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Spectral variations of 12 Lacertae 1976

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SPECTRAL VARIATIONS OF 12 LACEBTRE by Andrea Mary-Anne A l l i s o n E.Sc. , Loyola of Montreal, 1974 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOB THE DEGREE OF MASTER OF SCIENCE in the Department of GEOPHYSICS and ASTRONOMY He accept t h i s thesis as conforming to the reguirecl standard The University of B r i t i s h Columbia A p r i l , 1976 (c) Andrea Mary-Anne Allison, 1976 In presenting th i s thesis in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree l y ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of this thes is for f i nanc ia l gain sha l l not be allowed without my writ ten pe rm i ss i on . Department of GEOPfl^SlCS ftA>P flSTftOflOflly The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Abstract Observations of the Cephei star 12 Lacertae have been made at both high time and spectral resolution with an Image Isocon t e l e v i s i o n camera. The observations cover complete cycles of the pulsation on two consecutive nights i n October 1972. The spectral region covered includes the two Si i n c l i n e s at j 4553 and /J 4568. The l i n e p r o f i l e variations have been studied in d e t a i l ; in pa r t i c u l a r the l i n e doubling at certain phases of the pulsation. Wavelength s h i f t s of the separate components were measured and a discontinuous r a d i a l velocity curve was obtained. i i i fable of Contents page Introduction 1 The Observations 7 Preprocessing of the Data 9 Line P r o f i l e Variations 28 fiadial Velocity Measures 54 Discussion 66 Summary and Conclusions 73 Bibliography 76 Appendix A 78 i v L i s t of Tables page I., Observed Properties of 12 Lac 5 11(a). F i d u c i a l S h i f t Corrections for Oct. 15. 22 11(b). F i d u c i a l S h i f t Corrections for Oct. 16, 23 V L i s t of Figures page 1a. Mean Spectrum for October 15 data. 10 1b. Mean Spectrum f o r October 16 data. 11 2a. Time series seguence of spectra for October 15 13 data. 2b. Time series seguence of spectra for October 16 14 data. 3a. Variance v s ± Signal for October 15 data. 15 3b. Variance vs^ Signal for October 16 data. 16 4a. Contour plot of l e f t F i d u c i a l mark for October 17 15 data. 4b. Contour plot of right F i d u c i a l mark for October 18 15 data. 4c. Contour plot of l e f t F i d u c i a l mark for October 19 16 data. 4d. .. Contour plot of right F i d u c i a l mark for October 20 16 data, 5a. Time series of E e c t i f i e d spectra for Oct. 15. 26 5b. Time series of E e c t i f i e d spectra for Oct. 16. 27 6a. Time series of r e c t i f i e d spectra for Oct. 15, 30 6b. Time series of r e c t i f i e d spectra for Oct. 15. 31 7a, Time series of r e c t i f i e d spectra for Oct., 16. 32 7b, Time series of r e c t i f i e d spectra for Oct, 16. 33 8a, Contour plot of Si I I I \ 4553 l i n e for Oct. 15. 35 8b. Contour plot of S i III \ 4568 l i n e for Oct. 15. 36 8c. Contour plot of S i I I I \ 4553 l i n e for Oct. 16. 37 8d. Contour plot of S i III } 4568 l i n e for Oct. 16. 38 v i 9a. Bifference plots cf r e c t i f i e d spectra for Oct. 40 15. 9b. Difference plots of r e c t i f i e d spectra for Oct. 41 16. Second derivative plots for the data of Oct. 15. 4 2 Second derivative plots for the data cf Oct. 15. 43 Second derivative plots for the data cf Oct. 16. 44 Second derivative plots for the data of Oct. 16. 45 Line depth, half-width, and asymmetry v s A Time 49 for Oct. 15. Si III ^4553. Line depth, half-width, and asymmetry v s A Time 50 for Oct. 15. : Si III 4568. Line depth, half-width, and asymmetry v s A Time 51 for Oct. 16. Si I I I /I 4553. Line depth, half-width, and asymmetry v s A Time 52 for Oct. 16. Si III } 4568. Radial v e l o c i t y curve for S i III ) 4553 for Oct. 58 15. Radial v e l o c i t y curve for Si III ^ 4568 for Oct. 59 15. Radial v e l o c i t y curve for Si III ^ 4553 for Oct. 60 16. Radial v e l o c i t y curve for Si III 4568 for Oct. 61 16. Gaussian f i t to input data for block #10 of 62 Figure 5a. Gaussian f i t to input data for block #10 of 63 Figure 5b. v i i 15a. Badial velocity curve (centroid) for Si III 64 ^4553 for Oct. 15. 15b. Badial v e l o c i t y curve (centroid) for Si I I I 65 A 4553 for Oct. 16. A . 1 Deconvolved spectrum for block # 3 of Figure 5a. 80 A . 2 Deconvolved spectrum for block # 9 of Figure 5a. 81 A. 3 Deconvolved spectrum for block #10 of Figure 5a. 82 A.4 Deconvolved spectrum for block #12 of Figure 5a. 83 A . 5 Deconvolved spectrum for block # 1 of Figure 5b. 84 A . 6 Deconvolved spectrum for block # 3 of Figure 5b. 85 A . 7 Deconvolved spectrum for block # 5 of Figure 5b. 86 A . 8 Deconvolved spectrum for block #13 of Figure 5b. 87 A . 9 Deconvolved spectrum for block #14 of Figure 5b. 88 A . 10 Deconvolved spectrum for block #15 of Figure 5b. 89 v i i i Acknowledgments I would l i k e to thank my supervisor Gordon l a l k e r f or his patience and support. I would very much l i k e to thank John Glaspey and Greg Fahlman for taking part in many help f u l discussions. Drs. Fahlman and Glaspey also wrote many of the computer programs used and I thank them for making them available. I would also l i k e to thank Tad Ulrych for his assistance with part of the data analysis. I thank the National Research Council for support from a postgraduate sco l a r s h i p . I thank Graham H i l l of the Dominion Astrophysical Observatory for allowing me to use data he o r i g i n a l l y obtained. F i n a l l y I would l i k e to thank the graduate students of the department who were always w i l l i n g to answer my many guestions. 1 12 lacertae, also known as the variable star DD Lacertae, i s a member of the Cephei class of variables.... These are a group of stars which show short period variations i n both l i g h t and r a d i a l v e l o c i t y . The l i g h t variations 4m are usually only a few hundredths of a magnitude and the r a d i a l v e l o c i t y amplitudes 2K range from 5 km/sec to 150 km/sec giving, a large 2K/Am r a t i o . The £ Cephei stars range in spectral type from B0.5-B2 and have luminosity classes II-III-IV. The l i g h t and r a d i a l v e l o c i t y curves are approximately sinusoidal but many of the stars shon multiple p e r i o d i c i t y and some show a beat phenomenon. General review a r t i c l e s have been written by Struve (1955) , Underhill (1966), and Percy (1967) as well as others. The pulsation properties and spectral variations of the Cephei variables are guite d i f f e r e n t from the other i n t r i n s i c variables. Whereas the pulsational modes and exci t a t i o n mechanisms of such stars as the C l a s s i c a l Cepheids and BB Lyrae stars are well understood, those of the cephei stars are not yet known with any certainty. The suspected cause of pulsation for some of the cooler variables i s envelope i o n i z a t i o n . The flux modulation brought about by regions i n the s t e l l a r envelope containing abundant elements such as hydrogen and helium i n various stages of ion i z a t i o n can have the proper phasing for dr i v i n g of pulsations, and under the proper conditions can induce 2 pul s a t i o n a l i n s t a b i l i t y in a star. The /? Cephei stars are very hot (Te 20,000-25,000°K) ana so t h i s would not apply. There i s no H i o n i z a t i o n zone and the He* i o n i z a t i o n zone l i e s too close to the s t e l l a r surface to be e f f e c t i v e (Cox,1974). A good discussion of /? Cephei stars and th e i r l o c a t i o n i n the t h e o r e t i c a l H-fi diagram i s given i n Lesh and Aizenman (1973a). They f i n d that the Cephei stars occupy a very narrow s t r i p i n the H-R diagram and by model ca l c u l a t i o n s they f i n d that t h i s region i s traversed i n an S-shaped manner i n the course of post-main seguence evolution: once i n the core hydrogen burning phase, once i n the secondary contraction phase, and once i n the s h e l l hydrogen burning phase. It i s not known d e f i n i t e l y i n which of these 3 stages the cephei stars are, but i t i s suspected that the pulsation i s connected with the s t r u c t u r a l changes caused by one of these stages of evolution. I t has sometimes been thought that the ft Cephei stars were a 'peculiar' type of star but both Watson (1972) and Lesh and Aizenman (1973b) attempt to show that the Cephei stars are normal B type stars. Satson (1972) uses the (0e,log g) plane to define the /5 Cephei i n s t a b i l i t y s t r i p . He concludes as do Lesh and Aizenman (1973b) that the p Cephei stars d i f f e r from the nonvariable stars only i n their state of evolution. Various attempts have been made towards understanding the nature of the pulsation. A brief summary of t h i s work can be found i n Cox (1974). Radial and non-radial modes have been 3 examined as well as interactions between the two to explain the observed phenomenon. Two recently proposed e x c i t a t i o n mechanisms are found in Aizenman, Cox and lesh {1975) and Osaki (1974). Aizenman et al.. conclude that i n model calculations for a 10 H 0 s t a r , the lower gravity (g +) modes are unstable against non-radial o s c i l l a t i o n s i n the early s h e l l hydrogen burning stages. ( Considering only the adiabatic case, the eguations of motion for the non-radial case reduce to a fourth order problem where the eigenvalue v~ 2 enters ncn-linearly i n the c o e f f i c i e n t s . This results i n two d i s t i n c t spectra f o r t f 2 . For T 2 large the solutions correspond to accoustical or pressure (p) modes. A l l these p modes have discrete positive eigenvalues (T 2 >0) . For T 2 small, there are two p o s s i b i l i t i e s which a r i s e . If there i s convective s t a b i l i t y , the solutions correspond to gravity or g + modes. The T 2 are a l l positive and thus the star i s dynamically stable. I f there i s not convective s t a b i l i y , the solutions correspond to unstable convective modes g -; Vz becomes negative. There i s i n addition an intermediate fundamental or f mode with a positive eigenvalue T ^ 2 separating <T2 and <*32 ) . Non-linear mode coupling i s suggested whereby the long period g* modes can excite a sh'orter period stable mode: f, p, or r a d i a l harmonic modes which have been shown to agree with observations. Osaki (1974) assumes that the P Cephei stars are massive stars i n the late stages of core hydrogen burning. The basic mechanism i s t h i s : i f the large-scale convective motion i n the core (described as unstable g - modes ) i s o s c i l l a t o r y and i f the 4 frequency of one of the convective modes happens to coincide with the eigen-freguency of a non-radial o s c i l l a t i o n of the whole st a r , then t h i s resonance may excite a non-radial o s c i l l a t i o n of the star, which may be observable at the s t e l l a r surface. A few properties of 12 l a c are mentioned i n Table I. The symbols E1 and P2 correspond to the primary period 4h 38m and secondary period 4h 44m respectively. I t was f i r s t thought to be a spectroscopic binary (Adams, 1912) because of the broadening of the l i n e s at certain phases. It was found that the absorption l i n e s are double at maximum and minimum r a d i a l v elocity but the ef f e c t i s measurable only when the variable amplitude 2K i s large (Struve, 1950 ). In my work I w i l l just consider the variations over one cycle of the pulsation (of duration 4h 38m). The whole question of the multiple p e r i o d i c i t y of 12 Lac w i l l be reviewed i n the discussion section. Host of the work done on 12 Lac has been photometric, including an International Campaign i n 1952. These e f f o r t s have mainly been directed towards more accurate period determinations. Among these are Path (1938), Opolski and C i u r l a (1961) , De Jager (1962), Earning (1962), Opolski and C i u r l a (1962) , Jerzykewicz (1963), C i u r l a (1973), and Sato (1973). Some spectroscopic work has been done by Struve (1950), Struve and Zebergs (1955), De Jager (1957), Grabowski (1966), Opolski and Grabowski (1966), Beres (1965), Grabowski (1969) and Heard et a l (1976) . 5 Table Ij. Observed Properties of 1.2 Lac A | SYMBOL | ! RE MK Spectral Type i ! B1.5 III i 1 i i B2 I I I J 2 Visual Magnitude 1 V | 5.25 | 2 ! ! 5.18 | 4 Absolute Magnitude -3.61 I 1 I ! -4.0 | 2 i ! '4.15 | 4 Log of Period (days) 1 log P1| -.705 1 4 Radial Velocity Range I 2K | 36 (P1) J 2 (km/sec) I I 15 (P2) | 2 B-V Colour Index 1 B-V J -.135 | 4 1 1 -.140 | 2 Amplitude of Light Curve 1 A m 1 .074 (P1) | 2 ! ! .042 (P2)| 2 E f f e c t i v e Temperature .214 | 1 Parameter=5040/T K I I I 01tra-Violet Colour I U-B J -.945 | 4 Index i ! I Apparent Rotational I v s i n i j 79 I 4 Broadening (km/sec) j j 29 | 2 References 1. Lesh and Aizenman (1973) 2. -. Percy (1967) 3. Opolski and Grabowski (1966) 4. H i l l (1967) . 6 The primary aim of my investigation was to look at high dispersion, high time and high spectral resolution observations of 12 Lacertae and examine the l i n e doubling feature i n much more d e t a i l than had previously been done. Observations of BB Vul and ^Cephei, two other f> Cephei stars obtained with the same detection system were analyzed by Goldberg (1973). As i n 12 Lac, rapid spectral variations occurring at certain phases of the pulsation cycle were detected i n both stars. 7 The Observations The observations were taken using a refri g e r a t e d image Isocon t e l e v i s i o n camera as a detector. The spectrum i s imaged onto the photocathode of the Isocon tube and integrated for a predetermined time. An electron reading beam then scans the target normal to the spectrum., The data i s d i g i t i z e d by means of a 12 b i t analog-to-digital converter and stored on magnetic tape. Each scan gives 840 data points over 70 mm of the cathode. The system i s monitored by an Interdata Hodel 4 computer which displays the spectrum on an oscilloscope. Short period observations of the dark current over the same integration period are taken after the star data. A more detailed description of the system i s given by Walker et a l A (1972) . The observations were taken using the coude spectrograph of the 48 inch telescope of the Dominion Astrophysical Observatory, V i c t o r i a . The 96 inch camera with a mosaic of four 800 line/mm gratings was used i n the second order. In addition a transfer lens which magnifies a small region of the spectrum and matches the curvature of the f o c a l plane to the curvature of the the photocathode (Richardson, 1973 ) was used giving a dispersion of 0.5 A/mm. The useable area of the tube face between the f i d u c i a l marks gives a spectral coverage of -25 A. The integration time for 12 Lac was 30.87 sees. 576 i n d i v i d u a l frames were obtained on October 15,1972 O.T. and 688 8 on October 16, 1972 U.T. The t o t a l time of observations was 4h55m and 5h53m respectively. The primary period of 12 Lac i s 4h38m. Thus the observations cover more than one cycle of the pulsation. The wavelength region examined showed the two Si I I I l i n e s \ 4553 and } 4568. 9 ElgEfQggssing of the Data The average of the dark current f o r each night was smoothed by an 11 point (an a r b i t r a r y number) running mean. This was then subtracted from a l l of the spectra. The mean spectra for the two nights are shown in Figures 1{a) and 1(b). A 5-point running mean smoothing was applied to the data. The two dips at either end of the spectra are f i d u c i a l markers on the face of the tube. These are pieces of black tape which serve as reference markers to monitor any changes in the scanning raster pattern.. The numbers on the wavelength axis correspond to the 840 data points i n the spectrum. The tiny b l i p s at points -290 and 430 also show up on the spectrum of a c a l i b r a t i o n lamp. They are most l i k e l y due to "dust" specks on the face of the tube. In order to see the l i n e p r o f i l e changes during a cycle, the time-series spectra were grouped into blocks of equal time duration. The spectra i n each block were added together and the f i n a l sum was normalized such that the sum within a s p e c i f i e d data point i n t e r v a l was 1000.. This i n t e r v a l was chosen away from the spectral l i n e s to prevent any variations i n l i n e i n t e n s i t y from af f e c t i n g the normalization. In addition each block was divided by the average of the c a l i b r a t i o n lamp spectrum. A 5 point running mean smoothing was also applied to the data to improve the signal to noise r a t i o . 12 LACERTAE  12 The seguence of spectra for each night are shown i n Figures 2(a) and 2(b). Each spectrum i s a 20.6 minute mean of 40 i n d i v i d u a l exposures. The signal to noise r a t i o i s -75:1. The signal to noise c h a r a c t e r i s t i c s of the image isocon have been discussed for the U.E.C. system by Buchholz et a l (1973). They found that there i s a l i n e a r s i g n a l to noise rela t i o n s h i p from spectra with strong absorption l i n e s . This re l a t i o n s h i p i s shown to hold for the 12 Lac spectra as indicated in Figures 3 (a) and 3 (b). For the variance analysis, signal values were selected from a limited data point i n t e r v a l . Thus the addition of N frames of data gives a/F improvement i n the si g n a l to noise. The scanning pattern of the camera reading beam i s known to vary with time (Fahlman and Glaspey, 1973). This can produce an a r t i f i c i a l wavelength s h i f t i n the spectrum. I t i s important to have some estimate of t h i s s h i f t for measures of r a d i a l v e l o c i t y . The f i d u c i a l marks mentioned previously are used f o r t h i s purpose. A two-dimensional (wavelength,time) presentation i n contour form of the area around the f i d u c i a l marks i s shown in Figures 4(a), 4(b), 4(c) and 4(d). The data i s guantized into 20 minute blocks as for Figure 2 and the i n t e n s i t y d i s t r i b u t i o n of the l i n e s i s mapped. The labels on the contour l i n e s are a measure of i n t e n s i t y ; smaller numbers i n d i c a t i n g lower i n t e n s i t y , i e . absorption. From these figures one can see c l e a r l y the raster s h i f t during the period of observation.  14 Figure 3(a). Variance vs. signal for October 15 data. cn i n i x J - CE E r 2 IS -w »•• •- o I 2.4 I i — 4.8 7.2 S I G N A L 0.0 9.6 ~1 12 Figure 3(b). Variance vs. signal for October 16 data. 0) ~1 12 0.0 "1— 2.4 -1 "1 4.8 1.2 S I G N A L ~T"~ 9.6 17 Figure 4(a). Contour plot of left Fiducial mark for October 15 data. Figure 4(b). Contour plot of right Fiducial mark for October 15 data. o E~T 1 1 1 1 1 1 4.95 5.93 6.91 7.B9 8.B7 9.85 10. TIME UT MRS.) Figure 4(c). Contour plot of left Fiducial mark for October 16 data. Figure 4(d). Contour plot of right Fiducial mark for October 16 data. 21 The displacements are calculated using a program described by Fahlman and Glaspey (1973). The spectrum to be measured i s systematically s h i f t e d r e l a t i v e to a reference spectrum i n order to find that s h i f t which minimizes, in a least squares sense, the differences between the two spectra. The s h i f t i n g and differencing i s done in the Fourier domain. The correction to be applied to the absorption l i n e i s found by l i n e a r i nterpolation between the measured s h i f t s of the f i d u c i a l s . The mean spectrum was used as the standard. The s h i f t s of the right and l e f t f i d u c i a l marks as well as the slope of the straight l i n e f i t t e d through the s h i f t s are given i n Tables 11(a) and II (b). The values of the s h i f t s are given to three decimal points, however the accuracy of the technique can only be guaranteed to -0.1 of a sample point. I t i s necessary to assume a l i n e a r s h i f t across the face of the tube since we only have two reference points, Thompson (1974) found an apparent ^non- li n e a r expansion of the camera reading beam which necessitated putting upper l i m i t s to the r a d i a l v e l o c i t y variations. This point w i l l be returned to l a t e r i n the r a d i a l v e l o c i t y discussion. The s h i f t corrections were not applied u n t i l the actual r a d i a l v e locity measures were made. another instrumental problem associated with isocon data i s what appears to be microphonics. The glass target i s only fL2/4m thick, and i t i s suspected that the scanning by the reading beam could cause the target to vibrate. Shat r e s u l t s i s a sinusoidal variation imposed on the spectrum. Unfortunately the frequency of the microphonic variation i s rouqhly the same as 22 Table 11 F i d u c i a l S h i f t Corrections for Oct... : r T " 1 Block | i F i d u c i a l Shift j Slope I No. 1 Left J Bight I (Shift/Pt) I j (70-124) 1 (685-730) j 1 1 • I -2.472 1 -6.064 | -0.0059 I 2. J -0. 833 1 -4.287 | -0.0057 i 3. | 0.269 1 -2.743 J -0.004 9 I 4. | 0.791 | -1.907 J -0.0044 I 5. | 0. 683 J -0.396 J -0.0018 I 6. | 0. 653 I -0.252 I -0.0015 I 7. j 1.032 I 0.008 | -0.0017 1 8 . I 0.492 I 0.539 | 0.0001 J 9. | -0.016 l 1.255 | 0.0021 1 10. | 0.239 I 1.429 | 0.0020 111. J 0.207 l 1.739 I 0.0025 I 12. I 0.085 I 2.033 | 0.0032 I 13. I -0.521 | 2.635 I 0.0052 I 14. I -1.165 J 2.819 | 0.0065 t , J . j . _ x 23 Table II b. F i d u c i a l S h i f t Corrections for Oct. 16 . | Block | 1 F i d u c i a l S h i f t | Slope | Do. | l e f t (70-124) i , Bight (685-730) | (Shift/Pt) i J J 1 * — 1 I 1. I . 320 1 - 4 . 0 0 0 | - 0 . 0 0 7 1 1 2 . | . 0 3 5 1 -3 .500 | - 0 . 0 0 5 8 I 3 . I .287 ! - 2 . 2 2 0 | - 0 . 0 0 4 1 1 4 . | . 187 ! - 1 . 3 6 0 | - 0 . 0 0 2 5 1 5 . J . 212 ! - 0 . 7 6 0 | - 0 . 0 0 1 6 1 6 . 1 .183 1 0 . 182 | 0 ,0000 1 7 . | . 300 ! 0.820 J 0 .0008 1 8 . | , 1 6 2 •! 1.100 | 0 .0015 1 9 . I . 020 ! 1. 380 | 0 . 0022 1 10. J - . 0 6 1 ! 1.300 I 0 . 0022 I 1 1 . I .131 ! 1. 586 | 0 .0024 I 1 2 . | - . 0 4 1 1 2 .110 | 0 .0035 1 1 3 . | - . 2 3 2 ! 2 .180 | 0 ,0040 I 14 . J - . 3 1 5 ! 2 .240 I 0 .0042 I 15 . I - . 5 0 0 ! 2.400 I 0 .0048 | 16 . | - . 4 1 7 ! 2 .620 | 0 .0050 J 17 . J - . 8 4 2 ! 2.655 | 0 .0057 ' 1 A - „ J . i _ J 24 that of the spectral lines and so cannot tie f i l t e r e d out. fi visu a l inspection of the data was carried out and those frames which shewed a marked sinusoidal pattern were rejected. It was d i f f i c u l t to determine an objective c r i t e r i a for rejection as the amplitude of the ef f e c t was often hard to e s t a b l i s h , but the volume of data prohibited any other approach. The sequence of spectra as in Figure 2 were recomputed with the rejected records omitted, No s i g n i f i c a n t change was noted other than a decrease in the signal to noise r a t i o where lesser numbers of records were averaged together, and so i t was decided to use the o r i g i n a l data i n t a c t . For further analysis of the data, i t was decided to f i l t e r the data using a bandpass f i l t e r truncated with a lanczos window instead of using the running mean smoothing. The high frequency cutoff point of the f i l t e r was chosen as the frequency where the power spectrum of a single block of data f e l l below -32 db. For the data cf Oct. 15 t h i s was 25 per cent of the Nyguist freguency, and for Oct. 16 i t was 28 per cent. The low frequency cutoff was set at zero. The instrumental response i s removed from the spectra by a r e c t i f i c a t i o n procedure. The spectra were r e c t i f i e d to a continuum by a l e a s t squares f i t cf 4th order polynomials to points in the spectrum which represent r e a l continuum. new continuum f i t using the same points was determined for each of the 20 minute blocks to compensate for any fluctuations i n the instrumental response. The sequence of r e c t i f i e d spectra f o r 25 the two nights are shown in Figures 5(a) and 5(b).  Figure 5(b). Time series of Rectified spectra for Oct. 16. 28 Line P r o f i l e Variations The l i n e p r o f i l e variations over a cycle are quite remarkable. The d i q i t i z e d form of the data allows much freedom i n presentation. What we have e s s e n t i a l l y i s a two-dimensional set of data: a certain wavelength region i s sampled at discrete time i n t e r v a l s - r e s u l t i n g in a "time s e r i e s " of wavelength blocks. I f the spectral features are changing with time as i s the case here, i t i s often a problem to c o r r e c t l y v i s u a l i z e what i s r e a l l y going on. To aid i n overcoming t h i s d i f f i c u l t y , I have chosen a number of ways of i l l u s t r a t i n g the l i n e p r o f i l e variations, both q u a l i t a t i v e and quantitative. These are the following: 1) The "Time Series" Plot.- An example of t h i s i s shown in Fiqures 5(a) and 5(b). Such a diaqram i s useful in qiving a true "picture" of the spectral l i n e s . 2) The Contour Plot.- A preliminary description of t h i s technigue has already been given for the f i d u c i a l marks. This i s a p a r t i c u l a r l y e f f e c t i v e method of showing immediately the wavelength s h i f t s and i n t e n s i t y changes. 3) The Difference Plot.- From each block of data i s subtracted the mean spectrum of a l l the data and the r e s u l t i s given i n the form of a time series plot. This i s very useful i n showing the differences between in d i v i d u a l blocks. 4) The Second Derivative.- The second derivative of each block of the time series i s taken. This " l i n e sharpening" technique has the important character of sometimes beinq able to resolve 29 i n d i v i d u a l components from blended l i n e s . 5) F i n a l l y a quantitative description of the changes can be given in terms of measures of l i n e width, asymmetry and equivalent width. 1) The Time Series Plot. As stated i n the previous section Figures 5(a) and 5(b) are a series of r e c t i f i e d spectra: each spectrum a 20 minute mean of 40 i n d i v i d u a l exposures. In order to obtain better time resolution, a smaller time sampling i n t e r v a l was considered. Figures 6 and 7 are a series of r e c t i f i e d spectra: each block a 6 minute mean of 12 in d i v i d u a l exposures. Because lesser numbers of records are averaged together, there i s a corresponding decrease in signal to noise. Unfortunately when one averages t h i s few number of records, evidence f o r microphonics i s present i n some of the blocks (eg. #28 F i g . 6, and #48 Fig. 7.) nevertheless one sees c l e a r l y the development from sharp l i n e to broad l i n e phase. In p a r t i c u l a r , the presence of two components at certain phases i s very c l e a r l y indicated. I t i s also possible to monitor the r e l a t i v e growth and decline of the components. From looking at these l i n e p r o f i l e s one sees that the l i n e at )4568 appears broader than that a t J 4553. This i s due to a s l i g h t non-linearity i n dispersion across the face of the tube. This i s probably due to the use of the transfer lens. Around )4568 the dispersion i s -.045 A/pt. ; at J4553 i t i s closer to .051 A/pt. l i n e p r o f i l e depth, half- S i 111 4568 Si 111 4553  4568 Si 111 4553 I l l 4568 Si 1 1 1 ^ 4553 34 2) The Contour Plot. Contour plots of the data blocks of Figure 5 are shown i n Figure 8. Each spectral l i n e i s contoured separately. A great amount of information can be gathered from such a two- dimensional display. Consider Figure 8 (a): A) At the beginning of observation the l i n e i s sharp and symmetric. This i s indicated by the symmetric d i s t r i b u t i o n of the contour l i n e s on both sides of the minimum contour l i n e of 100. B) As the cycle progresses, the l i n e becomes broader—shown by a greater spread i n the equal intensity contour l i n e s . C) The nature of the asymmetry can also be seen from these plots. Around ^6 hrs. U.T. the contour l i n e s on the blue side of the absorption peak are more widely separated than those on the red side. Thus there i s an asymmetry with an extended wing to the blue. This i s confirmed from Figure 5(a) by looking at block numbers 5,6,7, and 8 . Towards the end of observation the asymmetry reverses i t s e l f showing an extended wing to the red. D) The wavelength s h i f t s are obvious from looking at the v e r t i c a l displacement of the contour l i n e s . E) A rough idea of the variations in l i n e depth can also be obtained. Take for example Figure 8(d). The path of l i n e minima i s indicated very approximately by the dashed l i n e . In the f i r s t half i t crosses successively lower valued contour l i n e s - i e . the l i n e gets deeper. As the cycle progresses the l i n e gets shallower again. 35 Figure 8(a). Contour plot of Si 111 4553 line for Oct. 15. 36 F i g u r e 8 ( b ) . Contour p l o t o f S i 111 4568 l i n e f o r Oct . 15. •37 Figure 8(c). Contour plot of Si 111 4553 for Oct. 16. Figure 8(d). Contour plot of Si 111 4568 for Oct. 16. 39 3) The Difference Plot. Difference plots for the blocks of Figure 5 are given i n Figure 9. The mean spectrum subtracted from each block, i s e s s e n t i a l l y that of Figure 1 with the instrumental response removed. The difference spectrum of a displaced spectral l i n e i s e s s e n t i a l l y an S shaped curve. I t resembles a sinusoid whose wavelength i s egual to the width of the l i n e . These diagrams show more c l e a r l y than any of the others, the i n d i v i d u a l block changes at discrete time i n t e r v a l s . The difference plots are b a s i c a l l y f i r s t derivative plots which are very sensitive to position changes. The second derivative plots described below are more sensitive to p r o f i l e changes. 4) The Second Derivative. The second derivative plots of the data blocks of Figure 5 are shown i n Figures 10 and 11. One modification was necessary. The data was f i r s t 5 point smoothed instead of f i l t e r e d to 25 or 28 per cent of the Nyguist. When you apply the second derivative to a set of data each s l i g h t change i n slope becomes greatly amplified. The noisier the data the worse the problem. I t was found that a 5 point smoothing gave the best compromise between resolution and noise l e v e l . The 2nd derivative was i n addition f i l t e r e d to 20? of the Nyguist. From looking at the plots one sees immediately the growth and development of the two components. Consider f o r example i n Figure 10 (a) A 4553: a) In the f i r s t 4 blocks only one component i s present. 40 Figure 9(a). Difference plots of r e c t i f i e d spectra for Oct. 15. Figure 9(b). Difference plots of r e c t i f i e d spectra for Oct. J6o     46 B) In the f i f t h block a component i s beginning to develop on the blue side. C) In blocks 7,8, and 9 th i s blue component continues to grow and increase in amplitude. D) In block 10 the redward component begins i t s decline and continues to decrease in amplitude in blocks 11,12, and 13. E) In block 14 you have e s s e n t i a l l y only one component present. The s i t u a t i o n i s a l i t t l e more confused when you lock at ^4568. At times three components actually seem to be present. This i s suspected to be more a character of the second derivative technique than any r e a l spectral feature. 5) Quantitative Measurements. The equivalent width, l i n e depth, half-width and an ind i c a t i o n of the degree of asymmetry were measured from the l i n e p r o f i l e s . The blocks of figures 6 and 7 were used to give a more detailed analysis cf the variations. Eguivalent widths for the l i n e s were determined by area integration between continuum and l i n e from the r e c t i f i e d spectra. This r e s u l t w i l l be influenced by inaccuracies in the continuum f i t . Nevertheless, r e l a t i v e measures of eguivalent width can be made to a very good approximation. For the two nights of observation, the eguivalent widths of the l i n e s appears tc remain constant over the entire cycle. The eguivalent width did not vary by more than - 1 % on the average during the observation period. This i s in agreement with 47 Huang's (1955) r e s u l t for T Sco (2K-80-120 km/sec) and Goldberg's (1973) r e s u l t for Bw Vul(2K*150 km/sec). The l i n e depth was defined as the difference between minimum in t e n s i t y and the continuum intensity set equal to 1. The r e s u l t s are shown i n Figure 12. Again, as for the equivalent widths, absolute measures are not possible but the i n t e r n a l consistency of the continuum f i t t i n g procedure allows very good r e l a t i v e comparisons. The extent of one period i s shown by the dotted l i n e s in Figure 12. The half-width was'defined as the f u l l width of the l i n e at the half-peak i n t e n s i t y l e v e l . Hhat i s plotted in Figure 12 i s the half-width normalized to the mean half-width for a l l the blocks of data. The mean half-width W for the Si III ̂  4553 l i n e was 1.21 A and 1;36 A for Oct. 15 and Oct. 16 respectively. W for Si I I I A 4568 was 1.16 A and 1.32 A for Oct. 15 and Oct, 16 respectively. The asymmetry i s defined as i n the paper by Heard et a l (1976). It i s taken as the displacement of the core from the midpoint of the width at half-depth. The result i s again normalized by the mean width at half depth. A positive measure of asymmetry means that the core i s to the red and a negative value means the core i s to the v i o l e t . The position of the core was taken to be that of minimum i n t e n s i t y . Naturally at those times when there are actually two components present the core position i s not r e a l l y meaningful and t h i s shows up as increased 4 8 scatter i n the points as shown most c l e a r l y in Figures 12 (c) and 12(d). Figure 12(a). Line depth, half-v/idth, and asymmetry vs. time for Oct. S i 111 4553 =3 •a U J a U J •—i i n —4 Q © o ® 0 o © a ^ (!) © Q © © ©®(b© © © CD G) © FFI © O © © © fflm © © © © ° © 0 © l — Q i—« cr. •_! L U - n: a a °©©e> 0 © Q © O O O © 0 © 0 ° © © © o© © © ffiOffl © O © © Q © © © Q,® © ©© © o 2 : >— CO c r 1 a © 0) © © * *> O f f l © V 0 © < V © „ © / * © © © Q Q) 0 ® © © ©~ ~ O o o © © © ©° © © © © ° o© 0 0 -1 1 1 — 4.81 5.79 6.77 TIME UTCHRS; ?.83 7.75 8.73 Figure 12(b). Line depth, half-width, and asymmetry vs. time for Oct Si 111 4568 9 13 , LU a U J i n —« © (Do ( D O © (!) © CD © Q CD » © a© © 'ffi © © © © © i n I— Q •—c GC --4 © © © a ©a a © © (5 © O © © w © a © " © © " fflfl) © © © © a a x i — Q : i n 0 0 a ^G^O ©a a © a© © "1 cn F— U J O z : >-. CO (X a. 0 a o © ° © ^ e a © a • © a © © © © © © ©a © a © w a, © ©o © a © © © © © ® o a ca a a ?.83 4.81 5.79 6.77 TIME UT(HRS) 7.75 8.73 51 9 i n m , m --̂ QL. O U J Figure .12(c). Line depth, half-width, and asymmetry vs. time for Oct. 16. Si 111 4553 i n —4 ©© © ® o ©ca © _ © © © ~ 0 © © © a© © ©o© a © © © © © © © © o © © © © © © v © © © © ©o © « . ~ 0 0 " © © O i i n I E " 4 r - a CE -_J UJ " m t— a . i n C9 0 ° © o © © © © © ©-© ~© © 0 © ^ . . o ©o© 0 O © 6 ©o© © © © © © © © © © V ffl a a ° " © © a © >- cn i — >— CO d ( n o . i © © © ° ° „ ^ Q © © o ©o 0 © o u o u o © o ©a © © © © © o i © © © © 0 a 0 0 © a© © © 4.95 5.93 -1 1 1 6.91 7.89 8.87 TI:E UTIHRS) 9.85 10.83 Figure 12(d). Line depth, half-width, and asymmetry vs. time for Oct. 16. 9 o m Si 111 4568 Q_ U J Q U J IT) —< , © Q ° © * 0 © Q © © (9 © © Q © 0 CD © © © © o© © © © © © © © o ° ' o m _© 0 o ° D © & Q © ° I © © in »— a CE i i i — 1 i — Q © © 0 © ^ ° O Q © , °© °© © © © © a © © © © © ° ~ © o © © °o © © ©^ •3 I— U J O >— CO <x CO CD. i © 0 © © © © © © © © a © ° o D © Q © 0 ° © Q © o ©o a © © © a c© | © o o© © ° © I 1 1 1 1 1 1 4.95 5.93 6.91 7.89 8.87 9.85 10.83 TIME UT(HRS) 53 It i s important to have some idea of the error associated with the points plotted in Figure 12, This can be obtained a number of ways; one i s from a consideration of the equivalent widths, I have said before that the equivalent width remains approximately constant throughout the cycle. There are of course small block to block variations and i t i s these that one can use to estimate the errors of the points i n Figure 12. The error bars are - 2 to 3 times the diameter of the symbols used i n the plot. 54 The Radial Velocity Measures The broadening of the l i n e s at certain phases i s assumed to be due to the presence cf two unresolved components. In BH Vul the r a d i a l v e l o c i t y amplitude i s on the order of 150 km/sec. The double nature of the spectral l i n e s at c e r t a i n phases i s very evident i n t h i s star (Goldberg, 1973). In 12 Lac the r a d i a l v e l o c i t y amplitude i s considerably l e s s . Radial v e l o c i t y curves obtained by various authors are a l l continuous. One would usually measure the wavelength s h i f t from the position of l i n e minimum. The observations described here however, have improved spectral and time resolution. In l i g h t of the discontinuous velocity curve obtained by Bruce Goldberg for BW Vul an attempt was made to measure the r a d i a l v e l o c i t i e s of the separate components. A new method of deconvolution developed by Rob Clayton and Tad Dlrych (Clayton and Olrych, 1S76) was applied to the data. A brief description of this method and diagrams shewing examples of the deconvolved spectra are given i n Appendix A. The r e s u l t s are guite amazing. The wavelength s h i f t s for each component were measured from the positions of l i n e minimum from the deconvolved spectra. This technigue was applied to the blocks of Figures 6 and 7. Corrections for s h i f t s and n o n - l i n e a r i t i e s in the scanning raster discussed in a previous section were applied to the data. A reference v e l o c i t y of -20 km/sec was somewhat a r b i t r a r i l y 55 taken as the s t i l l s t a n d velocity (the term ' s t i l l s t a n d 1 i s further described in the Discussion section). One must therefore view the actual numbers with caution and consider instead the amplitude changes, A wavelength c a l i b r a t i o n spectrum was taken but was not used since our primary i n t e r e s t i s i n motion within the framework of the star . Also, large raster s h i f t s in the c a l i b r a t i o n spectrum were found which would l i m i t the accuracy i n any case. The r e s u l t i n g r a d i a l v e l o c i t y curves are plotted i n Figure 13. Each l i n e was considered separately. In the r a d i a l velocity measures l i n e positions were determined only to the nearest instrumental channel. From a consideration of the raster s h i f t and the discrete sampling i n t e r v a l mentioned above, the errors in velocity are estimated to be - 7 km/sec. It should be noted that the errors w i l l be larger where there are two components present; the scatter i s very small when there i s only one component present. Because of the scatter i n the points there i s some guestion as to the exact location of the s t i l l s t a n d v e l o c i t y . For t h i s reason an average velocity curve for the two l i n e s ( ^4553 and ^4568 ) was not obtained. Further remarks cn the velocity curve are reserved t i l l the next section. The deconvolution technigue used to separate the two components uses the Maximum Entropy method cf spectral analysis ( a very brief description of t h i s i s given i n Appendix A). This gives guite accurate position measurements but does not 56 provide true values for the i n t e n s i t i e s of the components. A synthetic p r o f i l e f i t t i n g method developed by John Glaspey was used to determine the i n t e n s i t i e s of the components. Details of the technigue can be found in Glaspey gt a l (1S76). Each l i n e p r o f i l e was reconstructed by adding two Voigt components, both of the same width but with d i f f e r e n t v e l o c i t i e s and amplitudes, and convolving t h e i r sum with the Modulation Transfer Function of the isoccn. The width of the Voigt components was chosen as that which gave the best f i t when compared to a l i n e p r o f i l e at sharp l i n e phase. A Gaussian p r o f i l e gave a s a t i s f a c t o r y f i t . The doppler width of the components was taken to be constant throughout the cy c l e . The v e l o c i t i e s used were those obtained from the deconvolved spectra., Only two parameters, the amplitudes of the components, were then permitted to vary in order to obtain a *good» f i t . Because of the large volume of data to be analyzed an automatic f i t t i n g procedure was used. The method of least squares consists of determining the values of the amplitude parameters A1 and A2 which y i e l d a minimum fo r the function Of 2 ( whereof 2 i s e s s e n t i a l l y the sum of squares of the difference between the input data p r o f i l e and the f i t t i n g p r o f i l e ) . The q r i d search method described by Bevington (1969) was used. The l i n e depths of the separate components are shown in the upper plots of Figure 13. The symbols used in the l i n e depth plots correspond to the two components in the velocity curves. Examples of the Gaussian p r o f i l e f i t to the input data are shown 57 in Figures 14(a) and 14(b), The Gaussian curve i s indicated by the plus(+) signs. The re s u l t i n g difference curve ( observed- Gaussian ) i s shown below the l i n e p r o f i l e . The position of the component (s) i s marked by the arrows. In order to be able to more readily compare my velocity curve with that obtained by other observers, I have plotted the ve l o c i t y curve as obtained by measuring the wavelength s h i f t from the centroid of the l i n e (j c-(A1 +42 JL)/ (A1+A2) ; X and are'the wavelength positions of the two components). The r e s u l t i n g curves for A 4553 are shown i n Figures 15(a) and 15(b) for Oct. 15 and Oct. 16 respectively. 13(a). Radial velocity curve for Si 111 4553 for Oct. 15. I 1 I i 1 4.81 5.79 6.77 7.75 8.73 . TIME UT MRS) 13(b). Radial velocity curve for Si 111 4568 for Oct. 15, < « o ° o o X < ~ - * o°°o ° K * * « o ° o o oo o o < o « o o o ooo o O o O Oo O 4.81 5.79 6.77 7.75° 8.73 TIME UT MRS) Figure 13(c). Radial velocity curve for Si 111 4553 for Oct. 16. oO O o oo o o o ^ o o o o oo (V) IE •• I— a. a U J - J a ' oo o o ° o°o < o « °<^o 4 apx a a . in CJ UJ CO ' a . O Oo UJ2. CE i—i cn - ens i o in. i < 4 « 4 oo o do o o o o oo ooo o 0 o ° ° o o o o o o °o o oo 4 4 4 4 4 < 4 4 o 4 4 4 4.95 5.93 5.91 7.89 8.87 TIME UT MRS.) 9.85 10 Figure 13(d). Radial velocity curve for S i 111 4568 for Oct. 16. o° o ° °o _ o o o oo °o 0 o o 0 o uo O O o o ° o ° o°° > / oo 0 0 o « r ° o o « o o K < o o o o oo O o OO O O o ooo ° °o o o o ° o ° ° o O O 0 o oo oo o o ° I 1 1 1 1 R" 4.95 5.93 6.91 7.89 8.87< « 9.85 10 TIME UT MRS) Figure 14(a). Gaussian f i t to input data for block #10 Figure 5a. S i 111 4553 12 LRC 63 64 Figure 15(a). Radial velocity curve (centroid) for Si 111 4553 for Oct. 15. 01 o i n . Q i n U U J CO ' i n . i O Oa 0 , • D 03 03 a B3 Q ts a at • • Q Q a cn I— I l a i n r n . 3.83 4.81 C3 Q Q • • a 5.79 6.77 TIME UT MRS) m 1 7.75 8.73 i Figure 15(b). Radial velocity curve (centroid) for Si 111 4553 for Oct. 16. o in" a in" U UJ cn 2 ° i O Oa LuS . cn •a cn . 01 ft. i i n t n . 0 - •a a o o o c m CO CQQJ m CD CQQ a _tD o ta o o O D D o D D O CD 0 Q CO 1 1 1 1 1 1 95 5.93 6.91 7.89 8.87 9.85 10.83 TIME UT MRS) 66 Discussion Throughout t h i s discussion I w i l l use the velocity curve obtained by Bruce Goldberg (1973) for BH Vul for comparison. The two most prominent features of my observed velocity f c r 12 Lac are 1) i t s discontinuous nature and 2) evidence for a s t i l l s t a n d on the descending branch of the velocity curve. In BM Vul there i s a period of -40 minutes duration during which the r a d i a l velocity regains approximately constant. Because of the scatter i n the data points i t i s not clear i n the case of 12 Lac whether or not*the velocity during • s t i l l s t a n d 1 i s r e a l l y constant. Nevertheless as t h i s feature i s quite pronounced in BH Vul I w i l l use the term • s t i l l s t a n d * in my discussion. In describing the various features of the velocity curves I w i l l indicate in brackets [ ] which component I am re f e r r i n g to: x or o. Consider Figures 13(c) and 13(d): 1) The s t i l l s t a n d [o] f i r s t takes place at -5 hrs D.T. The discontinuity preceding the s t i l l s t a n d r esults from the s p l i t t i n g of the l i n e into two components. 2) There i s evidence f o r a blue s h i f t following s t i l l s t a n d £o] at -6.5 hrs O.T. This effect i s more obvious in Figure 13(d) for \ 4568. The •f l a t t e n i n g ' of the velocity curve near the phase of maximum blue s h i f t as observed.in BW Vul i s also guite evident i n V4568. For /i 4568 the amplitude difference between maximum blue s h i f t and the s t i l l s t a n d velocity i s -30 km/sec. For A 4553 i t i s only half t h i s amount. It i s not clear from these observations whether t h i s discrepency i s i n t r i n s i c or 67 instrumental, 3) The rapid r i s e of one component [o] to maximum r e d s h i f t i s seen pr i o r to a renewal of the cycle. 4) After the red-shifted component reaches a maximum velocity i t seems to experience a blue s h i f t before fading away. This i s seen p a r t i c u l a r l y i n Figure 13(c) for f x ] at - 6 hrs D.T. This e f f e c t i s not seen in EH Vul, i n fact the opposite occurs; i t continues to be red-shifted. It i s suspected that the blue s h i f t i s not r e a l and that what r e a l l y happens i s that the f i r s t few points have been overcorrected for raster s h i f t . This point i s further explained i n #6 below. 5) For t h i s night we have more than one cycle of the pulsation and are able to see the discontinuity reappear ( at -9.5 hrs D.T.) as a component [x] develops having the s t i l l s t a n d v e l o c i t y . Consider Figures 13(a) and 13(b): 6) I might note that i n Figure 13(a) the points at the beginning of observation at -4 hrs. D.T. seem abnormally high (by -10 km/sec). Se are seeing just s l i g h t l y more than one cycle of the pulsation and thus the v e l o c i t i e s at the beginning of the cycle on the ascending branch should be the same (at least for consecutive c y c l e s ) . The f i d u c i a l s h i f t corrections were greater than 2 sample points only near the beginning of the observing run. In pa r t i c u l a r the correction for the f i r s t few blocks were on the order of 15 km/sec - i e . the uncorrected points were at - -25 km/sec. It i s possible that the non-linear expansion of the camera reading beam might be more serious near 68 the s t a r t of observation on a par t i c u l a r object. 7) There seems to be evidence for a blue s h i f t following s t i l l s t a n d [o] p a r t i c u l a r l y in Figure 13(a) for /] 4553 at -7.75 hrs as noted in point (2) for the other night's data. 8) The scatter i n Figure 13(b) i s very large. a possible cause for the scatter i n the r a d i a l v e l o c i t y measurements i s the p r o f i l e shape assumed f o r one component. Perhaps the two components have di f f e r e n t widths and/or the width of the components changes with time. The wavelength s h i f t s determined by the deconvolution method would then be i n error, In a few cases, the deconvolution technigue yielded three components. This i s most l i k e l y not a r e a l physical s i t u a t i o n but more a function of the p r o f i l e shape assumed for the various components. The l i n e p r o f i l e and r a d i a l velocity changes are d i r e c t l y correlated. Compare Figures 12 and 13. Consider Figures 12(a), 12(b), 13(a) and 13(b). fit the beginning of the ascending branch of the vel o c i t y curve the l i n e s are sharpest, deepest and e s s e n t i a l l y symmetrical. The deepest phase occurs at -4 hrs. O.T. as the cycle progresses the l i n e s become shallower and broader with an extended wing to the red. At - 5.5 hrs. O.T. the l i n e s begin to double as a component £o] develops having the s t i l l s t a n d v e l o c i t y . This component increases in i n t e n s i t y while the redshifted component £x] fades away. During s t i l l s t a n d the.lines become sharper and deeper again (shown very c l e a r l y in Figures 12(a) and 12(b) at -6.5 hrs. O.T.) but less 69 so than at the beginning of the ascending branch. During the blue s h i f t following s t i l l s t a n d [o] there i s an asymmetry showing an extended wing to the blue. The l i n e s then get narrower and deeper prior to the renewal of the cycle. Much the same thing i s seen for the data of October 16. It i s very i n t e r e s t i n g to compare Figures 13(a) and 15(a) and Figures 13 (c) and 15(b). »hen the two components are of / egual i n t e n s i t y , the v e l o c i t i e s i n Figure 15 are a straight average of the v e l o c i t i e s for the separate components. As one component becomes stronger than the other, the r e s u l t i n g ( centroid ) v e l o c i t y i s weighted mere towards the ve l o c i t y of the stronger component. For Figure 13(a) the velocity amplitude i s - 70 km/sec; for Figure 15(a) i t i s - 45 km/sec. The velocity amplitude determined from a curve showing the wavelength s h i f t s from the centroid w i l l thus be much les s than the true value. There does not appear to be much evidence i n Figures 15 (a) and- 15 (b) for a s t i l l s t a n d on the descending branch of the velocity curve. The whole guestion of the multiple p e r i o d i c i t y of 12 Lac i s very complex. Because of i t s importance I w i l l present here a short summary of the period determinations i n 12 Lac. In 1953 De Jager (De Jager^ 195 3) t r i e d to explain the v a r i a b i l i t y of the star by means of a sum of two independent o s c i l l a t i o n s with frequencies being not far from each other. His components with periods of 0.19308883 days(Pl) and 0.197367 days(P2) give the beat period P=8.908 days. In consequence, one ought to expect 70 that the amplitudes of brightness and r a d i a l v e l o c i t i e s , and the values of [0-C] calculated for maxima or minima of brightness and r a d i a l v e l o c i t i e s from the period 0.19308883 or 0.197367 days should change according to the phase of the beat period. This did not occur, and in 1957 De Jager (De Jager, 1957) found the existence of the t h i r d o s c i l l a t i o n in the star with a period of 0.15583 days. Rakosch (1960) confirmed the presence of the three independent o s c i l l a t i o n s but with s l i g h t l y d i f f e r e n t values of the periods (0.19308997 days, 0.19737253 days and 0.15292 days). He also found a new period for colour index variations (0.118644 days). A large number of observations obtained during the "International Lacerta Weeks" compiled and published by De Jager (1963) were used by Barning (1963) for the determination of v a r i a b i l i t y of 12 Lac. His resu l t i s four independent o s c i l l a t i o n s ; two with previously known periods (0.19308883 days and 0.197358 days), and two addit i o n a l ones with periods 0.182127 days and 25.85 days. The longest period found by Barning i s probably not r e a l because of the many uncertainties involved. Opolski and Ciu r l a (1961, 1962), Jerzykiewicz (1963) and Ci u r l a (1973) found that the short period (0.19308858 days) varies p e r i o d i c a l l y with a period of 8.876 days. They suggest that the components of the l i g h t curve with the short periods P1 (0.19308883), P2 (0.197367) and P3 (0.15583) represent no more than a formal description of the observed v a r i a b i l i t y of 12 Lac. Any periods determined from previous r a d i a l v e l o c i t y curves ( such as Figures 15(a) and 15(b) ) should be re-examined. The 71 velocity curve i s c l e a r l y discontinuous and t h i s s p l i t t i n g might influence any period determinations. There have been a number of t h e o r e t i c a l investigations undertaken towards explaining the l i n e s p l i t t i n g at certain phases. - 1) The double l i n e s are formed in d i f f e r e n t parts of the star's disk (McNamara and Gsbbie, 1961; Huang, 1955). 2) They arise from two layers, one above the other (Odgers, 1955; Goldberg, 1973). 3) They are the ccnseguences cf non-radial pulsation (Osaki, 1971) . As mentioned before, Huang (1955) found that the eguivalent width in T Sco did not vary throughout the cy c l e . He interpreted t h i s to mean that the s p l i t t i n g of l i n e p r o f i l e s i s due to macroscopic motion: the two streams must be moving i n d i f f e r e n t parts of the star's disk instead of one above the other. In (2) the motion i s r a d i a l . The interpretation i s based on the following picture: An atmosphere i s ejected with high velocity which after t r a v e l l i n g outwards for a time then f a l l s back i n t c the general s t e l l a r photosphere at high speed. For a time at the s t i l l s t a n d phase the s t e l l a r surface proper i s v i s i b l e and then another atmosphere i s ejected, (Goldberg, 1973). The paper by Heard et a l (1976) mentions work by Hatson and 72 Stanford that extends Osaki's work and shows that at moderate to large amplitude, non-radial pulsation can produce l i n e s p l i t t i n g of the type observed in EH Vul and now seen in 12 Lac. Both the non-radial and r a d i a l explanations offe r a g u a l i t a t i v e description of the l i n e s p l i t t i n g . More work needs to be done to decide which i s the correct i n t e r p r e t a t i o n . 73 Summary and Conclusions The broadening of the l i n e p r o f i l e s at certain phases i s seen to be due to the presence of two unresolved components. The discontinuity i n the r a d i a l v e l o c i t y curve preceding s t i l l s t a n d i s due to the growth of a component at the s t i l l s t a n d v e l o c i t y . The l i n e s p l i t t i n g and consequent discontinuous velocity curve should prove a strong incentive f o r more the o r e t i c a l and observational investigations. There i s one very s i g n i f i c a n t difference between 12 Lac and BW Vul. BS Vul appears to be singly periodic. The l i n e p r o f i l e and r a d i a l velocity changes repeat from cycle to cy c l e . The amplitude of the velocity curve remains constant. 12 Lac has (supposedly ) a number of periods. The r a d i a l velocity amplitude i s variable. On Oct. 15 the t o t a l velocity amplitude i s -70 km/sec; on Oct. 16 i t i s - 80 km/sec. The r a d i a l v e l o c i t y curves and l i n e p r o f i l e variations do not repeat from cycle to cycle. Cn Oct. 16 the velocity amplitude separation at s t i l l s t a n d between the components i s much larger than that on Oct. 15. More observations should be undertaken to accurately determine the period (s) of r a d i a l v e l o c i t y amplitude variation. In choosing between any r a d i a l or non- r a d i a l theory this difference i n velocity amplitude must be considered. An interesting side r e s u l t of t h i s investigation i s that 74 though the velocity amplitude i s greater on Oct. 16 r e s u l t i n g i n broader l i n e s , the l i n e p r o f i l e s at the sharp l i n e phase ( one component present ) have the same width for both nights. The same Gaussian p r o f i l e gave a s a t i s f a c t o r y f i t for both sets of data. Another important result i s that the amplitude of the r a d i a l v e l o c i t y curve i s much greater than what was previously obtained. Heard et a l (1976) find that the amplitude of the velocity varies s l i g h t l y from 43 km/sec to 35 km/sec. The s p l i t t i n g of the components shows that the true r a d i a l velocity amplitude i s about twice that obtained previously. Hith regards to 12 lac i t i s proposed that a number of things be done: 1) As summarized in the Discussion section there has been a great amount of work done through the years, on examining the l i g h t and r a d i a l velocity curves for multiple p e r i o d i c i t y . Any researcher who attempts to use these periods i n a model explaining 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 should keep i n mind the l i n e s p l i t t i n g and consequent discontinuous velocity curve. 2) Hore observations should be undertaken at t h i s high spectral and time resolution. I f one had observations of a larqe number of cycles, one would have a clearer idea of whether the scatter ( p a r t i c u l a r l y at s t i l l s t a n d ) was i n t r i n s i c or instrumental. 3) Lines of other elements should be observed; in p a r t i c u l a r He lJi4471 and the hydroqen l i n e s . Two possible motivations for 75 t h i s l i n e of investigation are: 1) the presence of emission l i n e s i n some ft Cephei stars (underbill, 1966) and 2) Struve and Zebergs (1955) have shown that the hydrogen l i n e s lag on the descending branch of the velocity curve. Observations of other /§ Cephei stars should most d e f i n i t e l y be obtained at high spectral and time resolution i n order tc see i f any l i n e s p l i t t i n g occurs. A complete l i s t of the 18 c l a s s i c a l ft Cephei stars i s given i n Underbill (1966). A l l of the stars are very bright (>6th magnitude) and t h e i r periods are short(<6 hours)i A complete cycle (or more) of the pulsation could be observed in one night of observation. 76 Adams, W.S. 1912, Ap.J., 35 , 179. Aizenman, H.L., Cox, J.P. and Lesh, J.R. 1975, Ap.J., 197 , 399. Barning, F.J.fl. 1962, B.A.N., J7 , 22. Beres, K. 1966, Acta Astr., 16 , 161., Bevington, P.R. 1969, Data Seduction and Error Analysis f o r the Physical Sciences (McGraw-Hill). Buchholz, V.L., Walker, G.A.H., Auman, J.R. and Isherwood, B.C. 1973, i n Astronomical Observations with Television- type Sensors, ea . J.a . Glaspey and G.A.H. Walker (Oniv. of B r i t i s h Columbia, Inst, of Astronomy and Space Science), p. 199. C i u r l a , T. 1973, Acta Astr., 21 * 367. Clayton, B.W. and Dlrych, T.J. 1976, i n preparation. Cox, J.P. 1974, Reports cn Progress i n Physics, 37 , 563. De Jager, C. 1953, E.A.N., J2 , 81. . 1 9 5 7 , B.A.N., 13 , 149. . 1963, E.A.N., 17 , 1. Fahlman, G.G. and Glaspey, J.W. 1973, i n Astronomical Observations with Television-type Sensors, ed. J.W. Glaspey and G.A.H. Walker (Univ. of B r i t i s h Columbia, Inst, of Astronomy and Space Science), p. 347. Fath, E.A. 1938, Pop. Astr., 46 , 241. Glaspey, J.W., E i l e k , J.A., Fahlman, G.G. and Auman, J.R. 1976, Ap.J., 203 , 335. Goldberg, E.A. 1973, PhD. Thesis, University of B r i t i s h Columbia. Grabowski, B. 1966, Acta Astr., 16 , 309. . 1 9 6 9 , Acta Astr., J9 , 23. Heard, J.F., Hurkens, R.J., Percy, J.R. and Porco, M. 1976, to be published. ' H i l l , G. 1967, Ap.J. Suppl., 14 , 263. Huang, S.S. 1955, P.A.S.P., 67 , 22. 77 Jerzykiewi.cz, M. 1963, Seta Astr., 13 , 253. Kanasewich, E.B. 1973, Time Sequence Analysis in Geophysics (The University of Alberta Press) . Lesh, M.I. and Aizenman, J.B. 1973a, Astron. and Astrophys., 22 , 229. • — . 1973b, Astron. and Astrophys., 26 , 1. flcNamara, D.fl. and Gebbie, K.B. 1961, P.A.S.P., 73 , 56. Odgers, G.J. 1955, Publ. B.A.O., 10 , No. 9, 215. Opolski, A. and C i u r l a , T. 1961, Acta Astr., JM , 231. . . 1962, Acta Astr., 12 , 269. Opolski, A. and Grabowski, B. 1966, J6 , 303. Osaki, Y. 1971, Pub. Astr. Sec. Japan, 23 , 485. . 1974, Ap.J., J89 , 469. Percy, J.B. 1967, J.B.A.S.C., 61 , 117. Bakosch, K. 1960> A.N., 285, 211. Bichardson, E.fl. 1973, in Astronomical Observations with Television-type Sensors, ed. J.W. Glaspey and G.A.H. Walker (Univ. of B r i t i s h Columbia, Inst, of Astronomy and Space Science), p. 433. Sato, N. 1973, Astrophys. and Space S c i . , 24 , 215., Struve, 0. 1951, ftp.J., 113 , 589. >. 1955, P.A.S.P., 67 , 135. Struve, 0. and Zeberqs, V. 1955, Ap.J., 122 , 134. Thompson, I.B. 1974, B.Sc. Thesis, University of B r i t i s h Columbia. Underhill, A.P. 1966, The Early Type Stars (Dordrecht, Holland: D. Beidel Co.). Walker> G.A.H., Auman, J.B., Buchholz, V.L., Goldberg, B.A., Gower,*A.C, Isherwood, B.C., Knight, B. and Wright, D. 1972, Advances in Electronics and Electron Physics, 33E , 819. Watson, B.D. 1972, Ap.J. Suppl., 24 , 167. 78 ADjjendix A In any method of d e c o n v o l u t i o n one of the requirements i s t h a t you have a knowledge of the p r o f i l e shape of the components. A Gaussian p r o f i l e which qave the best f i t to a data p r o f i l e a t the sharpest l i n e phase was used. The widths of the two components are assumed to be e q u a l ; as w e l l the assumption i s made t h a t the components though they d e c r e a s e / i n c r e a s e i n i n t e n s i t y , s t i l l r e t a i n the same p r o f i l e shape throughout the c y c l e . The new method i s e s s e n t i a l l y t h i s : 1) The data i s t r e a t e d as a power spectrum. 2) The a u t o c c v a r i a n c e f u n c t i o n i s o b t a i n e d from t h i s 'power spectrum*. 3) The autocovariance i s weighted by the amplitude spectrum of the assumed p r o f i l e shape f o r one component. <4) The Maximum Entropy Power spectrum i s then o b t a i n e d where the a u t o c o r r e l a t i o n c o e f f i c i e n t s computed above are used d i r e c t l y to o b t a i n the p r e d i c t i o n - e r r c r c o e f f i c i e n t s . One of the drawbacks of c o n v e n t i o n a l d e c o n v o l u t i o n t e c h n i g u e s which employ a smoothing of the a u t o c o r r e l a t i o n f u n c t i o n by a time domain window or a smoothing of the sguared magnitude of the F o u r i e r Transform i s t h a t they do not design a window which i s based on the t r u e spectrum. T h i s l e a d s to leakage of power through s i d e lobes i n the t r a n s f e r f u n c t i o n and a l s o puts a l i m i t cn the r e s o l u t i o n . The Maximum Entropy Method 79 estimator retains a l l the estimated lags without smoothing and uses Weiner optimum f i l t e r theory to design a prediction f i l t e r which w i l l whiten the input power spectrum. From the whitened output power and the response of the prediction f i l t e r i t i s possible to compute the input power spectrum (Kanasewich, 1969). The accuracy of the positions depends on the number of sinusoids i n the autocorrelation function. It i s estimated that the positions are determined to t 1 sample point. Some results are shown i n the following Figures. The.data must be inverted to be considered as a power spectrum which accounts f o r the pseudo-emission l i n e spectrum. The bottom graph i s a plot of the input data. The middle plot shows the deconvolved spectrum. The top graph i s the reconstructed p r o f i l e . , This i s obtained by convolving the deconvolved spectrum with the assumed p r o f i l e shape for one component. The data blocks considered are those of Figures 5(a) and 5(b) for \ 4553. Figure A.l Deconvolved spectrum for block // 3 of Figure 5a. Figure A.2 Deconvolved spectrum for block // 9 of Figure 5 0.0 50.0 100.0 150.0 200.0 POINTS Figure A.3 Deconvolved spectrum for block # 10 of Figure 5a Figure A.4 Deconvolved spectrum for block # 12 of Figure 5a. Figure A.5 Deconvolved spectrum for block # 1 of Figure 5b. o O.D 50.0 100.0 POINTS 150.0 200.0 Figure A. 6 Deconvolved spectrum for block // 3 of Figure 5b I 1 1 1 I O.D 50.0 100.0 150.0 200.0 POINTS 86 Figure A. 8 Deconvolved spectrum for block // 13 of Figure 5b. I 1 1 1 1 0.0 50.0 100.0 150.0 200.0 POINTS Figure A. 9 Deconvolved spectrum for block it 14 of Figure 5b. Figure A.10 Deconvolved spectrum for block # 15 of Figure 5b.

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