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Alpha virginis : line profiles and spectroscopic binary orbit Moyles, Katherine-Ann 1982

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C i ALPHA VIRGINIS - LINE PROFILES AND SPECTROSCOPIC BINARY ORBIT by KATHERINE-ANN MOYLES B.Sc, Auckland University, 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1982 (c) Katherine-Ann Moyles, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date DE-6 (3/81) ABSTRACT In t h i s work, spectroscopic observations of o V i r g i n i s have been reduced. The l i n e p r o f i l e variations observed in the He I (6678 A) l i n e show that inspite of concern expressed by Lomb (1978) that the p Cephei v a r i a t i o n was no longer active, the star i s s t i l l pulsating - possibly in a non-radial mode. The binary orbit has been recalculated at the epoch of the observations and an analysis of this and e a r l i e r r a d i a l v e l o c i t y data confirms the existence of apsidal motion and also suggests possible v a r i a t i o n in the apsidal period. CONTENTS Abstract .. i i Contents . . . . . . . . . . . . . . . i i i L i s t of tables iv L i s t of figures v Acknowledgements v i i Chapter I Beta Cephei Stars.... 1 Chapter II The Spectroscopic Observations. . 24 Chapter III The Spectroscopic Binary Orbit and Apsidal Motion 43 Chapter IV The Line Profiles....63 Chapter V Conclusions 85 Bibliography 89 Appendix A Physical C h a r a c t e r i s t i c s . . . . 94 Appendix B Summary of Observations 95 Appendix C Radial V e l o c i t i e s 97 Appendix D Simplex Program 109 iv LIST OF TABLES Table.1. S t a t i s t i c a l Surveys of Beta Cephei Stars 4 Table.2. Possible Pulsation Mechanisms ...12 Table.3. Light and Radial Velocity Amplitudes 1 7 Table.4. Published Observations 22 Table.5. Alpha V i r g i n i s - O r b i t a l Solutions ... 51 Table.6. Rotational Velocity ..60 LIST OF FIGURES Fig.1. Short Dark 23 Mar 79 UT 29 Fi g . 2. Lamp 23 Mar 79 UT . 30 Fig.3. He I 6678, F i l t e r e d . 31 Fig.4. He I 6678, F i l t e r e d , smoothed .32 Fig. 5. He I 6678, Continuum F i t 33 Fig.6. He 6678, R e c t i f i e d 34 Fig.7. Sample scan, 20 Mar 79 UT 38 Fig.8. Sample scan, 21 Mar 79 UT 39 Fi g . 9. Sample scan, 22 Mar 79 UT ..40 Fi g . 10. Sample scan, 23 Mar 79 UT 41 Fig.1 1. Sample scan, 24 Mar 79 UT ...42 Fi g . 1.2. 6Jo vs time ....50 Fi g . 13. k vs MK Spectral type ....59 F i g . 14.. " " 20 Mar 79 UT 65 Fig.15. " " " 66 Fi g . 16. " " 21 Mar 79 UT 67 F i g . 17. " " 22 Mar 79 UT 68 F i g . 18. " " 23 Mar 79 UT 69 F i g . 19. " " " " " ..70 Fi g . 20. " " 24 Mar 79 UT 71 Fig.21. Difference plot, 23 Mar 79 UT 73 Fig.22. He I 6678, l i n e depth, 20 Mar 79 UT ..,.76 Fi g . 23. •" " " " " 21 Mar 79 UT ..,77 F i g . 24.. " '" " " " 22 Mar 79 UT ...78 Fig.25. " " " " " 23 Mar 79 UT ...79 Fig.26. " " " " " 24 Mar 79 UT ...80 ACKNOWLEDGEMENTS In completing t h i s work, I would l i k e to f i r s t of a l l express greatest thanks to Stephenson Yang for a l l his help with the data reduction, teaching me how to use the computer (and Reticent of course!) and also for reducing and making available the observations of Spica in 1981. Many thanks also to Gordon Walker for being patient enough to wait for the results and to Greg Fahlman for assistance with the position finding methods. The UBC Computing Centre consultants, in part i c u l a r Mike Patterson, were also very h e l p f u l . I would also l i k e to thank Michael Ovenden for his encouraging discussions. 1 CHAPTER I BETA CEPHEI STARS The fi Cephei stars are pulsating stars of early spectral type ( 09 - B3 ). They are massive ( 8 - 2 8 solar masses ), luminous ( classes II - V ) stars and many are brighter than apparent magnitude 7,. Light and ra d i a l v e l o c i t y variations are observed with a timescale of a few hours. The amplitude of the l i g h t variation i s not large - often less than about 0.1 mag. The ra d i a l v e l o c i t y may show excursions of up to 25 km/s, although « Sco and BW Vul both vary by twice as much. Unlike the c l a s s i c a l Cepheids or RR Lyrae stars, the l i g h t and r a d i a l v e l o c i t y curves are 90° out of phase. The maximum in l i g h t occurs on the descending branch of the r a d i a l v e l o c i t y curve. If a r a d i a l pulsation mode was appropriate - and there i s some indication that non-radial modes may be involved - maximum brightness would occur at minimum radius. Since these stars are s l i g h t l y bluer when brightest, maximum temperature also coincides with minimum radius. While some p Cephei stars appear to pulsate with only one period, for example r Peg, others show much more complex behaviour such as multiple periods and beat e f f e c t s , presumably due to the interference of two similar pulsation frequencies. 2 The beat e f f e c t i s detected as a long period modulation of the va r i a t i o n . When the f i r s t members of this class of stars were assigned i t appeared that they had abnormally low rotational v e l o c i t i e s compared to most B type stars. Some of those found l a t e r including c V i r g i n i s and some of those in H i l l ' s l i s t ( H i l l 1967) show high rotational v e l o c i t i e s . It has been pointed out that since the e a r l i e s t i d e n t i f i c a t i o n s were made on the basis of spectroscopic observations, sharp lined (slow rotators) would be p r e f e r e n t i a l l y selected. On the other hand, since some of these stars show complex l i n e p r o f i l e and equivalent width variations, the observed broadening in the spectral l i n e s may be due to the same processes responsible for the va r i a t i o n rather than to rapid rotation. The van Hoof effect has been observed in some p Cephei stars. Maximum li n e width occurs f i r s t in the Si III and 0 II li n e s and then in the H or He l i n e s . This i s generally interpreted as a vel o c i t y gradient in the s t e l l a r atmosphere. The most recent reviews of these stars are those by Melnikov and Popov (1975), Lesh and Aizenman (1978) and the proceedings of the "Workshop on Pulsating B Stars" (Nice Observatory 1981). It i s s t i l l uncertain at which stage in their evolution massive stars become p Cephei stars - or i f indeed a l l such stars dp so. The p Cephei stars appear to occupy a well defined region on the H-R diagram. They are found in a s t r i p p a r a l l e l to and about 1.5 mag above the main sequence between Mv = -3 and Mv = -5 (Lesh and Aizenman 1978). If the evolutionary tracks of 3 stars between 10 and 15 solar masses are plotted on the HR diagram i t becomes apparent that the so c a l l e d "S bend" in the evolutionary track coincides with the i n s t a b i l i t y s t r i p where the p Cephei stars are located. The S bend occurs just after the star leaves the main sequence. This means then, that the stars are in either (i) hydrogen core burning, ( i i ) core collapse or ( i i i ) early hydrogen s h e l l burning. This has .been found so for a variety of assumed theoret i c a l models - for example work by Schmalberger (i960), Watson (1972), Lesh and Aizenman (1973a) and Shobbrook (1978b). So far i t has been impossible to distinguish between these states for p Cephei stars and as a result the way in which these stars pulsate i s poorly understood. In an attempt to answer this question several s t a t i s t i c a l surveys of p Cephei stars have been done, the intention being in each case to compare the percentage of p Cephei stars r e l a t i v e to normal B type stars with the percentage of i t s l i f e t i m e spent in the S bend by an evolving 10 or 15 solar mass star. Lesh and Aizenman ( 1973a ) find that this time represents approximately 3% of the l i f e t i m e of such a star. They find that 3% of the stars in their sample are p Cephei stars and hence favour having p Cephei stars in contraction or early hydrogen s h e l l burning. Percy (1974) and Shaw (1975) find a higher percentage of their sample stars to be variables and as a result do not consider that these are solely in contraction or s h e l l burning unless the the o r e t i c a l timescale i s in error. The results of these various surveys are summarised in Table 1 . 4 Table.!. S t a t i s t i c a l Surveys of Beta Cephei Stars 1. Schmalberger (1960) Sample s i z e : 15 £ Cephei stars Compares observational ZAMS and theoret i c a l evolutionary tracks on H-R diagram. 2. McNamara and Hansen (1961) Sample s i z e : 96 stars,BO.5 - B2,I - V m > 5.6, north of -30 No. 0 Cep stars/sample = 23/96 No. stars with v s i n i < 50km/s/sample = 23/96 No. 0 Cep stars/stars with v s i n i <. 50km/sf 11/23 Concludes: 11% of bright BO.5 - B2 stars are fi Cephei stars. 65% of those stars with v s i n i < 50km/s (luminosity class III - V) are I Cephei stars. 5 3. Lesh and Aizenman (1973b) o Sample si z e : 278 stars, BO - B3, III - V, out to a distance of 350 pc. (V - M - 7.7) No. £ Cep stars/sample = 10/278 = 3% Sample s i z e : 207 stars, BO.5 - B2, III - V out to a distance of 500 pc. (V - M = 8.5) No. > Cep stars/sample = 11/207 = 5.3% Note: contains stars not examined for v a r i a b i l i t y . Incomplete for class B2 V. 4. Percy (1974) Sample s i z e : 42 stars, B1 - B2, out to distance of 200 pc. No. fi Cep stars/sample =19% Note: giant stars over-represented. 5. Shaw (1975) Sample s i z e : 134 stars, 4.3<logT <4.5, out to a distance of 500 pc. Includes a l l stars in i n s t a b i l i t y s t r i p and class V stars. No. p Cep stars/sample = 16/134 = 12% No. suspected p Cep/sample = 6/134 Note: not a l l stars examined for v a r i a b i l i t y . 6 6. Lesh and Aizenman (1978) Addition of 7 confirmed variables to 1973 sample B. No. f Cep stars/sample = 14/207 = 8.7% 7.. Shobbrook (1978) ubvy-0 photometry Sample s i z e : 17 stars, V - M < 8.0 Stars within the i n s t a b i l i t y s t r i p . A l l variables known. No. fi Cep/sample = 14/17 = 82% + 22% Sample s i z e : V - M < 9.0 May include undetected variables (perhaps one or two). No. fi Cep stars/sample = 69% ± 17% Note: at V - M > 8.0 some stars not c a r e f u l l y checked for v a r i a b i l i t y . At V - M > 9.0 ubvy-p data may not be complete for fai n t stars. 8. Shaw (1975) Sample s i z e : 48 stars, out to a distance of 700 pc. Includes stars in the i n s t a b i l i t y s t r i p only where M (faintest) = -3.0. No. i Cep stars/sample = 16/48 = 33% No. suspect 0 Cep stars/sample = 1/48 Note: excludes a l l class V stars. Not a l l stars checked for v a r i a b i l i t y . 9. Balona and Englebrecht (1981) Sample s i z e : 28 stars in NGC 3293 No. 0 Cep stars/sample = 9/28 = 32% 7 Unfortunately the results of these surveys are somewhat inconclusive due to the following: (i) not a l l the stars may have been checked for v a r i a b i l i t y ( i i ) the samples may not be volume limited - they may be incomplete for luminosity class V ( i i i ) giant stars may be overrepresented (iv) the i m p l i c i t assumption of steady state star formation may be in error since there i s evidence that the p Cephei stars may have formed at discrete times as members of 0 or B type associations. Most of these are within the Gould belt ( a l o c a l grouping of stars and i n t e r s t e l l a r material at an i n c l i n a t i o n of about 20° to the galactic plane) accounting for their tendancy to be found at high g a l a c t i c l a t i t u d e r e l a t i v e to the more distant disc population B stars. (Lesh and Aizenman 1973b). The Gould system i s apparently expanding - the expansion age being approximately 7.107 years (Lesh 1968,1973) and since t h i s i s greater than the age of the more massive stars two waves of star formation are postulated at 6.107 years and at 2.107 years ago (Frogel and Stothers 1977). A s l i g h t l y d i f f e r e n t s t a t i s t i c a l approach was taken by Shobbrook (1978b). ubvy-p photometry of stars in the i n s t a b i l i t y s t r i p was used to make conclusions regarding the r e l a t i v e numbers of stars in each of the arms of the S-bend since stars w i l l evolve at d i f f e r e n t rates along the d i f f e r e n t arms. He finds that 80% of stars in the i n s t a b i l i t y s t r i p are variable and suggests that the extreme end of core burning i s the most appropriate phase. The problems associated with surveys of f i e l d stars were 8 nicely avoided in .a study of 0 Cephei stars i n the galactic cluster NGC 3293 by Balona and Englebrecht (1981). Of the 28 stars in the cluster which were observed, 9 turned out to be 0 Cephei stars. It i s s i g n i f i c a n t that a l l these stars occupy a position in the cluster colour-magnitude diagram just below the gap between the upper main sequence and the super giant branch. This and the fact that quite a high .proportion of the stars are I Cephei variables suggests that they are in fact in late core hydrogen burning just prior to g r a v i t a t i o n a l contraction. The d i s t i n c t i o n between pulsating and non-pulsating B type stars appears to be subtle. It does seem that the 0 Cephei stars occupy a quite well defined " i n s t a b i l i t y s t r i p " in the H-R diagram. If we are interested in what causes v a r i a b i l t y in some B stars and not others i t becomes important to know i f variable and non-variable B stars can coexist in the same region of the H-R diagram. I t seems that variable and non-variable stars may coexist (Shobbrook 1978a) - however th i s i s dependent on the extent to which the i n s t a b i l t y s t r i p can be resolved. Shobbrook (1978b) notes that the i n s t a b i l i t y s t r i p may not be resolved by the ubvy-0 or the log Te/M observations and Watson (1972) disputed coexistance on the basis of (©e/log g) measurements. However his sample of 18 0 Cephei stars includes only MK spectral types B1 and B2 whereas a complete sample would range over types 09 to B3. The 0 Cephei stars were the f i r s t group of variable B stars to be i s o l a t e d ( 0 Cep was discovered to be variable by Frost (1902)) but since then i t has become apparent that v a r i a b i l i t y is common in B stars (see the "Workshop on Pulsating B stars", 9 Nice 1981) and i t is not e n t i r e l y clear that membership in the 0 Cephei class can be uniquely defined.. For instance some 0 Cephei stars (a V i r and 16 Lac) been observed to decline markedly in amplitude of v a r i a t i o n (le Contel e t . a l . 1981). Approximately 50% of 0 Cephei stars are binaries as are normal B type stars and they are no more l i k e l y to belong to groups or associations than are B type stars (Lesh and Aizenman 1973b). I n i t i a l l y i t was thought that 0 Cephei stars were characterised by low rotational v e l o c i t i e s . McNamara and Hansen (1961) suggested that the s t a b i l i t y of B stars was dependent on t h e i r having high rotational v e l o c i t i e s . However, the discovery of apparently rapidly rotating 0 Cephei stars does not support t h i s . The most s a t i s f y i n g suggestion i s that these stars are at the end of core hydrogen burning and that the i n s t a b i l i t y and pulsation are the v i s i b l e consequence of the structural changes as the core contracts. This would also account for the apparent gap between the i n s t a b i l i t y s t r i p and the main sequence which would not occur i f the stars were s t i l l burning core hydrogen.. A number of pulsation models have been developed and are summarised in Table 2 . Stothers and Simon (1969) discuss the so c a l l e d it and 0 mechanisms. The n mechanism requires that the 0 Cephei star be i n i t i a l l y the secondary in a close binary system. Transfer of increasingly He r i c h material from the primary res u l t s in a reversal of the mean molecular weight gradient causing pulsational i n s t a b i l i t y . Unfortunately for t h i s theory only 50% known 0 Cephei stars are members of binary .systems. The 10 I mechanism also involves mass accretion and a resul t i n g high radiation.pressure. Osaki (1974) proposes that the p Cephei stars are actually in core H burning and that o s c i l l a t i n g ( or overstable ) convection in a rapidly rotating core i s in resonance with one of the non-radial eigen frequencies of the star. Some problem i s encountered then in accounting for the low rotational v e l o c i t y of some stars. Also i t i s not clear how angular momentum i s transferred from the core to the surface under these conditions. Surface e f f e c t s have also been c i t e d , for example Cox and Stellingwerf (1979) suggest l o c a l changes in opacity as a function of temperature at a given density ( the "bump" mechanism ) and Fi t c h (1967) mentions the structure of the second He ionisation zone as a source of i n s t a b i l t y . Cox e t . a l . (1981) suggest that sudden mixing of hydrogen into a hydrogen depleted core may cause pressure pulses resu l t i n g in pulsation. The mixing may be via a semi-convective region outside the core. Observationally they predict a rapid increase in pulsation amplitude with observable decay of either r a d i a l or non-radial modes in the intervals between mixing. The predicted decay time for o V i r appears to be about 6 years. Marked decline in pulsation amplitude was observed for thi s star in 1969-1972. One more suggestion which may be more appropriate for the 53 Per stars i s that by Pringle and Papaloizou(1978) who propose that interaction between a rapidly rotating core and a slowly rotating envelope may result in wave motion leading to non-ra d i a l pulsations. 11 Pulsational modes can be i d e n t i f i e d by comparing observed and t h e o r e t i c a l l i n e p r o f i l e s and by comparing the theoreti c a l pulsation constant d~((>/po)X' with that obtained from measurements of mass and radius of a given star. Table.2. Possible Pulsation Mechanisms 1. McNamara and Hansen (1961) - Rotational Velocity High rotational v e l o c i t i e s in ordinary B type stars thought to i n h i b i t pulsation, p Cep stars were then thought to be slow rotators. However, rapidly rotating fi Cep stars have since been found. 2. F i t c h (1967) - 2nd He Ionization Pulsation may depend on the structure of the 2nd He ionization zone. Observed c h a r a c t e r i s t i c s dependant conditions in the surface layers. 3. Stothers and Simon ( 1969) - it and fi mechanisms Nuclear energy from the core supplies the energy for pulsation. JI mechanism - reversal of mean molecular weight ( reduces the r e l a t i v e central condensation) as a result of mass exchange in binary systems. However, not a l l fi Cep stars are members of bina r i r y systems. Also, Odell (1974) already finds the conventional s t e l l a r models to be i n s u f f i c i e n t l y condensed to account for the apsidal motion of o V i r . n mechanism - high radiation pressure. Result of complete mixing induced by mass accretion. 13 4. Osaki (1974) - Over-stable Convection Upper MS stars have convective cores. Rapid rotation of tthe core may mean that the convective motions in the stas are overstable ( o s c i l l a t o r y ) . Over-shooting from the convective core i s not usually expected to be observed at the surface. Resonance may occur with one of the eigen-frequencies of non-radial pulsation and an excited non-radial pulsation mode may be observed. Possibly some means of mass transfer from the core to envelope and d i f f e r e n t i a l rotation should be considered. 5. Cox and Stellingwerf (1979) - Envelope Ionization Local change in opacity as a function of temperature at a given density. Radiation pressure contributes to maintaining the pulsation. 6. Odgers (1955), Goldberg (1973) - Shell Ejection Moving shel l s or shock waves in the s t e l l a r atmosphere. Perhaps peculiar to BW Vul. 7. Papaloizou and Pringle (1978) Interaction between rapidly rotating core and slowly rotating envelope may produce waves leading to non-radial pulsation. 8. Cox et. a l . (1981) - H mixing into convective core H from the envelope mixed into the convected core which is exhusted or depleted of H to produce periodic pressure pulses. 14 The i d e n t i f i c a t i o n of the correct pulsation mode i s often ambiguous. The mean Q values calculated suggest either a f i r s t harmonic r a d i a l or :p1 non-radial mode Lesh and Aizenman (1978). It i s also possible that the observed surface mode might be coupled in some way to an underlying and d i f f e r e n t mode which i s actually responsible for the pulsation. If so then i t would v i r t u a l l y impossible to identify the cause of pulsation from surface phenomena. After studying the l i n e p r o f i l e and r a d i a l v e l o c i t y variations in BW Vul, Goldberg et. a l . (1976) suggest that there are moving she l l s or shocks in the s t e l l a r atmosphere of t h i s star. However BW Vul i s a somewhat extreme case in terms of the amplitude of the variations and t h i s model does not appear to have been developed for p Cephei stars other than 12 Lac (Leung 1965). More recently, with the development of theories of non-r a d i a l pulsations i t has been possible to construct model l i n e p r o f i l e s which can be compared with observed p r o f i l e s . It appears that non-radial pulsation may most accurately duplicate the l i n e structure observed in p Cephei stars. The e f f e c t of r a d i a l pulsation on r o t a t i o n a l l y broadened l i n e s has been investigated by, among others, Campos and Smith (1980) and Stamford and Watson (1980). 15 a V i r g i n i s was added to the l i s t of known 0 Cephei stars in 1969, however i t had long been regarded as an int e r e s t i n g spectroscopic binary and even in the early observations there was some hint of the complexity of the system. It was f i r s t recognised as a binary by Vogel (1890, 1892) although the f i r s t recorded spectroscopic observations are apparently those from Greenwich in 1876. Vogel was able to measure r a d i a l v e l o c i t i e s and calculate an orbit for the system. Further r a d i a l v e l o c i t i y measurements were made between 1906 aand 1908 by Baker (1909) at the Allegheny Observatory and by Struve and Ebbighausen (1934) who also noted variations in the l i n e i n t e n s i t i e s of both components. Luyten and Ebbighausen (1935) also analysed the observations from the early 1930's and found that the l i n e of the apsides was advancing. Radial v e l o c i t i e s were again measured in 1945 by Kao and Struve (1949) - Struve noticed l i n e variations in these observations also (Struve 1948). Struve e t . a l . (1958) after analysis of spectroscopic observations at Mt. Wilson in 1956, 1957 and 1958 recomputed the orbit and measured the apsidal period to be 133 years. Line structure was observed at a l l phases of the o r b i t , often complicated by blending. They attributed t h i s as possibly being due to non-synchronisation of rotation and revolution. They class the primary and secondary as B2 and B3 respectively on the appearance of the spectral l i n e s but fin d that the mass-luminosity data suggests that the secondary i s a B7 star. To resolve the discrepancy they suggest that r e f l e c t i o n e f f e c t s may be important. The e a r l i e s t photometric observations are those by Stebbins 16 who observed o V i r g i n i s in 1912 (Stebbins 1914). He suggested then that the star might be variable. Other more recent photometric observations include those from Magalachvili and Kumsichvili (1961). In 1969 Shobbrook e t . a l , found that the primary was a p Cephei variable. The i d e n t i f i c a t i o n was made on the basis of the star's regular short period l i g h t v a r i a t i o n s . Since then most of the interest in c V i r g i n i s has centred on i t s p Cephei nature rather than the c h a r a c t e r i s t i c s of the system as a whole, o V i r g i n i s was the f i r s t p Cephei stars found to be a member of a close binary system. The pulsation period was found to be 4.17036 hr with a mean amplitude of 0.016 mag (Shobbrook e t . a l . 1969). The e a r l i e r spectroscopic data are then discussed in terms of the p Cephei v a r i a b i l i t y by Smak (1970). Using the r a d i a l v e l o c i t y data from Baker (1909), Struve and Ebbighausen (1934), and Struve e t . a l . (1958) he computes a period for the pulsation at each epoch and compares th i s with the result from the photometric observations. The l i g h t and r a d i a l v e l o c i t y curves are found to be sinusoidal with minimum r a d i a l v e l o c i t y occurring at approximately 0.2P after maximum l i g h t as seen in other p Cephei stars. After comparing the position of o V i r g i n i s on the H-R diagram with evolutionary tracks for 9 and 5.4 solar mass stars i t appeared that c V i r g i n i s was either in core hydrogen burning or gr a v i t a t i o n a l contraction as expected for a p Cephei star. Smak also finds evidence for a period decrease of 5s/century which could be assumed to indicate g r a v i t a t i o n a l contraction. However later work by Shobbrook e t . a l . (1972) finds 17 that the periods determined by Smak are in fact a l i a s e s of the true period and that o V i r g i n i s i s not a stable,, singly periodic star. The analysis by Shobbrook e t . a l . i l l u s t r a t e s the application of periodogram and Fourier methods to a search for p e r i o d i c i t i e s in astronomical data. Although they d i s c r e d i t the 5s/century period decrease, i t is apparent from t h e i r analysis that neither the period nor the amplitude of the pulsation has been constant. They suggest that in 1906-1908 the star was pulsating in a fundamental mode but by 1934 was s h i f t i n g to a f i r s t overtone mode. As can be seen in the following table, the decline in amplitude i s quite marked. Table.3. Light and Radial Velocity Amplitudes Year 1969 1969 1934 1908 1890 Period (days) Rad. Vel. Amp.(km/s) 0.17380 0. 173796 0. 173790 0.25219 0.1961 or 0.2439 16 17 18 31 22 Year Magnitude Range 1 970 1969 0.0143 mag 0.0290 Ref: Shobbrook e t . a l . (1972) Herbison-Evans e t . a l . (1971) used the Narrabri 18 interferometer to observe a V i r g i n i s in 1966 and 1970. They present the most complete set of o r b i t a l and physical-parameters for the system including a value for the radius independant of magnitude-luminosity data. They suggest the p o s s i b i l i t y -of shallow ( about 1% ) eclipses. Although a V i r g i n i s also appears in Kopal's l i s t of e c l i p s i n g binary systems showing apsidal motion (Kopal 1978) the eclipses, i f they occur are s l i g h t and the variations in magnitude might be due to modulation by t i d a l l y induced e l l i p t i c i t y of the stars. The spectroscopic observations by Dukes (1974) were made in order to investigate the ef f e c t of variations in t i d a l potential on the amplitude of the pulsation. After re-determining the orbit and performing a least squares f i t to the periodogram calculated at frequencies near the published period a frequency of 5.7542 c/d was found. When the observed l i g h t and r a d i a l v e l o c i t y variations were compared with the t i d a l potential due to the secondary at the centre of the primary he found that the largest v e l o c i t y amplitude occurred at maximum t i d a l potential although there was no clear c o r r e l a t i o n with the l i g h t variations which had declined in amplitude between 1968 and 1970. Dukes i d e n t i f i e d four periods in the data and after a comparison with various the o r e t i c a l models assigned the following modes. f(1) = 5.7542 c/d ra d i a l - f i r s t overtone f(2) = 5.7854 non-radial ( p? ) f(3) = 4.1158 ra d i a l - fundamental f(4) = 3.6171 non-radial ( g? ) The apparent preference for f i r s t overtone r a d i a l mode i s contrary to work by Watson (1972) which indicates that the 19 fundamental mode i s the most l i k e l y . The appearance of a strong overtone could then account for the i n s t a b i l i t y of the system. Since c V i r g i n i s i s a massive early type star i t should also radiate in the UV. It was observed in 1975 by Hutchings and H i l l with the Copernicus 0A0-2 f a c i l i t y (Hutchings and H i l l 1977). These observations were made at 1550 A and at 1110-1130 A. Due to Spica's high rotational velocity only two l i n e s in the 1110 A region were suitable for measurement. Taking into account errors in measurement and the low S/N of some of the data, they found no systematic residuals from the o r b i t a l v e l o c i t y with amplitude greater than 12 km/s with either a four or six hour period. It i s possible that variation was occurring but could not be detected. Variation in the strength of the continuum was seen. New data, including scans at 1105-1117 A (Hutchings and H i l l (1980 ) showed strong assymmetry in the l i n e p r o f i l e s although again there were no detectable p e r i o d i c i t i e s in the data. In the early 1970's the behaviour of o V i r g i n i s was extremely confusing. Lomb (1978) discussed both the e a r l i e r published observations and also unpublished photometric observations by Shobbrook in 1971, 1972, 1973 and 1976. F i r s t of a l l the amplitude of pulsation appears to have declined s i g n i f i c a n t l y during 1970-71 continuing the trend f i r s t seen in Shobbrook's published observations. Between 1971 and 1973 the largest amplitude was only 0.0024±0.001 mag. The l i g h t variations in 1968 and 1969 are modulated by the four day o r b i t a l period whereas those in 1970 and 1971 are much more random and the same modulation curve i s not apparent. Lomb also 20 considers the r a d i a l v e l o c i t y v a r i a t i o n . After frequency analysis of Dukes' observations he finds only three periods one of which i s ambiguous. A period of 0.1738 days i s present. The photometric data was also -searched for Dukes' periods. There did not appear to be any peaks in the frequency spectrum corresponding to Dukes' periods. In the case of the e a r l i e r spectroscopic observations a period of 0.1738 days was present in the data sets from Shobbrook (1972), Struve e t . a l . (1958) and Struve and Ebbighausen (1934) and with some uncertainty a period of 0.252 days in the data from Baker (1909). The d i s p a r i t y between the results of various frequency analyses i s in part due to the nature of the data. The spacing of the observations i s t y p i c a l l y an integral number of days which may cause a l i a s i n g problems since the o r b i t a l period i s very close to four days. The v e l o c i t i e s measured by Dukes include measurements for nights on which the l i n e s were quite severly blended. The manner in which corrections were made to the blended v e l o c i t i e s may have introduced systematic errors. Also the increasing randomness of the variations may result in the i d e n t i f i c a t i o n of spurious periods. Although the observations of c V i r g i n i s were undertaken without a complete awareness of the complexity and i n s t a b i l i t y of t h i s star they were made at an appropriate time. During the eary 1970's i t i s clear some kind of changes were taking place such that the p Cephei pulsation had either almost ceased or had become obscured by some other peculiar event in the v i c i n i t y . This should not have been e n t i r e l y unexpected in view of the evidence of l i n e p r o f i l e and intensity variations even in the 21 e a r l i e s t observations. The observations made in V i c t o r i a in 1979 are p a r t i c u l a r l y interesting in that they show ( on the basis of the l i n e p r o f i l e s ) that the 0 Cephei variation i s s t i l l active. 22 Table.4. Published Observations Year Type Reference 1876 1890 1907 1908 1912 1929 1930 1933 1934 1945 1950 1956 1957 1958 1954 1957 1956 1 966 1970 s s s p s s s Interf Struve et. a l . (1958) Vogel (1892) Baker (1909) Stebbins (1914) Struve (1930) Struve and Ebbighausen (1934) Struve and Ebbighausen (1934) Struve (1948) Kao and Struve (1949) Petrie (1950) Struve e t . a l . (1958) Magalachvili and Kumsichvili (1961 Herbison-Evans e t . a l . (1971) Year Type Reference P P Speckle s UV P p UV s S UV s 5 Shobbrook e t . a l . (1969) Shobbrook e t . a l . (1972) Shobbrook e t . a l . (1972) Dukes (1974) Gezari e t . a l . (1972) i Hutchings and H i l l (1977) Lomb (1978) (unpub. obs. by Shobbrook) Bless (1972) thi s work Hutchings and H i l l (1980) Walker e t . a l . (1981) Walker e t . a l . (1981) p = photometric s = spectroscopic 24 CHAPTER II THE SPECTROSCOPIC OBSERVATIONS c V i r g i n i s was observed at the Dominion Astrophysical Observatory by Walker, Fahlman and Yang in 1978, 1979, and 1981. In 1978 and 1979 the 1024 Reticon detector was used with the 1.2m telescope to record spectra at a dispersion of approximately 10 Amm"1. In 1981, the 1024 Reticon was replaced by an 1871 Reticon. With a projected s l i t width of 60»i the 1024 Reticon gave a resolution of 0.6 A. The wavelength region observed included the He I l i n e at 6678.149 A and the Ha l i n e at 6562.8 A. The Reticon detector i s a linear array of 1024 l i g h t sensitive diodes. An individual diode has dimensions of 25»i by 425»J which gives a s p a t i a l resolution comparable with photographic plates where the t y p i c a l dimension of a single grain in the emulsion i s about 20n. A description of the UBC Reticon system can be found in Walker et.al.(1976). Advantages of the Reticon over photographic plates include l i n e a r i t y of response, wide dynamic range (up to 5.106 photons can be recorded), s p a t i a l s t a b i l i t y and minimal e f f e c t s due to scattered l i g h t . When not saturated, lag i s minimal. The electronics of the Reticon are described in d e t a i l by Bucholz e t . a l . (1974) and by Walker e t . a l . (1974). B r i e f l y though, the 25 Reticon diode consists of a layer of N-type s i l i c o n on a gold substrate with a thin upper layer of P-type s i l i c o n . The diode is back biased to 5V creating a depletion zone. When l i g h t i s incident electron-hole pairs are produced in addition to those produced thermally and the depletion region i s discharged. In readout, the diodes are each rebiased in turn. The recharging current i s converted to a voltage pulse which can be integrated and d i g i t i s e d . Since the amplifier also picks up the pulses regulating the s h i f t register responsible for readout, a double readout i s necessary to eliminate the resulting fixed up-down baseline pattern. The detector i s cooled to -90° C with l i q u i d nitrogen to almost e n t i r e l y eliminate the dark current which i s T-3/2. -~?OlT/T dependent on temperature ( Xdq.-k o c I C ). The dark current i s also a function of the amount of l i g h t f a l l i n g on the detector. The readout noise has a lower' l i m i t set by the c h a r a c t e r i s t i c s of the readout electronics of 3600 charge c a r r i e r s . The quantum e f f i c i e n c y of the Reticon i s high in the near infra-red making i t p a r t i c u l a r l y suitable for observations at or near Ho. In 1981, the 1024 Reticon had been superceeded by the 1872 array. The 1872 Reticon i s a 1-D array of 1872 diodes. Each diode has dimensions of 750»» by 15»» and there i s no dead space between the diodes. As before each diode i s back biased by 5V and the readout process i s similar except that there are four separate video channels for readout. Since the four channels process the signal independently and each channel may have a s l i g h t l y d i f f e r e n t gain, the o r i g i n a l spectrum appears very rough. D i v i s i o n by the lamp spectrum compensates for gain 26 differences between channels but w i l l not remove the effect of differences in l i n e a r i t y between the responses of d i f f e r e n t channels. The s l i g h t differences between the ways in which the 1978/79 data and the 1981 data were reduced r e f l e c t the di f f e r e n t c h a r a c t e r i s t i c s of the two detectors. DATA REDUCTION 1979 OBSERVATIONS Time series of spectra were obtained on each of the f i v e nights in March (March 20, 21, 22, 23, 24 UT). One complete o r b i t a l period (4 days) i s covered by the observations. The data was processed using RETICENT - a command language s p e c i f i c a l l y designed to manipulate spectrophotometry data. A complete description of RETICENT can be found in Yang (1980b). Each frame of data (consisting of one spectral record) was processed in the following manner: (1) A base l i n e was subtracted. The base l i n e consisted of the average of several short dark exposures made on the same night. (2) Each frame was divided by the average lamp spectrum for that night. The lamp was simply the spectrum from an incandescent lamp. The idea was to remove differences in diode to diode response. The process was not e n t i r e l y successful (small features above the noise l e v e l were s t i l l present in each spectrum) possibly due to the fact that since the lamp was placed just outside the s l i t during exposure, the f - r a t i o for 27 the lamp spectrum d i f f e r s from that of the s t e l l a r observations. (3) Due to the way in which the Reticon i s read out there may be a regular variation in signal with a four diode baseline superimposed by the electronics of the system. This was smoothed out using the RETICENT routine SPTNA which i s described in Yang (1980b). (4) A 20 point f i l t e r with a Papoulis window was applied to f i l t e r the data to 45% of the Nyquist frequency. (5) Each frame was r e c t i f i e d by dividing by a t h i r d order polynomial f i t to the true continuum points. In order to obtain the best f i t to the instrumental response curve the Ho and He I lin e s were treated separately. The following figures i l l u s t r a t e the appearance of the data at each stage of the reduction. 1978 OBSERVATIONS The 1978 observations made on May 8 UT had s i g n i f i c a n t l y lower S/N. Nonetheless they were reduced in the same manner. 1981 OBSERVATIONS The 1981 observations were made on A p r i l 14 and 16 UT with the 1872 Reticon and so were treated somewhat d i f f e r e n t l y . Rather than subtracting a short dark, long dark exposures were made with exposure times which were the same length as that of the observations. An average dark was subtracted from the observation of the corresponding length. 28 The lamp exposures were made by looking at the l i g h t from an incandescent lamp re f l e c t e d from the i n t e r i o r of the dome to better approximate the optics of the s t a r l i g h t . 29 30 suun ope 32 Fig.4. He I 6678, 23 March 79. Smoothed using the SPTNA rountine. si;un ope 33 Fig.5. He I 6678, 23 March 79. 3rd Order polynomial f i t to the continuum. suun ope Fig.6. He I 6678, 23 March 79. The R e c t i f i e d Spectrum 35 WAVELENGTH CALIBRATION Since c V i r g i n i s was observed at large a i r mass ( i t does not r i s e more than 30° above the horizon in V i c t o r i a ) sharp t e l l u r i c l i n e s were present in the spectra. They were p a r t i c u l a r l y evident in the wavelength region round Ha. The t e l l u r i c l i n e s could be used to define a rest frame for r a d i a l v e l o c i t y measurements, o Boo ( and o Aur on March 20 UT when a Boo was not observed) were chosen as standard stars. Since these stars are of later spectral type (K2 III and G8 III respectively) t h e i r spectra display many li n e s in the wavelength region observed. Once the position and wavelength of these l i n e s were determined a dispersion r e l a t i o n of the form X.=a+bx+cx2 . . .. could be found by the method of least squares f i t t i n g a polynomial to the position-wavelength data. Since the spectrum of the standard star i s Doppler sh i f t e d with respect to the rest frame defined by the t e l l u r i c l i n e s i t is possible to calculate the factor needed to convert the standard star dispersion re l a t i o n to that of the o V i r g i n i s spectrum. That i s , I COHERE flN*> (CL+ 6X + CUX * • • • ) /* THE 7>I5PEKS>V,V oc GQO s pec TRUM. 36 This was not e n t i r e l y straight forward. Problems occured due to the nature of the position finding routine used to measure accurate l i n e positions. This involved least squres f i t t i n g of a parabola to the l i n e p r o f i l e and is described in the RETICENT documentation (Yang 1980b). The parabola f i t t i n g r e l i e s only on points at the core of the l i n e and so i s ideal for finding the position of sharp narrow li n e s such as the t e l l u r i c l i n e s . The routine does not work well for blended or assymmetrical l i n e s and so not a l l l i n e s on the standard s t e l l a r spectrum were used. Since the qu a l i t y of the observations varied from night to night d i f f e r e n t l i n e s were used on some nights to define the dispersion r e l a t i o n . An average spectrum was calculated on which to measure the position of the t e l l u r i c l i n e s which were assumed not to s h i f t in position during the night. Wavelength s h i f t s of the t e l l u r i c l i n e s due to atmospheric pressure changes are so small as to be i n s i g n i f i c a n t here. 1978 OBSERVATIONS In 1978 a Boo was again observed as the standard star. It was observed in the same wavelength region as the t e l l u r i c l i n e s in the c V i r g i n i s spectrum so that i t was possible to obtain the wavelength scale on the a V i r g i n i s spectrum d i r e c t l y . 1981 OBSERVATIONS A more f l e x i b l e position finding routine had been developed which weights the l i n e s according to their r e l a t i v e i n t e n s i t i e s before f i t t i n g a dispersion r e l a t i o n . This i s described in the 37 current RETICENT documentation (Yang 1981). a Boo was again used as the v e l o c i t y standard. PIXEL NUMBER 00 HR5056 '. 21 MARCH 1979 UT 530 .6 474.4+ 418.3 + 362.1 + 306.0 CO P3 IB io 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 600 850 900 950 1000 PIXEL NUMBER PIXEL NUMBER ID HR5056 : 23 MARCH 1979 UT 551 .6 494.8+ 438.0 + 381 .1 + 324 .3 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800 850 900 950 1000 PIXEL NUMBER PIXEL NUMBER 43 CHAPTER III THE SPECTROSCOPIC BINARY ORBIT AND APSIDAL MOTION On several nights (namely May 8 1978 UT, March 21, 23 1979 UT, and A p r i l 14 and 16 1981 UT) the spectral l i n e s from the two components were c l e a r l y separated and r a d i a l v e l o c i t i e s could be measured for both stars. Velocity measurements were made for the He I l i n e only as the Ho l i n e was extremely broad and was blended with t e l l u r i c l i n e s such that i t was not possible to measure the position of the secondary component. A vi s u a l method was used to measure the position of the primary and secondary components. F i r s t the time series of spectra were plotted on a large scale (25 pixels per inch). Templates were then constructed by tracing standard primary and secondary l i n e p r o f i l e s onto transparancies. The templates were aligned v i s u a l l y with the actual p r o f i l e s and the position of the centre of the standard p r o f i l e was measured in p i x e l number. Knowing the dispersion of the spectra, position in p i x e l number could be converted to wavelength and thence to v e l o c i t y using the Doppler s h i f t formula. The method i s analogous to computerised position finding routines which also ali g n a standard l i n e p r o f i l e with the actual. One such method i s described in Fahlman and Glaspey (1974) and in Fahlman (1980). Here the spectrum i s shifted with respect to a standard spectrum 44 and one looks for the s h i f t which minimises the least squares difference between the two spectra. The s h i f t i n g and differencing process i s done in the Fourier domain. The success of such a method depends strongly on the assumption that the shape of the l i n e p r o f i l e s i s constant. Since in t h i s case the li n e p r o f i l e s show very large variations i t was f e l t that a vi s u a l " f i t t i n g by eye" method might be just as successful. In determining the best f i t v i s u a l l y care must be taken be as objective as possible. The method also r e l i e s to some extent on the experience of the observer. With some practice i t was found that measurements of li n e position were repeatable to within ±2 pixels (±0.48 A). Several problems were encountered. F i r s t , the rest wavelength from which the Doppler s h i f t v e l o c i t y was calculated was assumed to be the laboratory value. In fact the determination of rest wavelengths i s a well known problem in ra d i a l v e l o c i t y measurements. The actual wavelength of a l i n e may depend on a number of factors - the l i n e may be blended with other fainter l i n e s or else the wavelength measured may depend on the spectral type of the star or even on the spectrograph used. Since the intention was to la t e r calculate v e l o c i t y differences rather than absolute v e l o c i t i e s t h i s did not appear to be a source of error. Secondly, t h i s method of measuring v e l o c i t i e s s t i l l assumes that the l i n e p r o f i l e s do not change s i g n i f i c a n t l y during the night. Even a quick glance at the time series (say on March 23) shows that the l i n e p r o f i l e s are highly variable. This requires that some q u a l i t a t i v e judgement be made as to what constitutes the l i n e centre. In practice when aligning the standard and test spectra greatest weight was given 45 to f i t t i n g the wing of the p r o f i l e furthest from the other component on the assumption that the most severe i n t r i n s i c p r o f i l e v a r i a t i o n occurs at the l i n e centre (e.g. s p l i t t i n g into two components) and that the blending would be most evident in the wing closest to the other component,. The qua l i t y of these observations was high and the two components were s u f f i c i e n t l y d i f f e r e n t i a t e d that blending would not d i s t o r t the ra d i a l v e l o c i t y measurements. V e l o c i t i e s were measured to within ±5-10 kms - 1 in 1979. On March 20, 22, and 24 1979 UT the primary and secondary were blended but i t was s t i l l possible to to see and measure sharp features which corresponded to the primary and secondary and i t was f e l t that acceptable r a d i a l v e l o c i t y measurements could be made. Blending between l i n e s from two or more s t e l l a r spectra i s known to d i s t o r t the r a d i a l v e l o c i t y measurements such that v e l o c i t i e s near zero v e l o c i t y crossing may be shifted towards the r v e l o c i t y axis. This i s a well known effect and i s mentioned in Struve e t . a l . (1958). Observations were also made on A p r i l 15 and 17 1981 UT but blending of the two components was complete and no v e l o c i t y measurements were made. The v e l o c i t i e s were also corrected for the Earth's o r b i t a l motion. The r e l a t i v e orbit of a spectroscopic binary can be defined by eight parameters which specify the dimensions and orientation in space of the o r b i t . These are: e c c e n t r i c i t y (e), period (P days), apsidal period (Pa years), r a d i a l v e l o c i t y amplitude of the primary (Kp kms - 1) and of the secondary (Ks kms" 1), longitude of periastron (o°), time of periastron passage (To,JD) 46 and the r a d i a l v e l o c i t y of the centre of mass (r kms" 1). The o r b i t a l parameters are defined and the mechanics of binary o r b i t s are discussed in Smart (1979) and also in Batten (1973). When solving the orbit of a binary system, values for the o r b i t a l parameters are sought such the the calculated r a d i a l v e l o c i t i e s agree with the observed v e l o c i t i e s at any given epoch. Usually the o r b i t a l solution is given appropriate to a s p e c i f i e d epoch since some of the parameters (e.g. period) may be time dependent. If they are found to be so then t h i s may provide important information about the system, for example the presence of circumstellar material might be suspected. The apsidal period i s of t h e o r e t i c a l interest since apsidal motion depends on the internal structure of the stars and so i s an important test of s t e l l a r models. The apsidal period i s usually long - in most cases at least one or two hundred years. To obtain a good value for the apsidal period a long data base i s required. For a V i r g i n i s since observations are available from as far back as the turn of the century i t was decided to f i r s t e s t a b l i s h a value for the apsidal period using a l l the available observations and then to obtain o r b i t a l solutions appropriate to each epoch at which observations were made. The problem then i s how to combine a number of subsets of observations - each by d i f f e r e n t observers who each used d i f f e r e n t equipment and techniques of v e l o c i t y measurement. The e a r l i e r - observations are a l l photographic while the l a t e s t were made with a reticon detector. In each case the quoted r a d i a l v e l o c i t i e s are based on the measurement of d i f f e r e n t spectral l i n e s . These are factors which may introduce systematic offsets 47 between d i f f e r e n t groups of observations. Rather than try to introduce "correction factors" into any of the observations i t was decided to attempt using a l l the observations but f i t the o r b i t a l parameters by considering the vel o c i t y difference (Vp-Vs) rather than the absolute v e l o c i t y . The o r b i t a l v e l o c i t i e s of stars in a binary orbit are represented by the functions where v i s the true anomaly (Batten 1973). A simplex search method was used to minimise the function R N P + R N 5 where (Vp-Vs)obs i s the observed v e l o c i t y difference, (Vp-Vs)calc i s the ve l o c i t y difference calculated, RNP i s the number of observations of the primary v e l o c i t y and RNS i s the number of observations of the secondary. The function was minimised with respect to the eight o r b i t a l parameters defined e a r l i e r . The simplex search works by f i r s t setting up a simplex in eight dimensions and evaluating the function at each vertex of the simplex. The worst vertex (the largest function value since we are minimising the function) i s discarded and replaced with i t s mirror image in the centroid of the remaining vertices before recomputing the function at each vertex. It i s possible to scale the simplex search in the neighbourhood of a solution. The simplex routine i s documented by the University of B r i t i s h 48 Columbia Computing Centre (Patterson 1978). The simplex search is somewhat prone to false convergence although since only eight variables are involved t h i s was not thought to be a problem. In practice i t was found that the search was sensitive to the scaling factors chosen for each parameter (these were set to indicate the maximum expected change in the value of each parameter between i n i t i a t i o n and conclusion of the search) and also to the i n i t i a l values chosen. The scaling factors were set by considering the variation in the previously published values for the o r b i t a l parameters. The solution published by Shobbrook e t . a l . (1972) was used for the i n i t a l solution. It was found that even with a good i n i t i a l solution and car e f u l choice of the scaling factors the simplex search required a large number of iter a t i o n s and expenditure of computing time. This may have been a result of trying to force a solution to a very large data set of variable q u a l i t y . The function value at the conclusion of the search i s a measure of the success of the search. Large function values indicate a poor f i t to the data. The r a d i a l v e l o c i t y data could be subdivided into the following sets: Baker (1909), Struve and Ebbighausen (1934), Struve e t . a l . (1958), Shobbrook e t . a l . (1972), UBC-1978, UBC-1979, and UBC-1981. UBC-1978 consisted of only one night of data so t h i s was not used as a separate subset of the data. Only observations where both primary and secondary were observed were used. A large number of observations were made by Dukes (1974) however only v e l o c i t i e s for the primary were published. It i s not clear why the secondary was not observed - as Dukes says " . . . I t was very d i f f i c u l t to see the l i n e s of the secondary and 49 consequently they were not measured....In addition the v e l o c i t i e s were of very poor q u a l i t y . " The r a d i a l v e l o c i t y data used i s l i s t e d i n Appendix C. Using the entire c o l l e c t i o n of data the apsidal period was found to be.149 years which i s in agreement with Dukes' ( 1974) value of 143±20 years but larger than Shobbrook et.a l . ' s (1972) value of 128±12 years. Since the v e l o c i t y differences were to be minimised the r v e l o c i t y was set to 0 kms"1 for the simplex search (which i s in agreement with other solutions e.g. Shobbrook e t . a l . (1972)). The data was then broken down i n t o the subsets and the apsidal period fixed at 149 years before repeating the simplex search at each epoch. The longitude of periastron (u 0) a s a function of time i s shown in fig.12. Not only i s the steady advance of the l i n e of apsides c l e a r l y shown but the rate of change in u appears to be only approximately l i n e a r . The apsidal period from the slope of t h i s graph i s 136±8 years. The question of a variable apsidal period has been raised. There i s the p o s i b i l i t y of a deviation from a straight l i n e by the f i r s t couple of points. However the data i s sparse and there are no observations between 1910 and 1930 nor between 1935 and 1950. Supposing that there i s some var i a t i o n in the apsidal period a possible explanation might be v a r i a t i o n in n (the longitude of the nodes of the orbit) which can be shown to occur in binary systems where t i d a l d i s t o r t i o n and s t e l l a r rotation occur (Kopal 1978). 50 F i g . 1.2. Longitude of periastron as a function of time for alpha V i r . WO /9'0 ttio /93o iv+o I«TO /f(o wo ytro year Table.5. Alpha V i r g i n i s - Orb i t a l Solutions I n i t i a l F i n a l 1906 T (JD) 2417686.914 2417686.391 P (days) 4.01454 4.01491 (km/s) 0 0 e 0.14 0.08 Kp (km/s) 124 1 22 Ks (km/s) 1 97 219 oJo (deg.) 319 338 Pa (years) 149 149 function value 385.37 1 3.96 no. i t e r a t i o n s 417 1933/34 T (JD) 2427504.028 2427502.526 P (days) 4.01454 4.01443 (km/s) 0 0 e 0. 14 0.10 Kp (km/s) 124 1 23 Ks (km/s) 197 205 coc (deg.) 49 47 Pa (years) 149 149 function value 77.06 11.11 no. i t e r a t i o n s 294 I n i t i a l F i n a l 1956 T (JD) 2435603.674 2435603.674 •P (days) 4.01454 4.01430 (km/s) 0 0 e 0.14 0.11 Kp (km/s) 124 100 Ks (km/s) 197 211 Do (deg.) 105 105 Pa (years) 149 149 function value 45.03 9.10 no. i t e r a t i o n s 313 1969 T (JD) 2440284.76006 2440284.57293 P (days) 4.01454 4.02253 (km/s) 0 0 e 0.14 0.19 Kp (km/s) 124 1.01 Ks (km/s) 197 201 u)o (deg.) 142 142 Pa (years) 149 149 function value 47.23 6.70 no. it e r a t i o n s 390 I n i t i a l F i n a l 1979 T (JD) 2443954.05000 2443954.05001 P (days) 4.01454 4.01454 (km/s) 0 0 e 0.14 0.14 Kp (km/s) 124 126 Ks (km/s) 197 191 afe (deg.) 170 171 Pa (years) 149 149 function value 209.24 11.96 no. it e r a t i o n s 185 1981 T (JD) 2444708.75300 2444708.54919 P (days) 4.01454 4.01458 (km/s) 0 0 e 0.14 0.14 Kp (km/s) 124 123 Ks (km/s) 197 201 oJo (deg.) 176 178 Pa (years) 149 149 function value 440.72 3.46 no. it e r a t i o n s 165 54 As already indicated, the apsidal motion in a binary system i s of p a r t i c u l a r interest since i t i s one of the few means available for investigating s t e l l a r i n t e r i o r s . The apsidal motion i s the secular advance (or regression) of the l i n e of the apsides as measured by the change in the longitude of periastron. Apsidal motion may be due to any one of a number of causes a l l representing a departure from the case of point masses, Keplerian o r b i t s and Newtonian mechanics. Although the quantity observed i s the apsidal period, discussion i s usually in terms of the apsidal constant k which i s defined separately for the primary (k 2 P ) and for the secondary (k 2 s ) stars and which describes the d i s t o r t i o n of the o r b i t by a disturbing g r a v i t a t i o n a l p o t e n t i a l . The apsidal constant i s sensitive to the internal structure of the stars being 0 for point masses and increasing to 0.75 for homogeneous stars. The subscript '2' indicates that k arises from a second order term in the Legendre polynomial expansion for the g r a v i t a t i o n a l p o t e n t i a l . The subscripts 'p' and 's' refer to the primary and the secondary components respectively. Mathis and Odell (1973) calculated the apsidal constant for various s t e l l a r models and compared the result with the "observed" apsidal constant for a V i r g i n i s in the hopes of learning about the internal structure of the primary. They found that k 2p(model) was consistently greater than k 2 P(observed) by a factor of 2 i . e . the s t e l l a r models are not s u f f i c i e n t l y c e n t r a l l y condensed. In order to explain this and also to f i t the observed flux r a t i o at 2905/V A they found i t necessary to include a r b i t r a r y mixing in the central regions of the models or 55 to increase the metal opacity in the outer layers. This suggested that i t might also be worthwhile to look at the assumptions made in cal c u l a t i n g k 2 (observed). The observed apsidal motion i s due to several factors which interact in a f a i r l y complex manner. The most complete discussion i s in Kopal's "Dynamics of Close Binary Systems" (1978). E f f e c t s to be considered include t i d a l d i s t o r t i o n , r otational d i s t o r t i o n , r e l a t i v i s t i c corrections and perturbations due to a t h i r d body or circumstellar material. Of these t i d a l e f f e c t s dominate while s t e l l a r rotation i s less s i g n i f i c a n t unless the angular v e l o c i t y of rotation i s much greater than that of the o r b i t a l motion. Unless the system contains very massive condensed components, the r e l a t i v i s t i c correction i s so small that i t can be disregarded. A t h i r d body in an orbit at some distance from the binary (the most stable configuration) may result in observable perturbations in the binary o r b i t . It would be interesting to know i f any of these cases applies to a V i r g i n i s . Kopal (1978) gives as the rate of change in u in the case of equilibrium tides in a viscous medium ( i . e . no mass movement in the coordinate frame moving with the radius vector of the o r b i t ) : 56 In t h i s case the l i n e of apsides w i l l advance. For the change in u due to s t e l l a r rotation he gives (/ ~ z*'"Y&s Safesti )/«n2 i ) (v i s the rotational velocity,^? i s the mean s t e l l a r density, G i s the g r a v i t a t i o n a l constant, a i s the semimajor axis of the o r b i t , R i s the s t e l l a r radius and M i s the s t e l l a r mass). The contribution to apsidal motion from s t e l l a r rotation depends on the i n c l i n a t i o n of the axes of rotation to the plane of the o r b i t . I f , as expected, the s t e l l a r equator i s coplanar with the orbit (9=i=0) t h i s would result in a positive do/dt. As the angle of i n c l i n a t i o n increases do/dt decreases and may become negative. The t o t a l apsidal advance ( to a f i r s t approximation) i s Mathis and Odell (1973) use a s l i g h t l y s i m p l i f i e d form which is 57 i . e . P/Pa = c,k 2p + c 2 k 2 s , c, and c 2 being the c o e f f i c i e n t s in the expansion for P/Pa, In their c a l c u l a t i o n of the apsidal constant Mathis and Odell make a number of assumptions. The stars are assumed to be r i g i d l y rotating with axes of rotation perpendicular to the o r b i t a l plane. Also to calculate k 2 P(observed) from their equation 6 i t i s necessary to know k 2 S . This they obtain by assuming that the secondary i s on the main sequence and that there i s a linear logk-logM relationship on the main sequence. The following comments could be made. The apsidal advance i s due to the interaction of both components and the observed apsidal constant i s actually the weighted sum of the contribution from each star (Kopal 1978). The c o e f f i c i e n t s c 1 and c 2 can be determined from a knowledge of the physical c h a r a c t e r i s t i c s of the system but a simple measurement of P/Pa gives only a mean value k where k = (C]k2P + c 2 k 2 6 )/(c,+c 2) and k=(1/c,+c 2)/(P/Pa). The assumption that k 2 P can be isolated by assuming that the secondary i s on the main sequence i s a s i m p l i f i c a t i o n . However the spectral c l a s s i f i c a t i o n of the secondary has been in some doubt and on the basis of the l i n e p r o f i l e s discussed in Chapter IV, the secondary may also be variable. In addition the v a r i a t i o n in k 2 with spectral type i s not p a r t i c u l a r l y well defined. It seems that k 2 decreases with l a t e r types. Fig.13. shows k 2 as a function of the spectral class of the primary in systems where k 2 has been determined. The data i s from Kopal (1978). Theoretical models show that k 2 decreases as a star evolves away from the main sequence. As the outer layers expand the star becomes r e l a t i v e l y more.centrally c e n t r a l l y condensed. 58 However Mathis and Odell point out that assuming a later evolutionary stage for the secondary does not lead to better agreement for k 2p(observed) and k 2 P(model) since as the stars leaves the main sequence (R/a) 5 increases at a much greater rate than k 2 decreases. 59 Fig.13. Mean value for k as a function of spectral type of the primary component in binaries exhibiting apsidal motion. 60 Another problem i s that of synchronisation of rotation and revolution. The usual development of the theory of apsidal motion assumes synchronisation in which case equilibrium tides form. More r e a l i s t i c a l l y dynamical tides in a viscous medium should be considered. Over a long period of time the action of t i d a l lag and other d i s s i p a t i v e forces (viscosity and matter-radiation interactions) w i l l tend to induce synchronisation. Evidence for t i d a l effects in a V i r g i n i s i s discussed by Smak (1970) and by F i t c h (1967). The spectrum of a V i r g i n i s shows very broad l i n e s . If the l i n e broadening i s ascribed solely to rotation the angular v e l o c i t y must be very high, much higher than the o r b i t a l veloc i t y . Table 6. Rotational Velocity l i n e A Vp s i n i km/s Vs s i n i km/s Ref. C 4267 Si 4552-4575 He I 4387-4471 He I 6678 He I 6678 0 II 6721 1 62 200 155 251 ± 5 258 ± 5 265 ± 10 62 100 105 Struve e t . a l . 1958 Watson 1972 70 ± 5 UBC March 1979* 97 ± 5 UBC A p r i l 1981* UBC A p r i l 1981* * These are f u l l width half maximum v e l o c i t i e s The rotational v e l o c i t y as measured from the width of the spectral l i n e s i s given in table 6. Not only does i t appear that 61 the rotational velocity i s much greater than expected for synchronisation, i t appears that there has been considerable va r i a t i o n in v e l o c i t y . The v e l o c i t i e s measured from the UBC observations are considerably greater than those made e a r l i e r . This may be in part due to measuring d i f f e r e n t l i n e s (the amount of broadening due to s t e l l a r processes other than ro t a t i o n may vary with wavelength) and the use of the reticon rather than photographic spectra. Two interpretations are possible. If the l i n e width is ascribed solely to rotation then non-synchronisation w i l l cause dynamical tides to form which w i l l influence apsidal motion, t h i s might occur in a young system where d i s s i p a t i v e forces have yet to result in synchronisation. The fact that the primary star in t h i s case i s a p Cephei variable mitigates against t h i s since i t has already been shown that the p Cephei stars are l i k e l y to be evolving off the main sequence A l t e r n a t i v e l y the l i n e p r o f i l e s are already known to show complex variations and i t i s more l i k e l y that the observed l i n e width i s due to both rotation and the i n t r i n s i c s t e l l a r v a r i a t i o n s . Variation in s t e l l a r a c t i v i t y at d i f f e r e n t epochs could also account for the var i a t i o n in measured v e l o c i t y . The observed p r o f i l e s are in fact consistent with non-radial pulsation modes (Campos and Smith 1980). In passing i t might be noted that the effect of pulsation on apsidal motion has not been investigated. Regarding the p o s s i b i l i t y of a t h i r d component, Mathis and Odell (1973) were unable to eliminate the p o s s i b i l i t y of a t h i r d star in the system. Features in the UBC spectra which were at f i r s t thought to be possibly due to a t h i r d component were found 62 to be moving through the primary l i n e with measureable v e l o c i t y inconsistent with a slow moving t h i r d body. That i s the t h i r d star would be expected to occupy a large orbit about the binary and would not appear to move rapidly with respect to the binary. Speckle interferometry does not show a t h i r d star (Gezari e t . a l . 1972). No conclusions regarding possible circumstellar material were made on the basis of the spectroscopic observations. 63 CHAPTER IV THE LINE PROFILES The intention in 1979 was to observe a V i r g i n i s for l i n e p r o f i l e v a r i a t i o n s . Long time series were taken on a l l nights over four or f i v e hours which i s about the expected period for the fi Cephei pulsation. The individual spectra displayed here have been smoothed, f i l t e r e d and r e c t i f i e d by d i v i s i o n by the t h i r d order polynomial f i t to the continuum as described in Chapter I I I . On two nights (March 21 and 23 1979 UT) the primary and secondary He I l i n e s were c l e a r l y separated. Even a cursory glance shows that the l i n e p r o f i l e of the primary i s varying in a complex manner. The l i n e p r o f i l e variations are at least s u p e r f i c i a l l y similar to those seen in other fi Cephei stars. The primary p r o f i l e appears to s p l i t into two components at certain phases - or i t might be that the l i n e core i s f i l l e d with emmision at these phases. This has been observed in other fi Cephei stars, for example BW Vul (Goldberg 1973) and 12 Lac ( A l l i s o n 1976). The d e t a i l s of these observations required d i f f e r e n t interpretations for each of these stars. In the case of BW Vul (which is an extreme member of the class as far as i t s large velocity amplitude goes) th i s was explained by the existance of moving s h e l l s or shock waves in the s t e l l a r atmosphere. As far as 12 Lac i s concerned, i t 64 appeared that there were at least two unresolved components present. a V i r g i n i s i s p a r t i c u l a r l y complex - i f only because any i n t r i n s i c v a r i a t i o n must be modulated by the presence of the close and quite massive companion. The apparent doubling of the primary He I l i n e was also noted by Struve e t . a l . (1958) in their photographic spectra, "...the l i n e at He I 6678 shows a s t r i k i n g l y double primary at phase +0.10 day and a barely noticeable d u p l i c i t y on the following spectrogram, phase +0.23 day. The same figure shows that Ho i s free of emission." It was not clear from the Reticon observations i f the observed l i n e s p l i t t i n g could be assigned a four hour period. In addition to the v a r i a t i o n occuring at the l i n e centre, there i s structure in the wings of the l i n e . For example, on March 23 1979 UT i t appears that there i s a s l i g h t feature, i n i t i a l l y present on the short wavelength side which by the end of the night (after about four hours) had s h i f t e d to the long wavelength side of the p r o f i l e . On March 21 a similar pattern i s seen except that the faint feature appears to move from long to short wavelength. These features are l a b e l l e d a and b respectively in fig.21 and 22 and in f i g 18. In each case the feature i s moving away from the wing adjacent to the secondary. It i s p a r t i c u l a r l y interesting to note that there appear to be l i n e p r o f i l e variations in the secondary also. Early observations, for example those by Struve and Ebbighausen (1934) indicated that there were quite large periodic variations in the intensity of the secondary although Struve e t . a l . (1958) found l i t t l e evidence for this in t h e i r 1956-58 observations. 65 Fig.14. Time series, He I 6678, 20 March 1979 UT. Time scale in fractions of a Jul i a n Day. 0 .772 8 : ^ 0 .798 0 .806 0 .813 0 .821 0.829 0.837 0 .845 0 .852 0 .859 0 .866 0 .873 0 .878 0 .885 0 .892 0 .898 0 .905 0 .913 P IXEL # 66 Fig.15. Time series, He I 6678, 20 March 1979 UT cont. Time scale in fractions of a Jul i a n Day. 1 0 0 150 200 - P IXEL # 250 67 Fig.16. Time series, He I 6678, 21 March 1979 UT. Time scale in fractions of a Julian Day 0 .824 0 .838 0 .852 0 .865 0.893 0 .904 0 .916 0.927 0 .940 0 .952 0 .963 0 .975 0 .987 0 .998 1 .010 1 .024 165 215 P IXEL # 68 Fig.17. Time series, He I 6678, 22 March 1979 UT. Time scale in fractions of a Jul i a n Day. 0.896 0 .905 0.913 0 .922 0 .931 0 .939 0 .948 0 .957 0 .966 0 .974 0 .983 1 .009 1 .018 1 .044 1 .053 100 150 200 250 300 350 P IXEL # 69 Fig.18. Time series, He I 6678, 23 March 79 UT. Time scale fractions of a Julian Day. 0.805 0 .814 0.822 0.829 0.835 0 .841 0.846 0.852 0.858 0 . 864 0.869 0.875 0.881 0.887 0.893 0.898 0 . 904 200 250 300 350 400 P IXEL # 70 Fig.19. Time series, He I 6678, 23 March 1979 UT cont. Time scale in fractions of a Julian Day. 200 0.910 0.916 0.922 0.928 0.933 0.939 0.945 0.951 0.956 0.962 0 .968 0 . 974 0.979 0 .985 0.991 0.998 1 .008 400 71 Fig.20. Time series, He I 6678, 24 March 1979 UT. Time scale in fractions of a Julian Day. 0.800 0 .809 0 .820 0 .830 0 .840 0.850 0 .860 0 .871 0 .882 0 .892 0 .902 0 .912 0 .923 0 .934 0 .944 0 .955 0 .965 0 .975 0 .986 0 .996 1 .007 1 .089 1 .030 1 .042 200 250 300 P IXEL # 350 72 In order to better follow the features in the primary, difference plots were constructed. The spectra were aligned with respect to the primary to compensate for o r b i t a l motion, the position of each He I p r o f i l e was found with respect to the f i r s t spectrum in the series using the method described by Fahlman (1980) and then sh i f t e d by the amount required to al i g n i t with the f i r s t spectrum. The mean p r o f i l e was then calculated and subtracted from each p r o f i l e . The individual spectral records were averaged in groups of three to improve the d e f i n i t i o n of the signal. (Fig.23). 73 Fig.21. Difference p l o t , He I 6678, 23 March 79 UT. Time scale in fractions of a Julian Day. p = primary component, s = secondary component. c,d refer to the features i d e n t i f i e d by Walker et.al.(1981). 5.9042 5.9389 g563 5.9736 5.9915 800 850 900 Jo-5/ PIXEL # 0 -24A/p ixe l 74 It then became obvious that there were at least three s i g n i f i c a n t and persistent features present at or above 0.5% of the continuum l e v e l . These features were c l e a r l y real since they were also present i n the observations made in A p r i l 1981 (Walker e t . a l . 1981, 1982). The two sets of observations were made two years apart with d i f f e r e n t detectors and with the wavelength scale running in the opposite sense in each case. The wavelength and hence r e l a t i v e v e l o c i t y of these features could be measured with respect to the l i n e centres. The work by Walker et.al,. shows that these features have an acceleration of, on average, 0.0065 km • s - 1 . Assuming that these features are displaced only as a result of s t e l l a r r o t a t i o n . Walker e t . a l . give vsini=(a r y-x. s i n i ) . For i=65.9° and r=stellar radius=7.84R 0, they f i n d that vsini=l80km s" 1. This i s much less than the v e l o c i t y given by the f u l l width half maximum of the primary p r o f i l e which was 250 km s" 1. If the lower v e l o c i t y is correct then rotation and revolution are at least nearly synchronised and the observed l i n e width must be due to processes other than rotation. For synchronisation one would expect v s i n i to be of the order of 100 km s" 1. E a r l i e r i t was mentioned that there was some ambiguity regarding the spectral c l a s s i f i c a t i o n of the secondary. On A p r i l 14 1981 UT i t was possible to see the secondary at Ho. The r a t i o of the primary to secondary l i n e depth at He 6678 and Ho was approximately 3. This is consistent with a B3 spectral c l a s s i f i c a t i o n for the secondary. At class B3 the Ho l i n e depth r a t i o should be similar to or less than the He I l i n e depth rat io 75 As a matter of interest, the l i n e depth of the He 6678 A p r o f i l e was measured for the 1979 series of observations The primary and secondary components appear to be constant in depth. The change in l i n e depth on March 20, 22 and 24 i s due to -the blending of the two components. 76 Fig.22. He I 6678 l i n e depth, 20 March 1979 UT. m d a p au;i Fig.23. He I 6678 l i n e depth, 21 March 1979 UT. o-ny • ' . . p r i m a r y o*ty C-07V a> T3 a> c • • • • • o-osy • • • e-nJr • * s e conda r y j u l i an day 2443950 + Fig.24. He I 6678 l i n e depth, 22 March 1979 UT. . • * * . * * b lended l ine • # « s-oo Julian day 2 443950-79 Fig.25. He I 6678 l i n e depth, 23 March 1979 UT. on 6-10 " • • • • . , . # • • -pr imary • • • • ov9\ o-og •*-» Q. a> •o CO c — <?<4 0-o3\ • • • • • • • • # • • secondary • * S~ffO f?0 60O i/O , Jul ian day 2443950+ Fig.26. He I 6678 l i n e depth, 24 March 1979 UT. on* .c o-lo\ a a> a> c 0<H\ blended l ine 6*0 J^fO TOO * Julian day 2443950 + 81 It now seems that the most useful information regarding the way in which p Cephei stars pulsate w i l l come from analysis of the l i n e p r o f i l e variations. It i s the l i n e p r o f i l e s which after a l l contain the information on the v e l o c i t y f i e l d at the s t e l l a r surface. It should at least be possible to distinguish between r a d i a l and non-radial pulsation. Unfortunately pulsation analysis i s complicated by the fact that there may be a mixture of modes involved. For example coupling between r a d i a l and non-r a d i a l modes may be possible. Also the variation in pulsation c h a r a c t e r i s t i c s between d i f f e r e n t members of the p Cephei class may be due to the selection of s p e c i f i c modes in response to some external influence - perhaps membership in a binary system. The motion of a mass unit in a pulsating star i s described m loot * k(r by £r(r,Q ,<{),{) - SA^ S/X (&,<0)^ (u = frequency of the pulsation, k = the time constant of the pulsation and -m<l<m), The eigenfunctions of the pulsation are degenerate in m although t h i s degeneracy can be removed by s t e l l a r rotation or weak magnetic f i e l d s . To obtain a solution, the above equation of mass motion can be separated into the hydrodynamic equation of motion, Poisson's equation, the equation of continuity and the adiabatic energy equation. These then form a set of four simultaneous d i f f e r e n t i a l equations which, with appropriate boundary conditions, can be solved numerically. A solution for which 1=0 corresponds to r a d i a l motion. 1=1 indicates side to side motion of the whole star and i s non-physical. 1=2 represents non-radial o s c i l l a t i o n . There are non-radial modes associated with two types of restoring force, p (pressure or acoustical) modes are those in which pressure acts as a 82 restoring force and g (gravity) modes are those for which gravity i s the restoring force. As an example of the type of surface motion expected consider the case of 1=2. m then takes on the values -2, -1, 0, 1, 2. When m i s negative there i s a t r a v e l l i n g wave in the same sense as the s t e l l a r rotation. When m=2 we expect to see an e l l i p s o i d a l surface region with axis p a r a l l e l to the polar axis rotating about the polar axis.. For m=1, the e l l i p s o i d a l surface i s in c l i n e d at angle of 45° to the polar axis and precesses about th i s axis. A detailed discussion of the theory of non-radial pulsations can be found in Unno e t . a l . (1979). It i s tempting to associate the features found by Walker e t . a l . (1981,1982)'with such surface zones or t r a v e l l i n g waves. Once the surface c h a r a c t e r i s t i c s of the non-radial pulsation mode has been worked out, i t i s possible to model the expected l i n e p r o f i l e . Line p r o f i l e modelling for p Cephei stars has been done by Stamford and Watson (1980), Campos and Smith (1980) and Stamford (1980). Radial pulsations might be expected to result in quite symmetrical l i n e p r o f i l e s however Stamford and Watson (1980) show that interaction between rotation and s t e l l a r expansion and contraction may introduce assymmetry into the l i n e p r o f i l e s . Hence r a d i a l pulsations have not e n t i r e l y been disregarded as an explanation for the p Cephei stars. For a V i r g i n i s i t seems that i t i s the non-radial pulsations which most successfully duplicate the observed p r o f i l e s . By introducing the appropriate parameters into the l i n e p r o f i l e calculations (mode, rotational v e l o c i t y and 83 macroscopic turbulance) i t seems possible to describe features such as the severe l i n e assymmetry and l i n e doubling. The only c r i t i c i s m of t h i s approach seems to be that i f you introduce enough parameters you can describe any thing you l i k e . There are other tests for non-radial pulsation. If the d i s t o r t i o n i s non-spherically symmetric then i t may be possible to observe po l a r i s a t i o n of the star l i g h t which varies with the same period as the l i g h t and r a d i a l v e l o c i t y v a r i a t i o n s . No pola r i s a t i o n would be observed i f the pulsation were r a d i a l (Odell 1979). Polarisation of star l i g h t occurs as a result of electron scattering in the s t e l l a r atmosphere and the observation of po l a r i s a t i o n i s dependent on the geometry of the star. Light from the centre of the s t e l l a r disc appears unpolarised but l i g h t from the limb may be seen as having up to 12% p o l a r i s a t i o n . If the star i s spherically symmetrical then l i g h t of a l l polarisations i s seen from the limb but non-spherically symmetric d i s t o r t i o n s w i l l result in the observation of ove r a l l p o l a r i s a t i o n . Local temperature ef f e c t s w i l l also be important in determining the contribution from a spec i f i e d region on the surface or the stage of ionisation and hence amount of scattering in that region. As yet polarimetric observations of c V i r g i n i s are not avai l a b l e . As mentioned e a r l i e r the pulsation constant Q i s indeterminate being consistent with either a f i r s t harmonic r a d i a l mode or non-radial p, or p 2 mode. On occasion more than one period has been observed for c V i r g i n i s . Looking at the periods found by Dukes (1974), the ra t i o between successive periods i s between 1.10-and 0.88 which 84 suggests non-radial modes rather than a series of harmonics. Since the l i g h t and r a d i a l v e l o c i t y curves are 90° out of phase, the r e l a t i v e phase of l i g h t and r a d i a l v e l o c i t y i s not a sensitive test for mode typing. 85 CHAPTER V CONCLUSION o V i r has had a p a r t i c u l a r l y interesting recent history. Not long after Shobbrook (1969) i d e n t i f i e d the star as a p Cephei variable, Lomb (1978) pointed out that the amplitude of the v a r i a t i o n had been declining for a number of years and had become "almost or completely undetectable". Around 1969-1970 the decline in amplitude was accompanied by increasing randomness or disorder in the p Cephei pulsations (that is disorder in the phase and amplitude). In l i g h t of t h i s , the present observations are of interest even i f only to show that variation i s s t i l l occuring. The l i n e p r o f i l e s display changes in shape reminiscent of the type seen in other p Cephei stars such as a Sco or p Cep which are also members of binary systems. Analyses of the pulsation and searches for pulsation frequencies at t h i s time were confused by the randomness of the pulsations and modulation by the close binary companion. This apparently lead to the i d e n t i f i c a t i o n of spurious periods (Lomb 1978) or multiple periods (Dukes 1974). The only other p Cephei star to come to mind for which a similar decline has been observed i s 16 Lac. The amplitude of 16 Lac appears to have declined by a factor of two (Jerzekowski e t . a l . 197.9). The fact that 16 Lac i s also a 86 member of a binary system i s probably s i g n i f i c a n t . The only pulsation mechanism presented so far which allows for a dying away of the pulsations i s that by Cox e t . a l . (1.981:) •which suggestes that sudden mixing of hydrogen into a convective core i n i t i a t e s an episode of pulsation which decays after a number of years. The only p Cephei star for which pulsation amplitudes have been observed to increase seems to be 6 Cet (Ciurla 1979). There i s some controversy as to whether a similar increase has been observed for BW Vul (Sareyan e t . a l . 1980). Since the p Cephei stars were f i r s t i d e n t i f i e d as a group, v a r i a b i l i t y among B type stars has turned out to be quite common. It i s possible that some stars which are now thought to be spectrum variables might at some e a r l i e r time have been c a l l e d p Cephei stars. Among these other variables the Be stars have also been observed to "switch o f f " their v a r i a t i o n every now and then. The p Cephei stars are a rather disparate group. Since both variable and non-variable stars are found in and around the p Cephei i n s t a b i l i t y s t r i p on the HR diagram, appropriate position on the HR diagram does not necessarily imply membership in the group. Not a l l the p Cephei stars are slow rotators as was at f i r s t thought. Multiple p e r i o d i c i t y , beat frequencies, the van Hoof eff e c t and membership in a binary system are a l l observables which are not seen in a l l p Cephei stars. Concerning the fundamental nature of the pulsations both r a d i a l and non-radial models have been quite successfully applied, often to the same star. For example, Odell (1980) considers o V i r to be pulsating in the non-radial g1 mode whereas Dukes (1974) concludes that a r a d i a l f i r s t overtone mode 87 i s most appropriate. As far as a V i r i s concerned, the l i n e p r o f i l e s seem to be strong evidence for non-radial pulsation. See, for example the results of l i n e p r o f i l e modelling of non-r a d i a l pulsations in Unno e t . a l . (1979) and Watson(l98 ). The moving features measured by Walker e t . a l . (1981) are not inconsistent with non-radial modes. One might question whether the fi Cephei stars have been defined as a homogeneous group. Po s s i b i l y there i s something l i k e a spectrum of v a r i a b i l i t y among B type stars with some overlap between c l a s s i f i c a t i o n s . For instance the 53 Per stars may be related to the fi Cephei stars but d i f f e r in that they are generally considered to be non-radial pulsators and show sudden period changes rather than gradual period changes as do the fi Cephei stars. c V i r is a close binary system, although probably not so close that mass transfer can occur between the components. T i d a l e f f e c t s are, however, undoubtedly important. Photometric observations are modulated by the o r b i t a l period as a result of the e l l i p t i c i t y of the components (Rucinski 1970). The presence of a close and massive star must also influence the fi Cephei pulsation although i t may not have i n i t i a t e d or drive the pulsation. Its presence may have been responsible for the recent disruption in the fi Cephei phenomenon. 16 Lac i s also a member of a binary system but which a r e l a t i v e l y less massive companion of l a t e r spectral type, a Vir should probably be looked at as an interacting binary system since the secondary also appears to vary. The apsidal effects measured also r e f l e c t the mutually interacting nature of the system. The survey of the l i t e r a t u r e on a V i r has shown that the ±fi 88 Cephei stars are more than anything else a group of exceptions From both a theore t i c a l and observational point of view c V i r i s an inte r e s t i n g but d i f f i c u l t case to deal with. Inspite of the many observations available, the interpretation of these observations i s s t i l l ambiguous. Further observations at the DAO are planned with the reticon at blue wavelengths. Simultaneous photometric and spectroscopic observations are not available but might prove valuable i f they could be c a r r i e d out. It would also be interesting to look at the p o s s i b i l i t y that singly and multiply periodic p Cephei stars are i n t r i n s i c a l l y d i f f e r e n t or that membership in binary systems results in uniquely c h a r a c t e r i s t i c pulsations. 89 BIBLIOGRAPHY A l l i s o n , A., 1976, Unpublished MSc Thesis, University of B r i t i s h Columbia. A l l i s o n , A., Glaspey, J.W., Fahlman, G.G., 1977, Astr.. J., 82, 283. Baker, R.H., 1909, Pub. Allegheny Obs., ±j_ 65. Balona, L.A., Englebrecht, C , 1981, Proc. Workshop on Pulsating B Stars, Nice Observatory. Balona, L.A., Feast, M.W., 1975, M.N.R.A.S., J_72, 191. 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Names RA Dec Rr MP Ms log g P Te Fv log(Lp/Lc) Mv,o Mvf V B-V brightness r a t i o spectrum prim, spectrum sec. a distance parallax o V i r g i n i s Spica HR 5056 HD 116858 13h 22m 33.301s -10 54' 3.36" 8.1 ± 0.5 R© 10.9 ± 0.9 MG 6.8 ± 0.7 M© 3.7 ± 0.1 cgs 22400 ± 1 000 °K _^  H (2.75 ± 0.24)10 erg cm s Hz 4.17 ± 0.10 -3.5 ± 0.1 -1.5 10.2 0.97 -0.23 6.4 ± 1.0 B1 .5 IV B3 -(0.90 ± 0.04)10 " (91.54 ± 10.05)10 " (1 .93 ± 0.06) 107km 84 ± 4 pc (0.019 ± 0.006)" 1 1 1 1 1 1 1 1 1 3 3 1 3 4 1 1 1 1 2 5 5 References: 1. Herbison-Evans et. a l . (1971) 2. Jenkins (1963) 3. Lesh and Aizenman (1978) 4. Smak (1970) 5. H o f f l i e t (1964) 95 Appendix B : SUMMARY OF THE UBC OBSERVATIONS. Observatory Location: The Dominion Astrophysical Observatory, Vancouver Island, B.C., Canada. lati t u d e 48 31' 15.70" longitude 123 25' 0" elevation 229m 1978 May 08 UT Telescope 1.22m Detector 1024 Lestercon Grating 3200 1/mm f i r s t order UBC Tape RE0325 P07478 Wavelength c a l i b r a t i o n = 7928.3 - 0.12x (A) Observers G.A.H.Walker, S.Yang 96 1979 March 20,21,22,23,24 UT Telescope Detector Grating UBC Tape Wavelength c a l i b r a t i o n 1 .22m 1024 Lestercon 1000 1/mm f i r s t order RE0325 P07478 20 March 21 March 22 March 23 March 24 March Observers 6717.6 - 0.25x (A) 6722.2 - 0.25x (A) 6738.2 - 0.25x (A) 6751.3 - 0.25x (A) 6746.0 - 0.25x (A) G.A.H.Walker, S.Yang, G.G.Fahlman 1981 A p r i l 14,16 UT Telescope Detector Grat ing UBC Tape 1 .22m 1872 Reticon 1200 1/mm f i r s t order RE0460 Wavelength c a l i b r a t i o n 14 A p r i l = 6490.8 - 0.15x (A) 16 A p r i l = 6488.9 - 0.15x (A) Observers S.Yang Appendix C: RADIAL VELOCITY DATA Radial Velocity - Alpha V i r - Baker (1909) year | Julian day | Vp (km/s)| Vs (km/s 1907 2417000+ 686.17500 107 -1 32 686.22014 89 -159 714.18889 76 -133 714.21528 70 -148 717.07917 -118 179 717.10417 -1 1 1 159 717.13681 -118 179 735.06806 1 34 -205 735.08542 155 -181 741.07708 -117 175 741.09583 -105 206 741.11806 -1 03 166 747.07431 1 04 -217 747.10000 125 -210 747.14028 131 -234 775.09444 140 -198 1908 955.38194 112 -209 955.40000 143 -216 955.40625 134 -213 957.38750 -1 12 177 957.40556 -1 13 178 957.41319 -120 181 960.36875 57 -98 966.35833 -63 1 50 966.37778 -60 136 992.26736 93 -130 992.30000 119 -1 33 992.31250 1 1 5 -101 1010.27778 -1 19 170 1010.29583 -1 14 177 1010.30347 -105 178 1011.31944 88 -111 1028.20556 1 30 -180 1036.23611 139 -1 99 1038.15625 -98 197 1038.17708 -99 191 1044. 18472 109 -214 1044.19028 1 32 -188 Radial Velocity - Alpha V i r - Baker (I909)cont. year | Julian day | Vp (km/s)| Vs (km/s) 1907 2417000+ 1044.20139 138 -205 1054.16944 -113 185 1054.17292 -1 1 1 .204 1054.18542 -125 198 1068.12361 138 -258 1068.13472 143 -242 1072.09861 118 -222 1072.10278 134 -220 1072.11389 109 -218 1074.07222 -120 186 1074.08472 -102 192 1074.10000 -109 188 1100.06042 138 -215 1100.06528 153 -213 1100.07222 140 -208 Radial Velocity - Alpha V i r Struve and Ebbighausen (1934) year | Julian day | Vp (km/s)| Vs (km/s) 1930 2426000+ 43.83681 66.77847 88.69444 88.74028 88.78681 90.68611 90.73264 92.64236 92.68958 102.65347 104.62083 104.66736 1933 1433.93889 1433.95972 1934 1453.97917 1457.88889 1467.98889 1468.00278 1489.89167 1491.89792 1491.91042 1495.82431 1505.83125 1505.92014 1509.79306 -124.0 196.3 -88.8 200.2 122.2 172.8 106.5 -170.2 101.8 -182.4 -111.7 188.5 -113.0 181.3 111.8 -176.9 1 06.2 -167.4 -109.5 176. 1 83.3 -135.5 107.0 -181.2 124.3 -221.7 1 30.8 -211.9 144.0 -203.5 1 37.5 -195.1 -111.2 181.2 -111.2 181.9 130.2 -198.2 -114.0 177.4 -115.2 179.6 -116.2 174.8 116.1 -192.2 136.6 -215.3 1 17.0 -193.8 Radial Velocity - Alpha V i r Struve and Ebbighausen (l934)cont. year | Julian day | Vp (km/s)| Vs (km/s) 1934 2427000+ 509.82778 120. 1 -207.6 51 5,. 81181 -115.9 197.0 515.82361 -114,0 182.9 537.72778 107.2 173.9 537.73750 113.0 -174.7 539.76250 -119.8 188.9 543.79306 -106.5 170.8 557.70069 83.9 -161.9 557.71458 88.0 -140.4 559.74514 -110.7 173.8 559.75625 -111.0 162.7 561,75486 90.8 -165.8 565.68125 66.6 -98.2 565.69167 64.3 -96.4 571.66597 -102.8 165.6 571.68264 -115.2 1 39.2 575.66528 -102.1 154.6 576.66667 -88.0 1 23.3 584.58056 -114.2 151.1 599.58056 -54.9 82.9 618.57847 1 34.6 -196.8 618,58403 139.8 -215.8 618.59306 1 33.5 -211.7 618.60069 1 38.5 -208.8 618.60833 1 24.8 -217.8 618.62083 1 33.8 -190.3 620.58611 -108.2 178.8 620.59514 -122.2 1 58.4 620.60347 -110.6 153.6 622.59028 131.6 -214.0 630.58611 139.0 -194.9 630.59306 1 36.2 -206.8 632.58819 -113.4 165.6 632.60417 -107.7 171 .6 Radial Velocity - Alpha V i r - Struve et.al.(1958) year | Julian day | Vp (km/s)| Vs (km/s) 1956 2435000+ 560.805 -101 149 560.834 -86 144 560.858 -84 142 560.881 -73 142 562.790 125 -167 562.812 125 -178 562.833 121 -178 562.856 100 -173 562.886 86 -167 562.907 94 -163 588.698 -100 176 588.750 -93 171 588.774 -100 168 588.797 -1 02 171 592.678 -1 18 183 592.704 -105 185 592.726 -110 183 592.747 -1 04 179 592.769 -111 180 592.792 -120 172 594.736 1 20 -178 594.760 124 -172 594.783 1 19 -182 618.692 1 06 -168 618.719 1 18 -176 618.749 125 -168 620.664 -111 192 654.667 1 05 -160 656.675 -112 207 656.732 -128 204 1028.678 86 -149 1028.719 88 -137 1029.667 -110 182 1029.708 -127 181 1185.025 1 15 -172 1212.992 122 -181 101 Radial Velocity - Alpha V i r Shobbrook et.al.(1972) year | Julian day | Vp (km/s)| Vs (km/s) 1969 2440000+ 283.0008 92.9 -145.8 283.0175 96.0 -167.1 283.0279 96.8 -131.6 283.0376 107. 1 -159.9 283.0467 101.1 -152.1 283.0557 1 09.2 -161.6 283.0675 108.8 -159.2 283.0779 107.7 -150.2 283.0925 96.8 -170.3 283.1029 87.0 -166.1 283.1113 94.4 -142.3 283.1217 94. 1 -170.7 283.1300 91 .3 -162.8 283.1404 92.5 -165.3 283.1495 95.0 -160.5 283.1592 97.5 -163.3 283.1668 90.4 -155.7 283.1786 103.7 -164.2 283.1890 96.6 -165. 1 283.1960 107. 1 -167.0 283.2036 105.2 -162.6 283.2126 104.9 -167.9 283.2231 105.0 -153.8 283.2335 1 04.3 -157.8 283.2453 102.9 -169.2 283.2543 107.8 -132.8 283.2627 1 00.7 -162.5 283.2717 96.6 -163.9 283.2821 89.7 -159.5 283.2883 97.8 -166.4 317.1800 -148.4 -191 .7 344.9312 -103.6 169. 1 364.9004 -70.0 127.3 364.9101 -74.0 127. 1 365.8434 -91.0 136.6 365.8475 -84.7 128.6 365.8434 -87.3 136.3 365.8691 -89.6 126. 1 365.8871 -86.3 121.1 365.8934 -75.3 132.6 Radial Velocity - Alpha V i r May 08 1979 UT year | Julian day | Vp (km/s)| Vs (km/s) 1978 2443000+ 636.72836 -124.7 196.8 636.72951 -131.2 201 .6 636.73125 -125.2 201 .1 636.73356 -125.8 199.5 636.73588 -128.0 200.0 636.73819 -126.3 197.3 636.74051 -127.4 202.2 636.74693 -128.5 200.5 636.74925 -129.1 201 .6 636.75156 -131.3 202. 1 636.75388 -130.7 202.7 636.75619 - 131 . 3 202.7 636.75851 -130.2 203.7 636.76082 -125.9 204.3 636.76314 -125.9 203.2 636.76545 -127.5 204.8 636.76777 -128.0 205.9 636.77008 -130.2 201 .0 636.77239 -126.4 197.3 636.77470 -132.4 203.7 636.77702 -127.0 203.7 636.77934 -125.9 207.4 636.78166 -132.4 208.0 636.78397 -132.5 205.2 636.78628 -128.6 207.4 636.78860 -128.7 207.9 636.79091 -131.4 210.1 636.79323 -130.8 205.8 636.79554 -136.8 205.2 636.79786 -131.4 207.9 636.80017 -131.4 206.8 636.80249 -135.2 207.9 636.80480 -126.0 208.4 636.80700 -129.2 209.0 636.80943 -137.4 207.9 636.81175 -133.0 203. 1 636.81406 -130.9 207.4 636.81638 -131 .4 204.7 636.81869 -136.9 209.5 636.82563 -136.9 207.9 636.82795 -132.5 207.3 Radial Velocity - Alpha V i r March 20 1979 UT y e a r | J u l i a n day | Vp (km/s)| Vs (km/s) 1979 2443000+ 952.77219 85.8 -105.0-952.78116 79.3 -109.4 952.78955 85.7 -74.4 952.79766 79.1 -73.3 952.80561 80.2 -64.5 952.81345 74.7 -47.0 952.82111 79.1 -42.6 952.82894 79. 1 -32.8 952.83690 78.0 -41 .5 952.84456 76.8 -52.5 952.85208 79.0 -50.4 952.85932 69.1 -63.5 952.86626 69.1 -52.6 952.87263 71 .3 -61 .4 952.87842 66.9 -54.8 952.88483 63.6 -63.9 952.89176 68.0 -57.0 952.89841 66.8 -56.0 952.90478 66.8 -53.8 952.91303 62.4 -60.4 952.91968 51 .4 -53.8 952.92634 56.9 -53.8 952.93242 56.9 -58.2 952.93821 56.9 -63.7 952.94399 34.9 -64.8 952.95007 52.5 -58.3 952.95644 47.0 -85.7 952.96309 56.8 -52.8 952.97003 48.0 -52.8 952.97698 54.6 -51 .8 952.98392 56.8 -46.3 952.99087 45.8 -50.7 952.99810 44.7 -50.7 953.00562 46.9 -51 .8 953.01315 51 .2 -49.6 953.02067 45.9 -43. 1 953.02819 40.2 -40.9 953.05337 29.2 -43. 1 953.06205 31.4 -42.0 953.07131 23.7 -36.6 953.08057 43.4 -35.5 Radial Velocity - Alpha V i r March 21 1979 UT year | Julian day | Vp (km/s)| Vs (km/s) 1979 2443000+ 953.82361 -131.1 176.4 953.83756 -133.3 185.1 953.85150 -135.5 178.5 953.86539 -140.0 187.3 953.89260 -142.2 188.3 953.90418 -141,2 196.0 953.91576 -143,4 198. 1 953.92734 -148.9 203.6 953.93950 -147.8 199,2 953.95166 -151.2 196.9 953.96323 -150, 1 204.6 953.97495 -152,3 209,0 953.98667 -154.5 218.8 953.99824 -153.5 218.8 954.01039 -1 52,4 220.0 954.02370 -153.5 215.5 Radial Velocity - Alpha Vi r March 22 1979 UT year | Julian day | Vp (km/s)| | Vs (km/s) 1979 2443000+ 954.89608 -20.8 37.4 954.90476 -19.7 42.9 954.91344 -17.5 40,7 954.92212 -16.5 30.7 954.93080 -15.4 30.7 954.93948 -11.0 21.9 954.94815 -15.4 21 .9 954.95683 -6.7 19.7 954.96551 -7.8 18.6 954.97419 -10.0 18.6 954.98287 -15.5 20.7 954.99155 -6.7 1 1.9 955.00023 -5.7 15.2 955.00892 -0.2 15.2 955.01761 -5.7 18.5 955.02630 -4.6 16.3 955.03512 4.2 15.1 955.04395 5.2 4.1 955.05284 5.2 -6.9 955.06323 16.1 9.6 Radial Velocity - Alpha V i r March 23 1979 UT year | Julian day | Vp (km/s)| Vs (km/s) 1979 2443000+ 955.80507 -171.1 81.8 955.81375 -176.7 84.0 955.82214 -171.1 86.2 955.82909 -174.5 78.5 955.83487 -171.2 80.7 955.84066 -172.3 80.6 955.84645 -172.3 76.2 955.85194 -171.2 77.3 955.85767 -177.9 85. 1 955.86369 -171.2 80.6 955.86948 -172.3 85.0 955.87527 -175.7 83.9 955.88105 -174.6 86. 1 955.88684 -176.8 86. 1 955.89263 -177.9 86.1 955.89841 -180.2 86. 1 955.90420 -179.1 87.2 955.90999 -180.2 87. 1 955.91578 -178.0 88.2 955.92156 -180.2 88.2 955.92735 -174.7 89.3 955.93314 -179.2 96.0 955.93892 -179.2 95.9 955.94471 -180.3 97.0 955.95050 -180.3 99.2 955.95628 -178.1 107.0 955.96207 -180.3 107.0 955.96786 -179.2 108. 1 955.97365 -181.5 108. 1 955.97943 -180.4 109.2 955.98522 -182.6 109.2 955.99101 -181.5 1 14.7 955.99841 -180.4 111.3 956.00796 -181.5 108.0 Radial Velocity - Alpha V i r March 24 1979 UT year | Julian day | Vp (km/s)| Vs (km/s) 1979 2443000+ 956.80034 63.4 -79.3 956.80931 64.5 -79.3 956.82972 63.4 -66. 1 956.83956 63.4 -57.4 956.83998 63.3 -58.5 956.85039 62.2 -46.5 956.86081 54.5 -47.6 956.86081 54.5 -47.6 956.87123 61 . 1 -42. 1 956.88171 57.8 -46.5 956.89206 53.3 -40.0 956.90248 57.7 -38.9 956.91289 52.2 -35.6 956.92331 52.2 -41.1 956.93373 51 .0 -35.7 956.94414 51.0 -34.6 956.95456 45.5 -32.4 956.96498 46.6 -37.9 956.97539 43.3 -38.0 956.98581 42.1 -31 .4 956.99623 39.9 -36.9 957.00722 35.5 -36.9 957.01880 35.5 -35.9 957.03037 32.2 -37.0 957.04209 31.1 -27. 1 Radial Velocity - Alpha V i r A p r i l 14 1981 UT year | Julian day | Vp (km/s)| Vs (km/s) 1981 2444000+ 708.77961 -144.0 227. 1 708.78696 -144.3 226.4 708.79287 -144.7 223.2 708.79877 -142.4 225.8 708.80467 -141.8 225.8 708.81058 -141.1 .221 .8 708.81648 -137.9 219.2 708.82239 -136.6 22 1.2 708.82829 -136.3 221 .1 708.83419 -135.7 220.5 708.84009 -133.7 221 .1 708.84600 -133.4 221 .1 708.85190 -130.8 218.5 708.85780 -130.5 219.1 708.86371 -131.8 219.8 708.86961 -127.3 219.7 708.87551 -130.5 219.7 708.88141 -130.2 219.1 708.88732 -130.3 219.7 708.89322 -130.3 219.0 708.89912 -129.0 219.7 708.90502 -127.7 215. 1 708.91093 -127.7 219.0 708.91683 -127. 1 216.4 708.92273 -123.8 219.0 708.92864 -124.2 215.7 708.93454 -124.8 215.7 708.94044 -124.2 216.3 708.94634 -126.2 214.4 708.95225 -122.5 213.1 708.95959 -122.6 213.0 708.96839 -122.3 213.0 708.97719 -124.58 210.4 708.98598 -123.3 207.8 Radial Velocity - Alpha V i r A p r i l 16 1981 UT year | Julian day | Vp (km/s)| Vs (km/s) 1981 2444000+ 710.71475 115.7 -164.0 710.72355 114.4 -161.4 710.73528 121.5 -152.3 710.74408 119.5 -157.5 710.75288 130.6 -157.6 710.76167 1 23.4 -160.2 710.76492 114.3 -157.6 710.91756 112.7 -162.4 710.92636 113.9 -159.3 710.93516 112.6 -159.9 710.94396 113.9 -161.9 710.95276 129.5 -163.9 710.96155 117.1 -163.9 710.97035 130. 1 -158.. 1 710.97915 1 13.8 -162.6 710.98795 129.4 -159.4 710.99675 117.0 -160.7 711.00555 117.7 -161.4 109 Appendix D: SIMPLEX OPTIMISATION PROGRAM This i s a l i s t i n g of the program used to f i n d the optimum o r b i t a l solution by a Simplex method. Many thanks to Stephenson Yang for th i s program. The example given i:s for the Struve e t . a l . (1958) data set. C C SOLVEOB C C TO SOLVE FOR THE ORBITAL PARAMETERS OF AN OSCILLATING BINARY SY C IMPLICIT REAL*8 ( A-H, 0~Z ) REAL*8 DX(13,12),DSCALE(12) COMMON /OBPAR/ TPO, TSO, SCAL NAMELIST /SOLVE/ NV EXTERNAL DFUNCT C C( N = 08 ) C TO = DX(1,1) C P = DX(1,2) C RVO = DX(1,3) C ECC = DX(1,4) C RKP = DX(1,5) C RKS = DX(1,6) C WO = DX(1,7) C PA = DX(1,8) C AP = DX(1,9) C PP = DX(1,10) C AS = DX(1,11) C PS = DX(1,12) C N = 08 C Q ************************************** C ** INITIAL COORDINATE VALUES ** C ** DATA SET : STRUVE ET.AL. (1958) ** C ************************************** C DX(1,1) = 2435603.674D0 DX(1,2) = 4.014540D0 DX(1,3) = 00.00D0 DX(1,4) = 0.14D0 DX(1,5) = 124.00D0 DX(1,6) = 197.00D0 DX(1,7) = 105.0D0 DX(1,8) = 149.00D0 NDIMX=N+1 NP1=N+1 C Q *************************** C ** SCALE FACTORS ** Q *************************** C DSCALE(1)=1.0D-4 110 DSCALE(2)=1.0D-4 DSCALE(3)=0.0D0 DSCALE(4)=0.005D0 DSCALE(5)=05.0D0 DSCALE(6)=05.0D0 DSCALE(7)=05.0D0 DSCALE(8)=0.0D0 IFSCAL=2 MAXIT=500 READ(5,SOLVE) CALL READAT( NV ) DEPSLN =+1.OE-6 LOG = 1. CALL DSIM( DF,DX, 13, NP1, DSCALE, IFSCAL, MAXIT, * LOG,DEPSLN, * DFUNCT, M100 ) STOP 100 STOP1 END SUBROUTINE ORBRV( TJUL, TO, P, RV0, ECC, RKP, RKS, W0, PA, * RVP, RVS ) C C**************************************************************** C TO CALCULATE RADIAL VELOCITIES OF OSCILLATING BINARY COMPONENTS C C TJUL = TIME IN JULIAN DATE C TO = EPOCH OF ORBIT (JD) C P = ORBITAL PERIOD (DAYS) C RV0 = CENTRE-OF-MASS RADIAL VELOCITY (KM/S) C ECC = ECCENTRICITY C RKP = AMPLITUDE OF RADIAL VELOCITY (PRIMARY) (KM/S) C RKS = AMPLITUDE OF RADIAL VELOCITY (SECONDARY) (KM/S) C W0 = LONGITUDE OF PERIASTRON AT EPOCH TO (DEGREES) C PA = PERIOD OF ROTATION OF THE LINE OF APSIDES (YRS) C AP = AMPLITUDE OF OSCILLATION (PRIMARY) (KM/S) C TP0 = EPOCH OF ZERO PHASE OSCILLATION (PRIMARY) (JD) C PP = FUNDAMENTAL PERIOD OF OSCILLATION (PERIOD) (DAY) C AS = AMPLITUDE OF OSCILLATION (SECONDARY) (KM/S) C TS0 = EPOCH OF ZERO PHASE OSCILLATION (SECONDARY) (JD) C PS = FUNDAMENTAL PERIOD OF OSCILLATION (SECONDARY) (DAY) C IMPLICIT REAL*8 ( A-H, 0~Z ) REAL*8 TWOPI/6.283185307179586D0/, PI/3.141592653589793D0/ COMMON /KEP/ RM, ECCE EXTERNAL KEPEQN ECCE = ECC DT = TJUL - TO W = TWOPI * W0 / 3.6D0 IF ( PA.GT.1.0D-5 ) W = W + TWOPI * ( DT / PA / 365.2421988D0 IF ((1.0D0-ECC). GT. 1.0D-5 ) REOE = DSQRT( (1.0D0+ECC) / (1.0D0-ECC) ) ECOSWP = ECC * DCOS( W ) ECOSWS = ECC * DCOS( W + "PI ) 111 C IF ( P.GT.1.0D-5) RM = TWOPI * DT / P ESINM = ECC * DSIN( RM ) ECOSM = ECC * DCOS( RM ) C IF((1.0D0+ECC**2-2.0D0*ECOSM).GT.1.OD-5) C E=(1.0D0- (ESINM**2)/2.0D0/(1.0D0+ECC**2-2.0D0*ECOSM)) C IF((1.0D0-ECOSM).GT.1.OD-5) E=RM+ESINM/(1.0D0-ECOSM)*E E = RM + ESINM / (1.0D0-ECOSM) * * ( 1.0DO - (ESINM**2)/2.0D0/(1.0D0+ECC**2-2.0D0*ECOSM) ) DO 10 J=1,50 10 E = RM + ECC * DSIN( E ) CALL NDINVT( 1, E, F, ACCEST, 50, 1.0D-12, KEPEQN ) V = 2.0D0 * DATAN( REOE * DTAN( E / 2.0D0 ) ) RVP = RVO + RKP * ( DCOS( V + W ) + ECOSWP ) RVS = RVO + RKS * ( DCOS( V + W + PI ) + ECOSWS ) RETURN END SUBROUTINE KEPEQN( E, F ) C IMPLICIT REAL*8 ( A-H, 0~Z ) COMMON /KEP/ RM, ECC C F = RM - E + ECC * DSIN( E ) RETURN END REAL FUNCTION DFUNCT*8(DX, N, J ) C C*************************************************** * ************ C FUNCTION TO CALCULATE THE RESIDUAL OF THE VELOCITIES C IMPLICIT REAL*8 ( A~H, 0"Z ) REAL*8 DX(N,12) COMMON /RV/ T1(1000), VP1(1000), VS1(1000), * T2(1000), VP2O000), RNP, RNS, NV1 , NV2 COMMON /OBPAR/ TP0, TS0, SCAL c C( N = 08 ) C TO = DX(1, 1) C P = DX(1, 2) C RVO = DX(1, 3) C ECC = DX(1, 4) C RKP = DX( 1 5) C RKS = DX( 1 6) C W0 = DX( 1 7) C PA = DX( 1 8) C AP = DX(1 ,9) C PP = DX( 1 ,10) C AS = DX( 1 ,11) C PS = DX( 1 ,12) C WRITE (6, 12) (DX (J, FCNP = 0.0D0 TO = DX(J, 1) P = DX(J, 2) DO 10 1=1,NV1 CALL ORBRV( T1(I),TO,P,DX(J,3),DX(J,4),DX(J,5), 1 12 * DX(J,6),DX(J,7),DX(J,8), * RVP, RVS ) FCNP = FCNP + ((VP1(I) - VS 1(I))-(RVP-RVS))**2 1.0 CONTINUE DFUNCT = DSQRT( FCNP/(RNP+RNS)) WRITE (6, 12) DFUNCT 12 FORMAT (1X,1P9G14.6) RETURN END SUBROUTINE READAT( NV ) C £************************************************************* C PROC TO INPUT TIME AND OBSERVED VELOCITIES C IMPLICIT REAL*8 ( A-H, 0~Z ) REAL*8 DAT(6000) COMMON /RV/ T1(1000), VP1(1000), VS1(1000), * T2(1000), VP2O 000), RNP, RNS, NV1 , NV2 C CALL FREAD( ~2, 'ENDLINE', 'STREAM' ) CALL FREAD( 5, 'R*8 VECTOR:', DAT, 6*NV ) NV1 = 0 NV2 = 0 DO 10 I=1,NV TT = DAT(6*1-5) EV = DAT(6*I-2) VCORP = DAT(6*1-1) VCORS = DAT(6*1) VPP = DAT(6*1-4) - EV + VCORP VSS = DAT(6*I-3) - EV + VCORS IF ( VSS .LT. -1.0D3 ) GO TO 5 NV1 = NV1 + 1 T1(NV1) = TT VP1(NV1) = VPP VS1(NV1) = VSS GO TO 10 5 NV2 = NV2 + 1 T2(NV2) = TT VP2(NV2) = VPP 10 CONTINUE RNP = DFLOAT( NV1 + NV2 ) RNS = DFLOAT( NV1 ) RETURN END $SIGNOFF 

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