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Statistical study of the orbital elements of spectroscopic binary stars. Olowin, Ronald Paul 1971

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A STATISTICAL STUDY OF THE ORBITAL ELEMENTS OF SPECTROSCOPIC BINARY STARS by RONALD PAUL OLOWIN B.Se., Gannon College, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of GEOPHYSICS We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1971 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of Br it ish Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geophysics The University of Brit ish Columbia Vancouver 8, Canada Date 16 A p r i l 1971 i i Abstract A s t a t i s t i c a l study of the o r b i t a l elements of Spectroscopic Binary Systems i s undertaken with attention given to the d i s t r i b u t i o n of u, the longitude of periastron, and the observed d i s t r i b u t i o n of o r b i t a l e c c e n t r i c i t i e s and periods. Selection e f f e c t s are taken into account i n both cases. Test r e s u l t s indicate that the major non-uniformities previously found by other investigators for the d i s t r i b u t i o n of u have been reduced or eliminated for c e r t a i n categories of data. A p o s i t i v e c o r r e l a t i o n i s discovered for the ob-served tabulation of periods and e c c e n t r i c i t i e s . The l a t t e r r e s u l t i s viewed with caution, due to c e r t a i n c r i t e r i a established i n the a n a l y s i s . i i i Table of Contents page Introduction 1 D e f i n i t i o n of O r b i t a l Elements 2 Problems Associated with the O r b i t a l Elements 8 S t a t i s t i c a l Studies a. Longitude of Periastron 11 b. Pe r i o d - e c c e n t r i c i t y C o r r e l a t i o n 13 The Analyses a. Longitude of Periastron 15 b. P e r i o d - e c c e n t r i c i t y Correlation 35 Conclusions a. Longitude of Periastron 57 b. P e r i o d - e c c e n t r i c i t y C o r r e l a t i o n 58 Bibliography 60 Appendix I 62 Appendix II 69 Appendix III 91 Appendix IV 103 Appendix V 106 i v L i s t of Figures page 1. Periodic s h i f t of. absorption l i n e s i n HD 171978 3 2. D e f i n i t i o n of O r b i t a l Elements 5 3. O r b i t a l Configurations of Spectroscopic Binaries 10 4. Observed D i s t r i b u t i o n of to for V i s u a l Binaries 18 5. Observed D i s t r i b u t i o n of co for Spectroscopic Binaries of a r b i t r a r y o r b i t a l q u a l i t y 19 6. Observed D i s t r i b u t i o n of to for Spectroscopic Binaries of high o r b i t a l q u a l i t y 20 7. Smooth Waves ori a Theoretical Frequency Function 22 8. D i s t r i b u t i o n of Mean E c c e n t r i c i t y vs. Log P 37 9. Scatter Plot of Observed E c c e n t r i c i t y vs. Log P 38 A - I : l Smooth Deviations from a Theoretical P r o b a b i l i t y Function 64 A - I I:1 Radial V e l o c i t y Measurements at A r b i t r a r y Points i n the Orbit of a Spectroscopic Binary 72 A-II:2 % Detection for S i n g l e - l i n e d Spectroscopic Binaries. 75 A-II:3 % Detection for Double-lined Spectroscopic Binaries. 76 A - I I:4 Scatter Plot Showing K = CP" 1/ 3 80 A - I I:5, 13 Selection E f f e c t s i n Spectroscopic Binaries ... 81-89 A - I I I:1 Scatter Plot of e vs. log P 93 A - I I I :2 Hypothetical Orbit for Binary Stars to Determine the Point of Closest Approach 95 A-III:3 Determination of the Constant K 98 V List of Tables page 1. Observed Distribution of u> for Various Categories 25 2. Observed Distribution of High Quality Systems 26 3. Test Results for Table 1 27 4. Test Results for Table 2 28 5. Contingency Table for Observed Data 39 6. x 2 Test for Observed Data 40 7. Contingency Table for Selection Corrected Data 41 8. x 2 Test for Corrected Data 42 9. x 2 for Various levels of Significance 46 10. Results for the x 2 Test 46 11. Linear Regression for e and log P 51 12. Regression Coefficient for e and log P Data 53 A-I:l Table of the Function Gamma(j) 67 A-I:2 Percentage Points,of the Chi-Squared Distribution .. 67 A-II:1 % Discovery of Single-lined Spectroscopic Binaries . 77 A-II:2 I Discovery of Double-lined Spectroscopic Binaries . 78 A-111:1 Table of Unusual Systems 100 A-V:l Listing of Spectroscopic Systems with Selection .... 108-1 v i Acknowledgements I should l i k e to thank Professor M. W. Ovenden for the guidance and counsel which he has provided during the course of t h i s research. My gratitude also extends to Drs. J . R. Auman, J r . , H. B. Richer, and G. A. H. Walker for the encouragement they have shown i n the l a t t e r part of t h i s undertaking. Also, to Mr. B. I. Olson whose patience withstood my many discussions of the problem. F i n a l l y , I should l i k e to thank my wife who has made th i s endeavour much more rewarding by her understanding and patient encouragement. v i i D e d i c a t i o n to Mary Introduction Certain s t r i k i n g c h a r a c t e r i s t i c s are noticed i n a tabulation of the o r b i t a l elements of spectroscopic binary systems. In p a r t i c u l a r , two d i s t r i b u t i o n s have been of in t e r e s t to astronomers. These are the univariate d i s -t r i b u t i o n of the longitudes of p e r i a s t r a and the b i v a r i a t e d i s t r i b u t i o n of the observed periods and e c c e n t r i c i t i e s . Previous analyses have shown that there exist c e r t a i n anomalies i n the above d i s t r i b u t i o n s . The f i r s t , namely, what i s c a l l e d the "Barr e f f e c t , " deals with the apparent concentration of the values of to, the longitude of p e r i -astron, i n a p a r t i c u l a r quadrant of the c i r c l e . The second deals with the so-called " p e r i o d - e c c e n t r i c i t y re-l a t i o n s h i p " which suggests a systematic increase of av-erage e c c e n t r i c i t y with increasing o r b i t a l period. These d i s t r i b u t i o n s are reviewed i n l i g h t of the most current compilation of the o r b i t a l elements of spec-troscopic binary systems (Batten,1967). An attempt to i n -clude the additional factor of s e l e c t i o n within the analy-s i s i s also made. D e f i n i t i o n of O r b i t a l Elements The term spectroscopic binary arises from the fact that the physical d u p l i c i t y of such a system i s discovered by a periodic s h i f t of the spectral absorption l i n e s due to the Doppler e f f e c t . An example of t h i s phenomenon i s shown i n figure 1 for the double-lined spectroscopic bin-ary HD 171978. The term "double-lined" refers to the fact that the spectra of both components, primary, or more mas-siv e , and secondary, are v i s i b l e . We are dealing here with v a r i a t i o n s of the v e l o c i t y component of the system along the l i n e of sight, t h i s velo-c i t y being defined as the r a d i a l v e l o c i t y . This v a r i a t i o n of v e l o c i t y i s assumed to be related to the Keplerian motion of one star about the other. The more massive star i s generally described as the primary component about which the less massive secondary revolves. Assuming that the motion i s Keplerian, we define six elements of the or b i t i n the plane of motion. The f i r s t three define the motion i n the o r b i t and are c a l l e d dynam-i c a l elements; the l a s t three determine the siz e and o r i -entation of the o r b i t and are c a l l e d the geometrical e le-Figure 1 ments. P = Period of revolution in days. The mean motion is n • 2ir/P. T = Time of periastron, or closest approach, passage. e = Eccentricity of the orbital e l l i p s e . a = Semi-major axis of the orbital ellipse co = Angular distance of periastron from the ascending node, measured in the direction of the orbital motion (longitude of peri-astron) . i = Inclination of the orbital plane to the plane of the sky. Figure 2 illustrates the conditions of the problem. Let the x-y plane be tangent to the cel e s t i a l sphere at the centre of motion, and let the z-axis, perpendicular to the x-y plane, be parallel to the line of sight. When the primary star is at any point in i t s orbit, i t s distance z from the x-y plane w i l l be given by z = -r sin(v+o)+180°) sin i (where v is true anoma • r sin(v+w) sin i . The radial velocity at this point w i l l be expressed as the time derivative of the above expression. Hence, j| - • sin(v+w) sin i j £ + r cos(v+w) sin i dv From the equation of an ellipse in polar coordinates we determine dr : - 5 -to earth -Z Definition of Orbital Elements Figure 2 - 6 -_ a (1 - e z ) 1_ _ l + e c o s v _ 1+e cosv ' r ~ a ( l - e z ) D i f f e r e n t i a t i o n of 1/r y i e l d s : 1 dr e sinv dv r 2 eft = a ( l - e 2 ) d T dr _ r 2 e sinv dv h e sinv 2TT ab e sinv o T - a(l-eO o T = a(l-e*) " ~ ~ P ~ a(l-e^) - «K E S J - N V _ „, //1 _2T" e sinv . nae sinv where the rate of change of the true anomaly, dv/dt was calculated from Kepler's laws: r a r " h ~ p — n a b dv _ n ab n ab (1+e cosv) nb (1+e cosv) r d T " T ~ a ( l - e 2 ) = I T F 2  na / ( l - e 2 J (1+e cosv) na (1+e cosv) lTp 3 • ( l - e * ) We also have: cosco = cos ( (v+u>) -v ) » cosv cos(v+u) + sinv sin(v+w) . Hence, our expression for the r a d i a l v e l o c i t y becomes: dz _ nae sinv sin(v+w) s i n i o T " / ( l - e z ) ^ na (1+e cosv) <• . > + - — \ / ( i . e 2 ^ cos(v+w) s i n l -7-Or: (e cosco + cos(v+co).). Now, the observed r a d i a l v e l o c i t y consists of a constant part, namely the r a d i a l v e l o c i t y y of the centre of mass, and also the variable portion dz/dt. Hence we may write the equation i n i t s usual form: V"r = y + K(e cosco + cos(v+co) ) where the semi-amplitude of the motion i s given by K = na s i n i n = 2ir Problems Associated with the O r b i t a l Elements A n a l y t i c a l l y , the derivation of the o r b i t a l elements of a spectroscopic binary presents no serious problem. Ob-s e r v a t i o n a l l y , however, we are forced into a best f i t t i n g compromise of forcing a Keplerian o r b i t onto r a d i a l v e l o c i t y data, and the p r o b a b i l i t y of committing an error i n the determination of the actual o r b i t a l elements i s high. To add to the problem, there e x i s t c e r t a i n combinations of o r b i t a l elements which motivate the s e l e c t i v e i d e n t i f i a b i l i -ty of c e r t a i n systems over others. For example, astronomers discover only those systems whose or b i t s are orientated at a considerable angle from the plane tangent to the c e l e s t i a l sphere. There i s also a d e f i n i t e preference for the d i s -covery of those systems that have large masses and whose orbit s are e c c e n t r i c , since both of these factors increase the observed range i n r a d i a l v e l o c i t y . The observed d i s -t r i b u t i o n of o r b i t a l elements are indeed affected by these factors and i t i s d i f f i c u l t , i f not impossible, to deal with them a l l . Certain questions ari s e which are very important, for instance, i n studies of s t e l l a r evolution, to explain the r e l a t i v e l y high number of multiple systems. Can the obser-ved systems give us a clue to the o r i g i n of binary systems? To answer t h i s question, and ones l i k e i t , we must be rea-sonably sure that those systems we do see a c t u a l l y represent a random sample from the parent population of a l l such binary s t a r s . I f , indeed, t h i s i s the case, then any ob-servable systematic trend i s of the utmost importance, for i t implies an a s t r o p h y s i c a l l y s i g n i f i c a n t set of conditions which may e f f e c t the o r i g i n and subsequent evolution of binary systems. The search for these trends leads us to a discussion of the d i s t r i b u t i o n s of the o r b i t a l elements of the observed spectroscopic binary systems. Two that have been singled out are the d i s t r i b u t i o n of co, the longitude of periastron; and the d i s t r i b u t i o n of period and e c c e n t r i c i t y . The f i r s t i s to determine i f there exists any pre-ferred o r i e n t a t i o n of spectroscopic systems with respect to the p o s i t i o n of the Sun. Some possible values of co and subsequent o r b i t a l configurations as seen from the sun are shown in figure 3. Secondly, the b i v a r i a t e d i s t r i -bution of periods and e c c e n t r i c i t i e s i s discussed because of i t s obvious astrophysical importance. Orbit -10-V e l o c i t y Curve to earth d O r b i t a l Configurations for Spectroscopic Binaries Figure 3 S t a t i s t i c a l Studies Longitude of Periastron In 1908, J . M i l l e r Barr, a Canadian amateur astron-omer, c a l l e d attention to an in t e r e s t i n g d i s t r i b u t i o n of the values of co, the longitude of periastron, for spectro-scopic b i n a r i e s . On the basis of 23 or b i t s he found a concentration near co=90° with a marked absence of systems having 180°<co<360°. He concluded that the e f f e c t was due to "some neglected source of systematic e r r o r " i n the ob-served r a d i a l v e l o c i t i e s (Barr, 1908). In more recent times, t h i s question has been discus-sed by a number of inv e s t i g a t o r s , some of whom offered t h e o r e t i c a l explanations for the unequal d i s t r i b u t i o n which they regarded as r e a l , and others who refuted these explanations. The addition of new data from the publica-t i o n of several catalogues of the o r b i t a l elements of spectroscopic binary systems has not eliminated the con-troversy. On the basis of 275 o r b i t s , Aitken (1935) found a maximum near co=45°; i n 1948, Struve (1948) found a max-imum near to=15° for 419 systems. The F i f t h Catalogue of the Orbital Elements of Spectroscopic Binary Stars by Moore and Neubauer (1948) provided the s t a t i s t i c s for the deter-mination of the d i s t r i b u t i o n of w for 415 or b i t s by Blanco and Williams (1949). These investigators r e f e r r e d to the concentration of the longitudes of p e r i a s t r a i n the f i r s t quadrant as the Barr Effect and summarized t h e i r conclusions as follows: "The Barr e f f e c t i s present i n both e c l i p s i n g and non-eclipsing systems; i t i s more conspicuous i n spec-troscopic binaries with primary stars of early spectral types, e s p e c i a l l y those e a r l i e r than class F2; i t i s par-t i c u l a r l y noticeable i n systems with small e c c e n t r i c i t y having moderately short periods (from two to ten days), f a i r l y small values of a sin i (1.0 to 3.0 x 10 6km), and large values of the semi-amplitude (K>50 km/sec). In short, the Barr e f f e c t i s correlated with a l l the parameters men-tioned i n a manner consistent with t h e i r own i n t e r r e l a t i o n s . " Soon afterwards, Scott (1949) conducted a more rigourous s t a t i s t i c a l analysis of the data. Her analysis demonstra-ted that "for most categories of b i n a r i e s , i t i s safe to assert that the d i s t r i b u t i o n of to i s not uniform." The most recent analysis, a f t e r the p u b l i c a t i o n of the Sixth Catalogue of the Orbital Elements of Spectroscopic Binary Systems compiled by Batten (1967) , was performed by Batten and Ovenden (1968) . These investigators found that to was concentrated i n the range 0°<io<180°, i t s d i s t r i -bution being more pronounced for e c l i p s i n g systems with a large primary semi-amplitude and having a short period. It was also noticed that the concentration to the f i r s t quad-rant i s most pronounced for stars with periods between 3 and 10 days. Attention was drawn to a minimum at ^250° which i s common to almost a l l categories of binary i n the catalogue. P e r i o d - e c c e n t r i c i t y C o r r e l a t i o n For some years i t had been considered as well estab-l i s h e d that the e c c e n t r i c i t i e s i n binary systems, spectro-scopic and v i s u a l , increase i n the mean with the periods. Largely because of the accumulation of increased amounts of data, the study of the p e c u l i a r i t i e s of t h i s apparent r e l a -t i o n has been continued. Heintz (1969) suggests that a close inspection of the data reveals that the apparent d i s t r i b u t i o n should not be interpreted d i r e c t l y i n terms of a c o r r e l a t i o n . Varsavsky (1962) t r i e d to connect an upper e c c e n t r i c i t y l i m i t rather than mean e c c e n t r i c i t i e s to the periods. But a main feature as pointed out by Walter (1950) i s that the or b i t s form two d i s t i n c t groups rather than a continuous d i s t r i -bution i n the P-e diagram: the short period n e a r - c i r c u l a r o r b i t s , and the o r b i t s d i s t r i b u t e d over nearly a l l eccen-t r i c i t i e s which form the majority for periods longer than about 50 days. Walter makes an i n t e r e s t i n g hypothesis at explaining t h i s d i s j u n c t i o n by considering the influence of d i f f e r e n t i a l r o t a t i o n and t i d a l v i b r a t i o n on non-cir-cular o r b i t s , thus connecting the e c c e n t r i c i t y problem with that of r o t a t i o n a l synchronism. According to Heintz (1971) these arguments have been refuted on t h e o r e t i c a l grounds, however. THE ANALYSES D i s t r i b u t i o n of the Longitudes of Per i a s t r a As r e l a t e d i n an e a r l i e r part of t h i s paper, a study of the r e l a t i o n of the observed d i s t r i b u t i o n of the o r b i t a l elements of binary systems with that of a t h e o r e t i c a l model simulating the actual d i s t r i b u t i o n of these stars i s of i n -terest i n the problem of s t e l l a r evolution. For example, the problems of r o t a t i o n a l synchronism, mass l o s s , and the motions of cir c u m s t e l l a r material a l l may involve a s t a t i s -t i c a l study of o r b i t a l elements. Theoretical considerations suggest that the value of co, the longitude of periastron, which describes the orien-t a t i o n of the l i n e of apsides of the o r b i t a l e l l i p s e , must be uniformly d i s t r i b u t e d between 0° and 360° . It has been shown, however, from previous analyses by Blanco and Williams (1949), Scott (1949) , and most recently by Batten and Ovenden (1968) that, at l e a s t for some categories of s t a r s , the d i s -t r i b u t i o n of the observed values of co i s not uniform. It i s obvious, though, that the actual d i s t r i b u t i o n of co could be d i s t o r t e d by many causes. For example, systems with cer-t a i n values of co are easier to i d e n t i f y as binary stars than are others. These s e l e c t i o n e f f e c t s , as they are c a l l e d , impose a s e l e c t i v e i d e n t i f i a b i l i t y on the discovery of spec-troscopic binary systems. The problem i s further compounded by the fact that the ordinary procedure of computing the o r b i t a l elements from observational data may show some prefer ence for c e r t a i n values of to at the expense of others. This may also be true of the method used to reduce the data. The most frequently used method of o r b i t computation i s that of Lehmann-Filhes (1894). Also employed (Aitken, 1935) are those of Schwarzschild, Wilsing-Russell, and Zurhellen and the graphical methods of King, Laves-Pogo, and R u s s e l l . Com-parison of these methods, however, i s beyond the scope of t h i s paper. It i s important to note that a f f i r m a t i o n of the aggre-gated e f f e c t of these factors on the o r b i t a l elements i s not s u f f i c i e n t to explain the observed d i s t r i b u t i o n of to. I f the d i s t r i b u t i o n s t i l l remains non-uniform a f t e r these e f f e c t have been removed, the necessity of providing some addition-a l hypothesis i s obvious. The most recent study of the d i s t r i b u t i o n of the lon-gitudes of p e r i a s t r a by Batten and Ovenden (1968) u t i l i z e d the complete set of data, that i s , data of a r b i t r a r y q u a l i t y , from Batten's (1967) Sixth Catalogue of the Orbital Elements of Spectroscopic Binary Systems. This was contrary to the suggestion of Batten (1967) that i t be "recommended that elements of categories (d) and (e) be omitted from most sta-t i s t i c a l i n v e s t i g a t i o n s . " The conclusion drawn from t h e i r analysis was that co i s non-uniformly d i s t r i b u t e d for c e r t a i n categories of stars and t h i s implies that spectroscopic bin-a r i e s are p r e f e r e n t i a l l y orientated towards the Sun. This an a l y s i s , however, was done by es t a b l i s h i n g a crude indi c a -t i o n of s t a t i s t i c a l s i g n i f i c a n c e to the data without re-course to s e l e c t i o n . On the basis of the above suggestion, that spectro-scopic binaries are p r e f e r e n t i a l l y orientated towards the sun, i t was decided to study t h i s d i s t r i b u t i o n once more, using higher q u a l i t y data and a more sophisticated s t a t i s -t i c a l a nalysis. E f f e c t s of s e l e c t i o n were also to be taken into account. Figures 4,5, and 6 show the d i s t r i b u t i o n s of the observed data. In these f i g u r e s , the observed absolute and r e l a t i v e frequencies of the univariate d i s t r i b u t i o n of co are plotted at 10° i n t e r v a l s for both v i s u a l and spectro-scopic binary systems. The v i s u a l b i n a r i e s were used i n t h i s analysis as a control group. The same d i s t r i b u t i o n was also plotted at 20° i n t e r v a l s for the high q u a l i t y spectroscopic o r b i t s . These i n t e r v a l s are i d e n t i c a l to Ax^ i n the f i g u r e s . An analysis s i m i l a r to that used by Scott (1949) was used to test the departure from uniformity of the d i s t r i -bution of co. What i s meant by a departure from uniformity is simply t h i s : that the observed d i s t r i b u t i o n does not Absolute frequency of to for v i s u a l b i n a r i e s . F. = 1 n: Ax, Ax, = 10* Relative frequency of u for v i s u a l b i n a r i e s . 1 N = 772 = # stars Figure 4 Absolute frequency of to for spectroscopic binaries = D.i_ Ax^ i i on° Relative frequency of to for specrtoscopic b i n a r i e s . $ i = - i - N = 542 = # stars Figure 5 300° 360° Absolute frequency of to for spectroscopic binaries with high q u a l i t y o r b i t a l elements. 1 100° Relative frequency of u> for spectroscopic b i n a r i e s with high q u a l i t y o r b i t a l elements. *• = N = 3 3 4 1 N = # stars Figure 6 TuMp 36 ^ conform to a t h e o r e t i c a l l y uniform d i s t r i b u t i o n formed by small perturbations acting on a smooth frequency function. This basic assumption admits that i f the observed probabi-l i t i e s are not a l l equal to N/v, where N i s the t o t a l num-ber of observations and v i s the i n t e r v a l of i n t e r e s t , then t h e i r deviations from t h i s value w i l l be smooth. This idea may g r a p h i c a l l y be represented as i n figure 7. In t h i s f i g u r e , the sequence of points represents the t h e o r e t i c a l p r o b a b i l i t i e s , * y = 1/v . The combined e f f e c t s of errors of observation and s e l e c t i o n d i s t o r t t h i s horizontal se-quence by creating a few smooth waves, as suggested by the continuous curve. The d e t a i l s of t h i s analysis w i l l be found i n Appendix I. S u f f i c e i t to state here that a t e s t i s used that u t i l i z e s the number of expected maxima and minima i n the continuous curve shown i n figure 7 and equates t h i s number, say k, to the order (as explained i n Appendix I) of the t e s t . The technique of applying the k-th order test consists of computing the test c r i t e r i o n and r e j e c t i n g the hypothesis that the observed d i s t r i b u t i o n 2 of co i s uniform whenever the computed value of Afc exceeds a c e r t a i n c r i t i c a l value x 2(k»«) . The problem of s e l e c t -ing the order, k, of the test i s by no means t r i v i a l . Recall Smooth Waves on a Theoretical Frequency Function that k must be selected to allow for the number of maxima and minima which are l i k e l y to occur i n the sequence of p r o b a b i l i t i e s * v i f the hypothesis made i s f a l s e . If we assume that spectral l i n e d i s t o r t i o n s can add, for instance, one maximum and minimum and that the se l e c t i v e i d e n t i f i -a b i l i t y of spectroscopic binary systems may add another two maxima and minima, we may assume that i n the i n t e r v a l 0° < to < 360°, there may be three maxima and minima i n the sequence of p r o b a b i l i t i e s * v . We then choose the fourth-ordered test to make the tes t s e n s i t i v e to small departures from uniformity, as suggested by Scott (1949). We thus conclude that 2 where the \^ are functions of the observations and are computed i n Appendix I. Now, the c r i t i c a l values of x 2 ( k > ° 0 depend on the order k of the tes t and on the desired l e v e l of s i g n i f i -cance, a. They are generally described as the percentage points of the x 2 d i s t r i b u t i o n which i s given i n table A-I:2 in Appendix I for several degrees of freedom and le v e l s of si g n i f i c a n c e . Note that the order of test used i s i d e n t i c a l with that of the number of degrees of freedom. The table i s an excerpt from an extensive table given by Pearson and Hartley (1956). The p r o b a b i l i t i e s i n table A-I:2 are the p r o b a b i l i t i e s of committing an error i n r e j e c t i n g the hy-pothesis that the d i s t r i b u t i o n of to i s uniform at a p a r t i c u -l a r l e v e l of s i g n i f i c a n c e . Two groupings of data were used i n t h i s a n a l y s i s . The f i r s t contains data relevant to a l l stars i n the catalogue regardless of q u a l i t y . These stars numbered 542. The se-cond group consisted of systems whose q u a l i t y of o r b i t was considered high by Batten (1967), that i s , those with qua-l i t y c l a s s i f i c a t i o n a, b, and c. These numbered 334. The reason for t h i s segregation was to q u a l i t a t i v e l y eliminate those systems with spectral l i n e d i s t o r t i o n , blending, and other spectral anomalies that have a high p r o b a b i l i t y of introducing spurious errors into o r b i t a l element computa-t i o n . For example, the evidence that spectral anomalies are a c t u a l l y responsible for d i s t o r t i o n of v e l o c i t y curves, and consequently for the de r i v a t i o n of spurious o r b i t a l elements - - i n p a r t i c u l a r the element to -- i s strongest i n e c l i p s i n g systems l i k e U Cephei. In t h i s case, the photo-metric elements are i r r e f u t a b l e . The primary and secondary eclipses are separated by equal i n t e r v a l s , which can only ar i s e when the o r b i t a l e c c e n t r i c i t y i s zero or when to = 90° or 270°. It seems clear that the spectroscopic o r b i t for th i s system, which gives to = 10°, must have been derived from a v e l o c i t y curve whose ordinates do not r e f l e c t or-b i t a l motion alone. Hence, an attempt was made to at least o O CM 0 O sr 0 O vO o O 00 0 O o iH V V V V V 3 3 3 3 3 V V V V V N o O o O CM e O sr 0 O SO o O 00 542 45 39 33 31 40 110 10 14 7 7 9 432 35 25 26 24 31 99 6 11 4 6 12 144 12 11 12 9 6 147 9 10 7 8 9 152 18 7 10 8 13 180 16 10 9 7 16 196 16 12 11 12 10 98 7 8 6 10 8 67 6 9 7 2 6 189 17 16 11 13 16 18 3 14 11 13 9 13 100 9 8 4 6 7 70 5 4 5 3 4 153 14 12 11 8 12 389 31 27 22 23 2 8 38 2 5 0 5 3 64 4 5 7 1 4 98 7 10 8 6 7 190 17 17 12 13 11 72 4 5 6 3 6 181 17 7 7 9 16 o O CM rH o O sr fH e O vO rH e O 00 rH 0 O o CM o O CM CM V V V V V V 3 3 3 3 3 3 V V V V V V o O O fH e o CM rH o o sr rH e O vO rH o O 00 rH o O o CM.1 33 28 26 27 27 32 9 4 6 4 5 4 24 24 20 23 22 28 9 6 6 9 6 5 10 10 7 4 7 4 8 8 5 8 6 11 6 4 8 6 8 12 8 10 8 7 10 11 12 9 10 9 10 12 6 6 4 6 6 3 7 2 4 5 1 6 11 11 1 0 12 8 8 11 11 11 8 7 12 7 5 2 5 8 5 4 1 3 2 4 7 13 9 4 10 6 11 2 0 19 2 2 17 21 21 3 5 2 0 2 3 7 1 1 6 3 9 10 5 3 7 4 3 10 12 11 7 9 14 2 4 3 3 7 4 11 7 9 10 7 f 11 0 O sr CM o O sO CM « o oo CM 0 O o CO e O CM co o O sr CO V V V V V V 3 3 3 3 3 3 V V V V V V e O CM CM 0 o <r CM 0 o SO CM e O 00 CM o O O CO e O CM CO 23 t 9 36 21 29 27 1 1 3 5 6 6 22 8 -33 16 23 21 2 0 4 4 4 2 3 L 13 7 9 7. 9 6 7 5 9 11 9 2 12 5 7 7 12 4 13 7 8 12 8 4 17 6 13 12 3 1 3 5 5 3 0 0 3 3 3 0 7. 1 9 7 9 7 9 4 16 11 10 8 4 3 9 2 6 5 3 1 2 1 4 7 3 3 9 7 6 4 20 6 27 14 23 23 0 0 2 0 2 2 1 1 3 1 3 3 2 0 8 2 7 4 5 3 12 8 11 7 2 1 3 6 5 2 14 5 12 5 6 14 o VO CO V 3 V e O sr co 36 ALL GROUPS 9 ECLIPSING 27 NON-ECLIPSING 3 00 < P < 3 0 12 3 D < P < 1 0 D 11 10D < P < 1 0 0 C 10 100D< P 12 00 < Kl < 25 13 25 < Kl < 6 0 8 60 < Kl < 1 0 0 3 100< Kl 16 0.0 < E < 0 . 1 5 0.1 < E < 0 . 3 5 0.3 < E < 0 . 5 10 0.5 < E 11 DOUBLE-LINE 25 SINGLE-LINE 2 DM .GT. 1 . 0 M 4 DM .LT. 1.0M 5 0 < SPT < 85 11 B5 < SPT < A5 6 A5 < SPT < F5 14 F5 < SPT Observed Distribution of to for Various Categories Table 1 o CM o sr o vO o oo o o r-t o CM r-l o sr rH o vO r-l o 00 r-l o o CM o CM CM o sr CM o vO CM o 00 CM o o co o CM CO o sr co o vO co V V V V V V V V V V V V V V V V V V 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 V V V V V V V V V V V V V V V V V V N o O e O CM o o sr o o vO e O 00 o o o e O CM o o sr e O vO e O 00 o O o CM 0 O CM CM e O sr CM 0 o vO CM 0 o 00 CM 0 O O CO e O CM CO e o sr CO 339 24 26 22 20 26 22 19 20 17 16 18 13 6 22 11 15 20 22 ALL GROUPS 54 3 7 6 2 4 3 4 6 2 1 2 1 0 2 2 3 1 5 ECLIPSING 285 21 19 16 18 22 19 15 14 15 15 16 12 6 20 9 12 19 17 NON-EC LI PS ING 60 6 6 4 2 8 6 4 6 3 3 4 0 0 1 -2 3 0 2 00 < P < 3D 98 4 9 9 5 5 6 8 7 4 3 2 2 0 10 5 5 6 8 3D < P < 10D 105 4 8 6 7 7 6 5 3 7 5 8 7 4 7 2 5 7 7 10D< P < 100D 76 10 3 3 6 6 4 2 4 3 5 4 4 2 4 2 2 7 5 100D< P 91 7 6 2 5 7 5 4 4 4 6 5 7 4 7 1 3 9 5 00 < K l < 2 5 99 5 3 6 6 6 5 5 8 6 5 5 5 1 9 2 6 9 7 25 < K l < 50 98 7 10 9 8 7 7 8 4 5 4 3 1 1 4 .7 4 2 7 50 < Kl < 100 51 5 7 5 1 6 5 2 4 2 1 5 0 0 2 1 2 0 3 100< K l 139 11 13 8 . 8 12 10 9 9 5 6 4 0 7 5 6 6 13 0.0 < E < 0 .1 107 7 6 8 5 8 8 5 8V 5 4 7 8 3 10 5 4 4 2 0.1 < E < 0.3 49 4 4 4 5 3 3 3 0 2 4 2 0 2 4 1 2 4 2 0.3 < E < 0.5 44 2. , 3 2 2 3. 4 u f 1. , 3 1 3 3 1 1 1 0 3 6 5 0.5 < E 16 5 5 8 6 4 7 6 2' " 3 6 3 2 0 0 4 1 4 0 4 0 < SPT < B5 114 9 11 8 7 6 6 9 10 3 5 7 3 1 10 4 5 5 5 85 < SPT < A 5 36 0 3 3 2 2 1 4 2 1 4 3 0 0 1 3 2 2 3 A5 < SPT < F 5 124 10 4 5 7 11 9 4 5 7 4 6 10 5 7 3 4 13 10 F5 < SPT Observed Distribution of High Quality Systems • Table 2 -27-_N_ ORDER 1 ORDER 2 ORDER 3 ORDER 4 CATEGORY 772 3 . 746 3. 926 4. 187 4 . 3 3 6 V ISUAL 542 8 . 541 17.6,60 1 8 . 4 4 9 1 9 . 3 1 9 ALL SPECTROSCOPIC 110 7. 701 1 9 . 7 0 2 2 2 . 0 8 6 2 2 . 1 1 2 E C L I P S I N G 432 3 . 170 4 . 8 9 4 , 5 . 6 3 0 5. 631 NON-ECL IPS ING 99 1 2 . 7 7 4 1 3 . 0 7 2 1 5 . 7 2 9 . 1 5 . 7 3 0 P < 3 DAYS 144 1 .693 10. 846 11 . 702 - 11 . 8 5 6 3D < P < 100 147 0 . 061 1.771 2 . 0 7 5 2 . 4 9 9 10D < P < 100D 152 2. 593 6 . 2 4 6 7 . 415 9 . 5 9 2 100D< P 180 0 . 3 6 0 2 . 5 5 6 2 . 589 3. 856 K l < 25 KM/SEC 196 0 . 3 1 9 3 . 8 6 0 3 . 860 4 . 0 7 4 25 < K l < 60 98 4 . 7 5 5 7 . 1 4 4 1 0 . 9 4 8 ( 1 0 . 9 4 8 60 < K l < 100 67 1 3 . 4 0 8 1 4 . 7 9 9 1 4 . 9 4 1 1 4 . 9 4 2 100< K l 189 8 . 1 4 0 14 . 829 1 8 . 2 0 5 . 1 9 . 7 1 8 0 . 0 < E <.0.1 183 3 . 0 46 3. 065 4 . 023 5. 195 0 . 1 < E < 0 . 3 100 1 .667 2 . 6 4 9 2 . 9 5 8 3 . 0 8 6 0 . 3 < E < 0 . 5 70 0. 849 8 .411 * 1 1 . 2 9 9 1 4 . 9 0 1 . 0 . 5 < E 153 7 . 7 3 7 1 0 . 5 1 4 1 1 . 1 7 4 1 1 . 6 8 4 DOUBLE-L INED 389 2 . 9 0 8 9 . 2 5 5 9 . 5 4 5 9 . 9 7 2 S I N G L E - L I N E D 38 3 . 7 5 8 4 . 0 2 8 7. 015 7 . 0 7 3 A M .GT . 1.0 MAG 64 1.820 1 .936 2 . 1 9 4 2 . 9 7 0 AM . L T . 1.0 MAG 98 6. 407 8. 955 1 0 . 1 9 3 1 0 . 9 9 4 0 < S PT < B5 190 7 . 2 0 8 1 1 . 168 1 1 . 170 . 1 1 . 3 4 6 B5 < SPT < A5 72 0 .101 0 . 9 7 4 1 .296 1 .297 A5 < SPT < F5 181 0 . 0 9 5 , 2. 178 2 . 4 1 2 5 . 996 F5 < SPT Test Results for Table 1 Table 3 - 2 8 -N_ ORDER 1 ORDER 2 ORDER 3 ORDER 4 CATEGORY 339 6.822 10. 946 . '14.434 14.618 ALL S PE CT ROS COP I 54 4.514 7.115 10.281 10.292 FCLIPSING 285 3. 701 5.989 7.583 7.762 NON-ECLIPSING 60 14.118 14. 132 15.739 16. 117 P < 3 DAYS 98 0.491 3.757 6.982 8.062. 3D < P < 10D 105 0.047 0. 102 0. 668 * 0.669 10D < P < 100D 76 1.271 4.709 4.720 6.965 100D< P 91 0. 00 5 0. 376 0.406 1.323 K l < 2 5 KM/SEC 99 0.446 0.616 2.625 2.735 25 < Kl < 5 0 98 8.759 12.304 ' 15.072 . 15.717 50 < K l < 100 51 12.028 13.563 14.146 14.376 100< Kl 139 4.764 9.917 14.795 . 16.479 0.0 < E < 0 . 1 107 2.598 3. 546 4.097 4.636 0.1 < E < 0 . 3 49 2.252 4. 089 4. 156 4. 775 0.3 < E < 0 . 5 44 0.528 3.385 7.978 ' 8.719 0.5 < E 65 11.283 12.913 14.743 14.822 0 < SPT < B5 114 5.765 6.719 6.737 6. 737 B5 < SPT < A5 179 15.525 17.925 18.774 18.804 0 < SPT < A5 36 0.004 - 0.004 2. 504 2.519 A5 < SPT < F5 215 12.737 14.752 16.964 17.008 0 < SPT < F 5 124 0. 145 2.362 3.637 . 4.608 F5 < SPT Test Results for Table 2 ) Table 4 a l l e v i a t e the most overt candidates of thi s c l a s s . Tables 1 and 2 show the c l a s s i f i c a t i o n of the data. They d i f f e r only i n the respect that while table 1 contains the data from the f u l l catalogue, table 2 contains data only from q u a l i t y classes a, b, and c. Tables 3 and 4 show the re s u l t s of the test of the hypothesis that to i s uniform for d i f f e r e n t categories of star systems. Recall that i f the hypothesis i s to be ac-cepted, the test c r i t e r i o n must be less than the c r i t i c a l value of x 2 at the a l e v e l of s i g n i f i c a n c e . The re s u l t s are tabulated for both table 1 and table 2. Discussion of the Test Results As can be seen from table 1 and table 2, the data from the Sixth Catalogue was divided into several catego* r i e s . These included those systems found to be e c l i p s i n g or non-eclipsing, for example; and those i n various ranges of the o r b i t a l elements, and spectral types. These catego-r i e s were further subdivided into v ranges of to, where v = 18. That i s , the data was compiled i n 20° i n t e r v a l s of to, with 0° < to < 360° , for a l l categories of i n t e r e s t . -30-The choice of the i n t e r v a l of 20° simply provided a prac-t i c a l means to have a large enough i n t e r v a l with a reason-able population but yet small enough to possibly observe fi n e structure trends, should they occur. To demonstrate the use of the tables, we see that i f the fourth-ordered test i s used at a l e v e l of s i g n i f i c a n c e o = 0.001, then the hypothesis that the d i s t r i b u t i o n of to i s uniform should be rejected whenever 2 2 2 2 2 = X i + , X 2 + X 3 + X u exceeds the c r i t i c a l value of 18.465 which i s x 2(4,0.001). As an example, we f i n d from table 4 that f o r the fourth-2 ordered test Ai, = 7.762 for non-eclipsing systems. Since t h i s value i s less than the c r i t i c a l value of 18.465 at a l e v e l of s i g n i f i c a n c e a = 0.001, we have a p r o b a b i l i t y of approximately 0.11 of r e j e c t i n g the hypothesis that to i s uniform i n a case when i n fact the hypothesis i s true. In th i s case, therefore, the d i s t r i b u t i o n of to i s uniform for non-eclipsing systems with high q u a l i t y o r b i t a l elements. The r e s u l t s noted from the ap p l i c a t i o n of the test c r i t e r i o n to the population from table 1 show that for the 0.1% l e v e l of s i g n i f i c a n c e , the d i s t r i b u t i o n of to, the longitude of periastron, i s not uniform for the t o t a l num-ber of systems i n the Sixth Catalogue taken c o l l e c t i v e l y . As a comparison, the same test was applied to the data for v i s u a l b i naries from Finsen and Worley's (1970) Third Cat-alogue of Orbits of Visual Binary Stars and i t suggested that the d i s t r i b u t i o n of to i s uniform for that catalogue. This i s shown i n table 3. In t h i s case, however, care must be taken not to attach too much s i g n i f i c a n c e to t h i s r e s u l t since the test was designed around those e f f e c t s that might perturb the v e l o c i t y curves of spectroscopic b i n a r i e s . Thus, i n the case of v i s u a l b i n a r i e s , i t might be more appropriate to use a test of lower order. This does not change the con-c l u s i o n , nevertheless, since the d i s t r i b u t i o n s t i l l appears to remain uniform i r r e s p e c t i v e of the order of test used. Closer inspection of the d i f f e r e n t categories of the star systems i n Batten's Catalogue reveals that the follow-ing sample populations exhibit the major non-uniformities i n to at the a = 0.001 confidence l e v e l : a) e c l i p s i n g systems b) systems with 0.0 < e < 0.1 For l e v e l s of s i g n i f i c a n c e of 1% and 51, the c r i t i c a l values 13.277 and 9.478, re s p e c t i v e l y , must be exceeded for r e j e c t i o n of the hypothesis. The test was applied and i t was found that for the complete catalogue, to i s non-uniform for the following categories at the 1% and 5% s i g n i f i c a n c e l e v e l s : a) e c l i p s i n g systems b) systems with P < 3 days c) systems with Ki > 100 km/sec d) systems with 0.0 < e < 0.1 This trend follows through to a l l levels of s i g n i f i c a n c e and one may thus make the claim that ca i s non-uniform i n these categories for the catalogued data of a r b i t r a r y q u a l i ty. We s h a l l now proceed to review the r e s u l t s of the high er q u a l i t y data. Thg data from table 2 was used i n the second test s e r i e s . The test c r i t e r i o n was applied for several l e v e l s of s i g n i f i c a n c e and the r e s u l t s tabulated i n table 4. It i s i n t e r e s t i n g to note that the d i s t r i b u t i o n of co i s uni-form for the fourth-order t e s t at the a = 0.001 l e v e l of s i g n i f i c a n c e . At the l e v e l s of s i g n i f i c a n c e , a = 0.01 and a - 0.05, however, to i s again computed to be non-uniform for c e r t a i n categories and that the major contributions to the non-uniformity are a r e s u l t of the population included i n the following categories: a) e c l i p s i n g systems b) systems, with P < 3 days c) systems with 50 < Kj < 100 km/sec d) systems with 0.0 < e < 0.1 e) systems with 0 < Spectral Type < B5 These r e s u l t s confirm the contention of Ovenden (1969) that s e l e c t i o n cannot account for the observed non-uniformities i n to for these categories. As a r e s u l t of t h i s a n a l y s i s , i t i s r e a d i l y becoming apparent that the near c i r c u l a r , short period, e c l i p s i n g systems are re l a t e d to the observed non-uniformities i n to. This i s not, however, j u s t i f i c a t i o n for the establishment of a causal r e l a t i o n between the two. On the other hand, i t i s highly possible that these systems i n p a r t i c u l a r are more prone to experience errors i n r a d i a l v e l o c i t y obser-vations, e s p e c i a l l y near e c l i p s e . Problems of the e f f e c t s of gas streaming, ci r c u m s t e l l a r material, and r a d i a t i o n d i l u t i o n have not yet been taken into account. These e f f e c t s may be responsible for the observed non-u n i f o r m i t i e s , and since the magnitude of the non-uniformity increases with the addition of low q u a l i t y data (as shown by a comparison of tables 3 and 4) i t i s postulated that the apparent non-uniform d i s t r i b u t i o n of to i s not due to a i p r e f e r e n t i a l o r i e n t a t i o n of the l i n e of apsides of spectro-scopic binaries towards the Sun, but to a r t i f a c t s d i r e c t l y connected with o r b i t determinations themselves. It i s s i g n i f i c a n t to note that according to Savedoff (1951) the group of stars belonging to the category of systems having a period less than f i v e days and having a spectral type e a r l i e r than A5 shows the largest differences -34-between the spectroscopically and photometrically determined values of (e cosco) . This r e s u l t further confirms the sus-p i c i o n that the true d i s t r i b u t i o n of u may be indeed uniform and that the value of w determined from spectroscopic measure-ment does not always y i e l d the true longitude of periastron because of a d i s t o r t i o n of the v e l o c i t y curve, as the photo-metric values i n d i c a t e . P e r i o d - e c c e n t r i c i t y C o r r e l a t i o n In his b r i e f summary of most of the well-established s t a t i s t i c a l r e l a t i o n s among binary s t a r s , Aitken (1935) re-viewed the suspected c o r r e l a t i o n between mean e c c e n t r i c i t y and period. In that analysis, and ones subsequent to i t , i t was found that for an increase i n the mean e c c e n t r i c i t y there was a corresponding increase i n the o r b i t a l period. This was held as a support for the theory that binary systems, both v i s u a l and spectroscopic, constitute one family of c e l e s t i a l objects. The data for v i s u a l systems was recent-l y reviewed by Dommanget (1963) who concludes, with other in v e s t i g a t o r s , that the apparent c o r r e l a t i o n which was as-sumed to be r e a l , was i n fac t a c t u a l l y due to s e l e c t i o n . For spectroscopic systems, however, an apparent c o r r e l a t i o n exists that seems much more d i f f i c u l t to dismiss. Heintz (1969) suggests i n a review a r t i c l e that the apparent trend for mean e c c e n t r i c i t y to increase as the period should not be interpreted d i r e c t l y i n terms of a c o r r e l a t i o n . He men-tions e f f o r t s by Varsavsky (1962) to connect an upper ec-c e n t r i c i t y l i m i t rather than mean o r b i t a l e c c e n t r i c i t i e s to the periods. A main feature, however, as pointed out by Walter (1950), i s that the or b i t s form two d i s t i n c t groups rather than a continuous d i s t r i b u t i o n i n the P-e diagram: -36-the short period c i r c u l a r o r b i t s , and the or b i t s d i s t r i b u t e d over nearly a l l e c c e n t r i c i t i e s which form the majority for periods longer than about 50 days, i . e . , for semi-major axes greater than 0.4 astronomical u n i t s . The disagreement of these investigators demonstrates the need to encourage further questioning into the matter. For t h i s reason, the problem was reconsidered using the data of the Sixth Catalogue. Selection e f f e c t s were also taken into account i n an attempt to discover the nature of the. apparent d i s t r i b u t i o n . A casual inspection of figure 8 shows the trend to be studied: we see that the mean e c c e n t r i c i t y appears to i n -crease as the logio of the period increases. A scatter diagram of the raw data i n the e-log P plane i s presented i n figure 9. In this f i g u r e , two major boundaries are seen en-clo s i n g the data; the one on the l e f t being a dynamical l i m i t imposed by the minimum admissable distance between the two stars at the moment of periastron passage, and the right-most being that due to s e l e c t i o n as described e a r l i e r . These notions are developed more c a r e f u l l y i n Appendix I I I . In order to test i f the data i s corr e l a t e d the b i -variate d i s t r i b u t i o n of e c c e n t r i c i t i e s and log(Periods) i n the Sixth Catalogue was grouped into c l a s s i n t e r v a l s of 0,6 •H u •P c o o u o c ccj o e 0.4 ~4 0.2 1.6 0.0 1.6 3.2 4.8 log P D i s t r i b u t i o n of Mean E c c e n t r i c i t y vs. Log P Figure 8 Scatter Plot of Observed E c c e n t r i c i t y vs. Log P / 0 0 ** 9 J • • • > • • • • • 7,. . <. • • • i Figure 9 -1.6 -o.a i o.o i 0.8 ~1 ™ T -1.6 2.4 i nnrDFDTrtni T 3.2 4.0 4.8 - T — S.6 0 = observed E = e x p e c t e d -39-t a b l e e n t r y ( e x p e c t e d ) XiY.1 521 X i 126.6 265.0 61.5 88.0 42.0 38.0 29.0 29.0 29.0 17.0 Y j 0.45 Log (Period)-0.0 0.0 0.11 0.23 0.0 0.05 0.0 0.08 0.0 0.04 2.0 0.03 0.0 0.02 0.0 0.0 0.02 0.02 0.0 0.02 18.75 16.0 0.0 0.02 0.26 2.39 1.33 0.0 0.06 0.0 0.56 0.0 0.32 o:o 0.13 0.0 1.18 0.68 0.0 0.03 0.27 0.0 0.16 0.0 0.04 0.0 0.39 0.0 0.22 0.0 0.02 3.0 0.19 2.0 0.11 0.0 0.19 2.0 0.17 0.0 0.10 1.0 0.01 0.0 0.13 2.0 0.07 0.0 0.0 0.01 0.01 0.0 2.0 0.13 0.13 0.0 1.0 0.07 0.07 0.0 0.08 0.0 0.08 2.0 0.04 0.0 0.01 2.0 0.08 0.0 0.05 1.90 0.0 0.46 0.0 0.97 0.0 0.22 0.0 0.32 0.0 0.15 0.0 0.14 2.0 0.11 2.0 1.0 0.11 0.11 0.0 0.06 2.0 0.07 3.66 3.05 6.66 0.0 0.89 0.0 0.74 0.0 1.62 0.0 1.86 0.0 1.55 0.0 3.39 2.0 0.43 0.0 0.36 4.0 0.79 3.0 0.62 2.0 0.52 7.0 1.12 3.0 0.29 2.0 0.25 2.0 0.54 2.0 0.27 2.0 0.22 2.0 0.49 2.0 0.20 2.0 0.17 2.0 0.37 0.0 2.0 0.20 0.20 2.0 2.0 0.17 0.17 3.0 1.0 0.37 0.37 2.0 0.12 0.0 0.10 2.0 0.22 1.0 0.13 0.0 0.11 2.0 0.24 3.30 5.00 0.0 0.80 0.0 1.22 0.0 1.69 0.0 2.54 2.0 0.39 0.0 0.59 2.0 0.56 3.0 0.84 2.0 0.27 2.0 0.40 0.0 0.24 2.0 0.36 2.0 0.18 2.0 0.28 0.0 2.0 0.18 0.18 3.0 5.0 0.28 0.28 2.0 0.11 2.0 0.16 2.0 0.12 0.0 0.18 9.24 6.79 8.69 11.95 9.04 17.44 18.11 26.18 0.0 2.24 0.0 1.65 0.0 2.11 0.0 2.90 0.0 2.19 2.0 4.24 4.0 4.40 14.0 6.36 2.0 4.70 2.0 3.45 5.0 4.42 2.0 6.08 7.0 4.59 22.0 8.87 24.0 9.21 45.0 13.32 7.0 1.09 2.0 0.80 3.0 1.03 24.0 1.41 9.0 1.07 26*. 0 2.06 20.0 2.14 27.0 3.09 5.0 1.56 6.0 1.15 5.0 1.47 13.0 2.02 8.0 1.53 22.0 2.95 8.0 3.06 20.0 4.42 5.0 0.74 0.0 0.55 4.0 0.70 5.0 0.96 2.0 0.73 2.0 1.41 4.0 1.46 4.0 2.11 2.0 0.67 5.0 0.49 3.0 0.63 2.0 0.87 2.0 0.66 5.0 1.27 4.0 1.32 7.0 1.91 2.0 0.51 4.0 0.39 2.0 0.48 2.0 0.66 2.0 0.50 3.0 0.97 3. 0 1.01 2.0 1.46 3.0 2.0 0.51 0.51 2.0 2.0 0.39 0.39 3.0 2.0 0.48 0.48 2.0 3.0 0.66 0.66 3.0 0.0 0.50 0.50 3.0 3.0 0.97 0.97 2.0 4.0 1.01 1.01 2.0 2.0 1.46 1.46 2.0 0.30 3.0 0.22 0.0 0.28 4.0 0.39 0.0 0.30 2.0 0.57 0.0 0.59 0.0 0.85 3.0 0.33 2.0 0.24 2.0 0.31 0.0 0.43 2.0 0.32 0.0 0.63 0.0 0.65 0.0 0.94 E c c e n t r i c i t y Table 5 -40-0.11 0.23 0.05 0.08 0.04 U7,24 0.02 0.02 0.02 0.20 0.20 0.01 0.06 0.13 0.03 0.04 0.02 0.19 69.44 0.01 0.01 0.01 0.01 0.01 0.56 1.18 0.27 0.39 42.32 19.84 0.13 0.13 5.88 0.08 44.28 0.07 0.32 0.68 0.16 0.22 33.49 0.10 11.59 0.07 11.59 29.29 0.05 22.43 0.46 0.97 0.22 0.32 0.15 0.14 33.84 33.84 7.54 0.06 54.89 0.06 0.89 1.86 0.75 9.18 24.80 2.01 3.11 0.20 3.11 6.52 5.71 0.11 0.74 1.55 0.36 4.28 12.51 2.73 4.09 19.84 4.09 0.10 0.11 0.09 1.62 3.39 13.14 30.87 3. 98 4.72 7.15 18.63 1.07 2. 82 2.42 15.72 0.80 1.68 0.35 11.30 0. 24 17.92 0.18 17.92 33.14 6. 52 6.67 8.00 1.22 2.54 0.59 5.51 6. 33 7.32 1.88 26.65 80.21 20. 70 0.18 0.15 2.24 2.91 32.00 26.59 24. 30 0.16 4.29 12.02 4.29 9. 59 21.44 0.28 1.65 1.74 1.79 20.53 0. 55 41.00 34.71 6.96 1.02 34. 76 2.34 0.21 2.11 0.08 3.79 29.06 15. 53 8.83 4.75 13.08 4.75 0. 28 9.09 0.27 2.90 4.24 112.42 59.79 16. 92 0.02 0.17 2.68 8.19 33. 52 0.43 0.37 2.19 1.26 58.98 27.44 2. 22 2.73 0.49 12.39 0.50 0. 29 8.63 0.28 2.47 19.43 94.45 22.01 0. 25 10.93 4.24 4.24 4.24 0. 33 0.63 0.54 0.04 23.74 149.23 7.98 4. 32 5.43 3.94 0.98 8.88 0. 59 0.65 0.56 U2 75.39 185.01 7.04 • 1. 69 13.58 0.20 0.14 0.14 0. 85 0.94 0.80 Zcol=29.57 142.99 659.92 251.67 200.72 237.19 201.95 152.08 163.47 164.97 158.34 49.91 X2 = 2412. 796 with 187 degrees of freedom X 2 Test for Observed Data Table 6 0 = observed E • expected Log(Period) »-Xi 170.51 900.07 340.07 177.08 20.26 t a b l e e n t r y m X l Y j (expected) n n - 1409.25 51.24 58.30 20.81 232.76 35.03 198.51 193.14 Yj 87.44 52.94 23.67 232.09 81.10 15.86 1.91 10.56 55.85 21.10 10.99 1.26 5.85 0.45 0.19 9*59 3.18 3.62 1.29 14.44 1.79 60.20 49.58 9.86 2.02 2.30 2.10 6.41 33.81 12.78 6.65 0.76 1.92 2.19 0.0 0.0 0.0 2.17 12.32 11.98 0.87 84.66 0.0 0.0 0.0 0.78 8:74 1.32 7.46 7.26 44.56 0.11 57.75 31.33 15.44 0.67 3.88 2.00 5.39 28.46 10.75 5.6 0.64 1.62 1.84 1.64 58.15 0.17 0.0 0.0 0.67 7.36 1.11 6.28 6.11 12.82 0.0 6.22 10.18 7.76 0.69 0.86 0.16 2. 72 0.0 0.0 19.57 0.0 1.55 8.19 3.09 1.61 0.18 0.47 0.53 0.19 2.12 0.32 1.81 1.76 23.48 0.0 0.12 36.21 21.21 3.29 0.16 0.16 2.84 14.99 5.67 2.95 0.34 0.85 0.97 0.61 20.96 0.35 3.88 3.67 0.0 0.0 0.58 3.31 3.22 39.22 16.50 31.63 3.66 0.0 2. 78 0.88 4.74 25.05 9.46 0.0 0.11 0.42 1.99 10.54 3.98 0.0 0. 74 6.27 3.83 20.20 7.63 0.0 0.0 0.0 0.44 2.34 0.88 83.64 4.93 5.04 2.07 9.42 3.97 1.09 0.46 2.44 0.56 0.0 0.24 3.94 0.45 0.63 0.05 29.89 1.43 3. 77 0.60 0.14 1.15 0.69 0.13 0.54 1.62 44.04 0.68 2.75 1.31 0.21 0.15 2.43 19.76 0.58 6.48 0.0 1.92 0.0 0.97 5.52 5.38 0.94 0.24 2.54 0.47 2.86 0.05 0.59 2.72 0.80 5.22 6.59 0.60 5.70 0.11 0.0 0.41 2.32 2.26 3.13 86.66 0.77 4.46 4.33 1.29 0.0 0.0 0.09 0.52 0.50 11.83 0.0 0.0 0.53 0.11 0.40 0.0 2.08 0.0 19.70 19.62 0.22 0.0 1.43 7.56 2.85 1.49 0.17 0.43 0.49 0.17 1.95 0.29 1.67 1.62 4.13 14.18 0.0 0.0 2.19 6.33 0.48 0.74 0.89 2.63 0.42 0.49 0.36 0.30 0.50 2.64 0.99 0.52 0.06 0.15 0.17 0.06 0.68 0.10 0.58 0.57 0.0 0.0 0.0 0.40 0.60 0.10 0.81 1.31 9.60 0.0 0.0 38.39 1.72 9.06 3.42 1.78 0.20 0.52 0.59 0.21 2.34 0.35 1.99 1.94 4.06 0.0 0.0 1.95 0.91 0.90 0.21 0.22 0.0 0.30 0.84 9.59 0.0 0.49 2.59 0.98 0.51 0.06 0.15 0.17 0.06 0.67 0.10 0.57 0.56 6.47 0.33 8.65 0.0 0.0 0.0 0.0 0.0 0.0 1.08 2.07 0.6*3 0.0 20.05 0.78 4.13 1.56 0.81 0.09 0.24 0.27 0.09 1.07 0.16 0.91 0.0 0.0 0.0 0.0 0.40 0.0 0. 32 0.0 0.40 0. 22 0.0 0.04 2.11 0.08 0.04 0.005 0.01 0.01 0.005 0.05 0.008 0.05 0.0 0.0 0.0 0.0 1.86 1.19 0.0 0.0 0.60 0.0 20.31 1.05 5.52 2.09 1.09 0.12 0.31 0.36 0.13 1.43 0.22 1.22 0.0 0.89 0.0 0.04 1.19 0.12 0. 32 0.0 0.0 0.0 0.0 0.0 0.0 0.46 0.0 0. 0 0.0 0.0 0.01 0.08 0.03 0.02 0.002 0.004 0.005 0.002 0.02 0.003 0.02 0.0 0.0 0.0 0.0 0.0 2.43 0.0 0.0 0.0 0.0 0.0 0.04 0.2 0.08 0.04 0.005 0.01 0.01 0.005 0.05 0.008 0.04 0.0 0.02 0.0 0.04 E c c e n t r i c i t y Table 7 -42-x2 - I -^51 16.28 556.14 170.61 2.16 0.34 2.24 2.78 0.94 1.63 2. 17 12.32 3.33 20.60 105.97 1.55 2.09 0.08 0.004 0.01 659.48 1. 32 7.46 5.17 30.14 39.40 17.29 0.001 3.15 0.01 1.40 350.49 0. 79 6.28 1.55 0.47 16.27 23.49 1.44 0.32 0.26 33.69 2.12 0. 32 174.26 2.84 14.75 164.49 113.03 29.60 0.56 0.68 0.19 75.19 16. 46 3.31 4.74 19.80 7.8 1256.60 6.31 566.40 0.72 5.9 27.22 0. 97 2.35 1.99 10.32 3.18 4.26 0.24 16.8 2764.84 2.04 1.66 68. 25 2.11 3.83 18.75 0.24 7.48 27.06 0.89 1.58 9.11 3.74 7. 3 1514.99 0.44 2.34 0.88 0.86 6.73 2.41 0.02 157.92 59.80 16. 00 0.52 1.43 7.56 1.29 1.28 0.31 0.43 5.16 0.17 161.6 1288. 40 1.26 7.26 6.11 1.76 3.22 5.38 0.50 2.64 1.45 64.92 2.94 2.32 3.05 110.08 0.09 1.52 0.08 0.13 1.72 9.06 3.42 1.07 0.80 0.34 0.09 5.76 22.52 0.35 1.99 684.85 0.49 2.59 0.96 0.31 11.76 0.02 0.01 0.06 0.20 5.48 142.74 0.56 0.78 4.13 1.56 0.81 0.09 0.24 2.43 43.56 0.18 0.16 402.57 0.89 0.04 2.11 0.08 0.04 31.20 0.01 9.61 0.01 2.45 1.56 0.05 0.04 1.05 5.52 2.09 1.09 25.23 2.50 0.36 0.13 0.48 0.22 298.71 59.29 0.01 0.08 0.03 0.02 0.002 0.004 41.41 0.002 0.02 0.003 0.02 0.02 0.04 0.20 0.08 0.04 0.005 201.64 0.01 0.005 0.05 0.008 0.04 0.04 42.23 707.20 519.8 1496.30 146.15 800.3 2833.03 370.9.1368.9 1411.2 2571.1 790.24 X2 = 13061.376 with 187 degrees of freedom. X 2 Test for Corrected Data Table 8 0.05 i n e and 0.40 i n log P. A contingency table was set up for both the raw data and the data corrected for s e l e c t i o n . The weighting factor for s e l e c t i o n was calculated as the i n -verse of the p r o b a b i l i t y of discovery as determined i n Ap-pendix II m u l t i p l i e d by the r e l a t i v e frequency of points per unit area per c e l l . This was done for a l l points i n the diagram. The contingency tables were constructed for the purpose of studying the r e l a t i o n between the two variables e and log P since we are p a r t i c u l a r l y interested to know i f they are correlated and, i f so, to what degree. Both cases have been corrected for the presence of the two boundaries. By means of the Chi-squared t e s t , x 2> i t i s possible to t e s t the hypothesis that the two variables are indepen-dent. Thus, i n connection with tables 5,6,7, and 8 , the X 2 test can be used to test the hypothesis that there i s no r e l a t i o n s h i p between the e c c e n t r i c i t y and the logarithm of the o r b i t a l period for spectroscopic binary systems. The tables represent the data with and without s e l e c t i o n e f f e c t s being accounted for and are described i n more d e t a i l i n Appendix IV. Before considering how the Chi-squared test may be applied to t h i s problem, some general remarks concerning the test are i n order. Consider a general contingency table containing R rows and C columns. Let p ^ be the p r o b a b i l i t y that an i n d i v i d u a l selected at random from the population under consideration w i l l be a member of the c e l l i n the i - t h row and j - t h column of the contingency table. Let p be the p r o b a b i l i t y that the in d i v i d u a l w i l l be a member of the i - t h row and l e t p ' be the p r o b a b i l i t y that the i n d i v i d u a l • J w i l l be a member of the j - t h column. The hypothesis that the two variables are independent can then be written i n the form: u «T> — T \ -ri i = l , 2 , . . . , r H o * P i j = p i . p . j ' 4 . . . J J j = 1,2,. . . ,c If a sample of size N ind i v i d u a l s i s selected and n ^ of them are found i n the c e l l i n the i - t h row and j - t h column, then the c l a s s i c a l Ghi-squared defined by X 2 = 1 ^ i c i where k i s the number of pairs of frequencies to be compared and CK and denote the i - t h pair of observed and expected frequencies, w i l l assume the form x 2 - ! i i n i j - ^ i ^ i 2 i J N p i A j under the hypothesis H Q. Since the p r o b a b i l i t i e s p. and p . are unknown, i t i s necessary to estimate them from the sample. This i s done by c a l c u l a t i n g the marginal p r o b a b i l i t i e s of the sample. That i s , we calculate the observed f r a c t i o n of the population i n each column and row and determine the expectation value Np p by multiplying t h e i r product by the sample s i z e . In i t . j other words, „ _ R i _ C j p i . " ~FT~ » p . j _ T T « « R l C j  p i . p . j = Since Ep. = 1 and Ep = 1 , there are (R-l + C-l) -(R+C-2) parameters that need to be estimated. Hence, the proper number of degrees of freedom for te s t i n g independence i n a contingency table of R rows and C columns i s given by v = RC-1 - (R+C-2) = (R-l) (C-l) . In the case under consideration, there are 18 7 degrees of freedom. The c l a s s i c a l x 2 f ° r t h i s value i s derived from the expression X 2 = \ {X +/(2v-l)} 2 due to Wilson and H i l f e r t y as described by Pearson (1935). Table 9 shows the values of x 2 calculated for 18 7 degrees of freedom for various levels of s i g n i f i c a n c e . The object of the test i s to r e j e c t the hypothesis that e and log P are independent i f the computed value of x 2 exceeds the value i n the table of the percentage points of the x 2 d i s t r i b u t i o n for the o l e v e l of s i g n i f i c a n c e . The r e s u l t s -46-1 7 C X + / ( 2v - 1) ) v = 187 a 0.995 0.990 0.975 0.950 0.900 0.750 0.500 0.250 0.100 0.050 0.025 0.010 0.005 0.001 -2.5758 -2.3263 -1.9600 -1.6449 -1.2816 -0 .6745 0.0000 0.6745 1.2816 1.6449 1.9600 2.3263 2.5758 3.0902 X 2U,187) 140 .0704 144.2775 150.5669 156 .0846 162.5694 173.7007 186.5000 199.7542 212.0731 219.0731 226.2747 234.1342 239.5643 250.9563 Table 9 raw data s e l e c t i o n corrected data Results for x 2 Test X 2 = 2412.796 X 2 = 13061.376 Table 10 of t h i s test are tabulated i n table 10 for both data sets. It i s obvious from t h i s t e s t that the values of x 2 imply that the variables e and log P are not independent. It must be mentioned here that the nature of the dependence is not demonstrated by t h i s t e s t . For t h i s reason, an attempt was made to determine whether the data i s l i n e a r l y dependent. This was done by c a l c u l a t i n g the c o r r e l a t i o n c o e f f i c i e n t in order to deter-mine the i n t e n s i t y and nature of any l i n e a r c o r r e l a t i o n that may be present. The philosophy of the test i s simply t h i s : suppose we have N pairs of observations ( e i , p i ) , ( e 2 , P 2 ) » •••(e N>P N) We f i r s t f i n d the means e" and p of the e and log P values taken separately, that i s _' N p = log P = (X log P,)/N i 1 e - (I e. )/N . i x 2 2 We then cal c u l a t e S , S , the observed p and e variences, i . e . , S* = (I log P - p ) 2 / ( N - l ) P t i S 2 = (I e - i ) 2 / ( N - l ) e i 1 -48-We then f i n d what is. c a l l e d the covarience, C , of the e ' e p ' and log P values: C e P = ^ ( l 0 g P i ' P " H e i " e ) ) / ( N - l ) . F i n a l l y , the (observed) c o r r e l a t i o n c o e f f i c i e n t r i s de-fined thus: r - C /(S S ) . ep' e p ' The c o r r e l a t i o n c o e f f i c i e n t can be calculated d i r e c t l y from the equation N ][(log P - p)(e' - e) r = - i - : N •> N _ 9 l/o * t l d o g P « - VV I ( e ± - e ) 2 ] 1 / 2 1 1 Now, just as the varience i n s t a t i s t i c s i s a measure of the deviations from the mean of a single set of observations such as e i , e 2,...,e n, so the covarience i s a measure of how the log P and e deviations vary together. Thus, i f large p o s i t i v e log P deviations are associated with large negative e deviations and vice-versa, C e p w i l l be large and negative, since there w i l l be many terms i n the numerator of the expression for r which w i l l be the product of a large p o s i t i v e and large negative term. Or, i f the large p o s i t i v e log P and e deviations and the large negative log P and e deviations are associated with each other, the covarience w i l l be large and p o s i t i v e . If there i s no p a r t i c u l a r a s s o c i a t i o n , however, the terms in the numerator w i l l tend to cancel each other thus making the covarience small. Note that the c o r r e l a t i o n c o e f f i c i e n t i s a c t u a l l y the r a t i o of the covarience C to the geometric mean of the log P and e variences, that i s , to /(S*S p) . The c o r r e l a t i o n c o e f f i c i e n t i s a c t u a l l y used as an estimator of the t o t a l population c o r r e l a t i o n c o e f f i c i e n t , p I f , for example, the population c o r r e l a t i o n c o e f f i c i e n t p has a value between, say, +0.8 and +1 t h i s indicates that there i s a strong p o s i t i v e c o r r e l a t i o n . Values between -0.8 and -1 indicate a strong negative c o r r e l a t i o n . If p a c t u a l l y equals +1 (or -1) we say that there i s perfect p o s i t i v e (or negative) c o r r e l a t i o n . Values around 0.5 or -0.5 indicate a f a i r amount of c o r r e l a t i o n ; between -0.2 and 0.2 indicate only a weak c o r r e l a t i o n ; and p = 0 means that (within l i m i t s ) there i s no c o r r e l a t i o n at a l l . The c o r r e l a t i o n c o e f f i c i e n t r i s also useful i n estimating regression l i n e s from samples. We may cal c u l a t e the l i n e a r regression for our sample by estimating the equation e -y e = ( p a e / a p ) ( l o g P -u p) where p i s the population c o r r e l a t i o n c o e f f i c i e n t ; u , u -50-are the true means of the log P and e values, and a 2 ,a 2 * e p t h e i r true variences. Our estimates come from the observed sample quantities p, e~, S g, S p, and r as calculated e a r l i e r . Substituting these estimates into the above equation y i e l d s e - e" = (r S e/S p) (log P - p) The r e s u l t s of t h i s c a l c u l a t i o n are shown i n table l l for various data sets. On the assumption that each pair of observations ( e ^ log P i ) i s selected from a normal b i v a r i a t e d i s t r i b u t i o n , we can test whether r i s so large that the hypothesis that p = 0 ( i . e . , there i s no c o r r e l a t i o n between e and log P) can be rejected. The test function (Mack, 1966) i s given by C = |r|/(N-l) If C i s greater than 1.96, 2.58, or 3.29 we can r e j e c t the hypothesis at, about, the 5, 1, or 0.11 p r o b a b i l i t y l e v e l s , r e s p e c t i v e l y . The above c r i t e r i o n gives us a method of de-cidin g that very l i k e l y there i s some association or cor-r e l a t i o n between e and log P. We can test a hypothesis about the strength of t h i s association i n t h i s way: given the hypothesis that the true c o r r e l a t i o n has a s p e c i f i e d value, p ' , we f i r s t c a l c u l a t e -51-(e - e) = r (Se/Sp) (log P - p) data . group e r Se/Sp P a l l systems 0 .226 0 .51281 0.19235 1 .449 t a l l systems 0 .346 0 .88995 0 .12217 3 .006 t t a l l systems 0 .208 0 .82879 0.11438 1 .870 * a l l systems 0 .211 0 .50790 0.22962 1 .288 t * a l l systems 0 .327 0 .88077 0.12877 2 .720 t t * a l l systems 0 .147 0 .76124 0.08157 1 .494 *log P < 1.6 0 .168 0 .56907 0.45433 0 .824 t t * l o g P < 1.6 0 .086 0 .61842 0.13128 0 .665 *log P > 2.0 0 .345 0 .24203 0.39999 2 .719 t t * l o g P > 2.0 0 .262 0 .76707 0.08129 3 .095 * high q u a l i t y data + w e i S h t = ^ d e t e c t t + » e i S h t - " d e t e c t X NApAe Table 11 and the test function C» - z 0|/(N-3) and i f C* i s greater than 1.96, 2.58, 3.29 we r e j e c t the hypothesis at, about, the 5, 1, 0.1% l e v e l . The r e s u l t s of these tests are shown i n table 12. It i s important to point out, however, that a s i g n i f i -cant value of r does not nece s s a r i l y imply that there i s a causal r e l a t i o n between e and log P. Heintz (1971) suggests, for instance, that one can r e a d i l y derive a cor-r e l a t i o n between accidental death rate and body length (sim-pl y because men have a higher occupational accident r i s k and an average t a l l e r body than women); t h i s r e s u l t i s mathemati-c a l l y f u l l y correct and w i l l yet not make sense. On the other hand, though, while independent variables are necessarily uncorrelated, the converse i s not true --uncorrelated variables are not necessarily independent. As an example, consider the random variables C and tj given by s i n 6 and n = cos6 where 6 i s a random variable d i s t r i b u t e d as TT/2 0 < 6 < 2TT Me) - 0 otherwise -53-data group N a l l systems 525 0 .51281 11 .73876 12 .94381 t a l l systems 3375 0 .88995 51 .69376 82 .55574 t t a l l systems 1453 0 .82879 31 .59201 45 .11080 * a l l systems 334 0 .50790 9 .26831 10 .18641 t * a l l systems 1474 0 .88077 33 .80367 52 .89697 t t * a l l systems 1210 0 .76124 26.46883 34 .71263 *log P < 1.6 243 0 .56907 8 .85265 10 .01006 t t * l o g P < 1.6 1083 0 .61842 20 .34216 23 .74189 *log P > 2.0 72 0 .24203 2 .03938 2 .05115 t t * l o g P > 2.0 925 0 .76707 23 .31691 30 .76435 *high q u a l i t y data * w " 8 h t " I T S I S c t n NApAe Table 12 Note that oo. f = s i n 6 = J sine PQ(e) d 6 •ioo f 2 1 r l s i n e d e = 0 * o S i m i l a r l y , n = o. The covarience, C_ i s given by ' Cn 6 7 CSn = ^ n = s i n 6 c o s 6 = j s i n 2 6 • T | 4 F 2 TT I s i n 2 6 d 6 = 0 . Hence, C and n are uncorrelated. But i t i s well known that 5 2 + n 2 = 1 . So that the variables £ and n are n o t independent! This demonstration amply suggests the danger of estab-l i s h i n g a causal r e l a t i o n to the nature of a mathematical c o r r e l a t i o n . In our case, e and log P appear to be cor-related and the c o r r e l a t i o n , although weak i n some cases for unweighted data, i s increased i n the p o s i t i v e sense when se l e c t i o n e f f e c t s are taken into account. Unfortunately, the model does not have the power to e s t a b l i s h the exact nature of the c o r r e l a t i o n and i t i s important to r e a l i z e that the res u l t s presented can only be interpreted i n i t s framework. We may speculate as to the physical cause of th i s cor-r e l a t i o n by suggesting some sort of i n t e r a c t i o n of the com-ponents of binary systems as being responsible agents i n the case that the c o r r e l a t i o n found i s meaningful i n the physical sense. A recent analysis by Piotrowski (1965) with respect to the v a r i a t i o n of o r b i t a l elements that occurs with mass transfer suggests that under these conditions, an increase of e c c e n t r i c i t y i s always accompanied by a increase i n period and, consequently, by the average mutual distance of the components. This e f f e c t may be one of the reasons why most close binaries have near c i r c u l a r o r b i t s and why d i s -t i n c t l y eccentric o r b i t s are met most exclusively among well separated systems. That mass loss occurs i s t h e o r e t i c a l l y well-founded, and may be observable as i n the case of obser-vations of circ u m s t e l l a r material. Furthermore, the e x i s t -ence of an e c c e n t r i c i t y - m e t a l l i c i t y c o r r e l a t i o n found by Smak (1967) gives the argument more weight, since systems having undergone mass loss are apt to display abundance anomalies. It should be emphasized, though, that apparent d e f i c i e n c i e s or overabundances can often be caused by an anomalous atmosphere rather than by t r u l y d i f f e r e n t abun-dances, and that d i f f e r e n t physical conditions i n the s t e l l a r -56-atmosphere may be the cause of an apparent discrepancy. For systems with periods less than, say, 100 days, i t i s possible that the above explanation offers a sol u t i o n to the problem of the p e r i o d - e c c e n t r i c i t y r e l a t i o n s h i p . For systems with periods greater than t h i s value, the nature of the c o r r e l a t i o n i s unknown and may merely be a mathematical a r t i f a c t without true physical meaning. Conclusions Longitudes of P e r i a s t r a It has been shown that for c e r t a i n categories of binary systems, the d i s t r i b u t i o n of the o r b i t a l element to, the lon-gitude of periastron, i s non-uniform for o r b i t s of a r b i t r a r y q u a l i t y . However, i f only the systems having high q u a l i t y o r b i t a l elements are considered, i t i s found that the major non-uniformities are eliminated or reduced. In both cases, s e l e c t i o n e f f e c t s have been applied to the data. It i s conceivable, then, that the major contribution to the non-uniformity of to i s related to those systems whose or-b i t a l q u a l i t y i s low and hence suffer from badly reduced or erroneous elements. There i s a tendency, as Sahade (1960) suggests, for gas streaming to force to into a p a r t i c u l a r quadrant of the c i r c l e by means of a c h a r a c t e r i s t i c d i s t o r -t i o n of the v e l o c i t y curve. Thus, i t i s very probably cor-rect to assume that the major non-uniformities i n the d i s -t r i b u t i o n of to are related to d i s t o r t i o n s of the v e l o c i t y curve, as Struve (1948) contends. S t i l l unresolved i s the apparent low density of spectroscopic binaries with to *\» 250° which was observed by Batten and Ovenden (1968) , although recent computations of o r b i t a l elements of newly discovered systems y i e l d values -58-of to i n t h i s range. Further observations may resolve t h i s problem completely. Pe r i o d - e c c e n t r i c i t y Correlation Tests applied to the observed d i s t r i b u t i o n of o r b i t a l e c c e n t r i c i t i e s vs. the logarithms of the periods have shown that e and log P are correlated i n the p o s i t i v e sense. Values for the Calculated x 2 a r e overwhelmingly in favour of t h i s conclusion when s e l e c t i o n e f f e c t s are applied to a contingency table a n a l y s i s . It was noticed that the devia-tions from the calculated expectation value were always large and i n the d i r e c t i o n of increasing e and log P. Furthermore, a l i n e a r regression analysis indicates a f a i r l y strong p o s i t i v e c o r r e l a t i o n i n the same context. It i s thus concluded that the observed data points and these points corrected for s e l e c t i o n e f f e c t s show a tendency to be correlated i n the p o s i t i v e sense. This does not mean, however, that there exists a d i r e c t causal r e l a t i o n between the two observables, the o r b i t a l eccen-t r i c i t y and the logarithm of the o r b i t a l period. One must r e a l i z e , though, that the variables i n question are highly observationally entwined. It may be that a multivariate analysis involving such factors as the semi-amplitude and mass of these objects could produce -59-a deeper understanding of the problem, but t h i s i s , un-fortunately, beyond the scope of t h i s paper. -60-Bibliography Aitken, R. G. 1935. The Binary Stars, New York, McGraw-Hill, 213. Barr, J . M. 1908. Jour. R.A.S. Canada 2_, 70. Batten, A. H. 1967 . Pub. Dom. Astrophys. Obs. 1_3, 119. Batten, A. H. and Ovenden, M.W. 1968. Pub. A.S.P. 8p_, 85. Blanco, V. M. and Williams, A.D. 1949. Pub. A.S.P. 6_1, 93. Dommanget, J . 1963. Ann. Obs. Roy. Belgique 9, part 5. Finsen, W. S. and Worley, C.E. 1970. Rep. Obs. Johannes. C i r . 7_,1 Heintz, W. D. 1969. Jour. R.A.S. Canada 63, 275. Heintz, W. D. 1971. personal communication. Lehmann-Filhes, R. 1894. Astr. Nachr. 136, 17. Mack, C. 1966. Essential S t a t i s t i c s for S c i e n t i s t s and Tech-nologists, London, Heinemann Ed. Books, Ltd., 116. Moore, J . H. and Neubauer, F.J. 1948. Lick Obs. B u l l . 20_, 1. Neyman, J. 1937. Skandinavisk A k t u a r i e t i d s k r i f t 20_, 149. Ovenden, M. W. 1969. Committee meeting, I.A.U., 3,4,5 May. Pearson, K. 1935. Tables of the Incomplete Gamma Function, London. Pearson, E. S. and Hartley, H.O. 1956. Biometrica Tables for S t a t i s t i c i a n s , 1_, 131. Piotrowski, S. L. 1965. B u l l . Acad. Polon. S c i . , Ser. s c i . , math., as t r . et phys. 12_, 419. Sahade, J . 1960. Stars and S t e l l a r Systems, 6, 474. Sahade, J . 1970. personal communication. Savedoff, M. P. 1951. Astron. Jour. 56, 1. -61-Scott, E. L. 1949. Astrophys. Jour. 109, 194. 446. Scott, E. L. 1951. in Statistical Astronomy by Trumpler 5 Weaver, New York, Dover Pub., 209. Smak, J. 1967. Comm. Obs. Roy. de Belgique, Ser. B, No. 17, 195. Struve, 0. 1924. Astrophys. Jour. 6fj, 167. Struve, 0. 1948. Pop. Astr. 5£, 348. Varsavsky, C. M. 1962 . Symposium on Stellar Evolution, ed. J . Sahade (La Plata: Observatorio Astronomi p. 173. Walter, K. 1950. Astr. Nachr. 279_, 1. - 6 2 -A P P E N D I X I The conceptual background of the test to ascertain whether the d i s t r i b u t i o n of the longitude of periastron, co, i s uniform i s due to J . Neyman (1937). He suggested the u t i l i z a t i o n of a "smooth" test to determine the good-ness of f i t between sets of a l t e r n a t i v e hypotheses. The smooth test admits a c e r t a i n set of hypotheses that are represented graphically i n figure A - I : l . These hypotheses w i l l be the p r o b a b i l i t y functions, p(a,H), represented by smooth curves not very d i f f e r e n t from the curve represent-ing the hypothesis tested, say H°, s p e c i f i e d by the proba-b i l i t y function, p(a,H°). The t y p i c a l p o s i t i o n i s i l l u s -trated i n the f i g u r e , where the continuous curve represents the p r o b a b i l i t y function p(a,H°) s p e c i f i e d by the hypothe-s i s tested, H°, and the discontinuous one corresponding to an alternate hypothesis H. Of course, t h i s d e s c r i p t i o n of the al t e r n a t i v e hypothesis requires further s p e c i f i c a t i o n . We require that perturbing elements may d i s t o r t the continuous curve i n figure A - I : l by crea t i n g , say, a few smooth waves, as suggested by the discontinuous curve. However, we r e j e c t the p o s s i b i l i t y that the d i s t r i b u t i o n can be represented by an i r r e g u l a r , say, ramp function. Neyman suggests that every "smooth" d i s t r i b u t i o n may rea-sonably be approximated by the d i s t r i b u t i o n p v = exp(* k(v)) -64-Smooth Deviations from a Theoretical P r o b a b i l i t y Function Figure A-I : 1 where "^(v) represents a polynomial of order k i n v having a r b i t r a r y c o e f f i c i e n t s and subject only to the condition that the sum over a l l v of p v is i d e n t i c a l l y equal to unity. Now, i n order to force that the admissible hypotheses be smooth, the polynomials ^ ( v ) are of r e l a t i v e l y low order. The order, k, must be fixed i n advance and represents the greatest number of maxima and minima that are considered possible i n the p r o b a b i l i t y function p v . The choice of the order of the polynomial and, hence, the order of the test i t s e l f i s discussed i n more d e t a i l i n Appendix I I . Scott (1949) has applied t h i s technique to the d i s -crete case, that i s , with grouped data, instead of for the case of a continuous d i s t r i b u t i o n . The technique of apply-ing the k-th order smooth test to ascertain whether the d i s t r i b u t i o n of co i s uniform w i l l now be discussed. As Neyman (1937) suggests, one computes the t e s t c r i t e r i o n A k = \ X i 1 = 1.2,....k and r e j e c t s the hypothesis, which i n t h i s case i s that to, the longitude of periastron, i s uniformly d i s t r i b u t e d , when-ever the computed value of A2, exceeds a c e r t a i n c r i t i c a l value of x 2(k » ° 0 « I R t n e above equation, \^ represents a function of the observations. They are determined as follows Consider n binary systems grouped into v classes of -66-equal p r o b a b i l i t y . Let 3^ be the number of stars i n the i - t h c l a s s . We then compute Y i = A f H I X(2l-l-v)J . for j=2,4,6,8 3 23 ^ i 1 = 1,2,...,v Table A - I : l gives values of Yj f ° r several values of v. The Yj values for d i f f e r e n t v may be used to te s t the hypothesis that an angle i s uniform i n d i s t r i b u t i o n between 0° and 360° when the observations are grouped into i n t e r v a l s of width (360°/v). Next, the weighted averages are computed: h = • n ^ ? ( v + 1 - " ) i ( B v + w + ( - 1 ) i e ^ V for i=l,2,3,4. We then compute: \Z = J L 1  1 n Y 2 , - (g2-ny2)  2 n(Y„-Y§) X 2 = ( Y z g a - y > C 02 3 nY 2(Y 2Y 6- Y*) ( ( Y 4 - Y l ) - ( Y s - Y a Y»)C2» n(Y2Y6-Y 2)) 2 " ( Y ^ - Y j ) ( C Y % - Y j ) ( Y 8 - Y 2 ) " ( Y 6 - Y 2 Y „ ) 2 ) And, f i n a l l y , -67-T ABLE OF THE FUNCTION GAMMA(J ) INTERVAL (DEGREES ) 1 0 ° 2 0 ° 30° 45° 6 0 ° 9 0 ° 1 8 0 ° J = 2 0.083269 0.083076 0.082755 0.082031 0.081018 0.078125 0.031250 JLH_4 0.012468 0.012372 0.012212 0.011856 0.011365 0.010009 0.001953 0.002212 0. 002184 0.002125 0.001997 0.001826 0 .00 1392 0. 000122 0 .000430 0.000418 0.000399 0.000358 0. 000308 0.000208 0.000008 Table A - I : l PERCENTAGE POINTS OF THE CHI-SQUARED DISTRIBUTION 0.990 0.950 0. 900 0. 500 0.100 0.050 0.010 0 . 0 0 1 k 1 0.000 0.004 0.016 0.455 2.706 3.841 6.635 1 0 . 8 2 8 2 0.020 0.102 0.211 1.386 4.605 5.991 9.210 1 3 . 8 1 6 3 0.115 C.352 0.584 2.366 6.251 7.814 11.345 16.266 4 0.297 0.711 1.064 3.357 7.779 9.487 13.277 18.467 a Table A-I:2 -68-k A2. = I A? for i=l,...,k . The c r i t i c a l values x2(k>°0 depend on the order k of the smooth test used and on the desired l e v e l of s i g n i f i c a n c e , a. They are described as the percentage points of the x 2 d i s t r i b u t i o n as given i n table A-I:2. The computed prob-a b i l i t i e s represent the p r o b a b i l i t i e s of committing an error i n r e j e c t i n g the hypothesis that the d i s t r i b u t i o n of to i s uniform. -69-APPENDIX II -70-Ovenden (1969) and Scott (1951) are among the many investigators who have attempted to apply s e l e c t i o n e f f e c t s to the discovery of spectroscopic binary systems. Of the myriad number of ways avail a b l e to do t h i s , no one method can cope with a l l the possible e f f e c t s that influence the discovery of a spectroscopic binary. Both external e f f e c t s , such as distance and apparent magnitude, and i n t e r n a l e f f e c t s such as various combinations of o r b i t a l elements, can account for the s e l e c t i v e i d e n t i f i a b i l i t y of spectroscopic binary s t a r s . It i s obvious that i t i s extremely d i f f i c u l t to take into account a l l of these selection effects simultane-ously, and so, at t h i s stage, we must be content to operate on a low-ordered l e v e l of approximation. The e f f e c t s of s e l e c t i o n on the d i s t r i b u t i o n of the longitude of periastron are many, as we have mentioned ear-l i e r . It was decided i n t h i s analysis to r e l a t e the e f f e c t s of s e l e c t i o n to the p r o b a b i l i t y of discovery of a hypothe-t i c a l binary system given a c e r t a i n range of o r b i t a l elements. This was done as a function of e, the o r b i t a l e c c e n t r i c i t y , and n - K/A , where K i s the semi-amplitude of the r a d i a l v e l o c i t y i n the usual convention, and A i s the minimum r a d i a l v e l o c i t y difference for recognition (Ovenden, 1969). A value of 10 km/sec was chosen for A since t h i s number was thought to represent a value consistent with that obtained by available medium-dispersion spectrographs. The philosophy of the determination of the p r o b a b i l i t y of discovery i s simply t h i s . We would l i k e to determine, as a function of e, the o r b i t a l e c c e n t r i c i t y , and n = K/A , the s e l e c t i o n c r i t e r i o n , the percentage of a l l possible pairs of r a d i a l v e l o c i t y measurements and subject to the condition that: > 1 . This i s shown gra p h i c a l l y i n figure A-II:1. The c a l c u l a t i o n was performed for both s i n g l e - l i n e d and double-lined spectra. For s i n g l e - l i n e d p a i r s , we c a l c u l a t e the r a d i a l v e l o c i t y i n the conventional manner, but with the systemic v e l o c i t y y set i d e n t i c a l l y equal to zero. Thus, for a s i n g l e - l i n e d system, we have: P\ = K ( c o s C v i + i o ) + e c o s i o ) and pi = K ( c o s C v j + u ) + e c o s i o ) . Here, p\ and p 2 represent the r a d i a l v e l o c i t y of the primary star at points 1 and 2 i n the o r b i t . K and u, i n the usual convention, represent the semi-amplitude of the r a d i a l ve-l o c i t y and the longitude of periastron, r e s p e c t i v e l y . Since the p o s i t i o n of the primary star varies with time, and v 2 , the true anomaly at positions 1 and 2, are.the only variables (t-T) Radial V e l o c i t y Measurements at A r b i t r a r y Points i n the Orbit of a Spectroscopic Binary Figure A-11:1 -73-i n the equations. The value | P i-P2 I K ( C 0 S ( V 2 + W ) - C 0 S ( V 2 + W ) ) was calculated. The c r i t i c a l case for any given K, e, and to occurs when I P1-P2I • A . The actual s e l e c t i o n c o e f f i c i e n t was calcul a t e d as the per-centage of a l l possible p a i r s i , j such that I p i - p j l > 1 . A S i m i l a r l y , for double-lined systems, we c a l c u l a t e PJ = K! (cos (v+toj) + e costoi) P2 = K 2(cos (v+to2) + e costo2) = -K 2(cos(v+toi) + e costoi) since to2 = toj + IT. Next we compute l p i ~P21 = K x * K z (cos(v •«,) • e costoi) A A where, as before, the c r i t i c a l case i s I P i - P2 I • A . In t h i s case, the s e l e c t i o n c o e f f i c i e n t becomes the percentage of a l l possible observations such that: l p*" p*l > 1 . A The values for both of these percentages, that i s , the s e l e c t i o n c o e f f i c i e n t s for both single and double-lined systems, were calculated numerically for d i f f e r e n t values of K/A, e, and to. This was done by d i v i d i n g the period of the o r b i t into 100 segments, finding the p o s i t i o n of the primary by i t e r a t i n g Kepler's Equation, solving for the r a d i a l v e l o c i t y , t e s t i n g for the c r i t i c a l case, and computing the s e l e c t i o n c o e f f i c i e n t . The r e s u l t s are plotted as percentages i n figures A-II:2 and tabulated i n tables A-II:1 and A-II:2. These percentages are to be re l a t e d to the actual select ive i d e n t i f i a b i l i t y of a spectroscopic binary. That i s , given a c e r t a i n combination of o r b i t a l elements, t h i s model w i l l predict the percentage of systems observed. It i s im-portant to note, however, that the model i s r e s t r i c t i v e i n the sense that i t assumes that a graph of r a d i a l v e l o c i t i e s a c t u a l l y conforms to Kepler's Laws for the motion of two bodies about a common centre of mass. Secondly, we must also r e a l i z e that no small perturbations of the elements are allowed. This i s hardly the case for the u l t r a - s h o r t period systems, for instance, where the e f f e c t s of mass loss and t i d a l i n t e r a c t i o n c e r t a i n l y can e f f e c t the computation of o r b i t a l elements. % Detection for S i n g l e - l i n e d spectroscopic binaries • i Figure A-II:2 1.1 1.3 175 TT? T""Q % Detection for Double-lined spectroscopic b i n a r i e s . -II-Percent Discovery for Si n g l e - l i n e d Spectroscopic Binaries e 0.0 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 log K/A 5.98 5.87 5.57 5.05 4.35 3.57 2.64 1.75 0.97 11.19 11.02 10.47 9.53 8.37 6.85 5.26 3.58 2.00 15.78 15.56 14.77 13.61 12.04 10.00 7.74 5.36 3 . 0 5 19.87 19.57 18.76 17.32 15.40 13.00 10.13 7.15 4.14 26.81 26.48 25.48 23.77 21.36 18.38 14.73 10.65 6.4? 32.57 32.18 31.05 29.16 26.57 23.16 18.95 14.10 8 . 7 4 37.37 36.90 35.74 33.83 31.03 27.31 22.83 17.44 11 . 0 8 41.35 41.04 39.82 37.82 34.93 31.17 26.34 20.47 1 3 . 5 7 44.98 44.55 43.36 41.35 38.41 34.49 29.59 23.47 15.81 48.09 47.64 46.46 44.44 41.51 37.61 32.51 26.12 1 8 . 1 5 50.84 50.39 49.18 47.18 44.25 40.32 35.18 28.65 20.41 53.28 52.79 51.65 49.72 46.73 42.82 37.76 31.16 22.50 11 . 3 3 55. 34 55. 03 53.85 51.91 49.02 45. 13 40.01 33.34 24.55 12 . 7 4 59.12 58.77 57.61 55.77 52.97 49.15 44.07 37.39 28.46 15.75 62.18 61.95 60.84 59.01 56.32 52.55 47.63 41.07 31.94 1 8 . 5 6 72.76 72.17 71.26 69. 73 67. 45 64. 29 59. 96 54.04 45.45 31 . 1 7 78.02 77.93 77.17 75,77 73.84 71.08 67.33 62.10 5 4 . 3 6 4 0 . 7 4 81. 72 81.63 80. 91 79. 74 78.03 75.60 72.28 67.60 6 0 . 5 8 4 7 . 8 S 84.48 84.14 83.62 82.58 80.98 78. 88 75. 87 71. 62 65.25 5 3 . 5 0 87.69 87.67 87.17 86.24 85.01 83.23 80.73 77.16 71.71 6 1 . 4 9 90. 10 89.90 89.37 88.72 87. 61 86. 06 83.91 80.80 76 . 0 8 6 7 . 0 1 92.84 92.89 92.74 92.21 91.31 90.15 68.49 86.18 82.57 7 5 . 4 9 94. 97 94. 84 94.46 94.02 93.40 92.40 91.08 89.11 8 6 . 1 6 80 . 3 2 96.05 9 5 . 9 6 95.69 95. 19 94. 59 93. 82 92. 69 91. 10 88. 5? 8 3 . 5 3 96.73 96.67 96.51 96.08 95.46 94.79 93.84 92.38 90.?1 8 5 . 8 0 98.04 98.05 97.91 <57. 69 97. 42 96. 96 96.21 95.24 93 . 8 4 9 0 . 8 ? 99.04 99.01 98.95 98.87 98.72 98.51 98.20 97.71 9 6 . 9 4 9 5 . 0 5 99.54 99.51 99.46 99.44 99.36 99.24 99.13 98.88 9 8 . 5 0 9 7 . 6 7 Table A-11:1 -78-Percent Discovery for Double-lined Spectroscopic Binaries e 0.0 0.10 0.20 0.3C 0.40 0. 50 0.60 0.70 0.80 log K/A 0.259 0. 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0. 0 0. 0 0.0 0.11 0.22 0.0 0.0 0.0 0.0 0.0 0.0 0.44 0.78 0.78 0. 0 0.0 0. 0 0. 0 0.0 1.11 1.56 1 .44 1 .11 0.0 0.0 0.0 1.56 3. 33 3. 78 3. 78 3. 22 2. 00 0.0 0.0 4.67 6.78 7.22 7.22 6.11 4.44 2.89 0.0 9.33 11.33 11. 67 11. 22 10. 11 8.22 6.44 4.11 27.56 22.11 18.89 17.22 15.00 13.33 10.78 7.94 5.11 37. 33 35.44 26. 89 22. 56 19. 11 16.33 13.22 9.89 6.22 44.00 43.00 38.11 28.56 23. 56 19. 56 15. 67 11. 78 7.56 49.33 48.33 45.11 35.44 28.11 22.33 18.00 13.39 8.67 53. 33 53.CO 49.78 43.67 32. 78 26. 00 20.33 15 .44 9.89 56.89 56.33 54.00 49.11 37.78 29.56 ?3.44 17.22 11.44 62.67 62.11 60.11 56.56 50.00 36.44 28.56 21.11 13.89 66.67 66.CO 64.33 61.78 56.89 45. 78 33.67 25. 22 16. 89 78.22 78.11 77.11 75.42 73.00 68.56 61.56 44.50 30 . 8 9 17.00 84.00 83.11 83.CO 81. 56 80. 33 77. 00 72.22 63.78 43.67 25.94 P 7 . l l 86.78 86.56 85.44 84.33 82.00 77.89 71.67 55.67 33.44 89.33 89.56 88.67 87.89 86.56 85.00 81.78 76.89 66.56 40.89 92.00 96.56 91.44 91. 22 90.22 88. 78 86.67 83. 22 76. 00 5?.83 93.78 93.11 93.22 92.78 92 . 0 0 90.89 89.39 86.67 81 . 0 0 63.89 95.56 95. 89 95. 78 95. 33 94. 56 94. 22 93. 00 91 .22 87 .11 78 .44 96.89 96.67 96.33 96.33 95.89 95.44 95.11 93.67 90.78 83.89 97.33 97.00 97.33 97.00 97.00 96.22 96.00 95.06 92 . 7 ? 87.33 97.78 97.22 97.89 97.56 97.56 96.89 96.67 96.00 93.78 89.44 98.67 99.22 98.78 98.44 98.44 98.17 98.11 97.67 96 . 0 0 93.78 99. 56 99. 78 99.22 "59. 1 1 99. 33 99.22 99.00 98 .67 98 .00 96 .78 99.56 99.78 99.78 99.67 99.78 99. 44 99. 44 9^.44 99.2? 98.?? Table A-II : 2 We can be reasonably sure of our a p p l i c a t i o n of t h i s single type of s e l e c t i o n , however, by a simple consideration of figure A-II:4. Here we have graphed the semi-amplitude against the period for the systems i n Batten's catalogue. For a system that obeys Kepler's Laws, Struve (1924) suggest ed that the two observable quantities K and P are related by - 0 . 3 3 K = CP . H e demonstrated that a l l bodies i n Keplerian motion must obey th i s r e l a t i o n whereas, for instance, the v e l o c i t y curves and periods of Cepheid variables do not. Ob v i o u s l y , we are indeed dealing with systems i n Keplerian motion and the a p p l i c a t i o n of our s e l e c t i o n i s acceptable i n t h i s sense. We have not considered e f f e c t s of other obser-vables such as apparent magnitude, however, but the t r e a t -ment of these e f f e c t s , even though they introduce further s e l e c t i o n , i s beyond the scope of t h i s paper. The r e s u l t s of the a p p l i c a t i o n of s e l e c t i o n to the d i s -covery of spectroscopic b i n a r i e s , has v e r i f i e d c e r t a i n i n -t u i t i v e considerations. Consider figures A-II:2,3. The p r o b a b i l i t y of i d e n t i f y i n g a spectroscopic binary as such increases as the o r b i t a l e c c e n t r i c i t y decreases and, also, as K/A increases. We also f i n d that the p r o b a b i l i t y of i d e n t i f y i n g a p a r t i c u l a r system as a binary increases as to departs from 0° and 180°, with maxima located near 90° and 270° for both s i n g l e - l i n e d and double-lined systems. This statement i s v e r i f i e d by a series of perspective surfaces -80-0 3 Scatter Plot Showing K - CP - 1 / 3 a CM CM •—I LH U J Q . LDm Om. -la' i * L." • « 03 a. i Figure A-II:4 co — i . i i r~ n * n o ~1 i fi 1 -j n I i P i P CO •IP VP IP CO V i in i IP 9? V i a-s • % C P Vi in Si I. <p in IP <P i P . 8 1 ' tn I CO V i Oft iP U-U-. ui i P <P u-i P -90-(figures A-II:5,13) depicting the p r o b a b i l i t y of detection (expressed as a percentage) as a function of n = K/A and w , for a given e c c e n t r i c i t y . The values of u range from 0° to 180° and a c h a r a c t e r i s t i c maximum i s seen at 90°. The sur-face i s symmetrical about 180° so that another maximum oc-curs at 270° With these facts i n mind, we may set the order of the test to determine i f u> i s uniform on the range 0° < w < 360°. From the analysis of s e l e c t i o n , we may a t t r i b u t e at least two maxima and minima to the sequence of p r o b a b i l i t i e s of the t h e o r e t i c a l d i s t r i b u t i o n of u. We may expect another maximum and minimum from q u a l i t a t i v e considerations of the action of gaseous streams on the determination of u> from spectra on the basis of a d i s t o r t i o n of the v e l o c i t y curve as suggested by Sahade (1970). Hence, we can expect the superposition of at least three maxima and minima i n the sequence of p r o b a b i l i t i e s , p^. In order to make the test most discriminating with respect to small deviations from uniformity with these three extremes, we w i l l use the fourth-order t e s t . -91-APPENDIX III The s e l e c t i o n e f f e c t s that are involved with the para-meters, o r b i t a l period and e c c e n t r i c i t y , are highly entwined. That th i s i s the case becomes r e a d i l y apparent when one i n -vestigates the fundamental r e l a t i o n of, say, the r a d i a l veloc-i t y with these parameters. In considering a p l o t of the o r b i t a l e c c e n t r i c i t y vs. the logarithm of the period i n days, several things are im-mediately obvious. There seem to be two major cut-offs i n the diagram. One at low values of log P around log P = 1.6 and another at higher values of log P, namely log P = 4.1. This i s shown i n figure A-III:1. In t h i s f i g u r e , we have chosen log P for the abscissa, simply because the percent-age change i n period, dP/P = d ZnP , i s more s i g n i f i c a n t than the numerical change i n period for a given change i n e c c e n t r i c i t y . It i s of i n t e r e s t to determine the nature of these apparent c u t - o f f s , and an attempt s h a l l be made to o f f e r an explanation, however crude i t s assumptions may be. In scru-t i n i z i n g the l e f t hand portion of figure A-III:1, we see an apparent c l u s t e r i n g of systems to near c i r c u l a r o r b i t s . It i s postulated that t h i s i s a physical b a r r i e r , that i s , a Figure A-III : 1 -94-dynamic r e s t r i c t i o n on the possible o r b i t s available for systems of short period. Simply, i t i s t h i s : two f i n i t e , extended bodies can exhibit Keplerian motion only u n t i l such time that a c e r t a i n combination of parameters, namely the period and e c c e n t r i c i t y , reach c e r t a i n l i m i t s , a f t e r which the bodies dynamically i n t e r a c t , i . e . , c o l l i d e . We may obtain a rough estimate of t h i s combination of e and P by studying the following s i t u a t i o n . Consider two bodies under a mutual central force at-t r a c t i o n revolving about a common centre of mass. This i s by d e f i n i t i o n , Keplerian motion. Let us further assume, for s i m p l i c i t y , that the bodies are of equal mass, say nMQ. Now the o r b i t s the bodies w i l l describe are s i m i l a r to that shown i n figure A-III:2. The point of closest approach, 6, w i l l occur at periastron, the minimal distance to the centre of mass, or barycentre. From the geometry of the system we have: 6 = (a-ae) = a(l-e) <$• = (a'-a»e») = a'(l-e') A = 6 + 6' = a(l-e) + a'(l-e») If the e l l i p s e s are the same, a = a' and e = e', so that A = 2a(l-e) . Hence, the maximum radius for equal e l l i p s e s (orbits) i s Hypothetical Orbit for Binary Stars to Determine the Point of Closest Approach Figure A-III:2 -96-given by R = i = a(l-e) max 2 so that R < a(l-e) where R i s the physical radius of eithe r s t a r . Now, from Kepler's Laws, we have Let M = M , so that: a 3 ( 2 n M ) M = fr2 i n solar masses. Solving for a we obtain a = ( a m y 1 ' 3 M 1 / 3 P 2 / 3 . Now, for "normal" s t a r s , the radius i s a function of the mass that i s , r = f (M) « M 1 ' 3 since, c l a s s i c a l l y , M = p t irR3 We thus have M 1 / 3 « f ( t f ) < ( 2 n M j M 1 / 3 P 2 / ? l - e ) Let K = 2nM -1/3 so that K oc g(M) < P or P 2 / ? l - e ) > K (1-e) > K P 3/2 (e-1) < K P 3/2 e < K P 3/2 + 1 Taking the logarithm of the previous expression we obtain log e = log K + 1.5 log P a l i n e a r equation i n e and P. We may e s t a b l i s h the mag-nitude of K from the data of the Sixth Catalogue as shown i n figure A-III:3. Here, a series of l i n e s with slope 1.5 have been drawn. We may a r b i t r a r i l y choose that l i n e which best contains the data points i n the f i g u r e , and hence estab-l i s h the value of K. This number was found to be approx-imately equal to log K = -1.775 . With t h i s information, the equation log e - -1.775 + 1.5 log P was plotted on the e-log P diagram. Since the majority of the data f a l l s to the r i g h t of the l i n e , as shown in figure we have a q u a l i t a t i v e estimate of the minimum radius bound-ary. It was noticed that a large f r a c t i o n of those systems to the l e f t of t h i s l i n e are i n t r i n s i c a l l y variable or are p a r t i c u l a r l y unusual systems. Table A-III:1 l i s t s several of these stars and t h e i r c h a r a c t e r i s t i c s . The right-hand boundary w i l l now be cosidered. In Appendix I, we assumed that the minimum detectable r a d i a l v e l o c i t y s h a l l be set at 10 km/sec. We can apply t h i s as-sumption to the d i s t r i b u t i o n of e and log P i n the following manner. Consider the equation for the semi-amplitude of the r a d i a l v e l o c i t y of a spectroscopic binary. We have: v _ n a s i n i . 2IT „ „ • . R , n r l / 2 /(1-e ) p— a s i n 1 (1-e } Now, i f the stars are equally massive, a 3 . G f ^ J L P l ^ 4ir 2 In the absence of e c l i p s e s , we must estimate the mean value of ( s i n i ) . I f the o r b i t a l planes are d i s t r i b u t e d at random, Schlesinger (19 35) finds s i n i = £ f s i n 3 i d i = |^ = 0.849 -100-Identification Batten # Comments RX And 30 a U Geminorum star HD 16506 DO Cass* 67 variable period? BD+32° 1582 YY Gem 246 i n t r i n . var. (detached) HD 64511 U Gem 257 dwarf nova W Pupp 268 i n t r i n . var. HD 76805 H Velr 284 HD 104350 AG Virg* 345 contact eclip . (W U Maj HD 107325 355 HD 134687 e Lupi 414 HD 139815 RW Cor B* 427 AE Aqar 628 eruptive var.; "fl i c k e r HD 200391 ER Vulp* 641 HD 204038 BV 342* 649 rapidly rotating; var. BD+52° 3383a RT And* 716 i n t r i n . var.; contact s *eclipsing Table of Unusual Systems Table A-III:1 -101 -Therefore, In the cgs system of u n i t s , K assumes the following value K = 268.18967 g j " ^ . ^ ) with the period given i n days. Replacing s i n i with i t s mean value the expression becomes: K = A P- 1 / 3 ( l - e 2 ) - 1 7 2 where log A - 2.35726. I f K = 10 km/sec, as we have assumed, then the s u b s t i -t u t i o n of t h i s value i n the above equation y i e l d s a r e l a t i o n purely i n e and P: log P = 4.07178 - 1.5 l b g ( l - e 2 ) . P l o t t i n g t h i s curve, we obtain the r i g h t hand boundary i n figure 9. It i s seen that these l i n e s e f f e c t i v e l y con-t a i n a l l the points under consideration and suggest a qual-i t a t i v e envelope for the spectroscopic b i n a r i e s . Thus we see that the two assumptions made regarding the d i s t r i b u t i o n of e vs log P are f a i r l y reasonable. It should also be pointed out that those points within -102-these bounds are also subject to the s e l e c t i o n e f f e c t s determined i n Appendix I I . A l i s t i n g of a l l of the systems i n the Sixth Catalogue i s made i n Appendix V. The l a s t column i n the l i s t i n g i s of p a r t i c u l a r i n t e r e s t since i t gives the p r o b a b i l i t y of detecting that system as a per-centage as determined i n Appendix I I . These p r o b a b i l i t i e s are subject only to the conditions that the minimum detect-able r a d i a l v e l o c i t y difference i s 10 km/sec and that the o r b i t has a random or i e n t a t i o n toward the sun. For con-venience, the systems are tabulated i n order of decreasing period. -103-APPENDIX IV -104-The philosophy of the use of a contingency table has been amply discussed i n the section concerning the e-log P c o r r e l a t i o n , and w i l l not be treated at length i n t h i s ap-pendix. Instead, we w i l l concentrate our e f f o r t s on a d i s -cussion of the a p p l i c a t i o n of s e l e c t i o n to the data. Consider table 5 on page 39. The marginal p r o b a b i l i t i e s for t h i s data set (whose numbers i n i t a l i c s represent the observed data from the Sixth Catalogue) are simply c a l c u l a -ted as the sum over a l l points located i n that p a r t i c u l a r row or column, divided by the i n t e r v a l Ae or Alog P i n which they occur. This d i v i s i o n enables one to correct for the r i g h t and left-hand boundaries shown i n figure 9. One then calculates the expectation value i n the normal fashion and, i n l i k e manner, the value of x 2 as shown i n table 6. The underlined values i n t h i s table r e f e r to a point of greatest deviation between the observed and expected values. The t o t a l x 2 f ° r t n e test i s simply the sum of the values i n table 6. Table 7 presents somewhat more of a problem. The ob-served values are calculated with a series of weighting factors as the following example w i l l show: Consider the -105-s i x t h observed ( i . e . , i n i t a l i c s ) value i n the bottom row of table 7. This number i s 1.43 and refers to the number 2 i n the analogous p o s i t i o n i n the top row of table 5. The tables have been reversed merely to separate them as i n -d i v i d u a l t e s t s , and any inconvenience to the reader brought on by t h i s t a c t i c i s unintentional. We f i n d that these 2 points l i e i n the range (e:0.85,0.90) and (log P:l.80,2.20). It so happens that the p r o b a b i l i t i e s of detection for these points, given an a r b i t r a r y to, are given by: p^ = 15.62% p 2 • 94.671 Let the number of stars be replaced by the sum of the i n -verse percentage of these numbers: n , = 100% + ioo% m 7 > 4 5 8 3 P i P2 Multiply t h i s by the r e l a t i v e frequency of points comprising the c e l l which contained the o r i g i n a l points: n " = n ' X NApAe = 7 - 4 5 8 3 X (521) (fj .05) (0 . 4 ) = 1.43 which y i e l d s the new "observed" value. Notice that non-i n t e g r a l numbers of " s t a r s " are allowed i n t h i s c a l c u l a t i o n . The marginal p r o b a b i l i t i e s are calculated i n the same fashion as i n table 5 and, hence, x 2 i n t n e usual convention. -106-APPENDIX V -107-The table to follow (table A-V:l) contains a l i s t i n g of the binary systems from the Sixth Catalogue of the Orbital Elements of Spectroscopic Binary Systems. The stars are i d e n t i f i e d by t h e i r catalogue number under the l a b e l SYSTEM. The table i s arranged according to decreasing period, and values of the longitude of periastron and o r b i t a l e c c e n t r i c i t y are also presented. A unique feature of t h i s tabulation i s the rightmost column which gives the p r o b a b i l i t y of detecting that system as a percentage subject only to the conditions that the minimum detectable r a d i a l v e l o c i t y change i s 10 km/sec and that the or b i t s are d i s t r i b u t e d at random. -108-SYSTEM OMEGA PERIOD K/D LGG(PERIOD) ECCEN, 499 166.60 32036.99219 0.31 4. 50565052 0.499 404 52.00 29651.99609 1 .00 4.47 20 53 53 0.529 331 127.20 21864.99609 O.C 4. 33974934 0.413 50 211.90 20146.09375 0.48 4. 30419064 0.608 222 145.70 18319.99219 0 .24 4. 26292515 0.595 455 113.80 16618.99609 0.38 4 . 22060490 0.440 435 343.60 16326.996C9 0.37 4. 21290684 0.750 324 174.00 16071.00000 0 .20 4. 20604229 0.350 421 218.00 15186.C2344 0.45 4 . 18144321 0.270 249 65.70 14693.99609 0. 13 4. 16714001 0.310 402 141.00 14609.99609 0 .60 4. 16464901 0.250 313 5.00 11430.99609 0.32 4 . 05808449 0.660 37 307.20 11124.99609 0. 41 4. 04629898 0.639 131 197 .00 10469.99609 1 .28 4. 01994610 0.650 149 347.80 10121.99219 1 .47 4.. 00526619 0.172 248 202.60 9751.99609 1.71 3. 98909283 0.460 623 351.20 9714.99609 0 .36 3. 98744202 0.350 736 274.40 9593.49609 0.51 3. 98197651 C.380 254 74.70 8524.99609 0.35 3. 93069363 0.750 415 7.60 8163.496C9 0 .34 3. 91187668 0.350 285 211.80 8107.99609 0 .26 3. 90891266 0. 170 655 42. 10 8015.996C9 1 .13 3. 90395737 0.544 673 26.00 7445.996C9 5.81 3. 87215519 0.340 500 97.50 6611.49609 0 .14 3. 82029915 0.279 203 43.00 6391.996C9 1 .49 3. 80563641 0.760 279 86.80 5491.99609 0. 73 3. 73972893 0.610 L i s t i n g of Spectroscopic Systems Table A-V:l % DETEC 1.00 32.11 1 .00 1.00 1.0C 1.00 1.00 1.00 1.00 1.00 10. 00 1.00 1 .00 22.40 50.40 42.97 1.00 C.99 1 .00 1.00 1.00 26.16 90.11 l .OC 17.19 11 .45 -109-SYSTEM OMEGA PERIOD 82 344 .00 5349 .99609 127 286.00 5199.99609 687 48. 50 4197.69922 283 33.00 4027.99951 13 19.80 3848.82959 423 185.40 3833.49951 600 201.10 3784.29932 575 270.00 3724.99927 155 77.00 3709.99976 586 186.30 3605.99976 171 270.00 3360.29932 406 55.10 3319.99951 179 105.00 3068.09937 145 227.00 3056.99927 210 168.00 2982.99951 256 190.00 2659.99951 605 342.70 2439.99951 720 225.70 2323.59912 232 214.60 2238.59937 508 187.20 2213.99951 442 340.00 2149.99951 104 13.80 1911.49927 633 43.10 1781.99927 292 301. 10 1700.75928 90 155.60 1654.89941 45 85.00 1651.99976 K/D LCG(PERIOD) ECCEN. % DETE< 3.46 3.72835255 C.720 81.7< 1.51 3.71600342 0.241 49.35 0.72 3.62301159 0.385 17.1^ 0 .60 3.60508919 0. 360 9.52 0.58 3.58532906 0.335 7.4S 0.92 3.58359528 0.406 2 5.84 3.48 3.57798481 0. 222 82.67 0 .75 3.57112503 0.320 20. 10 3.15 3.56937313 0.310 67.54 0. 45 3.55702591 0.480 1.00 1 .75 3.52637768 0. 100 57.68 0.78 3.52113724 0.218 24.14 1.33 3. 48686886 0.220 45.68 1.81 3.48529530 0.460 44.95 0.88 3.47465229 0.530 19.58 1.18 3. 42488194 0.400 34.73 0 .96 3.38738823 0.506 23.05 0.55 3.36616135 0.082 5.80 2.71 3.34997654 0.353 61 .76 I .79 3.34517670 0. 264 54.28 1 .60 3.33243847 0.6C0 31.70 0.52 3.28137398 0.210 2.12 0.53 3.25090790 0. 240 3.90 0.44 3.23064232 0.060 1. 00 0.44 3.21877098 0.26 3 1.00 0.33 3.218C0995 0.586 1.00 -110 if STEM OMEGA PERIOD K/D LOG(PERIOD) ECCEN 58 270.00 1649.99951 1 .35 3.21748257 0.750 309 238.90 1585.79932 0.37 3.20024681 0.138 118 66.70 1576.43970 1.81 3.19767666 0.407 49 94.00 1567.65991 1 .08 3. 19525051 0.356 78 234.60 1515.59912 1 .90 3.18058 300 0.734 316 37.60 1509.99951 0.92 3. 1789 7606 0.650 543 6.30 1434.99951 0 .35 3.15685177 G.014 607 119.10 1374.12 939 4.19 3.13802624 0.417 353 300.00 1299.99976 0.68 3.11394310 0.300 105 213.00 1254.67944 1.45 3.09853268 0.137 74 165 . 20 1202.19922 I .44 3.07997513 0.460 319 270.00 1199.99927 0.40 3,07918072 0. 100 481 222.30 1169.99976 4.65 3.C6818581 0.445 603 216.60 1140 .79932 6 .36 3.05720806 0.274 224 64.00 1065.99976 0.41 3.02775574 0.088 290 349.40 1062.39941 0. 39 3.C2628708 0.478 159 17.90 1031.39966 0.S7 3.01342583 0.098 393 140.00 1024.99951 0.82 3.01072216 0.400 653 80.00 1019.99976 0. 80 3.00859928 0.400 150 335.20 971 .99976 2.40 2.98766518 0.403 92 326.30 959 .99976 0.82 2.98227024 0.397 267 140.00 929.99976 1.08 2.96848202 0.400 296 92.30 921 .99976 1 .00 2.96472931 0. 293 403 260.80 869.99976 2 .31 2.93951893 0.130 216 117. 10 868.77979 1.06 2.93890858 0.695 369 0.0 846.99976 0 .78 2.92788315 0.400 - I l l SYSTEM OMEGA PERIOD 618 8 3 . 7 0 8 4 0 . 5 9 9 8 5 528 3 3 . 9 0 8 3 3 . 9 9 9 7 6 700 5 . 6 0 8 1 7 . 9 9 9 7 6 471 0 . 0 785 . 99976 547 9 1 . 1 0 7 7 7 . 9 9 9 7 6 410 0 . 0 7 50 . 0CCCC 85 1 2 8 . 1 0 6 8 0 . 0 9 9 8 5 580 2 0 2 . 30 6 7 6 . 1 9 9 7 1 213 2 0 7 . 0 0 6 7 4 . 9 9 9 7 6 330 3 2 0 . 0 0 6 6 9 . 1 7 9 6 9 175 1 3 5 . 5 0 6 5 5 . 1 5 9 9 1 39 8 2 8 7 . 3 0 5 7 5 . 2 3 9 5 0 7 1 4 5 . 7 0 5 5 6 . 1 9 9 7 1 390 3 26 . 30 4 9 4 . 1 7 2 6 1 130 5 5 . 0 0 4 8 8 . 4 9 9 7 6 509 2 3 4 . 5 0 4 7 7 . 9 9 9 7 6 348 2 3 5 . 3 0 4 6 0 . 9 9 9 7 6 189 2 0 0 . 7 0 4 4 5 . 7 3 9 5 0 38 3 5 8 . 6 0 4 3 6 . 9 9 9 7 6 164 4 0 . 0 0 4 3 4 . 7 9 9 8 0 571 2 1 6 . 5 0 4 3 4 . 0 8 5 9 4 450 2 4 . 6 0 4 1 C . 5 7 4 7 1 715 2 4 0 . 8 0 4 0 9 . 6 1 3 5 3 151 8 4 . 0 0 3 9 1 . 6 9 9 7 1 240 1 0 7 . 4 0 3 8 8 . 9 9 9 7 6 517 2 7 7 . 0 0 3 8 5 . 9 9 9 7 6 K/D LOG(PER IOD) ECCEN. % DETE I .38 2 . 9 2 4 5 8 9 1 6 0 . 0 3 0 5 0 . 5 1 .66 2 . 9 2 1 1 6 5 4 7 0 . 3 5 0 4 8 . 5 1.42 2 . 9 1 2 7 5 3 1 1 0 . 1 5 5 49.71 1 .00 2 . 8 9 5 4 2 1 0 3 0 . 3 0 0 3 2 . 7: 1 .93 2 . 8 9 0 9 7 9 7 7 0 . 3 1 0 5 4 . 5 : 0 . 6 7 2 . 8 7 5 0 6 1 0 4 0 . 0 1 7 . 2 3 4 . 3 0 2 . 8 3 2 5 7 2 9 4 0 . 2 1 1 85 . 76 1 .50 2 . 8 3 0 0 7 4 3 1 0 . 2 4 6 4 9 . 0 3 1 .35 2 . 8 2 9 3 0 2 7 9 0 . 5 7 0 2 9 . 3 5 0 . 8 0 2 . 8 2 5 5 4 2 4 5 0 . 5 3 1 1 6 . 4 2 2 .05 2 . 8 1 6 3 4 6 1 7 0 . 1 71 6 0 . 9 1 1.16 2 . 7 5 9 8 4 7 6 4 0 . 0 7 0 42 . 97 2 . 3 0 2 . 7 4 5 2 2 9 7 2 0 . 2 81 6 0 . 8 3 0 .84 2 . 6 9 3 8 7 9 1 3 0 . 258 2 6 . 6 9 1 .85 2 . 6 8 8 8 6 2 8 0 0 . 0 9 4 5 9 . 3 3 1.61 2 . 6 7 9 4 2 7 1 5 0 . 3 9 8 4 4 . 8 3 1 .42 2 . 6 6 3 6 9 9 1 5 0 . 1 6 9 4 9 . 6 4 2 . 8 6 2 . 6 4 9 0 8 0 2 8 0 . 5 4 9 5 0 . 6 1 1 .23 2 . 6 4 0 4 8 1 0 C 0 . 1 3 4 4 4 . 9 1 1.48 2 . 6 3 8 2 8 8 5 0 0 . 100 52 . 22 5 . 47 2 . 6 3 7 5 7 5 1 5 0 . 5 1 6 9 1 . 1 1 1.28 2 . 6 1 3 3 9 1 8 8 0 . 5 5 0 2 9 . 1 5 1 .36 2 . 6 1 2 3 7 4 3 1 0 . 6 5 6 2 3 . 0 5 0 . 5 8 2 . 5 9 2 9 5 2 7 3 0 . 3 7 0 7 . 70 1 .86 2 . 5 8 9 9 4 7 7 0 0 . 3 1 0 5 3 . 4 3 4 . 7 0 2 . 5 8 6 5 8 6 0 0 C . 4 7 0 8 9 . CO -112-SY STEM OMEGA PERIOD 356 0.0 369.48950 458 80.70 363.56982 141 268.40 360.46973 709 10.00 356.56665 69 267.70 33C.99976 306 0.0 329.29980 422 301.00 298.74976 314 311.00 292.55957 117 266.5G 283.27173 343 312.00 282 .68994 518 126.00 280.68286 206 0.0 275.99976 560 152.70 266.54370 242 349.30 257.79980 137 282.00 251.20494 601 140.00 250.99997 541 171.00 245.29997 218 8.0C 232.49994 311 236.40 230.08893 434 90.00 227.59998 420 339.5C 226.94998 644 16.60 225.43996 401 223.40 211 .94992 626 321.40 204.99995 349 74.00 198.52994 225 0.0 195.31995 K/D LOG(PERIOD) ECC EN. % DETE 2.53 2. 56760216 0.602 43. 1 0.60 2. 56058693 0.347 9.3' 5. 14 2. 55686760 0.280 77. 9< 2.15 2. 55214119 0.7C0 29. 52 4.62 2. 51982689 0.670 89.7? 1.41 2. 51759148 0.0 51 .0<5 0 .73 2. 47530651 0.580 11. 9C I .45 2. 46621418 0.300 46.32 2.07 2. 45220280 0.057 63.01 2 .62 2. 45131016 0.265 64.63 1 .72 2.44821453 0.419 45.92 2.80 2. 44090748 0.0 78.00 2.99 2. 42576790 0.833 20.99 1 .85 2. 41128254 0.170 56.28 0.50 2. 40002823 0.334 I.00 2.68 2. 39967251 0.050 69.60 1 .60 2. 38969612 0. 116 54. 53 5.15 2. 36642170 0.020 81.92 0.74 2. 36189461 0.061 22.97 5.75 2. 35717106 0. 060 89.22 1 .40 2. 35592842 0.680 22.28 2. 19 2. 35302925 0.226 61 .35 1.80 2. 32623196 0.540 39. 13 0.98 2. 31175232 0.128 35.41 6.57 2. 29782486 0.048 85.47 2.48 2. 29074669 0.0 6 8.06 113-/STEM OMEGA PERIOD K/D LOG!PERIOD) ECCEN 39 0.0 193.78995 1 .60 2. 287 330 63 0.0 187 333.00 180.87593 2.24 2. 25738049 0.509 504 79.20 180 .44994 5.68 2. 25635624 0.395 376 4.30 175 .54996 0 .63 2. 24440098 0.463 565 79.70 164.63992 1.18 2. 21653366 0.118 234 107.70 154.89993 4. 95 2. 19005108 0.300 18 349.00 143.60693 16.49 2. 15717411 0.562 63 188.20 142.32997 1 .78 2. 15329552 0.292 129 49.10 140.728C0 3.10 2. 14838028 0.750 539 274.30 138.41997 2.35 2. 14119816 0. 114 562 16.80 137.95695 4.91 2. 13974285 0.060 253 247.30 137.76694 0.36 2. 13914490 0.290 432 156.60 137.54997 2.12 2. 13846016 0. 137 38 188.20 134.07794 1 .99 2. 12735653 0.310 186 318.50 132.90994 0.89 2. 12355614 0. 162 207 6.60 131 .21097 3.33 2. 11796856 0.644 42 270.00 126.62593 12.00 2. 10252190 0. ICO 134 285.00 120.99997 2.82 2. 08278561 0.019 616 288.90 117.77597 2 .26 2. 07105637 C.240 297 96.20 116.64998 4.65 2. 06688309 0. 190 26 166.4C 115.70995 3.77 2. 06336975 0.005 12 18.70 115.32996 1. 53 2. 06194115 0.334 568 87.00 108.57098 2.21 2. 03571320 0.C54 436 0.0 108.07494 4.28 2. 03372383 0.0 48 24. 20 106.99693 3.76 2. C2937031 0.892 424 30.00 105.79996 1.12 2. 02448559 0. 100 SYSTEM OMEGA 161 343.OC 405 186.80 100 0.0 643 0.0 263 CO 3 74.90 2 222.10 272 0.0 60 101.00 540 191.30 621 231.OC 590 0.0 29 345.20 43 359.40 333 218.70 286 90.00 1 337.70 668 238. 10 354 191.60 583 350.30 339 331.20 47 50.00 153 273.90 5 313.3C 466 351.10 476 67.50 -114-PERIOD K/D 104.02298 2.64 101.55998 1.92 100.45999 2.60 97.55998 2.40 97.49997 3.70 96.69899 3.02 96.40997 2.39 96.10779 4.26 93.49997 1.94 88.35199 3.97 87.68700 3.25 84.99997 9.80 81.12000 11.43 78.00728 2.01 74.86098 2.88 74.14688 1.78 72.92999 1.64 72.01619 0.78 71.89999 7.42 71.79997 3.61 71.69998 2.37 69.91997 2.96 58.30998 3.67 55.90399 30.10 52.1C997 1.71 51.57797 3.61 LCG(PERIOD) ECCEN. 2.01712799 C.015 2.00672150 0.042 2.00199318 0.522 1.98927021 0.0 1.98900414 0.0 1.98542023 0.538 1.98412037 0.124 1.98275757 0.0 1.97080994 0.445 1.94621563 0.370 1.94293499 0.761 1.92941761 0.0 1.90912724 0.898 1.89213467 0.017 1.87425423 0.029 1.87009144 0.050 1.86290455 0.272 1.85742855 0.034 1.8567276C 0.336 1.85612297 0.419 1.85551739 0.082 1.84460068 0.300 1.76574135 0.189 1.74744225 0.033 1.7169199C 0.149 1.71246338 0.022 -115 SYSTEM OMEGA PERIOD 394 24.50 51.41997 639 66. 80 48.60797 143 26.50 45 .45398 363 238.80 44.41370 326 330.00 4C.44S98 446 87.20 39.88799 467 323.50 39.48088 139 314.30 39.28067 384 326.70 39.27997 430 206.80 38.93700 487 329.90 38.59579 599 280.00 38.43929 367 210.00 38.32399 40 178.80 36.58798 4 10.00 36.56697 38 5 258.20 36.03999 167 0.0 35.49997 35 320.10 35.37099 469 287.10 34.81688 444 308.00 34.23000 23 269.7C 33.74997 281 85.20 33.31099 341 223.60 32.86398 229 82. 10 32.80917 83 0.0 32.31497 250 44.00 31 .49997 K/D LCG(PERIOD) ECCEN. X DETE 4.69 1.71113205 0.380 72.2. 4.22 1.68670750 0.099 78.6! 1 .36 1.65757084 0.391 39.9 ' 2 .59 1.647 51530 C. 250 64. 5! 1 .80 1.60691738 0.100 58.52 3. 87 1.6C084057 0.280 72.91 3.18 1.59638691 0.039 73.21 2.58 1.59417820 0.360 60.10 4.32 1.59417057 0.786 35.01 5.36 1.59036255 0.796 38.57 1.23 1.586 53927 0.271 41.68 11.50 1.58477402 0.250 94.67 4.80 1.58346939 0.074 80.67 3.00 1.56333828 0.2C3 69.88 5.00 1.56308842 0.500 67.27 11.98 1.55678463 0.490 96.CC 2.80 1.55022812 C O 71.13 7.95 1.54864693 0.627 94.44 10.41 1.54181385 0.423 95.11 3.40 1.53440571 0.360 66.73 1 .63 1.52827263 0.405 44.91 27. 17 1.52258587 0.634 98.33 3.73 1.51671982 C.087 76. 10 2.75 1.51599503 0.080 70.08 3.60 1.5C940323 0.0 76.22 9.76 1.49831009 0.208 94.00 SYSTEM OMEGA PERIOD 55 318.40 31 .38838 110 214.40 30.43378 181 299.50 30.27798 507 0.0 29.67497 185 116.90 29.13509 555 330.10 28.59CCG 205 178.40 28.27997 576 210.70 27.96997 188 335.00 27.86397 10 140.00 27.79997 449 333.70 27.21797 190 84.60 27.15457 523 222.00 26.38997 662 0.6 26.32997 480 14.50 26.27417 108 120.00 26.09998 395 150.80 26.00497 704 0.0 24.64999 346 358.90 24.48279 681 60.40 24.43098 335 0.0 23.54149 726 0.0 23.28558 473 129.90 23.24498 211 133.60 23.17558 552 92.70 21.99799 650 0.0 21 .72397 K/D LCG(PERIOD) ECCEN. 10.25 1.49676800 0.534 10.55 1.48335457 0.612 2.16 1.48112679 0.217 22.00 1.47238922 0.0 31.10 1.46441460 0.758 4.C3 1.4562130C 0.041 11.39 1.45147896 0.556 10.45 1.44669056 0.350 2.55 1.44504166 0.550 3.00 1.44404316 0.200 7.60 1.43485451 0.744 22.64 1.43384266 0.730 1.24 1.42143822 0.176 1.65 1.42044926 0.0 9.82 1.41952801 0.491 1.40 1.41663933 0.100 3.96 1.41505623 0.202 3.32 1.39181614 0.0 10.22 1.38885975 0.609 3.31 1.38794041 0.027 2.58 1.37183285 0.350 2.70 1.36708546 0.0 2.77 1.36632824 0.427 9.85 1.36503029 0.024 1.28 1.34238148 0.205 4.14 1.33693886 0.0 -117-YSTEM OMEGA PERIOD K/D LCG<PERIOD) ECCEN. % OETEl 521 12.70 21.70558 3.49 1.33657074 0.218 72.62 699 213.10 21.69968 11 .68 1.33645344 0.380 95.56 276 0.0 21.64297 11 .00 1.33531666 0.0 94.OC 615 0.0 21.63997 13.00 1.33525658 0.0 91. 9C 237 102.00 21.20697 3.22 1.32647705 0.170 72.34 184 151.00 21.03148 10.58 1.32286835 0.131 90.08 375 104.20 20.53859 13.64 1.31256962 0.537 96.67 724 313.60 20.52119 0.66 1.31220150 0.040 16.75 646 219.70 20.29997 12.47 I.30749512 0.441 96.11 373 120.00 20.00519 5.65 1.30114174 0. 120 83. 17 640 259.00 19.69827 3.21 1.29442692 0.186 71.66 251 0.0 19.6050C 3.42 1.29236603 0.0 75.25 291 252.50 19.45888 2.02 1.28911686 0.206 59.78 325 333.00 18.89220 4.00 1.27628136 0.282 73.23 519 78.00 18.84558 6.50 1.27520943 0.510 71.58 174 0.0 18.8CC87 2.20 1.27417660 0.0 64.91 109 317.40 18.78058 3.17 1.27370930 0. 198 71.29 264 208. 10 18.72197 13.61 1.27235031 0.174 95.44 182 110.00 18.64999 3.CO 1.27067757 0.300 66.28 631 0.0 18.59718 9.32 1.26944542 0.0 89.39 630 345.40 18.06679 11 .10 1.25687981 0.091 94.33 88 90.50 17.92197 1.87 1.25338459 0.310 53.88 241 214.80 17.91098 4.56 1.25311947 0.520 53. 86 412 57.20 17.83359 10.52 1.25123787 0.202 94.00 24 0.0 17.7692C 2.63 1.24966717 0.0 69.35 691 103.30 17.75497 4.02 1.24931908 0.023 77.94 -118-CSTEM OMEGA PERIOD K/D ICG(PERIOD) ECCEN 42 5 304.20 17.35997 3.48 1. 23954773 0.377 672 269.90 17.32629 27.41 1. 23870564 0.224 597 34.50 17 .12428 11 .47 1. 23361111 0.6C7 176 243.00 16.78899 5.50 1. 22502327 0. ICO 106 309.00 16.72577 3.50 1. 22338486 0.055 29 3 169.30 16.23819 3.48 1. 21053696 0.090 295 355.20 15.98599 13.70 1. 20373917 G.5C4 545 262.20 15.95260 5.51 1. 20283127 0.159 322 27.30 15.83069 2.22 1. 19949818 0.305 99 107.40 15.51320 4.69 1. 19069958 0.221 53 358.30 15.29379 6.98 1. 18451500 0.608 708 51.00 15.27669 33.60 1. 18402863 0.490 513 0.0 15.19020 3.65 1. 18156147 0. 0 579 130.70 14.98590 10.72 1. 17568207 0.563 56 5.40 14.73200 11 .35 1. 16826153 0.043 525 224.90 14.674CC 7.66 1. 16654778 0.211 302 0.0 14.49800 11 .71 1. 16130638 C O 221 22.40 14.39610 20.52 1. 15824318 0.011 524 296.00 14.345CG 14.03 1. 15669918 0. 210 274 220.80 14.29599 2.27 1. 15521336 0.276 265 0.0 14.16830 6.67 1. 15131760 0.0 501 31.00 14.1567C 6.60 1. 15096188 0.370 17 152.40 13.50400 2 .36 1. 13046074 0.5C7 732 0.0 13.41870 26.15 1. 12770939 0.100 121 0.0 13.19889 1.85 1. 12053585 C O 659 175.40 13.17360 2.30 1. 11970425 0. 158 SYSTEM OMEGA -119-PERIOD K/O LCG(PERIOD) ECCEN. 533 253.50 13.08100 2 .89 1 .11663914 0.391 445 274.50 12.97620 0.74 1 .11314583 0.060 200 0.0 12.9663C 12.86 1 .11281586 0.0 72 78.60 12 .92739 12 .42 1 .11151028 0.250 336 154.00 12.91669 2.88 1 .11114979 0.080 277 168.20 12.91169 12.96 1 .11098194 0.200 534 217.00 12.90800 18.50 1 .11085892 0.017 42 8 40.00 12.58420 24.55 1 .09982491 0.014 550 19. 10 12.46999 15. 86 1. ,09586525 0.053 577 127.20 12.42559 2 5.64 1 .09431553 0.219 146 341. 00 12.41990 3.60 1 .09411621 0.140 407 97.10 12.32200 13.05 1. . 09068108 0.394 706 7.80 12.31059 2.22 1 .09027863 0. 100 429 82.80 12 .25999 12.97 1 .088 48858 0.192 654 46.80 12.21C0C 11.72 1. .08671379 0.318 223 21.70 12 .2C9C0 2.55 1. .08667946 0.295 733 213.50 12.15500 14.45 1, .08475304 0.278 49 5 36. 50 12.00579 36.02 1. 07939053 0.080 463 267.70 11.848CC 2.85 1. 07364464 0.308 358 198.50 11 .78199 4.13 1. .07121754 0.060 614 0.0 11.75300 0.72 1. 07014847 0.0 34 0.0 11.72C3C 28.00 1. 06893826 0.0 81 286.00 11.66500 0.44 1. 06688309 0.290 312 171.90 11.58320 3.41 1. 06382847 0.381 723 236.40 11.22979 2.67 1. 05C37022 0.037 22 7 340.00 11.11259 3.22 1. 04581547 0. 130 -120 SYSTEM OMEGA PERIOD 622 50. 10 11.08800 261 103.80 11.07640 624 61.80 11.03899 64 271.30 10.99030 695 67.5C IG.91140 632 162.10 10.e83CC 114 130.00 10.66999 671 0.0 IC.62249 31 27.40 10.61569 604 0.0 10.59999 452 4. 10 10.56C00 537 343.50 10.55010 503 256.80 10.52170 27 20.90 10.41800 561 198.70 IC.39320 512 0.0 10.27030 282 314.00 10.25C39 684 171.00 10.21300 318 187.30 10.21C39 2 75 0.0 10.17300 230 180.00 10.C92C0 488 133.00 10.09CCC 382 137.70 9.94480 214 152.90 9.94400 59 21.50 9.92912 280 117.60 9.90470 K/D LCGlPERIOD) ECCEN. % DETEC 5 .87 1.04485226 C.284 79.78 12.27 1.04439640 Q.107 94.56 2 .60 1.04292870 0. 227 65. 60 9.65 1.04100895 0. 184 93. 56 21.27 1.03787899 0.253 97.22 3.14 1.03674696 0.386 63.66 2.00 1.02816391 0.2CC 59.45 12.85 1.02622509 0.0 94.00 10.24 1.02594662 0.220 93.67 29 .00 1.02530384 0.0 98.00 16.38 1.02366257 0.430 96.78 20.75 1.02325535 0.370 97.56 6.77 1.02208519 0.314 91.44 6.43 1.01778412 0.132 84.51 14.43 1.01674843 0.520 96.44 7.60 1.01158142 0.0 86.97 2 .45 1.01073933 O.ICO 67.13 4. 85 1.00915146 0.006 81.23 9.69 1.0C904179 0.560 95.22 3.40 1.00744820 0.0 75.25 18.CO 1.00397587 0.500 97.33 1 .34 1.CC389C04 0. 142 47.9C 0 .60 0.99759 597 0.247 1C.44 6.72 0.99756080 0.081 85.39 0.88 C.99691045 0.059 31.25 3.50 0.99584115 0.620 49. 15 121 SYSTEM OMEGA PERIOD K/D LCG(PERIOD) ECCEN. % DETEC 162 10.00 9.86C00 5.76 0.99387670 0.1C0 88.89 66 129.60 9. 851CC 2.25 0.99348027 0.146 64.18 526 155.50 9.81CC0 6.22 0.99166912 0. 222 89.89 247 100.00 9.70C00 7.40 0.98677146 0.450 77. 03 204 100.60 9.6595C 5.17 0.98495454 0.026 82.00 515 326.40 9.61200 2.85 0.98281366 0.468 56.38 396 273.00 9.60450 14.04 0.98247480 C. 169 95.78 574 0.0 9.55C00 30.00 0.98000306 0.0 98.00 144 165.00 9.51910 2.58 0.97859561 0.033 68.79 198 35.40 9.35530 5.71 0.97105765 0.208 81.60 598 0.0 9.31600 16.48 0.96922952 0.0 98.00 238 278.00 9.30C9C 2. 71 0.96852493 0.160 68.63 307 351.10 9.28300 1.52 0.96768832 0.048 53. 80 245 266.40 9.21280 1 .29 0.96439153 0.499 32.51 33 122.10 9.07504 5.11 0.95784831 0.033 81.92 638 276.20 9.07300 9.40 0.95775104 0.390 94.44 529 0.0 8.89610 26.30 0.94919956 0.0 98.00 439 265.40 8.85500 3.16 0.94718832 0.376 64.42 461 10.00 8.82CCC 21.25 0.94546825 0.240 97.33 608 34 3.20 8.67800 3.79 0.93841934 0.360 69. 19 727 0.0 8.52068 16.10 0.93047422 0.0 93.27 651 12.20 8.44550 16.00 0.92662531 0.039 96.00 602 105.00 8.43C27 3.40 0.92584127 0.080 74.85 126 189.20 8.41780 3 .76 C.9251985C 0. 101 76.IC 593 0.50 8.33425 32 .55 0.92086619 0.264 98. 11 77 212.70 8.25C40 I .29 0.91647476 0.227 44.46 122' SYSTEM OMEGA PERIOD K/D lOG(PERIOD) ECCEN. 107 0.0 8.22334 3. 52 C. 91504782 0.0 483 132.10 8.21590 3.39 C.91465479 0.333 569 262.20 8.11581 2.12 0.90933186 0.169 389 290.00 8.02400 27.01 C.90439069 0.500 170 42.30 7.98960 14.48 0.90252489 ,0.016 366 278.00 7.90400 14.57 0.89784676 C.030 680 0.0 7.8327C 0.68 0.89391124 0.0 193 67.00 7.82710 26.83 0.89360058 0. 250 72 8 0.0 7.75310 8.96 0.88947505 0.0 371 308.60 7.64965 3.88 0.80364136 0.353 572 46.70 7.6383C 18.09 0.8829965C 0.527 416 0.0 7.60605 14.30 0.88115883 0.050 334 71.60 7.39902 16.07 0.86917418 0.377 564 0.0 7.39C00 13.04 0.86864412 0.0 399 317.40 7.36900 3.44 0.86740822 0. 199 360 287.60 7.33660 14.36 0.86549473 0.213 140 125.00 7.33C00 2.30 0.86510366 0.390 33 7 0.0 7.32827 7.25 0.86500138 0.0 713 71.60 7.25105 23.46 0.860400 74 0.376 59 5 18G.0C 7.22962 3.60 0.85911512 0.060 97 226 .00 7.16025 2.70 0.85492784 0. 210 142 289.70 7.C5C70 5.78 0.84823179 0.033 327 319.8C 6.95340 4.12 0.8421968E 0.120 648 0.0 6.94C30 15.29 0.84137803 C O 388 147.20 6.92700 2.14 0.84054518 0.230 287 143.00 6.89200 24.89 0.83834505 0.016 -123 SYSTEM OMEGA PERIOD 73 334.00 6.86357 437 37.90 6.82814 485 116.30 6.79840 294 113.00 6.74469 730 324.80 6.72418 689 0.0 6.68640 79 199.00 6.63827 342 4.10 6.62540 619 0.0 6.60C00 591 34.50 6.51973 212 330.60 6.50130 183 191.00 6.50000 289 161.00 6.39316 660 0.0 6.37C2C 536 195.50 6.36240 96 313.00 6.22360 721 31.40 6.06628 323 150.50 6.03530 666 148.00 5.97152 192 0.0 5.96900 112 151.00 5.95367 441 306.00 5.93C00 484 35.70 5.91820 328 309.40 5.905CC 260 0.0 5.90460 177 358.70 5.73245 K/D LOGIPERIOD) ECCEN. % DE TEC 2.70 0.83655006 0.220 66.65 34.42 0.83430231 0.276 98. 00 5.02 0.83240658 0.069 81.35 2. 15 0.82896197 0.180 62.26 3.73 0.82763898 0.032 76.59 34.00 0.82519203 0.0 98.00 5.81 0.82205468 0.072 83 .90 5.76 0.82121181 C. 02C 83.9C 24.22 0.81954354 0.0 98.OC 38.28 0.81422931 0.038 98.22 3.17 0.81299996 C.019 73.43 8.10 0.81291300 0. 140 87.21 6.69 0.80571544 0.139 84.83 16. 84 0.80415285 0.0 98.00 1 .77 0.80362064 0. 116 57. 84 21 .00 0.79404151 0.200 97.11 5.62 0.78292221 0.190 81.92 3.27 0.78069848 0.087 73.96 4.17 0.77608448 0.031 79.09 11.99 C.7759015C 0.0 94.00 3.76 0.77478445 0. 100 76. IC 2 .87 0.77305442 0.1C0 7C.85 2.51 0.77218926 0.031 6 8.30 7.31 0.77121961 0. 176 85.43 20.00 0.77119023 0.0 98.CC 10.10 0.75834000 0.097 90.00 -124 SYSTEM OMEGA PERIOD 120 88.60 5.61CGC 656 15 .00 5.54834 87 60. 00 5.54348 152 335.90 5.52242 511 0.0 5.51460 168 74.60 5.43373 36 322.50 5.42908 357 229.00 5.41491 645 58.50 5.41364 620 57. 10 5.38275 558 348.70 5.38C3C 669 302.70 5 .30465 566 294.00 5.29510 490 0.0 5.27968 194 0.0 5.20325 678 0.0 5.07 392 255 115.00 5.049C1 122 124.30 5.01C50 365 128.00 5.G0030 391 0.0 4.99175 447 355.90 4.95 100 486 0.0 4.92000 707 346.00 4.90876 617 54.50 4.9C52C 21 269.30 4.82022 553 59.20 4.81200 K/C LOGiPERIOD) ECCEN. % DE TEC 5. 86 0.74896264 0.040 84.22 3.50 C.74416280 0. 240 72. 22 6.20 0.74378210 0.038 84.90 20. 70 0.74212921 0.074 96.78 17.18 0.74151373 0.0 98.00 2 .68 0.73509777 0.076 69.52 1 .45 0.73472577 0.378 42.65 13.72 G.73359090 C.073 95.56 4.00 0.73348910 0. 114 77.47 3.17 0.73100406 0.069 73.43 3.82 0.73080647 0.018 76.99 7.74 0.72465634 0.528 93.67 13.50 0.72387397 0.089 91.82 3. 61 0.72260737 0.0 76.22 4 .80 C.71627426 0.0 81. 15 17.85 0.70534337 0.0 98.00 18.00 C.70320618 0.200 96.78 12.86 0.69988084 0.014 95.78 16.24 0.69899571 0.034 96.00 18.72 0.69825268 0.0 98.00 1 .63 0.69469279 0.511 38.61 43.00 0.69196504 C O 98.00 7.05 0.69097143 0.063 86.16 8 .22 C. 69065624 0. 133 87.60 2 .61 0.68306655 0.024 69. 11 17.21 0.68232554 0.073 96.22 -125 S Y S T E M OMEGA P E R I O D 62 5 0.0 4.8C6CC 3 72 0.0 4.79780 674 263.30 4.72015 457 0.0 4.62958 567 315.00 4.62500 180 272.60 4.62390 594 20.00 4.57282 557 0.0 4.47732 585 294.50 4 .46960 22 132.00 4.46722 202 0.0 4.44746 41 295.60 4 .43474 664 49.80 4.42775 98 344.00 4.41916 464 0.0 4 .39677 233 42.20 4.39341 61 90.OC 4.3714C 52 7 0.0 4.29991 596 134.20 4.28760 440 82.50 4.28503 270 101.70 4.28500 25 0.0 4.28284 532 0.0 4.2435C 611 0.0 4.21238 128 0.0 4.19500 470 0.0 4.18351 K/C L0GIPER10D) ECCEN. Z DETEI 3. CO 0.68178344 0.200 70.04 17.57 0.68104208 0.0 98. C( 4.20 0.67395544 C.075 78.92 23.44 0.66554153 0.0 98.OC 7.25 0.66511154 0.09C 86.04 4.81 0.66500807 0.116 80.34 4.30 0.66018373 0.300 74.24 23.10 0.65101886 0.0 98.00 4.13 0.65026838 0. ICC 78. 32 7.30 0.65003729 0.004 86.57 3.08 0.64811170 0.0 72.91 17.01 0.64686793 0. 108 96.44 7.07 0.64618289 0.126 85.60 26. 86 C.64533973 0.045 97.89 3.00 0.64313334 0.0 72.42 22.25 0.64280158 0.090 95.45 5. 95 C.64062059 0.047 84.38 5.12 C.63345903 0. 0 81.80 10.65 0.63221407 0.045 9C.57 3.81 0.63195354 0.089 76.55 6.35 0.63195068 0. 109 84.63 7.56 0.63173157 0. 0 86.97 20.12 0.62772393 0.0 98.00 42 .50 C.62452734 C O 98.OC 3.13 0.62273186 0.0 72. 91 17.29 0.62154073 0.0 98.00 -126 SYSTEM OMEGA PERIOD 166 0.0 4.13459 737 0.0 4.12275 538 107.60 4.11750 697 356.00 4.08322 163 51.10 4.06570 468 138.00 4.02350 377 90.10 4.01416 199 266.00 4.00604 125 12.70 4.00000 497 0.0 3.99280 329 0.0 3.98C5G 718 0.0 3.96637 196 0.0 3.96004 14 233.20 3.95583 111 0.0 3.95295 491 74.30 3.89400 148 217.10 3.88460 592 175.20 3.87972 259 60.00 3.87358 76 78.30 3.8540O 160 19.70 3.78900 722 103.00 3.76590 482 0.0 3.758C5 665 82.70 3.74860 19 12.80 3.74180 658 57.40 . 3.71063 K/C LOGIPERIOD) ECCEN. % DETEC 22. 32 0.61643231 0.0 98.00 3.00 C.61518681 0.0 72.42 16.30 0.61463350 0.012 96.00 10.49 0.611C0245 0.010 94.00 45.36 0.60913491 0. 051 96.56 18.27 0.60460389 0.023 96.78 31.08 0.60359448 0.156 97.89 4.45 C.60271484 0. 160 78.44 3.61 0.60205954 0. 060 76.10 17.64 0.60127717 0.0 98.00 0.50 0.59993738 0.0 0.51 19.20 0.598 39284 0.0 98.C0 21.89 0.59769917 0.0 98.00 4. 17 C.59723735 0.152 77.84 5.73 0.59692127 0. 120 83.25 20.47 0.59039557 0.040 97.00 3.51 0.58934599 0.013 75.74 41.92 0.58880013 0.070 98.33 6.50 0.58811224 0.200 83.29 2.48 C.58591127 0.042 67.98 2.16 0.57852441 0.033 64.22 3.70 0.57586837 0.120 75.78 22.00 0.57496220 0.0 98.00 18.53 0.57386869 0. 189 96.56 17.16 0.57308060 C.0C5 96.00 7.53 0.56944740 0.188 85.60 -127 SYSTEM OMEGA PERIOD 147 0.0 3.70045 54 258.00 3.67400 226 42.00 3.58474 352 0.0 3.58210 419 24.50 3.57530 132 C O 3.57124 688 267.60 3.57027 408 0.0 3.55078 7 20.80 3.52355 418 23.80 3.45220 637 83.60 3.43607 165 0.0 3.43470 347 0.0 3.42800 451 355.20 3.39430 559 194.40 3.38062 694 0.0 3.37840 86 300.80 3.36897 587 0.0 3.36400 712 202.00 3.33700 581 55.90 3.32068 588 0.0 3.31770 195 0.0 3.30600 243 0.0 3.30553 634 294.00 3.30353 368 211.10 3.28655 510 66.80 3.27554 K/D LOG(PERIDD) ECCEN. % DE TEC 5.79 0.56825411 0.0 84.14 15.72 G.56513911 C.05C 96. OC 4.40 0.55445755 0.280 75.37 14.GO 0.55413741 0.0 92.22 5.86 0.55331230 0.C79 93.9C 7.63 0.55281872 0.0 86.97 3.62 0. 55270106 0.010 76.26 7.01 0.55032367 C 0 86.16 39.82 0.54698002 0.037 98.44 29.18 0.53809553 0.072 97.89 2 .34 C.53606194 0. 145 65.15 19.71 0.53588837 0.0 98.00 5.13 0.53504062 0.0 81.80 6. 82 0.53075016 0.025 85.88 6 .97 0.52899617 0.030 86. 12 18.00 0.52871072 0.0 98.00 43. 34 C.52749676 0. 102 98.44 19.98 0.52685571 0.0 98.CC 10.08 0.52335584 0.271 87.13 3. 71 0. 5212270C 0.026 76.67 6.80 0.52083689 C O 86. 16 7.70 0.51930279 0.0 86.97 3.50 0.51924103 0.0 75.90 6.38 0.51897806 0.018 85.23 23.72 0.51673985 0.041 97.56 2.27 0.51528281 0.120 65.07 -128-SYSTEM OMEGA PERIOD K/C LOGIPERIOD) ECCEN 448 271.50 3.2617C 25.54 G.51344383 0.125 647 350.00 3.24340 6.20 C.51100022 0. 023 310 111.00 3.24200 5.44 0.51081300 0.070 719 353.10 3.21957 7. 16 0.5C779784 0.025 492 29.00 3 .17000 33.50 C.50105911 0.040 68 0.0 3.16626 5.36 0.50054640 0.0 11 339.00 3.11276 21.43 0.49314553 0.026 460 0.0 3.10000 43.00 0.49136168 0.0 438 0.0 3.07080 2 .34 0.48725134 0.0 679 173.90 3.07051 30.02 0.48721033 0.024 320 356.40 3 .06332 5.18 C.48619235 0.029 123 193.80 3.05911 14.49 0.48559469 0.017 305 0.0 3.05459 20.57 0.48495269 0.0 69 2 0.0 3.03754 3. 30 0.48252183 0.0 304 3.0 3.01990 2.50 0.47999376 0.0 635 86.30 2.99633 48.65 0.476*5^920 0.13*J 675 218.40 2.98992 25.08 0.47565937 0. 195 609 110.00 2 .98474 25.60 0.47490638 0.070 220 39.00 2.97230 11 .90 0.47309238 0.100 344 68.80 2.96310 34. 55 0.47174633 0.057 138 126.70 2 .95655 23.26 C.47078496 C.051 191 3C.0C 2.93317 7.62 0.46733695 0.040 244 94. 70 2.92832 3. 19 0.46661842 0.002 498 11.50 2 .91694 6.52 0.46492732 C.022 544 30.20 2 .91225 7.93 0.46422857 0. 169 271 0.0 2.90429 19.82 0.46303988 0.0 97. 84. 82. 86. 98. 82. 97. 98. 66. 97. 82. 96. 98. 74 . 68. 98. 97. 97. 91. 98. 97. 86. 73. 85. 86. 98. SYSTEM OMEGA -129-PERIOO K/C LOG(PERIOD) ECCEN 381 286.00 2.8957C 172 198.80 2 .88840 472 116.70 2.88314 613 195.00 2.8815C 84 273.00 2 .86731 201 0.0 2.86550 530 130.9G 2.86384 636 0.0 2 .85482 258 320.00 2 .82600 49 3 0.0 2.82424 426 349.90 2.8C666 340 61.80 2.78180 729 0.0 2.77455 231 143.50 2.77035 197 183.00 2 .74050 710 57. 10 2.72916 62 112.00 2.72780 670 146.90 2.72574 364 10. 20 2.7C45C 397 0.0 2.69600 113 36 .50 2 .67881 52 175.20 2 .67000 94 0.0 2.66400 80 0.0 2 .64836 657 357.70 2 .62823 387 C O . 2.62516 5.C5 G.46175349 0.100 7.12 0.46C65718 0.12C 2.87 0.45986569 0.110 45.71 0.45961845 0.006 4.41 0.45747417 C.019 6.50 0.45719999 0.0 3.78 0.45694864 0.020 23.71 0.45557857 0.135 3.40 0.45117176 0.3C0 6.02 0.45090127 0.0 6.34 0.44818968 0.015 3.10 0.44432575 0.018 5.40 0.44319218 0.0 6.22 0.44253451 0.210 5.56 0.43782943 0.018 40.86 0.43602866 0.058 3.84 C 4 3 5 8 1 2 4 1 0.090 3.07 C.43548393 0.025 4.07 0.43208688 0.051 2.43 0.43071973 0.0 5.33 0.42794186 0.293 25.35 0.42651123 0.292 2.20 0.42553425 0.0 3.30 0.42297691 C O 8.61 0.41966325 0.030 2.06 0.41915578 0.0 -130-SYSTEM OMEGA PERIOD K/D LOGIPERIOD) ECCEN. 690 0.0 2.61639 17.C8 0. 41770244 C O 3 79 214.80 2 .61314 0 .95 0. 41716260 0. 040 378 123.00 2.54850 4.20 0.40628451 0.170 70 0.0 2.53636 5.51 C 40421075 0.0 173 93.00 2 .52596 37.79 0. 40242636 0. 070 219 C O 2.52501 23.86 0. 40226305 C O 232 173.4G 2.5CC0C 9.30 0. 39794010 0. 110 573 95.70 2.4968C 4.73 0. 39738375 0. 056 28 10.00 2.49293 8.50 0. 39671004 0. 150 103 107.70 2.46111 23.87 0. 39113092 0.018 563 0.0 2.45490 30.95 0. 39003360 0. 0 413 0.0 2.45000 6.97 0. 38916600 C O 315 295.90 2.44506 6.26 0. 38828933 0.060 731 167.00 2.44452 8. 58 C. 38819367 0.068 505 41.00 2.41568 8.51 0. 38303930 0.C90 548 37.00 2.40823 23.67 0. 38169777 0.018 101 0.0 2.4C79C 2.30 0. 38163835 C O 589 152.70 2 .34702 26.07 0. 37051672 0.235 693 27.00 2.34092 8.00 0. 36938655 0.023 725 76.OC 2.33733 2. 80 C. 36872C0C 0.140 409 153.40 2 .32735 7.83 0. 36686164 0. 048 456 C O 2.30757 20.93 0. 36315489 0.0 642 81.00 2.29883 4.77 0. 36150670 0.047 489 0.0 2 .29229 1 .93 0. 36026961 C O 288 0.0 2.28160 4.00 0. 358 2 39 35 C O 235 126.10 2.2596C 30.55 0. 35403132 0.076 -131 SYSTEM CMEGA PERIOD K/U LOG(PERIOD) ECCEN. 57 0.0 2.23650 19.64 0.3495689C 0.0 136 2.40 2.20750 36.07 0.34390062 0.076 520 0.0 2.19660 26.68 0.34175074 0.0 266 200.00 2. 19231 7. 50 C 34090185 0.100 711 189.40 2.18063 5.40 C.33858198 0. 120 556 0.0 2.17815 9.80 0.33808768 0.0 683 0.0 2. 17210 12.77 0.33687967 0.0 735 185.00 2 .15800 0 .65 0.33405125 0.350 169 C O 2.15165 24.31 0.33277160 0.0 124 307.00 2. 14328 2.66 0.33107883 0.040 701 340.00 2.14180 30 .00 C.33077896 C 0 3 0 702 107.40 2.11104 58.31 0.32449639 0.127 582 6.0 2.10514 1.59 0.32328069 0.0 16 280.00 2.08186 3.84 0.31845146 0. 110 479 0.0 2 .05979 26.12 0.31382293 0.0 135 C O 2.05630 6.80 0.31308633 0.0 478 168.00 2.051C3 38. 88 C 31197184 0.058 506 211.00 2 .04760 21 .29 0.31124490 0. 037 115 30. 10 2.02858 35.23 0.30719215 0.051 370 269.70 1.99498 8.00 0.29993838 C.300 542 167.00 1 .98575 33 .96 C 29792440 0. 017 685 0.0 1.98322 23.17 0.29737055 0.0 20 C O 1.96418 24.26 0.29318094 0.0 75 0.0 1 .95253 6.80 0.29059738 0.0 570 0.0 1.95027 37.22 0.29009461 0.0 102 0.0 1.94C56 19.60 0.28792685 0.0 -132-f STEM OMEGA PERIOD K/C LOG(PERIOO) ECCEN 239 0.0 1.93265 21. 14 0. 28615308 0.0 400 31.10 1 .93017 6 .30 C. 28559536 C 101 606 44.70 1.88550 71 .01 0. 27542645 0.039 228 0.0 1.88475 38.CO C 27525353 C O 610 112.70 1 .87420 41 .21 0. 27281559 0.056 516 0.0 1 .85052 19.00 0. 26729345 0.0 546 156.20 1.84908 39. 18 0. 26695544 0.033 8 0.0 1 .81264 8.79 0. 25831127 C O 374 0.0 1.81068 4.20 0. 25784153 C O 584 94.00 1. 80526 6.67 0. 25653940 0.110 321 0.0 1.8C52G 10.57 C 25652516 C O 522 0.0 1.77857 30.10 0. 25007069 C O 70 3 106.30 1.77476 48.65 0. 24913937 0.034 46 135.60 1.73652 1. 21 C. 23967963 0.121 661 0.0 1.72897 22 .00 0. 23778731 C O 453 221.60 1.70116 8.66 0 . 23074502 0.090 301 0.0 1.68608 6.CO c. 22687787 0.0 474 0.0 1 .68170 17.58 0. 22574842 C O 299 0.0 1.68154 23.56 0. 22570682 0.0 477 306.00 1.67735 37.94 c 22462332 0.012 6 76 103.20 1.67697 31 .99 0. 22452503 0. 122 209 90.00 1.67254 16.70 0 . 22337633 C 0 5 0 32 97.00 1.66976 36.84 0. 22265393 0.034 69 6 0.0 1 .64100 29 .50 0. 21510839 0. C 535 37.00 1.63742 8.26 0. 21415973 0.050 677 C O 1.60469 27.65 0. 20539099 0.0 -133-YSTEM OMEGA PERIOD K/C LOGIPERIOD) ECCEN. % DETEI 433 0.0 1.57C1C 32. 80 C. 1959269C 0.0 98.0( 269 123.90 1 .56298 3.03 C.1939532S C.051 72.5] 65 63.80 1.56145 3.78 0.19352776 0.046 76.8" 494 177.00 1.54885 5.05 C.19000906 0.080 81.52 554 0.0 1.54039 8.87 0. 18763036 0. 0 89.23 119 271.00 1 .52738 4.31 0.18394673 C.047 79.31 178 0.0 1.48538 27.64 0.17183721 0.0 98.00 362 86.80 1 .46047 18.82 0.16449237 C.088 96.78 262 116.00 1 .45450 35.60 0.16271341 0. C3C 97.19 459 0.0 1.44627 46.50 0.16024899 0.0 98.00 431 0.0 1.4380C 4.50 C.15775841 0.0 80.02 9 202.00 1.43233 5.34 0.15604270 0.024 82.57 71 0.0 1.42833 8.70 0.15482825 0.0 89.07 338 0.0 1.35742 4. 90 C.13271379 0.0 81.31 154 0.0 1.33273 44.29 0. 12474185 0.0 98.OQ 95 32.60 1.32639 6.37 0.12267095 0.024 85.39 549 0.0 1.30226 2.76 0.11469746 0.0 70.81 350 0.0 1.27100 6.32 0. 10414547 C O 85.27 698 307.00 1.25454 6.38 0.09848428 0.022 85.31 158 0.0 1.24730 7.CO 0.09597051 0.0 86.16 15 330.00 1.24352 9.30 0.09465253 0.200 88.02 156 0.0 1.21C00 49.40 0.08278489 0.0 98.00 465 307.00 1.19881 9.26 0.07875013 0.015 89.64 215 9.20 1 .19C33 43.63 0.07566708 0.036 98.78 578 0.0 1 .18287 10.40 0.07293653 0.0 9C.36 682 0.0 1.15221 21.80 0.06153136 0.0 98.00 -134-SYSTEM OMEGA PERIOD K/D LOGJPERIOD) ECCEN 443 94.00 1.13979 12. 83 C.05682467 0.017 236 0.0 1 .13594 3.15 C.05535523 C O 273 0.0 1.08263 24.00 0.03447972 0.0 551 20.00 1.03088 1.21 C0132077C 0.015 667 141.00 1.02277 7.08 0.00977754 0.012 51 0.0 0.97153 11 .00 -0.01254156 0.0 652 0.0 C.93617 35.90 -0.02864482 0.0 93 0.0 0.91719 9.57 -0.03754165 C O 284 44.30 C.91470 4.63 -0.03872132 0.131 462 0.0 C.91207 8.68 -0.03997042 0.0 414 0.0 0.90141 2 .20 -0.04507910 0.030 89 0.0 0.84941 5.10 -0.07088357 0.0 6 0.0 C 8 4 1 6 6 18.67 -0.07486433 0.0 392 0.0 0.81687 27.30 -0.08784646 C O 246 7.00 0 .81428 23.74 -0.08922511 0.072 252 0.0 C 7 9 4 8 5 30.00 -0.09971315 0.0 649 76.00 0.78586 6. 59 -0.10465372 0.115 361 0.0 0.74728 7.50 -0.12651670 0.0 427 160.00 C 7 2 6 4 2 6.10 -0.13881397 0.120 627 C O C.71777 28. 94 - C 14401644 0.0 62 8 160.60 0.70102 34.60 -0.15426707 0.031 641 62.60 0.69809 28.78 -0.15608853 0.017 300 C O C.68738 5.90 -0.16280431 C O 91 0.0 C.68500 17.40 -0.16430944 C O 67 80.00 0.68468 8.00 -0.16451162 0. 120 298 C O . C.64834 9.23 -0.18819380 0.0 -135 SYSTEM • MEGA PERIOD K/C LOGIPERIOD) ECCEN. Z DETEC1 345 90.00 C 6 4 2 6 5 8.CO - 0 . 19202542 0.100 87.58 386 0.0 0.64058 8.00 -0.19342929 0.0 87. 21 716 282.20 0.62893 33.75 -0.20139623 0.089 98.00 317 0.0 0.60008 31.C7 -C.22178727 0.0 98.00 217 0.0 0.59307 12.30 - C 2 2 6 8 9 3 3 1 0.0 91.58 514 0.0 0.55088 10.40 -0.25894541 C O 90.36 612 0.0 C.51419 40. 50 -0.28887212 0.0 98.00 355 180.00 0.49116 3.66 - C 3 0 8 7 7 7 0 3 0. 300 70.93 454 C O 0.45340 33.00 -0.34351379 0.0 98.00 157 0.0 C.42339 25.00 -0.37326157 0.0 98.00 475 0.0 0.42152 7.80 -0.37517983 0.0 86.97 133 0.0 0.41567 38.50 -0.38125026 C O 98.OC 49 6 C O 0.40964 33.00 -0.38759667 0.0 98.00 351 0.0 C 4 0 7 5 2 35. 50 -C.38985223 0.0 98.00 278 0.0 0.38288 32 .90 -0.41693628 0.0 98.00 734 C O 0.37478 37.00 -0.42622131 0.0 98.00 686 0.0 C.37C8C 24.90 - C 43086034 0.0 98.00 417 0.0 0.35038 36.70 -0.45546550 0.0 98.00 359 0.0 0.33851 40.00 -0.47043329 0.0 98.00 303 0.0 C.33364 39.CO -0.47672552 0.0 98.00 717 0.0 0.33189 43.00 - C 4 7 9 0 1 0 9 4 C O 98.00 116 C O 0.32149 13.00 -0.49282700 0.0 91.90 705 0.0 C.32072 37.50 -0.49387944 0.0 98.00 44 0.0 0.31685 39.00 -0 .49914354 0. 0 98. OC 30 8 0.0 0.28411 30.50 -0.54651493 0.0 98.CG 629 C O C 2 7 8 3 2 30.50 -0.55545533 0.0 98.00 -136. SYSTEM OMEGA P E R I O D K/D LOG(PERIOD) ECCEN. % DETECT 663 411 30 380 502 257 208 531 268 0.0 0.0 220.00 C O 0.0 16C0C 0.0 0.0 266.00 0. 27624 23. 70 - C 5587070C C O 0.26781 34.68 -0.57217479 C O C21173 7.75 -0.67421764 0.400 C 19667 25.00 -C7C625955 C O 0.19362 14.90 -0.71304739 C O 0.17398 26.50 -0.75949305 0.050 C 13900 7.50 -0. 85698497 C O 0.13854 3.75 -0.85842466 C O C.06975 43.70 -1.15647316 0.087 98.00 9 8. CO 80.14 96.26 92.63 9 6.55 86.81 76.95 97.56 

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