98

O0 90

O, 85O 77

O0 2500

0 93

O0 71O, 71

O 70O 640, 520 22

00 07

0 84O 6900 50O0 280, 16O0 240. 320 36

O. 58O 490. 18

-00 24-O 034

0. 190o 510O 69

O 32

0 260, 340, 310, 13

-0O 19O0 51OD 0

-0 54

-o.e 68-0 75-O 73-O o50

00 11

0 0 35

-0. 30-0o. 61-0. 80-0,93-Oe 96-0. 91 I-0. 77--0 51

-O 34-0 76-0085-00 95-0.98

-0, 90-0. 65

0,03

-0.0 1 1-Oe I -0, 16-00 10-00 44-0O 42-0025

00430o0

0.04- 00 1 I- 0,24- 0.25- 00 1 5

- 00 1 9- Oe58

- 0005-0O 03- O 0 1

0.020 030 0 10O 0

-0.05

0 26O 0

0.260 490 5 10O t 7

00 1 4004 1

0.42

0.560054O 3400530o0

09 3 1000310, 0

Table 3

C ORRELATINS FOR LIGHT RAT IC = 0.2

C 36 O 900 03 Oa 0OCo.04 O 87

-Co25 O0 95

Co 37 0 g,6

0 96 0 66

0.0 00 0

Oe 0 O, 0

0.57 Co 92-C069 0 q 6-c. 64 CD 97- o011 0a 95

Co 25 Os 4O,a 1 O, c-

0094 0,9;0 95 -Co 03

Oa 82 0o 96005 05 O 95

-00 15. 00 94-0025 09 94-0o 12 09 92

0. 15 0 920.47 0, 9400 80 0, 96

-01o9 0e 71-0920 0. 73-0068 0o 96-e0 97 lo 00-0. 52 0, 95

00 01 0. 930.66 O 980090 -0 I 1

-0.20 Oe 34-0 C 35 O 43-0.43 0049-0 71 0 75-0 O93 O0 ,9-0o23 0 21

0. 0 O0 0CO O 0 0

Oa 03-OeC7-0.060.25

0 670.840.0

00

0 1400260o 780, 750,770 800,91Ov 02

004800700G760O 790 77O. 740 9 7430,730 74

-0 180 070 77

0080O 65

0068

-0. 1 7

-0,0 0-001I3

0,010,590. 6700 110.01 10.03.e

-O. o9-0e 13-0. 16

00 050 24

O0 760.00 0

-0. 13-0. 21

0.000 05

-0 01-0 15-0. 61-0 14

0.03-00-03

-0. 30-O0 31-0e 35-00 39-0. 42-0o51

-0 70

-0 5*-0 352-0037-0. 9e-0084-0 03-00 9&-0 29

-0031-0 36

- 00 33

-Oe 53-0O 180a000

-00.89-0O.96-0C96-0099-0.99-1.00

0.00e0

-0.79-0.96-009 1-0093-0095-0.9e

-1e.00-to . 0

-1.00

-0.73-0.43

-0O60--075-0. 84

-00.88-0093

-0.5 5-- Oe70

0 320*74

-0.38-0.6 I-0085-0oQ99

-0080-0C*92-Co.9 1--Co97

-- 0076-0e750.00.0

-0 . 54

-0 054

-0 . 350 14

-0 e.32-0 450. OOe.O

-C · 5q

0 oO0.530 e.07

-C .25-0 a59-0 .92

0 e34

-0 .81-0 o67

-0 039-0e10-0 .06-0 . 25-0 e 52-0 .73

00070013

-0 *77-0 035-C . 30-0 .54-0 08e

0 916

0o29

0 o5?

0 0590 08510o.810 .770 00.0

0.0 100290.250.25

-C,33-0,75

0.0000

-0 100C 29Co020003

-0.22-0 483-0.83

oele

-00Co57-0052-0.29

-0.030.02

-0. 12-0033-0*.52

0027

0.49-0.61

0.81-0.08

-0a I S-0052

0 37

00/ 4 30.6500650083C. 570 60000

000C

0 070.3100270o 19

-0.21

-0o750000 0

0.09-0 009-0.02-0,02

0.010 150of62

0 13

-0 190.20

0 0.180.180 23

00340.49

0.71

0 0.130 230O290 O8006000770O930027

-0 * 00-0,20

-0.17-0,22-0 0 54-0 o 13

0,00.00

00. 63

005700 44

-0 080. 3900460.0

0.0

-0082-o. 77-0. 35-00 030. 44

0.91-0 31

0076-0 015-0 42-O. 55

-00.47-0. 200* 210,.68

-0 49

-0,61

-0,79-00 98

-0 7

0,.54-O 49

-C0 41- 063-, 6g -086~

-00 2

-Oeq7-0.92

0,00, C'

-00o C3-00 32-00 28-0o26Oo 36Oe 7600.0O.oC

-0. 01

-0.e 63- 0 65-Oe 31-Oe C02

C .3Eo.el81

-0 15

0 25-Co 25-O 46-0 .56-0 *

-0. 280E050.44

-0 .0 51-0e 76-Co 79-0 .99-o.e3-0 53

0. 17-0C 34

-0. 45-C a. 6-0O 66-OOFq-0O 68-0 56

0.e C0a0

- 0, IC 0. C2-0933 -0.23-0 31 - 0o 17-0.21 0.20

0.20 0066Oe 75 0. 710.0 0.C0.0 0e0

-0. 10

- 0.29-0 27O0 0

0C 220 42O0 0

0e 0

-0 13 0O 15-0 O 150.06

Q0.28 -0*21

-00.03-0*03

-CeO3-0104- 00 15

-0.62-O0 1 3

-00. I 0

- 0,09

-0.020.02

-0.05- 00. 1 8-0o18-0,36- 0.63

C*07

- 0.06

0. 120.94

0O33-0o 16

- 068-0, 20

0. e0,290.o270.@29

0.650e43

0. 00.0

0. 50

0O780. 790.a20 .92

00 60

0 0 380.730.51

0*860. 86

0.83

00.81

O0.1409450 75

0,82

0.780.065

0 C 340155

0o90

0073

000~0 . cO a o

- Co 01

0. 02- 0 03-0. 16-0 61- 0 04

- 0. 09- O0 32-oe 31- Oe 34- 0 34-0. 37-0.45- O0 64

-0. 61- 00 45-C0 44- 0. 98

Co 76-0. 72- o0 89

00 07

-0.57- 00 64- 0O 52-Oe. 31

o-0. 62- Co 15

C. 0C 0

(1) (2) (3) (4) (5) (6) (7) (8) (S) (10) (11I (121 (13) (14) (15) (16) (17)

See text for explanation of tabular material.

O 2 ' C

OFOE

0.500.50

Oo&.0

00.500.050.800.50

0 0. 80

0.p0

1.00008C

eO0100

1.0001e.001e.00

1o001e.2

1.25

1.251.251.25

102510.25

1 25

1 a 2

1 0 25

102.1 0025

12.02.CO

2.000

2 a 5C

2 002.00

2.00

I D 2 C

20002 00

1 E561. 521 481 0 441o401 o 361 321,28

1 E6I o E2l a 481 e 44

1036

1 * 321023

1 56

1 52

1,451 441 l40O1 . 361.321 .28

1 E52

1 o48

1o4410401 361 o32lo28

1 E6e10 2

1 .481 * 441 4CI 36

1 321 a 28

-0.170 040 C60 31

-0.31

-O. 9600 C0 .0 C

-00. ,

00 63Co40

-0 1I

-0e E50- 0075-COe ;5-0s ,5

-0 o82-0, 7t-0.61

-0.40

-C 0 FS8

-0.53

-0.75-0o89

-00906

-00188-0 H-0057-0079-0 a 96-0.03

0o 170.330. 370067

0050

-0004000OoC0 a C

0o 980. 9S

0. 990 97O0 86

0 .93

OD 000 00, 0

0. 930O 870, 580 63O, 57OD 41

-. o 26O 75

0, 88O, 38

O0 280. 16

O0 16

0O 190O 12

-00 09

0. 63

0. 530, 25

-00 94

-0. 43-0 21-0, 42

O 79

00 320. 15

O, 2109 05OD C7OD 4-°0,490. 00.a C

Table 4

CORRELATIONS AS A FC,. CF RAO FF IM

PAD PRIM = 0C15

I CC1o001

I. C I 0o

1 0 C10*01 001.00

1 e561 * .21,¢8I a 441 o401 o 361,22

1 , 28

-020'S. l ,

)o 27

-0 050e S2

-0 3092-OoE9- 0,,30

-C 21

- Ci 70

-Co 28

- 0C, 31-0,43C. 27

-Co 05O-C. 5 1

0, 4 3C. e-sO, 25O0 77

O 4cC0, 21

-Oe 33-0. 33

-0,0 .

0.82

0.1.'O . 4 0

O. 17-0. 1 7-0. 56

-0. a .9- C, 11

C, 67* 14

-. 12-0 32-0C 42

-0 24

-C= I 7-Oe e 7-0 1 7

-0.37-0 a7 8-0o45

-Co ~t-0426-0.67

-0 . 2

0 500 .380 73

00230 e 14-0020-o .06'

C , 3_0,570a020a67

0, 27O, 160.C3

-0O35

fAD PPIM = 0.25

1.001.0CO

I C CCl00I.COI 0 3I1 0OIe0 0

1.59I .521,a 81e 441040

1 .361 . 321.28

-037-0. 61

0 0 15.e 42

0. 15

-0.40

-O.62-c0. e 2

- 0. 1 - 0o 30-3 51-O 75-C. 91-0O 77-0.54-003.

0i 320, 3.

0. 4F0O 740C 91O, 760, 550. 47

-3, 250.19co 150.500e770 92Oe 60Co 61Oo44

-0e29-O 07

0O 09O0 12Co, 80. 10O 05

-0. 14

-00 t I-Oa 2 9-0C4 9-0074-0,9 1-0o7 6-0. 5-00 45

-0 .320 C30.390 720 .900 .740 .47O 027

0. 19C 1a0C 3300660.8800 6 90O390G29

RAC PPIM = 0035

I0 C01 00I .CC1s C O1.CO1001. 0010 C O

I a 5eI a 521 052

1 ,441 0401e5lo321.28

-305 03-0C 74-0 76

-0, 66-0043-0. CS

C * C7

-co 15

-0, 06-0 1 6-0.26- 0,38- Oo' 52

-0,67-0. 72

-0,63

0. 27C, 240, 250O 3_O, 500o 6tC, 72O0 S4

-Oa 1 3c. 090. 3 10s,470.6000720o 76oe69

-0o21-O 02

00 150 230 2&002000 15O0 12

-0 0 9-30 I 9-0.27-0.C38

-Ce52-0 0 7-0,74

-Oa6; 7

-? · 28-0 .07

' el5^ *32O Oa-

C 650 0720 .65

C. 12

0.oc005C 0. 70.37005,C,67

0.57

11 ) (2) (3) (4.) (E) {6) (7) lB) (9) (lO) { 11) (12) (13) (1~.) { 15) (16)

See text for explanation of tabular material.

0*430 .06

-0.67-0 o 10

0.140 * 32

00250 025

-C * 34

-0, 87

-C e 98-0 095

-O. E -0 31

0 84C e 4

-O 29

-OeF5

-Oo;2-0. c40-O 17 4

-0 54-0 30

0 e820 79

C, 020.02OC 2

-0.03

- 0. 1-0 1 0-0.01-0.05

-0.460. 600 870 0900.590.ee860O 790 * 40

- 0. 84- 0. 43

- (Is 16-Oe 43

-00 16 7-Oe 17-C* 41- O 23-Co 31- Co 56

0. 86O .45

Oa 35

O 28

O. 48O. 28

O. 34O. 54

0e26

C e05-0O0I-0o 13-o0e90

-0. 10-0 .0o

0 .15

-O 27-C 0 68-oe ,D-0c 97-0. 35-O0 98-0. 32-0 74

-O0 28-0.65

-Oe86-0096

-0099

-Oe 97

-0Ce 0-0 72

- 0. 0 1- 0 0 1

0. 0OeO00.0

0.00O 00. 0

-0 370 150.65

0. 91Oe.;300950Ce70 76

-0 o 80-00 61-0 39-0e 18- 0e c8-C. 13- 0 22-0 32

O 84O0 670. 450, 23O. 11O. 180 290O 36

0ol 1

O 01

-0. 1A-0023

-0O.2

-002')-0.15

-0.12

-0O 22-O0 57-0. 77-0, 8c

-0 G6-0Oo P-0, 33-0.9e

-0Ce24-0e 5-0o 74-0. 86-Coq4

-00oC7-009s

-0097

- 002- 0.0 1

- 00 0 1- 0. 0 1- 0 0 1

0,00.0

0.31

-0C 35

Os 36-OeCI

Oe . 60,67

0,85

0 0 S450.340.360.35

-Cs 79- O. 67- co 51

-0.35-0 22-0G 14- s 11-00 13

0, 830. 740. 60Oa 450, 310. 2C0. 150,17

(17)

Table 5

CORRFELATICNS AITH LIE DARK VAR

LIMB DARK = 4 = 0. 1 .eclipsing 0.6-eclipsed * ecllpslng

I Co C1 o. 01 oO0I CO.

1aC 0I .00

I0C01 O00

I s 6

lo 521 ,40

1740

19261 32I a28I o 28

-0237

--OoA;.

--3064-0. 40

-0, 7

-00-70035

- Oo 05

C0 02Co 23

C I 5 5C, 65s

Co c, 50o 91OD 91

Oo e 30. 81c, 57

0, 57

0.710,620,650o690,620:5400520,50

-0o 31-00 27-O0 o33

-00 34

-00o 32-0033-0035-0044

o0023

-0o6 3-0O75-0o86-0093-0,95

-Ca,,f

-0097

-0 o51-0 o 59-0 e37

0 o020O 1C 003

-0 o 13-0 043

-0o 48-0044

-00 2 5

0011Co270o200005

-0, 10

L I ARK = Peclipsed = eclipsing = O

I oC,lr DI 0C01. * 0 3

1.C0leCO1.001.00

1 * Z61 a 521.85

1 z,

10401,36

1 0321*2-

-0. 2i--Co 72

0 o 59

C e 2'3-do C6-0 .30-0. ce-oa:43-0,51

-O, 10

-C.a 21

-OJ41

- L 79- C 56-0 ,4.3-Co 35

0a 30.0. 39

C. 69

N 67

Oa 530. 'e.

-O 0290O 01C 35C 67

0o800,680.53Co 39

-C , 37-0o27-30 14-0 05

-0o C100 020.01

-Oe 09

-0003-0 24-O4 1-0.63-0C8 1e-0 70-0057-0Co4

-0 ,36-0 13

0.2?0 .60C .770 e6,+0e45

0 23

0.200.22Oo 330,61C07700620,430O33

(1 ! (2) (21 (41 (5) {61 (7) (B ) (q) (101 ( lit ( 121 ( 131 (14~) (15) (t6) (t?)

See text for explanation of tabular material.

0o11

0o17

0022

o0020002300290o36Oo46

-0 52

Co07-00 22-0049-0 50-0o 32-Go 02

Oo 38

-0049

-0o 08-0o 30-0o 52-0055-0041-0o 19

00 09

0 10-0 o15- 0I1 2-0, 08- 0 1 4- 0o21

- 028- 0o38

0o 740o670o770 .83O E20o770.720.69

- 00 34

- O 29- Oo 32-O 29- OD 23-0 22-0o 25

-0 33

09 350 48O0 33

O 27O0 35OD 43O 46O0 43

0 320.250 120,03

-0.01-0 e03-0.0l

0 10

-0 23-0.63-0.87-0. s6-0. 98-0.96-0089-0 68

-0,23-0e. 62-0o e85-0. 96-0. 99-00 95

-0.87-0 . 66

0.o0-0. 01

O. o0.01

00 1O0. 0

0. 00.0

-Oe 390009Oef5200 *S

0.84.0oe74Oo@74

- 0 82- O 65- O 42- O0 22-0 15-0. 18- 0 25-C 33

0 840 69O 45Os 23O, 15O. 22O 310, 39