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Application of the HF precision-velocity technique to the study of [delta] Scuti variables 1985

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c APPLICATION OF THE HF PRECISION-VELOCITY TECHNIQUE TO THE STUDY OF 6 SCUTI VARIABLES by STEPHENSON YANG M . S c . , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1980 B . S c , U n i v e r s i t y of B r i t i s h C o l u m b i a , 1976 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES Department of Geophys i c s and'Astronomy We accept t h i s t h e s i s as conforming to the reqjaiire'd s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA October 1985 © Stephenson Yang, 1985 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia 1956 Main Mall Vancouver, Canada Department V6T 1Y3 DE-6(3/81) • ABSTRACT In c o n v e n t i o n a l r a d i a l - v e l o c i t y techniques, wavelength c a l i b r a t i o n i n s t e l l a r s p e c t r a i s l i m i t e d by c o l l i m a t i o n and g u i d i n g e r r o r s . These e r r o r s can be l a r g e l y e l i m i n a t e d by imposing a b s o r p t i o n l i n e s of known wavelengths on the s p e c t r a . The most s u i t a b l e n a t u r a l a b s o r p t i o n l i n e s belong to the R-branch (3-0) v i b r a t i o n - r o t a t i o n band of HF. The c a l i b r a t i o n i s achieved by p l a c i n g an a b s o r p t i o n c e l l f i l l e d with HF gas i n t o the s t e l l a r beam. The monel c e l l and the sapphire windows are maintained at 100°C i n order to avoid p o l y m e r i s a t i o n of the HF molecules. Connected to the c e l l i s a r e s e r v o i r of l i q u i d HF maintained at 0°C and t h i s would then maintain a constant pressure i n s i d e the c e l l . Meanwhile, the Retico n d e t e c t o r i s u t i l i s e d to p r o v i d e the necessary high s/n s p e c t r a . In reducing the s t e l l a r s p e c t r a with the imposed HF l i n e s , the e f f e c t of l i n e b l e n d i n g between the two s e t s of l i n e s can be minimised by performing numerical l i n e c a n c e l l a t i o n s u sing standard s t e l l a r or HF s p e c t r a . The r e l a t i v e p o s i t i o n s of s p e c t r a l l i n e s are measured by means of the Fahlman-Glaspey d i f f e r e n c e technique. A m o d i f i e d Fahlman-Glaspey d i f f e r e n c e f u n c t i o n can a l s o be used to minimise the e f f e c t of small o f f s e t and s c a l e d i f f e r e n c e s between the l i n e p r o f i l e s . S i m u l a t i o n s t u d i e s with a r t i f i c i a l s p e c t r a have confirmed that the accuracy of l i n e - p o s i t i o n measurement would i n c r e a s e l i n e a r l y with both s/n of the s p e c t r a and depth of the l i n e p r o f i l e s . Furthermore, the accuracy was found to decrease with i i i n c r e a s i n g l i n e w i d t h s when the e q u i v a l e n t widths were h e l d c o n s t a n t . A method has been d e v i s e d to measure the HF gas temperature d i r e c t l y from the observed HF l i n e s . T h i s i n v o l v e s c a l c u l a t i n g the r a t i o s between the observed HF l i n e s t r e n g t h s . T h e o r e t i c a l l i n e s t r e n g t h s f o r the (3-0) HF l i n e s have a l s o been c a l c u l a t e d i n order to study t h e i r dependence on the gas temperature. These t h e o r e t i c a l v a l u e s can a l s o be used with the observed HF l i n e s t r e n g t h s to c a l c u l a t e the r e l a t i v e gas p r e s s u r e . A c u t o f f - f r e e ATC theory on c o l l i s i o n a l l i n e broadening was used to c a l c u l a t e HF l i n e w i d t h s and l i n e s h i f t s . Good agreement has been achieved between the c a l c u l a t e d and p u b l i s h e d (1-0), (2-0), and (3-0) l i n e w i d t h s . T h i s a l s o enables a study on the temperature dependence of the (3-0) HF c o l l i s i o n a l s e l f - b r o a d e n e d l i n e w i d t h s . Poor agreement was obtained between the p u b l i s h e d and c a l c u l a t e d l i n e s h i f t s . N e v e r t h e l e s s , p r e s s u r e - s h i f t c o r r e c t e d wavelengths of the r e f e r e n c e (3-0) l i n e s have been d e r i v e d . The 5 S c u t i s t a r s are d i s t i n g u i s h e d by p u l s a t i o n p e r i o d s of l e s s than 0.3^ and s p e c t r a l type A or F. The small-amplitude 5 S c u t i s t a r s t y p i c a l l y have a l i g h t amplitude Amv of 0.05 m and a v e l o c i t y 2K amplitude of l e s s than lOkms" 1. Since the 2K/Amv value i s about 92kms~ 1mag~ 1 f o r most 5 S c u t i s t a r s , a v e l o c i t y p r e c i s i o n of j u s t ±0.1kms" 1 would a l r e a d y be e q u i v a l e n t to a p r e c i s i o n of about ±0.00l m i n the l i g h t curve. Hence one can improve the study of 6 S c u t i p u l s a t i o n s with the use of p r e c i s i o n r a d i a l - v e l o c i t y techniques. The HF technique would a l s o enable the study of i n d i v i d u a l s p e c t r a l l i n e s f o r p r o f i l e v a r i a t i o n s . The s t a r 20 CVn i s a 0.122 d 6 Del type 6 S c u t i v a r i a b l e observed u s i n g the HF technique at CFHT. A 2K value of 1.4kms _ 1 was o b t a i n e d f o r the Ca II X8662 l i n e w h i le a value of l ^ k m s " 1 was ob t a i n e d f o r the other l i n e s . These give a 2K/Amv of about 40kms~ 1mag~ 1 f o r 20 CVn which c o u l d imply n o n r a d i a l p u l s a t i o n . The s t a r p Pup i s a 0.141 6 Del type 8 S c u t i v a r i a b l e observed at CFHT. The l i n e i n t e n s i t i e s of the s t e l l a r l i n e s have been found to vary i n phase with the l i g h t c u rve. These v a r i a t i o n s are at l e v e l s between 0.5% and 1% of the continuum. The l i n e s are s t r o n g e s t near maximum l i g h t and weakest near minimum l i g h t . The v a r i a t i o n s can be c o n s i d e r e d as s p e c t r a l - t y p e v a r i a t i o n s caused by the v a r i a t i o n s of the e f f e c t i v e temperature of the s t a r over the p u l s a t i o n c y c l e . The s t a r o 1 E r i i s a 0.082 b r o a d - l i n e 6 S c u t i v a r i a b l e observed at CFHT. A 2K amplitude of 4.3kms"1 was measured fo r the Ca II X8662 and H I X8750 l i n e s . L i n e - p r o f i l e v a r i a t i o n s which are at a l e v e l of about 2% of the continuum, have been observed i n the Ca II l i n e . The v a r i a t i o n s can be c h a r a c t e r i s e d as the temporal movement of " f e a t u r e s " a c r o s s the broadened l i n e p r o f i l e . T h i s would suggest the e x i s t e n c e of n o n r a d i a l p u l s a t i o n i n the s t a r . The s t a r 0 Cas i s a 0.104 b r o a d - l i n e 5 S c u t i v a r i a b l e observed at CFHT. The Ca II X8662 l i n e has a 2K value of about 7kms"1 while the value f o r the H I X8750 and the other i v m e t a l l i c l i n e s i s about 4.5kms~ 1. The H I v e l o c i t y curve was found to l a g the other v e l o c i t y curves by about 2% i n phase. L i n e - p r o f i l e v a r i a t i o n s which are at a l e v e l of 1% of the continuum, have been observed i n the Ca II l i n e . The v a r i a t i o n s can be c h a r a c t e r i s e d as l i n e w i d t h v a r i a t i o n s . The b r o a d - l i n e phase c o i n c i d e s with the v e l o c i t y minimum while the n a r r o w - l i n e phase c o i n c i d e s with the v e l o c i t y maximum. v Table of Contents 1. I n t r o d u c t i o n 1 1.1 The measurement of s t e l l a r r a d i a l v e l o c i t i e s 1 1.2 Conventional r a d i a l - v e l o c i t y techniques 4 1.3 Modern p r e c i s i o n r a d i a l - v e l o c i t y techniques 6 1.3.1 Coude o p t i c a l f i b r e feed 6 1.3.2 I n f r a r e d heterodyne technique 6 1.3.3 Measurement of the s o l a r o s c i l l a t i o n s 7 1.3.4 I n f r a r e d F o u r i e r Transform spectrometer ....9 1.3.5 M o d i f i e d Michelson i n t e r f e r o m e t e r 10 1.3.6 Fabry-Perot techniques 11 1.3.7 Use of t e l l u r i c l i n e s 15 1.3.8 Imposing a r t i f i c i a l c a l i b r a t i o n l i n e s 16 1.4 D e l t a S c u t i v a r i a b l e s 18 1.4.1 I n t r o d u c t i o n 19 1.4.2 Maia sequence ? 20 1.4.3 6 D e l p h i n i anomalies 20 1.4.4 Coexistence of p u l s a t i o n and m e t a l l i c i s m ..21 1.4.5 Models of 6 S c u t i s t a r s 22 1.4.6 P e r i o d - l u m i n o s i t y - c o l o u r r e l a t i o n 24 1.4.7 L i g h t and v e l o c i t y amplitudes 24 1.4.8 Systematics of 6 S c u t i s t a r s 25 1.5 O s c i l l a t i o n modes -..26 1.5.1 Mode c l a s s i f i c a t i o n 26 1.5.2 Mode i d e n t i f i c a t i o n techniques 30 1.5.2.1 P e r i o d r a t i o s 30 1.5.2.2 Use of the p u l s a t i o n constant Q ...31 1.5.2.3 L i n e p r o f i l e a n a l y s i s 32 v i 1.5.2.4 Use of si m u l t a n e o u s l y observed data 33 1.5.2.5 Use of p o l a r i s a t i o n measurements ..38 1.6 S p e c t r o s c o p i c o b s e r v a t i o n s of D e l t a S c u t i s t a r s .38 1.6.1 D i f f i c u l t i e s with low-amplitude v a r i a b l e s .38 1.6.2 Recent o b s e r v a t i o n s 40 1.6.3 Use of p r e c i s i o n r a d i a l - v e l o c i t y techniques 42 2. The HF a b s o r p t i o n c e l l system 45 2.1 I n t r o d u c t i o n 45 2.2 Choosing the d e t e c t o r 45 2.3 The Ret icon d e t e c t o r 47 2.3.1 I n t r o d u c t i o n 47 2.3.2 Dark c u r r e n t 49 2.3.3 L i n e a r i t y i n response ...50 2.3.4 F i x e d l i n e p a t t e r n 51 2.3.5 Reduction of readout n o i s e 52 2.3.6 The incomplete readout phenomenon 54 2.3.7 The p e r s i s t e n c e phenomenon 59 2.3.8 Cosmic-ray events 60 2.4 The gas a b s o r p t i o n system 61 2.4.1 Choosing the abs o r b i n g gas 61 2.4.2 P h y s i c a l and chemical p r o p e r t i e s of HF ....66 2.4.3 Sa f e t y p r e c a u t i o n s on working with HF 66 2.4.4 The a b s o r p t i o n c e l l 67 2.4.5 The c e l l windows 71 2.4.6 Operation of the gas ha n d l i n g system 75 2.4.7 Placement of the c e l l 80 v i i 3. The HF spectrum 82 3.1 I n t r o d u c t i o n 82 3.2 Mol e c u l a r c o n s t a n t s f o r HF 83 3.2.1 Basic equations and constants 83 3.2.2 D e r i v a t i o n s of new cons t a n t s and wavelengths 86 3.3 The temperature and pre s s u r e of the HF gas 94 3.3.1 I n t r o d u c t i o n 94 3.3.2 HF l i n e s t r e n g t h 96 3.3.2.1 B a s i c equations 96 3.3.2.2 The Herman-Wallis f a c t o r s 97 3.3.2.3 D e r i v a t i o n of gas temperature ....101 3.3.2.4 D e r i v a t i o n of the gas p r e s s u r e ...104 3.4 The c o l l i s i o n a l l y broadened l i n e w i d t h s 109 3.4.1 I n t r o d u c t i o n 109 3.4.2 T h e o r i e s on c o l l i s i o n a l l i n e broadening ..111 3.4.3 The Anderson-Tsao-Curnutte (ATC) theory ..113 3.4.3.1 B a s i c approach 113 3.4.3.2 C o l l i s i o n c r o s s s e c t i o n 115 3.4.3.3 The c o l l i s i o n e f f i c i e n c y f u n c t i o n 116 3.4.3.4 C u t o f f procedure f o r smal l impact parameter 117 3.4.3.5 The c u t o f f - f r e e theory 118 3.4.3.6 B a s i c f o r m u l a t i o n s of s i m p l i e d theory 120 3.4.4 Survey of experiments and c a l c u l a t i o n s ...121 3.4.5 The c a l c u l a t i o n of l i n e w i d t h s and l i n e s h i f t s 122 3.4.6 C a l c u l a t e d l i n e w i d t h s f o r the HF l i n e s ...124 v i i i 3.4.7 C a l c u l a t e d l i n e s h i f t s f o r the HF l i n e s ..127 3.4.8 S h i f t - c o r r e c t e d r e f e r e n c e wavelengths ....130 3.4.9 Doppler and c e l l - w a l l broadening 130 4. HF data r e d u c t i o n 133 4.1 I n t r o d u c t i o n 133 4.2 P r e p r o c e s s i n g of R e t i c o n s p e c t r a 133 4.2.1 B a s e l i n e s u b t r a c t i o n 133 4.2.2 Use of "extra-readout" p o i n t s 135 4.2.3 R e l a t i v e gain c o r r e c t i o n 136 4.2.3.1 L i n e - n o r m a l i s a t i o n procedure 136 4.2.3.2 Use of step lamps 137 4.2.4 F l a t - f i e l d i n g 138 4.3 Reduction of HF data 141 4.3.1 Continuum r e c t i f i c a t i o n 141 4.3.2 L i n e - p o s i t i o n d e t e r m i n a t i o n 145 4.3.2.1 Li n e c a n c e l l a t i o n s 145 4.3.2.2 Li n e p o s i t i o n i n standard s p e c t r a 150 4.3.2.3 Use of the d e r i v a t i v e of the l i n e p r o f i l e 151 4.3.2.4 The Fahlman-Glaspey d i f f e r e n c e technique 156 4.3.2.5 L i n e - p r o f i l e v a r i a t i o n s 160 4.3.2.6 O p t i m i s i n g d i f f e r e n c e f u n c t i o n ...162 4.3.2.7 E r r o r e s t i m a t i o n 164 4.3.2.8 D i s p e r s i o n r e l a t i o n 165 4.3.3 E f f e c t i v e r e s t wavelengths 169 4.3.4 B a r y c e n t r i c c o r r e c t i o n s 171 4.4 S i m u l a t i o n s t u d i e s 176 i x 4.4.1 B a s i c approach 176 4.4.2 Noise gene r a t i o n 176 4.4.3 Reduction of the a r t i f i c i a l s p e c t r a 177 4.4.4 The e f f e c t of s/n 179 4.4.5 The e f f e c t of l i n e depth 179 4.4.6 The e f f e c t of l i n e w i d t h 182 4.4.7 T h e o r e t i c a l l i n e - p o s i t i o n accuracy 184 5. The D e l t a S c u t i v a r i a b l e 20 CVn 188 5.1 I n t r o d u c t i o n 188 5.2 V a r i a b i l i t i e s of 20 CVn 189 5.3 The o b s e r v a t i o n s 192 5.4 The data r e d u c t i o n • 193 5.5 The r a d i a l v e l o c i t i e s 197 5.6 D i s c u s s i o n 202 6. The D e l t a S c u t i v a r i a b l e p Pup 212 6.1 I n t r o d u c t i o n ....212 6.2 V a r i a b i l i t i e s of p Pup 212 6.3 The o b s e r v a t i o n s 217 6.4 The data r e d u c t i o n 221 6.5 The l i n e - p r o f i l e v a r i a t i o n s 222 6.6 The r a d i a l v e l o c i t i e s 242 6.7 D i s c u s s i o n 263 7. The D e l t a S c u t i v a r i a b l e o 1 E r i 265 7.1 I n t r o d u c t i o n 265 7.2 The o b s e r v a t i o n s 265 7.3 The data r e d u c t i o n 267 7.4 The r a d i a l v e l o c i t i e s 273 x 7.5 The l i n e - p r o f i l e v a r i a t i o n s 279 7.6 D i s c u s s i o n 281 8. The D e l t a S c u t i v a r i a b l e 0 Cas 284 8.1 I n t r o d u c t i o n 284 8.2 The o b s e r v a t i o n s 286 8.3 The data r e d u c t i o n 288 8.4 The r a d i a l v e l o c i t i e s 290 8.5 The l i n e - p r o f i l e v a r i a t i o n s 298 8.6 D i s c u s s i o n 303 BIBLIOGRAPHY 305 INDEX 326 x i L i s t of Tables 2. 01 The UBC-built 1872-Reticons 54 3. 01 Publ i s h e d molecular co n s t a n t s f o r HF 87 3. 02 Adopted molecular c o n s t a n t s f o r the (3-0) band 91 3. 03 Vacuum wavenumbers f o r the (3-0) band of HF 91 3. 04 SSTP wavelengths f o r the (3-0) band of HF 93 3. 05 The a ^ s and the M.'s f o r HF 98 3. 06 Herman-Wallis f a c t o r s f o r the (3-0) band of HF 100 3. 07 Lin e s t r e n g t h s of the (3-0) band of HF 105 3. 08 Linewidths of the (1-0) band of HF 125 3. 09 Linewidths of the (2-0) band of HF 125 3. 10 Linewidths of the (3-0) band of HF 126 3. 1 1 Li n e s h i f t s of the (3-0) band of HF 1 29 3. 1 2 S h i f t - c o r r e c t e d wavenumbers and wavelengths 131 3. 1 3 Lin e s h i f t s f o r the R-branch of the (3-0) band 131 5. 01 Parameters f o r 20 CVn 190 5. 02 Mid-exposure times f o r the 20 CVn s p e c t r a 1 92 5. 03 R e l a t i v e r a d i a l v e l o c i t i e s of 20 CVn 202 6. 01 Parameters f o r p Pup 213 6. 02 Mid-exposure times and exposures f o r p Pup 219 6. 03 R e l a t i v e r a d i a l v e l o c i t i e s of p Pup (I) 255 6. 04 R e l a t i v e r a d i a l v e l o c i t i e s of p Pup ( I I ) 257 7. 01 Parameters f o r o 1 E r i 266 7. 02 Mid-exposure times f o r the o 1 E r i s p e c t r a 269 7. 03 R e l a t i v e r a d i a l v e l o c i t i e s of o 1 E r i 274 8. 01 Parameters for /J Cas 285 8. 02 Mid-exposure times f o r the 0 Cas s p e c t r a 289 8. 03 R e l a t i v e r a d i a l v e l o c i t i e s of 0 Cas 295 x i i L i s t of F i g u r e s 2. 01 R e s i d u a l s from incomplete readout 56 2. 02 The p e r s i s t e n c e phenomenon 57 2. 03 Time decay of the p e r s i s t e n c e phenomenon 58 2. 04 S t e l l a r s p e c t r a i n the region of X8700 63 2. 05 F r i n g e p a t t e r n from d e f e c t i v e c e l l window 73 2. 06 Attempts to f l a t - f i e l d the f r i n g e s 74 2. 07 Schematic of a g e n e r a l i s e d HF system 77 3. 01 The (3-0) v i b r a t i o n - r o t a t i o n band of HF 84 3. 02 The temperature dependence of HF l i n e s t r e n g t h s 108 3. 03 The temperature dependence of the l i n e w i d t h s 128 4. 01 The f i r s t d e r i v a t i v e of the HF spectrum 1 52 4. 02 The f i r s t d e r i v a t i v e of the p Pup spectrum 1 53 4. 03 The f i r s t d e r i v a t i v e of the 38 E r i spectrum 154 4 . 04 The e f f e c t of s/n on accuracy 180 4. 05 The e f f e c t of l i n e depth on accuracy 181 4. 06 The e f f e c t of l i n e w i d t h on accuracy 183 4. 07 s/n as a f u n c t i o n of s i g n a l ( i n adcu) 185 5. 01 The 20 CVn spectrum 194 5. 02 The Ca II X8662 v e l o c i t y curve of 20 CVn 198 5. 03 The Fe I X8689 v e l o c i t y curve of 20 CVn 199 5. 04 The mean v e l o c i t y curve of 20 CVn from weak l i n e s 200 5. 05 V e l o c i t y d i f f e r e n c e ( Ca II curve - mean curve ) 204 5. 06 V e l o c i t y d i f f e r e n c e ( mean curve - Fe I curve ) 205 5. 07 V e l o c i t y d i f f e r e n c e ( Fe 1 curve - S i 1 curve ) 206 5. 08 V e l o c i t y d i f f e r e n c e ( Fe I curve - S I curve ) 207 6. 01 The p Pup spectrum 218 6. 02 The unoptimised Ca II X8662 v e l o c i t y curve 223 x i i i 6. 03 The unoptimised Fe I X8689 v e l o c i t y curve 224 6. 04 U n c e r t a i n t i e s i n the Ca II X8662 l i n e p o s i t i o n s 226 6. 05 U n c e r t a i n t i e s in the Fe I X8689 l i n e p o s i t i o n s 227 6. 06 The unoptimised Fe I X8757 v e l o c i t i e s 228 6. 07 The Ca II X8662 l i n e p r o f i l e s and t h e i r r e s i d u a l s 231 6. 08 The Fe I X8689 l i n e p r o f i l e s and t h e i r r e s i d u a l s 232 6. 09 The S i I X8752 and Fe I X8757 l i n e p r o f i l e s 233 6. 10 The S i I X8752 and Fe I X8757 r e s i d u a l s 234 6. 1 1 The S i I and Fe I r e s i d u a l s from BJD2445356 235 6. 1 2 The optimised Ca II X8662 u n c e r t a i n t i e s 244 6. 13 The optimised Fe I X8757 u n c e r t a i n t i e s 245 6. 14 The optimised Ca II X8662 v e l o c i t y curve 246 6. 15 The optimised H I X8750 v e l o c i t y curve 247 6. 16 The optimised Fe I X8689 v e l o c i t y curve 248 6. 17 The mean optimised Fe I v e l o c i t y curve 249 6. 18 The mean optimised S i I v e l o c i t y curve 250 6. 19 The mean optimised S I v e l o c i t y curve 251 6. 20 The optimised Ca II v e l o c i t i e s from the 22nd 252 6. 21 The optimised Ca II v e l o c i t i e s from the 25th 253 6. 22 E r r o r s i n the HF d i s p e r s i o n f i t s 260 6. 23 V e l o c i t y d i f f e r e n c e ( Ca II curve - Fe I curve ) 261 7. 01 The o 1 E r i spectrum 268 7. 02 The Ca II X8662 v e l o c i t y curve of o 1 E r i 270 7. 03 The H I X8750 v e l o c i t y curve of o 1 E r i 27 1 7. 04 The v e l o c i t y curve of o 1 E r i from weak l i n e s 272 7. 05 The Ca II X8662 l i n e p r o f i l e s 277 7. 06 The r e s i d u a l s of the Ca II X8662 l i n e p r o f i l e s 278 8. 01 The P Cas spectrum 287 x i v 8.02 The Ca II X8662 v e l o c i t y curve of 0 Cas 291 8.03 The H I X8750 v e l o c i t y curve of 0 Cas 292 8.04 The Fe I X8689 v e l o c i t y curve of 0 Cas 293 8.05 The v e l o c i t y curve of 0 Cas from weak l i n e s 294 8.06 U n c e r t a i n t i e s i n the Ca II X8662 l i n e p o s i t i o n s 299 8.07 The Ca II X8662 l i n e p r o f i l e s and t h e i r r e s i d u a l s 301 xv Acknowledgements I would l i k e to thank my sup e r v i s o r Dr. Gordon Walker f o r h i s a s s i s t a n c e , guidance, support, and p a t i e n c e . He i s i n v o l v e d i n every p a r t of the HF p r o j e c t . His p a r t i c i p a t i o n ranges from h i s design of the DAO HF system to performing most i f not a l l the image g u i d i n g at the te l e s c o p e f o r the data d i s c u s s e d i n t h i s t h e s i s . I would a l s o l i k e to thank Dr. Bruce Campbell f o r h i s a s s i s t a n c e . He and Dr. Walker are r e s p o n s i b l e f o r the whole HF p r o j e c t . Dr. Campbell designed, b u i l t , and operated the CFHT HF system with which the data d i s c u s s e d i n t h i s t h e s i s was based. Dr. Campbell a l s o d i s c o v e r e d and sol v e d many of the problems a s s o c i a t e d with the data r e d u c t i o n of the s p e c t r a . I would a l s o l i k e to thank Mr. John Amor and Mr. D i e t e r S c r e i b e r f o r t h e i r a s s i s t a n c e . Mr. John Amor has been r e s p o n s i b l e some of the d a t a - r e d u c t i o n techniques d i s c u s s e d i n t h i s t h e s i s while Mr. D i e t e r S c r i e b e r b u i l t the DAO HF a b s o r p t i o n c e l l and the c u r r e n t gas-handling systems f o r both DAO and CFHT. I would a l s o l i k e to express my a p p r e c i a t i o n to Mr. Ron Johnson f o r h i s t e c h n i c a l support of the i n s t r u m e n t a t i o n s . I would l i k e to thank Dr. Friedhelm Aubke and h i s graduate students f o r t h e i r h e l p with the DAO HF system. Dr. Aubke has made a v a i l a b l e , on many occa s i o n s , h i s l a b o r a t o r y and performed some of the dangerous tasks of r e p l a c i n g or removing HF from the a b s o r p t i o n - c e l l system. x v i I would l i k e to express my a p p r e c i a t i o n to Drs. Anne U n d e r h i l l , Greg Fahlman, Anthony Merer, Michael Ovenden, Jason Auman, and Harvey Richer f o r t h e i r guidance and h e l p f u l d i s c u s s i o n on t h i s t h e s i s . A l l the f a c u l t y , graduate students, and s t a f f s i n the department have o f f e r e d t h e i r h e l p and encouragement to the HF p r o j e c t and t h i s t h e s i s . I would e s p e c i a l l y l i k e to thank Zoran Ninkov, C h r i s M i l l w a r d , Garry J o s l i n , John N i c o l , D a n i e l T h i b a u l t , P h i l Bennett, Grant H i l l , E n r i c o K i n d l , Peter Whaite, Dennis Crabtree, and Gerry G r i e v e . Many of them have a s s i s t e d with the observing at DAO. Once, Zoran Ninkov c a r r i e d the HF l e c t u r e b o t t l e from Vancouver to V i c t o r i a i n h i s own c a r . U n f o r t u n a t e l y , the b o t t l e was empty by the time we r e c e i v e d i t . I would l i k e t o express my sympathy to John N i c o l who has been l o s i n g one beer f o r every week that I have been working on t h i s t h e s i s . I would l i k e to acknowledge the work of P h i l Bennett on the theory of s p e c t r a l - t y p e v a r i a t i o n s . I would a l s o l i k e to thank Mr. B i l l S i e p who l e t us borrow some of h i s equipment to go on obs e r v i n g t r i p s . I would l i k e to thank Dr. Tad U l r y c h and C o l i n Walker f o r the use of t h e i r power-spectrum-analysis program. I would a l s o l i k e to thank the t e c h n i c a l s t a f f s i n the Ph y s i c s department who, on many o c c a s i o n s , l e a k - t e s t e d our equipment. I would l i k e to thank the Dominion A s t r o p h y s i c a l Observatory i n V i c t o r i a and the Canada-France-Hawaii Telescope C o r p o r a t i o n i n Mauna Kea f o r e x t e n s i v e use of t h e i r f a c i l i t i e s . I would a l s o l i k e to thank the s t a f f s at both DAO and CFHT f o r t h e i r support. Murray F l e t c h e r has xvi i been very h e l p f u l to our ob s e r v i n g at DAO. On numerous o c c a s i o n s , he has to re-coat the red coude t r a i n before our o b s e r v i n g run. He has a l s o p r o v i d e d very h e l p f u l d i s c u s s i o n on the p r o j e c t . Dr. A l l a n Batten of DAO has a l s o provided the l i t e r a t u r e r e f e r e n c e s f o r the s t a r 0 Cas. I would a l s o l i k e to acknowledge the work of Dr. Michael De R o b e r t i s on the b a r y c e n t r i c - c o r r e c t i o n program. I would l i k e to acknowledge my l a t e parents f o r t h e i r support and pa t i e n c e over the y e a r s . I would l i k e to d e d i c a t e t h i s t h e s i s to the memory of them. x v i i i memory of my parents Chapter 1 INTRODUCTION 1.1 THE MEASUREMENT OF STELLAR RADIAL VELOCITIES In 1842, C h r i s t i a n Doppler d e r i v e d formulae r e l a t i n g the ocean wave frequency t h a t i s seen by a s t a t i o n a r y observer who i s on shore and that seen by an observer who i s on a moving s h i p . He noted that the d i f f e r e n c e i n the observed f r e q u e n c i e s depends on the v e l o c i t y of the s h i p . T h i s phenomenon has been commonly known as the Doppler e f f e c t . I f one c o n s i d e r s the d i f f e r e n c e i n the l i g h t frequency as emitted by a s t a r from that a c t u a l l y observed on E a r t h , the Doppler e f f e c t g i v e s : f = f 0 ( 1 - v/c ) (1.1) The term f i s the observed frequency, and f 0 i s the emitted or r e s t frequency. The v e l o c i t y of l i g h t i s c, and v i s the r e l a t i v e v e l o c i t y i n the l i n e of s i g h t or r a d i a l d i r e c t i o n between the observer and the s t a r . T h i s i s g e n e r a l l y c a l l e d the observed r a d i a l v e l o c i t y of the s t a r . I f v i s negative, the frequency i s blue s h i f t e d and the s t a r would be moving towards the observer. I f v i s p o s i t i v e , the frequency i s red s h i f t e d and the s t a r would be moving away from the observer. As an a p p l i c a t i o n of the p r i n c i p l e given by Doppler [1842], F i z e a u [1870] o u t l i n e d the method to measure the motions of c e l e s t i a l b o dies. There were many u n s u c c e s s f u l attempts between 1863 and 1887 to produce a s i n g l e t r u s t w o r t h y v e l o c i t y measurement of any s t a r . The low d i s p e r s i o n instruments a v a i l a b l e then would i n t r o d u c e e r r o r s 1 2 many times l a r g e r than the v e l o c i t i e s to be measured. I t was not u n t i l 1890 that the f i r s t r e l i a b l e v i s u a l l y determined s t e l l a r v e l o c i t i e s were ob t a i n e d . Based on the displacements of the s t e l l a r sodium D l i n e s , the f i r s t s t e l l a r v e l o c i t y measurement was"made on A p r i l 10th 1890 on the b r i g h t s t a r a B o o t i s (Keeler [1894]). In 1905, A l b e r t E i n s t e i n p o s t u l a t e d that the v e l o c i t y of l i g h t i s the same i n a l l v e l o c i t y frames of r e f e r e n c e and d e r i v e d the Doppler e f f e c t of s p e c i a l r e l a t i v i t y : f = f 0 ( 1 - (v/c)cos<* ) / /( 1 - ( v 2 / c 2 ) ) (1.2) The v e l o c i t y of the s t a r i s v while tf> i s the angle between the v e l o c i t y and the observer's l i n e of s i g h t d i r e c t i o n to the s t a r . Equation 1.2 w i l l g ive a non-zero Doppler s h i f t even when 0 i s 90°: The r a d i a l v e l o c i t y of a s t a r can be estimated by comparing the observed frequency of a s t e l l a r s p e c t r a l l i n e a g a i n s t i t s corres p o n d i n g l a b o r a t o r y r e s t frequency. G e n e r a l l y , the t r a n s v e r s e component of a s t e l l a r v e l o c i t y c o u l d c o n t r i b u t e the eq u i v a l e n c e of a few tens of metres per second to the observed Doppler s h i f t . T h e r e f o r e , the v e l o c i t y i n f e r r e d d i r e c t l y from the observed Doppler s h i f t does not represent the t r u e r a d i a l v e l o c i t y of the s t a r . C o r r e c t i o n f o r t h i s t r a n s v e r s e Doppler e f f e c t would have to be made i f the p a r t i c u l a r a p p l i c a t i o n r e q u i r e s the p r e c i s e value of the true r a d i a l s t e l l a r motion. In most a p p l i c a t i o n s , one i s only i n t e r e s t e d i n the r e l a t i v e change in the r a d i a l v e l o c i t i e s ; t h e r e f o r e , the second-order Doppler e f f e c t would be unimportant. 3 The r a d i a l v e l o c i t y of a s t a r has t r a d i t i o n a l l y used the c e n t r e of the Sun as the v e l o c i t y frame of r e f e r e n c e . Hence the observed value has to be c o r r e c t e d f o r the r e l a t i v e motion between the observer and the s o l a r c e n t r e . Moreover, the value f o r f 0 used i n Equations 1.1 and 1.2 i s g e n e r a l l y not the r e s t l a b o r a t o r y v a l u e . An e f f e c t i v e f 0 value which takes i n t o account many minor t e c h n i c a l and p h y s i c a l e f f e c t s i s normally used. These i n c l u d e s p e c t r a l l i n e b l e n d i n g e f f e c t s , s t e l l a r g r a v i t a t i o n a l r e d s h i f t , and d e t a i l e d s t e l l a r photospheric c o n d i t i o n s (Dravins [1982]). Minor z e r o - p o i n t c o r r e c t i o n s may s t i l l need to be a p p l i e d to the measured v e l o c i t i e s i n order to place them on the same system with the r a d i a l - v e l o c i t y standard s t a r s . There are s e v e r a l systems of r a d i a l - v e l o c i t y standard s t a r s . In a d d i t i o n to the I.A.U. system (Pearce [1955]), there are the e a r l i e r L i c k system and the v a r i o u s recent p h o t o e l e c t r i c systems ( F l e t c h e r et a l . [1982], Beavers et a l . [1979], Neese et a l . [1985]). The observed r a d i a l - v e l o c i t y estimate of a s t a r i s e s s e n t i a l l y d e f i n e d by the Doppler e f f e c t d i s p l a y e d by i t s s p e c t r a l l i n e s . But i t may not n e c e s s a r i l y imply a p h y s i c a l motion of the s t a r ' s c e n t r e of g r a v i t y r e l a t i v e t o the observer by the i m p l i e d v e l o c i t y . The Doppler e f f e c t as experienced by the s t e l l a r l i n e s i n v a r i a b l e s t a r s may, i n p a r t , be the r e s u l t of s t e l l a r atmospheric motions caused by s t e l l a r p u l s a t i o n . 4 1.2 CONVENTIONAL RADIAL-VELOCITY TECHNIQUES The c o n v e n t i o n a l d e t e r m i n a t i o n of r a d i a l v e l o c i t i e s i n v o l v e s simply the measurement of s p e c t r a l l i n e p o s i t i o n s as w e l l as the d i s p e r s i o n r e l a t i o n which t r a n s l a t e s these p o s i t i o n s i n t o wavelengths. The d i s p e r s i o n r e l a t i o n i s g e n e r a l l y d e r i v e d from the p o s i t i o n s of r e f e r e n c e s p e c t r a l l i n e s e.g. hollow cathode atomic emission l i n e s . The accuracy of a d e r i v e d d i s p e r s i o n r e l a t i o n depends on how w e l l the r e f e r e n c e s p e c t r a l l i n e s can be used t o remove the v a r i o u s systematic yet u n p r e d i c t a b l e e f f e c t s which can a l s o a l t e r s p e c t r a l l i n e p o s i t i o n s . The most s e r i o u s of these i s the e f f e c t of g u i d i n g e r r o r i n terms of uneven i l l u m i n a t i o n a c r o s s the entrance s l i t of the spectrograph. I n c o n s i s t e n t g u i d i n g can be caused by t r a c k i n g problems of the t e l e s c o p e and unavoidably by the changing atmospheric r e f r a c t i o n and se e i n g . A s l i g h t l y o u t - o f - f o c u s s t e l l a r image on the spectrograph's s l i t w i l l cause nonuniform i l l u m i n a t i o n of the c o l l i m a t o r . T h i s with focus e r r o r in the camera w i l l a l s o cause s p e c t r a l l i n e displacements ( P e t r i e and F l e t c h e r [1967]). Other problems i n c l u d e v a r i a t i o n s i n the d i s p e r s i o n and focus of the spectrograph, m i s c o l l i m a t i o n i n the t e l e s c o p e and spectrograph, as w e l l as i n s t a b i l i t i e s i n the spectrograph and d e t e c t o r . In c o n v e n t i o n a l Coude spectroscopy, the s t e l l a r and comparision beams may f o l l o w s l i g h t l y d i f f e r e n t o p t i c a l paths and the c o l l i m a t o r w i l l be i l l u m i n a t e d d i f f e r e n t l y . Zonal i m p e r f e c t i o n s i n the spectrograph camera m i r r o r w i l l cause v a r i a b l e s p e c t r a l s h i f t s ( T u l l [1969]). The use of emission r e f e r e n c e l i n e 5 sp e c t r a w i l l a l s o s u f f e r from severe i n t e n s i t y dependent displacement of the s p e c t r a l l i n e s i f the i n s t r u m e n t a l p r o f i l e of the spectrograph e x h i b i t s some degree of asymmetry ( G r i f f i n and G r i f f i n [1973]). Reviews of the problems i n accurate c o n v e n t i o n a l d e t e r m i n a t i o n of r a d i a l v e l o c i t i e s are d i s c u s s e d by Prevot [1967], P e t r i e and F l e t c h e r [1967], G r i f f i n and G r i f f i n [1973], Serkowski [1976,1978], Campbell et a l [1981], Campbell [1983], and Campbell and Walker [1985]. The h i g h e s t p r e c i s i o n ever a t t a i n e d i n the measurement of s t e l l a r r a d i a l v e l o c i t y by the c o n v e n t i o n a l photographic method i s about ±l00ms" 1 as r e p o r t e d by P e t r i e and F l e t c h e r [1967]. T h i s was achieved by narrowing the spectrograph's s l i t and hence reducing the g u i d i n g - i n d u c e d e x t e r n a l e r r o r per p l a t e of about ±0.3kms~ 1 to the same value as the i n t e r n a l e r r o r of about ±0.07kms~ 1. Higher p r e c i s i o n can be achieved using a Retico n d e t e c t o r as i n Gray [1983]. P h o t o e l e c t r i c methods using a c r o s s - c o r r e l a t i o n technique between the spectrum and a s p e c t r a l mask at the f o c a l plane of the spectrograph have a c h i e v e d a p r e c i s i o n of about 1100ms" 1. Reviews of these p h o t o e l e c t r i c c r o s s - c o r r e l a t i o n techniques can be found i n van C i t t e r s [1974], G r i f f i n and Gunn [1974], Baranne et a l . [1979], Beavers and E i t t e r [1977], and F l e t c h e r et a l . [1982]. The o b j e c t i v e of v a r i o u s p r e c i s i o n r a d i a l v e l o c i t y methods i s to achieve b e t t e r than ±l00ms~ 1 i n p r e c i s i o n . In f a c t , most methods aim to achieve about 110ms" 1, or i n some cases, 11ms"1 i n p r e c i s i o n . 6 1.3 MODERN PRECISION RADIAL-VELOCITY TECHNIQUES Non-conventional r a d i a l - v e l o c i t y methods can be c l a s s i f i e d i n t o roughly two main c a t e g o r i e s depending on whether a spectrograph i s u t i l i s e d or not. Most of the p r e c i s i o n r a d i a l - v e l o c i t y techniques i n v o l v e imposing wavelength c a l i b r a t i o n marks on the s t e l l a r l i g h t i t s e l f while many others r e l y on the s t a b i l i t y of the instruments over a s h o r t time s c a l e . In almost a l l of these techniques, the o b j e c t i v e i s to measure a c c u r a t e r e l a t i v e v e l o c i t i e s . Dravins [1975] has p o i n t e d out that small s c a l e inhomogeneities i n the s t e l l a r l i n e formation region would l i m i t the the measurement of a b s o l u t e v e l o c i t i e s to about ±500ms" 1. Convective motions would e s p e c i a l l y cause l i n e asymmetries and wavelength s h i f t s (Dravins [1982]). 1.3.1 COUPE OPTICAL FIBRE FEED Heacox [1983,1984] e l i m i n a t e d the g u i d i n g and c o l l i m a t i o n e r r o r s by scrambling the image of the s t e l l a r input to the spectrograph. T h i s was accomplished by f e e d i n g the Coude spectrograph with an o p t i c a l f i b r e . The comparison lamp i s a l s o used at the t e l e s c o p e end of the f i b r e . The goal of t h i s technique i s to achieve a p r e c i s i o n of about tSOrns" 1. 1.3.2 INFRARED HETERODYNE TECHNIQUE The wind v e l o c i t i e s i n the atmosphere of Venus have been measured by i n f r a r e d heterodyne measurement of the C0 2 emission l i n e s at about 950cm" 1 (Betz et a l . [1976]). A 7 review of i n f r a r e d heterodyne d e t e c t i o n can be found in Blaney [1975]. B a s i c a l l y , the i n f r a r e d beam from the te l e s c o p e i s combined with a s t a b l i s e d C0 2 l a s e r . The combined beam i s then focused on a HgCdTe photodiode mixer which generates a d i f f e r e n c e frequency c u r r e n t over the r a d i o frequency bandwidth. The r a d i o frequency c u r r e n t i s then a m p l i f i e d , and c o r r e c t e d f o r the Doppler s h i f t caused by the r e l a t i v e motion between the observer and Venus. The s i g n a l i s then analysed i n t o 40 independent channels whose outputs are m u l t i p l e x e d i n t o a computer. A p r e c i s i o n of 6ms"1 can be achieved a f t e r c o r r e c t i n g f o r v a r i o u s systematic e r r o r s e.g. frequency i n s t a b i l i t y of the C0 2 l a s e r . One of the main disadvantages of t h i s method i s the crowding by t e l l u r i c l i n e s i n the s p e c t r a l r e g i o n such that unblended C0 2 l i n e s are only v i s i b l e at s p e c i f i c r a d i a l v e l o c i t i e s which vary with the b a r y c e n t r i c motion of the observer. 1.3.3 MEASUREMENT OF THE SOLAR OSCILLATIONS V a r i o u s methods have been used to measure the 160 minutes and the 5 minutes p e r i o d p u l s a t i o n of the Sun. These o s c i l l a t i o n s have v e l o c i t y amplitudes of 1ms"1 and 2ms" 1, r e s p e c t i v e l y . Severny et a l . [1976] observed a p o l e - t o - p o l e s t r i p a c r o s s the Sun. A c i r c u l a r p o l a r i s e r i s pl a c e d i n the s o l a r beam i n f r o n t of the spectrograph such that only the l i g h t from the c e n t r a l p o r t i o n of the Sun would become p o l a r i s e d while the l i g h t from the p o l a r rim would remain u n a f f e c t e d . Two p h o t o m u l t i p l i e r s are set on the 8 wings of a m a g n e t i c a l l y i n s e n s i t i v e l i n e . Doppler s h i f t s can then be measured by comparing the d i f f e r e n c e i n the output of the phot'omultipliers at d i f f e r e n t p o l a r i s a t i o n s . The i n t e r n a l accuracy of t h i s technique to compare the inner c i r c u l a r h a l f of the s o l a r d i s k a g a i n s t i t s outer r i n g was re p o r t e d to be about 0.5ms~1. Brooks et a l . [1976] used an o p t i c a l , r e s o n a n t - s c a t t e r i n g method to measure the s o l a r p u l s a t i o n . B a s i c a l l y , an a b s o r p t i o n l i n e i s f i r s t i s o l a t e d with an i n t e r f e r e n c e f i l t e r . The s o l a r l i g h t i s then passed through a p o l a r i s e r and an e l e c t r o - o p t i c a l l i g h t modulator which produces l e f t - or right-handed p o l a r i s a t i o n . The s o l a r l i g h t i s then passed through a temperature c o n t r o l l e d metal vapour c e l l c o n t a i n i n g Sr, Na, or K. The vapour c e l l i s p l a c e d i n a l o n g i t u d i n a l magnetic f i e l d which produces the a' and a* Zeeman components. The s o l a r l i g h t e x c i t e s the o p t i c a l resonance of the vapour and the l i g h t r e - e m i t t e d by the vapour i s measured by c o o l e d p h o t o m u l t i p l i e r s . The i n t e n s i t y of s c a t t e r e d l i g h t i s p r o p o r t i o n a l to the i n c i d e n t l i g h t i n t e n s i t y at the wavelength of the p a r t i c u l a r s h i f t e d Zeeman component. By v a r y i n g the magnetic f i e l d and hence the wavelength of the Zeeman components, one can sample the whole a b s o r p t i o n l i n e p r o f i l e . However, i f onl y the r a d i a l v e l o c i t y i s d e s i r e d , one can simply apply a constant magnetic f i e l d that would p l a c e the two Zeeman components on the wings near the maximum slope of the s o l a r l i n e p r o f i l e . The d i f f e r e n c e i n the i n t e n s i t i e s of the s c a t t e r e d l i g h t as sampled by the a' and a* i . e . the short and long wavelength 9 components, w i l l g i v e the r a d i a l v e l o c i t y of the a b s o r p t i o n l i n e . A p r e c i s i o n of b e t t e r than ± 1 m s _ 1 can be achieved using t h i s or a v a r i a n t of t h i s method. F u r t h e r a p p l i c a t i o n of t h i s method can be found i n Blamont and Roddier [1961], Fossat and Roddier [1971], Grec et a l . [1976], Snider [1970], Grec et a l . [1980], and Brooks et a l . [1978]. The main disadvantage of t h i s method i s that i t cannot be a p p l i e d to the 10 1 1 times f a i n t e r s t e l l a r f l u x s i n c e the d e t e c t a b l e s c a t t e r e d l i g h t would only be a very small f r a c t i o n of the input f l u x . 1.3.4 INFRARED FOURIER TRANSFORM SPECTROMETER F o u r i e r Transform spectrometers have been used to measure p r e c i s i o n r a d i a l v e l o c i t i e s . B a s i c a l l y , the s t e l l a r beam i s d i v i d e d i n t o two beams u s i n g a beam s p l i t t e r or dual entrance a p e r t u r e s . The two beams are l a t e r recombined a f t e r the o p t i c a l path of one of the beams has been v a r i e d . The path d i f f e r e n c e i s g e n e r a l l y v a r i e d c o n t i n u o u s l y with a moving m i r r o r and i s monitored by a l a s e r going through the same o p t i c a l path as the s t a r l i g h t . The r e s u l t a n t i n t e r f e r e n c e p a t t e r n i s e s s e n t i a l l y the F o u r i e r transform of the s t e l l a r spectrum. The s t e l l a r spectrum i t s e l f can be recovered by n u m e r i c a l l y a p p l y i n g the i n v e r s e F o u r i e r t r a n s f o r m to the output. Many of these i n t e r f e r e n c e p a t t e r n s can be added together to achieve the r e q u i r e d s i g n a l - t o - n o i s e r a t i o ( s/n ). Reviews on F o u r i e r spectroscopy have been given by Connes [1970] and Ridgway and B r a u l t [1984]. H a l l et a l . [1979] used the K i t t Peak 10 F o u r i e r spectrometer to observe the CO band (3900-4500cm~ 1) in b r i g h t l a t e type s t a r s . T e l l u r i c l i n e s of CH„, C0 2, H 20, and N 20 were used as wavelength standards. They obtained a r a d i a l - v e l o c i t y p r e c i s i o n of 20ms"1 f o r the K g i a n t a T a u r i . H a l l and H i n k l e [1981] in t r o d u c e d a N 20 a b s o r p t i o n c e l l i n t o the s t e l l a r beam to impose a r t i f i c i a l r e f e r e n c e a b s o r p t i o n l i n e s on the spectrum. They expected the p r e c i s i o n to be b e t t e r than I0ms~ 1. Apparently, even f o r b r i g h t s t a r s , the r e q u i r e d s/n (==400) can be achieved only a f t e r s e v e r a l hours of i n t e g r a t i o n with a 4m t e l e s c o p e . Peery [1978] has used F o u r i e r spectroscopy to study the s t e l l a r atmospheric v e l o c i t y g r a d i e n t i n 19 Psc. V e l o c i t i e s were measured fom the l i n e s of CO, HF, Fe I, and T i I. However, the e f f e c t of l i n e b l e n d i n g has l i m i t e d the p r e c i s i o n t o only about 0.4kms~ 1. 1.3.5 MODIFIED MICHELSON INTERFEROMETER F o r r e s t [1983] and F o r r e s t and Ring [1978] used a m o d i f i e d Michelson i n t e r f e r o m e t e r to search f o r small r a d i a l - v e l o c i t y v a r i a t i o n s i n b r i g h t s t a r s . The temperature s t a b l i s e d i n t e r f e r o m e t e r has a f i x e d path d i f f e r e n c e ; i n f a c t , a r e f e r e n c e l a s e r beam i s used i n c o n j u n c t i o n with a servo c o n t r o l u n i t to maintain the f i x e d path d i f f e r e n c e . A f i b r e o p t i c feed i s a l s o used to scramble the input image. An a b s o r p t i o n l i n e i s o l a t e d by an i n t e r f e r e n c e f i l t e r i s the input s i g n a l f o r the i n t e r f e r o m e t e r . One d e t e c t o r i s used to observe the output i n t e n s i t y of the f r i n g e p a t t e r n while another i s used to observe the i n t e n s i t y of the in p u t . A 11 r a d i a l - v e l o c i t y s h i f t of the a b s o r p t i o n l i n e w i l l produce a change i n the output i n t e n s i t y as f r i n g e s move through the output. The i n t e n s i t y d i f f e r e n c e between the two d e t e c t o r s w i l l then g i v e the phase of the f r i n g e s and hence the r a d i a l v e l o c i t y . The system was used on the 3.8m UKIRT t e l e s c o p e and the 1.5m T e n e r i f e t e l e s c o p e i n a program to search f o r s t e l l a r o s c i l l a t i o n s . A p r e c i s i o n of about 2ms"1 was o b t a i n e d f o r two s t a r s while that of about 20ms"1 was o b t a i n e d f o r twelve other s t a r s . The s t a b i l i t y of the system between n i g h t s and o b s e r v i n g seasons has s t i l l to be t e s t e d . Connes [1983,1984] proposed the use of a s e r v o - c o n t r o l l e d i n t e r f e r o m e t e r with a vacuum e c h e l l e s pectrograph and an image scrambler. A tunable l a s e r r e f e r e n c e d to a s t a b l e l a s e r by beat mixing would be used to monitor the i n t e r f e r o m e t e r . 1.3.6 FABRY-PEROT TECHNIQUES The PEPSIOS spectrometer has a l s o been used to search f o r s t e l l a r o s c i l l a t i o n s (Traub et a l . [1978]) as w e l l as to measure the r e t r o g r a d e wind v e l o c i t i e s on Venus (Traub and C a r l e t o n [1979]). The PEPSIOS system has been d e s c r i b e d by Mack et a l . [1963]. The spectrometer c o n s i s t s of three Fabry-Perot e t a l o n s of d i f f e r e n t s p a c i n g s . The e t a l o n s are l i n k e d i n a s e r i e s to i s o l a t e a s i n g l e order of t r a n s m i t t a n c e by s u p p r e s s i n g a l l the neighbouring p a r a s i t e o r d e r s . The i n c i d e n t s t a r l i g h t i s premonochromatised by an i n t e r f e r e n c e f i l t e r b e fore e n t e r i n g the t r a i n of e t a l o n s . The scanning of the spectrum i s accomplished by v a r y i n g the 12 pressure of the N 2 gas i n each e t a l o n . A co o l e d GaAs p h o t o m u l t i p l i e r i s used as the d e t e c t o r . To d e t e c t short-term r a d i a l - v e l o c i t y v a r i a t i o n s without scanning, the spectrometer i s .set at a h a l f - i n t e n s i t y p o i n t of a l i n e p r o f i l e e.g. the Fe I X6678 l i n e . A change i n the output i n t e n s i t y w i l l imply a r a d i a l - v e l o c i t y change. A p r e c i s i o n of 3ms"1 was achieved f o r a B o o t i s while values l a r g e r than 10ms"1 were found f o r other b r i g h t s t a r s . Reay et a l . [1983a,1983b], Atherton et a l . [1978], and Wells et a l . [1978] d e s c r i b e d a Fabry-Perot s t e l l a r r a d i a l - v e l o c i t y spectrometer that has a p r e c i s i o n of 10ms" 1. The input to the system i s fed by an o p t i c a l f i b r e from the t e l e s c o p e . An i n t e r f e r e n c e f i l t e r i s used to i s o l a t e an a b s o r p t i o n l i n e . The scanning of t h i s e t a l o n system i s accomplished by a s e r v o - c o n t r o l l e d v a r i a t i o n of the gap between the p l a t e s of the Fabry-Perot. In f a c t , the scanning i n t h i s case i s used i n r a p i d chopping mode between the two i n f l e c t i o n p o i n t s of the l i n e p r o f i l e . The change in the output i n t e n s i t y w i l l be i n t e r p r e t e d as a r a d i a l - v e l o c i t y change. The s t a b i l i t y of the system i s supposed to be b e t t e r than X/1000 on time s c a l e of days t o weeks while b e t t e r than X/10000 can be expected on the s h o r t e r time s c a l e of minutes to hours. The system was used on the 1.9m SAAO and the 1.5m IRFC t e l e s c o p e to search f o r r a p i d r a d i a l - v e l o c i t y o s c i l l a t i o n s i n Ap and 5 S c u t i s t a r s . A p r e c i s i o n of about 20ms"1 was achieved f o r the 6 S c u t i v a r i a b l e p Pup while about 4ms~1 was achieved f o r the b r i g h t s t a r Canopus. 13 Serkowski [1972,1976] proposed a p o l a r i m e t r i c method to measure r a d i a l v e l o c i t i e s . The s t a r l i g h t i s f i r s t l i n e a r l y p o l a r i s e d by a Wollaston prism. I t i s then passed through a temperature s t a b l i s e d phase r e t a r d e r which r o t a t e s the plane of p o l a r i s a t i o n . Upon emergence from the r e t a r d e r , the s t e l l a r l i g h t has a steep change i n the p o s i t i o n angle of p o l a r i s a t i o n as a f u n c t i o n of wavelength. Hence the wavelength of each r e s o l u t i o n element i n the spectrum i s encoded with a unique p o s i t i o n angle of p o l a r i s a t i o n . The s t a r l i g h t i s then passed through a r o t a t i n g half-wave p l a t e and a c a l c i t e Wollaston prism. A Fabry-Perot e t a l o n i s used to i s o l a t e the i n d i v i d u a l s p e c t r a l r e s o l u t i o n element while a t i l t i n g e c h e l l e spectrograph i s used to s o r t out the v a r i o u s e t a l o n o r d e r s . The output i s then scanned by a D i g i c o n image tube d e t e c t o r . One of the main disadvantages of t h i s method i s that both the o p t i c a l system and the data r e d u c t i o n s are very complicated. The accuracy i n the wavelength c a l i b r a t i o n i a a l s o dependent on the s t e l l a r f l u x . Serkowski [1977,1978] and Serkowski et a l . [1979a,1979b] m o d i f i e d the p o l a r i m e t r i c r a d i a l - v e l o c i t y spectrometer i n t o a Fabry-Perot r a d i a l - v e l o c i t y spectrometer. The Fabry-Perot i n t e r f e r o m e t e r p r o v i d e s the wavelength c a l i b r a t i o n s i n c e the wavelengths of the t r a n s m i t t e d maxima depend only on the s p e c i f i c a t i o n s of the e t a l o n . The s t a r l i g h t passes through a r o t a t i n g r e v e r s i o n prism before r e a c h i n g the Fabry-Perot. T h i s e l i m i n a t e s the dependence of r a d i a l v e l o c i t i e s on the p o s i t i o n of the s t a r 1 4 in the entrance a p e r t u r e . A l a t e r v e r s i o n of the spectrometer uses an o p t i c a l f i b r e image scrambler i n s t e a d of the r o t a t i n g r e v e r s i o n prism. The image scrambler w i l l a l s o p r o v i d e i d e n t i c a l i l l u m i n a t i o n when l i g h t from a c a l i b r a t i o n lamp, ra t h e r than a s t a r , i s i n c i d e n t on the Fab r y - P e r o t . The output from the Fabry-Perot i s then d i s p e r s e d by an e c h e l l e spectrograph such that the output spectrum c o n s i s t s of rows of b r i g h t s p o t s . The scanning of the spectrum i s r e a l i s e d by t i l t i n g the Fabry-Perot i n t e r f e r o m e t e r hereby changing the wavelength of the t r a n s m i t t e d maxima. An Fe-Ar hollow cathode source was used i n the e a r l i e r v e r s i o n of the spectrometer to c a l i b r a t e the t i l t s of the Fabry-Perot. With the image scrambler i n the l a t e r v e r s i o n of the system, an N0 2 a b s o r p t i o n c e l l i l l u m i n a t e d by a lamp i s "used i n s t e a d . A D i g i c o n image tube was o r i g i n a l l y used as the d e t e c t o r . An a r r a y of 42x342 p i x e l s i n t e n s i f i e d c h a r g e - i n j e c t i o n d e v i c e (CID) was used i n s t e a d i n a l a t e r v e r s i o n of the spectrometer. I t can re c o r d e i g h t e c h e l l e o r d e r s s i m u l t a n e o u s l y . The spectrum covers about 260A and i s c e n t r e d at about X4230. The spectrum i s then r e c o n s t r u c t e d from the i n t e n s i t i e s of the w a v e l e n g t h - c a l i b r a t e d Fabry-Perot maxima. The most recent v e r s i o n of the spectrometer uses a charge-coupled d e v i c e (CCD) of 512x320 p i x e l s as the d e t e c t o r and can r e c o r d eighteen e c h e l l e orders s i m u l t a n e o u s l y (Smith et a l . [1983], McMillan [1984]). A p r e c i s i o n of 5ms"1 i s expected f o r t h i s l a t e s t v e r s i o n of the spectrometer. I n i t i a l t e s t with d i s k i n t e g r a t e d s u n l i g h t has achieved a p r e c i s i o n of about 6ms"1 1 5 while short exposures of A r c t u r u s with a small t e l e s c o p e gave a p r e c i s i o n of 40ms~ 1. 1.3.7 USE OF TELLURIC LINES G r i f f i n and G r i f f i n [1973] proposed to use t e l l u r i c a b s o r p t i o n l i n e s as wavelength r e f e r e n c e s . Since the t e l l u r i c l i n e s are imposed on the s t e l l a r spectrum before the s t a r l i g h t e n t e r s the spectrograph, most of the problems i n c o n v e n t i o n a l r a d i a l - v e l o c i t y methods can be e l i m i n a t e d . The poor photometric c h a r a c t e r i s t i c s of photographic emulsions have l i m i t e d the p r e c i s i o n of the r e s u l t i n G r i f f i n and G r i f f i n [1973], Modern e l e c t r o n i c d e t e c t o r s , e.g. the Re t i c o n s , have been used with the t e l l u r i c - a b s o r p t i o n - l i n e technique to measure the wind v e l o c i t i e s on Venus (Young et a l . [1979], Barker et Cochrane [1980]), and to search f o r s t e l l a r o s c i l l a t i o n s i n the b r i g h t s t a r s a T a u r i and a B o o t i s (Smith [1982a,1983]). The most common set of t e l l u r i c l i n e s used i s the 0 2 band at X6300. One disadvantage of the t e l l u r i c - a b s o r p t i o n - l i n e technique i s that the t e l l u r i c spectrum ( 0 2, H 20 ) i s not s t a b l e enough as a wavelength c a l i b r a t o r i f very h i g h p r e c i s i o n i s r e q u i r e d . Weak l i n e s from other a b s o r p t i o n bands and v a r i o u s i s o t o p i c s p e c i e s of the elements may o v e r l a p the re f e r e n c e a b s o r p t i o n bands to produce a r a t h e r c o m p l i c a t e d spectrum. The wavelengths as w e l l as the s t r e n g t h s of d i f f e r e n t t e l l u r i c l i n e s w i l l a l s o vary d i f f e r e n t l y under d i f f e r e n t atmospheric c o n d i t i o n s . 0 2 l i n e s 1 6 w i l l grow with z e n i t h d i s t a n c e while the H 20 l i n e s w i l l a l s o grow with humidity. The bl e n d i n g of t e l l u r i c l i n e s with weak s t e l l a r l i n e s and t e l l u r i c l i n e s as w e l l as the b l e n d i n g of s t e l l a r l i n e s with weak t e l l u r i c l i n e s cause f i c t i t i o u s l i n e s h i f t s depending on the t o p o c e n t r i c v e l o c i t y of the s t a r s and atmospheric c o n d i t i o n s . T h i s e f f e c t has been demonstrated by Campbell [1983]. Changing atmospheric pressure w i l l a l s o a f f e c t the wavelength of the r e f e r e n c e t e l l u r i c l i n e s . Campbell and Walker [1979] have r e p o r t e d pressure induced s h i f t s of more than 10" 2 angstrom f o r the A-band 0 2 l i n e s . The b l e n d i n g problems caused by t e l l u r i c l i n e s can probably be reduced by n u m e r i c a l l y removing the t e l l u r i c l i n e spectrum with a t e l l u r i c l i n e spectrum o b t a i n e d by obs e r v i n g , say, an e a r l y type s t a r under the same observing c o n d i t i o n s . S i m i l a r l y , the b l e n d i n g problems caused by the s t e l l a r l i n e s can be reduced by n u m e r i c a l l y removing the s t e l l a r l i n e s with a 'cl e a n e r ' s t e l l a r spectrum. N e v e r t h e l e s s , the numerical removal of the blends i s d i f f i c u l t and can never be p e r f e c t i . e . one can only reduce the e f f e c t but not e l i m i n a t e the problem. The t e l l u r i c a b s o r p t i o n l i n e technique should s t i l l be u s e f u l f o r short time s c a l e c a l i b r a t i o n with an u l t i m a t e p r e c i s i o n of about 10ms"1 under e x c e l l e n t atmospheric c o n d i t i o n s . 1.3.8 IMPOSING ARTIFICIAL CALIBRATION LINES Cochrane et a l . [1982] and Cochrane [1984] d e s c r i b e d a method to impose a r t i f i c i a l c a l i b r a t i o n l i n e s on the 1 7 spectrum before the s t a r l i g h t e n t e r s the spectrograph. The a r t i f i c i a l a b s o r p t i o n l i n e s , i n t h i s case, are formed by two t e m p e r a t u r e - s t a b l i s e d Fabry-Perot e t a l o n s i n r e f l e c t i o n . An o p t i c a l f i b r e scrambler i s again used here to completely scramble the input image. T h i s e l i m i n a t e s the v a r i a t i o n s i n the i l l u m i n a t i o n caused by g u i d i n g or seeing v a r i a t i o n s . The scrambled s t a r l i g h t i s f i r s t d i v i d e d i n t o two beams. Each beam f a l l s on a separate Fabry-Perot e t a l o n . Each Fabry-Perot t r a n s m i t s the i n t e r f e r e n c e maxima which are absorbed by a black background while r e f l e c t i n g back the r e s t of the beam. The spectrum i n each r e f l e c t e d beam c o n t a i n s imposed a b s o r p t i o n minima caused by the l o s s of the t r a n s m i t t e d maxima. The two r e f l e c t e d beams are then recombined to form the output beam. Both F a b r y - P e r o t s are i d e n t i c a l except that one i s t i l t e d such that the t r a n s m i t t e d maxima are d i s p l a c e d i n wavelength by h a l f the f r e e s p e c t r a l range of the Fabry-Perot. Consequently, the output spectrum c o n t a i n s a b s o r p t i o n minima separated from each other i n wavelength by h a l f the f r e e s p e c t r a l range of the Fabry-Perot while the depth of each a b s o r p t i o n minimum i s only 50% of the continuum. The d e t e c t o r to be used i s an Oc t i c o n which i s e i g h t 1872-Reticons p l a c e d end-to-end. The spectrum covers about 1 500A and i s c e n t r e d at about X6820. The a v a i l a b i l i t y of s e v e r a l hundred s t e l l a r and r e f e r e n c e l i n e s i n the wide bandpass w i l l i n c r e a s e the p r e c i s i o n of the r e s u l t i f the e r r o r i n each l i n e measurement i s randomly d i s t r i b u t e d . The expected p r e c i s i o n of the technique i s 1ms"1. 18 An a b s o r p t i o n c e l l f i l l e d with hydrogen f l u o r i d e (HF) gas has been used to impose a r t i f i c i a l r e f e r e n c e a b s o r p t i o n l i n e s on the spectrum before the s t a r l i g h t e n t e r s the s p e c t r o g r a p h . The generated a b s o r p t i o n l i n e s at about X8700 are s t a b l i s e d by c o n t r o l l i n g the temperature and p r e s s u r e of the HF gas. D e t a i l s of the HF a b s o r p t i o n c e l l technique are d e s c r i b e d i n l a t e r c h a p t e r s of t h i s t h e s i s . Yang et a l . [1982] and Yang and Walker [1983] used the technique to measure the r e l a t i v e r a d i a l - v e l o c i t y v a r i a t i o n s of 8 S c u t i v a r i a b l e s . Irwin and Campbell [1983] a p p l i e d the technique to search f o r o s c i l l a t i o n s i n b r i g h t s t a r s . Campbell and Walker have a l s o been a p p l y i n g the HF technique i n the search f o r low-mass companions ( e x t r a - s o l a r p l a n e t s ? ) to s o l a r - t y p e s t a r s (Campbell [1983], Walker et a l . [1984], Campbell and Walker [1985], Campbell et a l . [1985]). The i n t e r n a l p r e c i s i o n of the technique i s about ±6ms" 1 f o r data o b t a i n e d on the same ni g h t (Irwin and Campbell [1983]). T h i s value was obtained from the s c a t t e r i n the measured v e l o c i t i e s of a time s e r i e s of a Boo s p e c t r a . The p r e c i s i o n f o r data observed over n i g h t s and observing seasons i s about ±12ms~ 1 (Campbell and Walker [1985]). T h i s value was d e r i v e d from the s c a t t e r i n the measured r e l a t i v e r a d i a l - v e l o c i t y curve of T C e t i . 1.4 DELTA SCUTI VARIABLES 19 1.4.1 INTRODUCTION The D e l t a S c u t i or U l t r a s h o r t - P e r i o d Cepheid v a r i a b l e s are one group of v a r i a b l e s t a r s d i s t i n g u i s h e d by p u l s a t i o n p e r i o d s of l e s s than 0.3 day and s p e c t r a l type A or F. The l i g h t amplitude f o r t h i s group of v a r i a b l e s ranges from s e v e r a l thousandths of a magnitude to 0.8 magnitude i n V. The l a r g e - a m p l i t u d e ( >0.3 magnitude ) v a r i a b l e s are g e n e r a l l y r e f e r r e d t o as RRs s t a r s , dwarf Cepheids, or AI V e l v a r i a b l e s while the small-amplitude 5 S c u t i v a r i a b l e s have a t y p i c a l V amplitude of about 0.05 magnitude. The r a d i a l - v e l o c i t y amplitude i s g e n e r a l l y l e s s than 10 kms" 1. The 5 S c u t i i n s t a b l i t y s t r i p i s a lower l u m i n o s i t y e x t e n s i o n of the Cepheid i n s t a b i l i t y s t r i p . I t ranges from j u s t below the main sequence to about 2.5 magnitudes above i t . The blue edge of the s t r i p has been determined by Breger [1977] to be 8000K on the ZAMS and 8400K at a b s o l u t e v i s u a l magnitude of 0.65. The red edge i s at 7500K on the ZAMS and 6950K at ab s o l u t e v i s u a l magnitude of 1.7. 5 S c u t i s t a r s have been found i n open s t a r c l u s t e r s e.g. the a P e r s e i c l u s t e r (Slovak [1978]). About one-quarter to o n e - t h i r d of the s t a r s i n s i d e the i n s t a b i l i t y s t r i p have been found to be v a r i a b l e with amplitude g r e a t e r than 0.01 magnitude. T h i s suggests that the 5 S c u t i p u l s a t i o n i s a common and normal phenomenon. In f a c t , a f t e r the white dwarf p u l s a t o r s ( ZZ C e t i s t a r s ), the 5 S c u t i s t a r s are the most numerous type of p u l s a t o r s i n the Galaxy. 6 S c u t i v a r i a b l e s are u s u a l l y P o p u l a t i o n I o b j e c t s . They are g e n e r a l l y c o n s i d e r e d to be two-solar-mass P o p u l a t i o n I s t a r s t h a t are j u s t 20 e v o l v i n g o f f the main sequence. There i s a subgroup of the v a r i a b l e s l e d by SX Phe which resembles P o p u l a t i o n II s t a r s in t h e i r kinematics and low metal abundances. In f a c t , s e v e r a l v a r i a b l e s are probably members of the g l o b u l a r c l u s t e r to Cen or are o b j e c t s i n the g a l a c t i c halo ( F r o l o v and Irkaev [1984a]). 1.4.2 MAIA SEQUENCE ? Struve [1955] has proposed the e x i s t e n c e of a sequence of Maia (20 Tau) v a r i a b l e s which would f i l l the space i n the HR diagram between the (5 Cep and 5 Set s t a r s . The e x i s t e n c e of n o n r a d i a l p u l s a t i o n s as c o o l as B5 (Smith [1977]) has in c r e a s e d the s p e c u l a t i o n that n o n r a d i a l p u l s a t i o n may be q u i t e common f o r a l l B and A s t a r s . At the present time, the e x i s t e n c e of the Maia v a r i a b l e s i s s t i l l not c o n c l u s i v e (Breger [1979]). S t u d i e s of v a r i a b i l i t i e s have c o n c e n t r a t e d on 20 Tau ( B 7 I I I ) , 7 UMi ( A 3 I I - I I I ) , 7 CrB (AOIV), r Peg (A5IV), and 0 Ser (A2V). One of the d i f f i c u l t i e s l i e s i n the f a c t t hat the s t a r s may only o c c a s i o n a l l y show d e t e c t a b l e photometric v a r i a t i o n s e.g. the case of 7 CrB (Percy [1970], T i p p e t s and Wilcken [1970], Veto and Kovacs [1981], Veto [1983]). T h i s i s a l s o the s i t u a t i o n with another probable c a n d i d a t e , a Lyr (A0V). A recent review on the v a r i a b i l i t y of Vega has been given by Kholopov [1984]. 1.4.3 6 DELPHINI ANOMALIES Some 5 S c u t i v a r i a b l e s a l s o have 6 Del type anomalies while not a l l 6 Del s t a r s are 8 S c u t i v a r i a b l e s . D e l t a 21 D e l p h i n i s t a r s are subgiant or g i a n t s t a r s of s p e c t r a l type A or F. The degree of m e t a l l i c i s m v a r i e s among the 6 D e l p h i n i s t a r s but g e n e r a l l y , they are underabundant i n Ca and Sc r e l a t i v e to Fe while Sr, Y, Zr, and the r a r e earths are overabundant. There may a l s o be a marginal underabundance i n T i , V, and Cr r e l a t i v e to Fe. Because of the s i m i l a r i t i e s i n m e t a l l i c i s m and other p r o p e r t i e s , Eggen [1976] and Kurtz [1979] have suggested that the 6 Del s t a r s are evolved Am s t a r s . The abnormal abundances i n Am s t a r s have g e n e r a l l y been e x p l a i n e d by the d i f f u s i o n theory ( V a u c l a i r and V a u c l a i r [1982]). In a q u i e s c e n t and slowly r o t a t i n g s t a r , the helium i n the He II i o n i s a t i o n zone may disappear due to i t s tendency to s e t t l e downward g r a v i t a t i o n a l l y . T h i s d e s t r o y s the He II c o n v e c t i o n which would normally produce the mixing of elements. Consequently, under the i n f l u e n c e of downward g r a v i t a t i o n a l s e t t l i n g and upward r a d i a t i o n p r e s s u r e , some elements w i l l r i s e while others w i l l sink i n the outer s t e l l a r p h o t o s p h e r i c l a y e r s . T h i s e x p l a i n s the observed under- and overabundance of elements. M e t a l l i c i s m , i n f a c t , can occur as soon as 10 3 years a f t e r the disappearance of the He II c o n v e c t i o n ( V a u c l a i r [1976]). '1.4.4 COEXISTENCE OF PULSATION AND METALLICISM The d e p l e t i o n of helium from the He II i o n i s a t i o n zone however a l s o i m p l i e s that the s t a r w i l l not p u l s a t e . T h i s agrees very w e l l with the o b s e r v a t i o n a l f a c t t h at c l a s s i c a l Am s t a r s do not p u l s a t e . S e v e r a l marginal Am s t a r s have been 22 found to p u l s a t e (Kurtz [1978]); i n f a c t , the marginal Am s t a r HR3321 has been found to be a 6 S c u t i v a r i a b l e with a p e r i o d of 55 minutes and an amplitude of 0.006 magnitude (Kurtz [1984]). S i m i l a r l y , the e x i s t e n c e of the p u l s a t i n g Am-like 5 Del s t a r s poses a problem f o r the t h e o r e t i c a l i n t e r p r e t a t i o n which has to e x p l a i n the c o e x i s t e n c e of both p u l s a t i o n and m e t a l l i c i s m . V a l t i e r et a l . [1979], S t e l l i n g w e r f [1979], Cox et a l . [1979b], and Saez et a l . [1981] have suggested that m e t a l l i c i s m and p u l s a t i o n can c o e x i s t f o r the 6 Del s t a r s which are more evolved than the c l a s s i c a l Am s t a r s . With m i l d m e t a l l i c i s m , there w i l l be enough d r i v e f o r p u l s a t i o n to occur from enhanced H as w e l l as r e s i d u a l He due to incomplete s e t t l i n g or recent upward mixing. 1.4.5 MODELS OF 6 SCUTI STARS 6 S c u t i p u l s a t i o n i s d r i v e n mainly by the He II i o n i s a t i o n zone at 5.2x10 4K and p a r t l y by the H i o n i s a t i o n zone at 1.2X10"K. S t e l l i n g w e r f [1979] p o i n t e d out that there i s another minor d r i v i n g zone at 1.5X10 5 K caused by the c o i n c i d e n c e of maximum photon f l u x with the frequency of the second helium i o n i s a t i o n edge. P u l s a t i o n amplitudes, p e r i o d r a t i o s , p u l s a t i o n c o n s t a n t s , and other observables are o f t e n computed from the t h e o r e t i c a l models f o r comparision with the observed v a l u e s . A wide range of models with s t e l l a r mass ranging from 0.2 to 3.0 s o l a r masses and the composition Z (the content of elements h e a v i e r than He) ranging from 0.01 to 0.001 have been claimed to agree with 23 the o b s e r v a t i o n s . N o n l i n e a r approaches i n the t h e o r e t i c a l models are o f t e n r e q u i r e d f o r the 5 S c u t i s t a r s . Current r e s u l t s , however, depend very much on the treatment of the o p a c i t i e s , the a r t i f i c i a l v i s c o s i t y parameters ( i . e . the d i s s i p a t i o n mechanisms), and other input p h y s i c s ( S t e l l i n g w e r f [1980], Percy [1980]). Some of the c u r r e n t t h e o r e t i c a l problems i n v o l v e f i n d i n g the mechanisms which govern the p u l s a t i o n amplitudes and the p a r t i c u l a r e x c i t e d modes ( r a d i a l and n o n r a d i a l ) , as w e l l as why some modes appear and d i s a p p e a r . For example, F i t c h [1980] has suggested that multimode p u l s a t i o n may be p r e f e r e n t i a l l y e x c i t e d i n s t a r s where there i s a resonance between r a d i a l and n o n r a d i a l modes. One of the problems i s to f i n d out why some s t a r s i n the i n s t a b i l i t y s t r i p are not v a r i a b l e s . A n t o n e l l o [1982] has suggested t h a t the s t a r s i n the lower p a r t of the i n s t a b i l i t y s t r i p are p u l s a t o r s i f they have a s u f f i c i e n t l y high r o t a t i o n a l v e l o c i t y or i f they are i n an a p p r o p r i a t e b i n a r y system. Other problems i n c l u d e a b e t t e r d e t e r m i n a t i o n of the edges of the i n s t a b i l i t y s t r i p s as w e l l as the r o l e of r o t a t i o n and d i f f u s i o n on the v a r i a b i l i t i e s . Recent d i s c u s s i o n s on the t h e o r e t i c a l aspects of 5 S c u t i s t a r s can be found i n Peterson [1975], S t e l l i n g w e r f [1979], Cox et a l . [1979a,1979b], S t e l l i n g w e r f [1980], Cox and Hodson [1980], F i t c h [1980], Dziembowski [1980], Kurtz[1980], Andreasen et a l . [1980], Percy [1980], F i t c h [1981], Tsvetkov [I982ab], Andreasen et a l . [1983], and Cox [1984b], The most recent e f f o r t to compare between the t h e o r e t i c a l and o b s e r v a t i o n a l p r o p e r t i e s of both 24 high-amplitude and low-amplitude 6 S c u t i v a r i a b l e s can be found i n Andreasen [1983]. I t a l s o i n c l u d e s a d i s c u s s i o n on the high-amplitude metal-poor, m e t a l - r i c h , and double-mode 5 S c u t i s t a r s . 1.4.6 PERIOD-LUMINOSITY-COLOUR RELATION There i s a p e r i o d - l u m i n o s i t y - c o l o u r r e l a t i o n (PLC) f o r the 8 S c u t i s t a r s . Breger [1979] g i v e s the f o l l o w i n g r e l a t i o n : M y = -3.052 l o g P - 8.456 ( b - y ) - 3.121 (1.3) Other d e t e r m i n a t i o n s of the PLC r e l a t i o n i n c l u d e those by Breger and Bregman [1975], A n t o n e l l o and Conconi [1982], and F r o l o v and Irkaev [1984b]. In f a c t , PLC r e l a t i o n s are given f o r each r a d i a l p u l s a t i o n mode by Gupta [1978] and Tsvetkov [ 1 982b]. The corres p o n d i n g r e l a t i o n s between a n c * l o g T ^ are a l s o g i v e n . The p e r i o d - g r a v i t y r e l a t i o n of the 8 S c u t i s t a r s g i v e s a slope between 0.030 day and 0.035 day. T h i s agrees w e l l with the expected r a d i a l fundamental mode p u l s a t i o n constant Q 0 of 0.035 day (Breger [1980b]). 1.4.7 LIGHT AND VELOCITY AMPLITUDES A n t o n e l l o et a l . [19813 have found a r e l a t i o n s h i p between the v i s u a l l i g h t amplitude and the narrow band c o l o u r s of the 8 S c u t i s t a r s : logAm v = 2.06(8c,)oM v + 7 . 5 4 ( 6 c , ) 0 ( b - y ) 0 " 1.91 (1.4) The term Amv i s the peak-to-peak v i s u a l l i g h t amplitude. The above r e l a t i o n w i l l be s i m p l i f i e d i f the c o l o u r s are r e p l a c e d by the p e r i o d u sing the PLC r e l a t i o n . 25 There i s a l s o a l i n e a r r e l a t i o n between Amv and the v e l o c i t y amplitude 2K. Breger [1969] has given a value of 92kms" 1mag- 1 f o r IK/hm^. Smith [1982b] has shown that t h i s i s a t y p i c a l value f o r the r a d i a l p u l s a t o r s . For the n o n r a d i a l l y p u l s a t i n g 6 S c u t i s t a r s , a more t y p i c a l value i s about 45kms~^ag" 1. In f a c t , one may be a b l e to use t h i s l a r g e d i f f e r e n c e between the two values to d i s t i n g u i s h between r a d i a l and n o n r a d i a l p u l s a t o r s . The p u l s a t i o n p e r i o d s i n almost a l l 6 S c u t i s t a r s are b e l i e v e d to be s t a b l e i n s p i t e of the complex s t r u c t u r e s found i n the l i g h t and r a d i a l - v e l o c i t y c u r v e s . These are g e n e r a l l y a t t r i b u t e d to the b e a t i n g e f f e c t s from the v a r i o u s e x c i t e d p u l s a t i o n modes. There i s s t i l l , however, a q u e s t i o n about the s t a b i l i t y of the p u l s a t i o n amplitudes. Breger [1982] has proposed that s t e l l a r r o t a t i o n i s r e s p o n s i b l e f o r the small amplitudes in HR6434, an AI V e l - l i k e 6 S c u t i s t a r . 1.4.8 SYSTEMATICA OF 6 SCUTI STARS From the s t a t i s t i c a l d i s t r i b u t i o n of the p u l s a t i o n c onstant Q, A n t o n e l l o and P a s t o r i [1981] found that about 25% of the low-amplitude ( i . e . < 0.3 magnitude) v a r i a b l e s p u l s a t e i n the r a d i a l fundamental mode. Based upon a l l the a v a i l a b l e r e s u l t s on 6 S c u t i s t a r s with determined p u l s a t i o n modes from m u l t i p e r i o d i c a n a l y s e s , Breger [1980b] was able to draw some c o n c l u s i o n about t h e i r g e n e r a l p u l s a t i o n p r o p e r t i e s . S e v e r a l r a d i a l modes are simultaneously e x c i t e d i n many s t a r s . For s t a r s p u l s a t i n g mainly i n the higher 26 r a d i a l o v e r t o n e s , the fundamental and lower over tone modes may a l s o be p r e s e n t . The r a d i a l fundamental mode i s on ly dominant for s t a r s i n the c o o l p a r t of the 5 S c u t i i n s t a b i l i t y s t r i p . N o n r a d i a l p u l s a t i o n can be found in s t a r s over the e n t i r e i n s t a b i l i t y s t r i p . Not a l l s t a r s have n o n r a d i a l modes a l t h o u g h n o n r a d i a l p u l s a t i o n o f t e n c o e x i s t s w i t h r a d i a l modes. The AI V e l s t a r s p u l s a t e ma in ly in the r a d i a l fundamental and the f i r s t over tone mode. Recent rev iews of the p r o p e r t i e s of the 8 S c u t i v a r i a b l e s can be found in Leung [1970] , B a g l i n et a l . [1973] , F r o l o v [1975] , F i t c h [1976] , Eggen [1979] , Breger [1979 ,1980a ,1980b ,1980c] , W o l f f [1983] , and Breger and Stockenhuber [1983] . 1.5 OSCILLATION MODES 1.5.1 MODE CLASSIFICATION The e i g e n f u n c t i o n s which c h a r a c t e r i s e the normal modes of s t e l l a r p u l s a t i o n i n the s p h e r i c a l p o l a r c o o r d i n a t e s (rr6r<p) are p r o p o r t i o n a l to f u n c t i o n s of r a d i a l d i s t a n c e r, the s p h e r i c a l harmonics Y^ ( 0 , < £ ) , and a t empora l dependence e x p ( / a i ) . The f u n c t i o n Y ' ( 0 , 0 ) i s e q u a l to p ' ( cos0 )exp( /m0) w i t h the P^(cos0) be ing the A s s o c i a t e d Legendre P o l y n o m i a l s of the f i r s t k i n d . The terms a are the e i g e n v a l u e s which have for the r e a l p a r t s , the o s c i l l a t i o n a n g u l a r f r e q u e n c i e s . The a n g u l a r quantum numbers, / = 0 , 1 , 2 , . . . , are the o r d e r s of the s p h e r i c a l harmonics w h i l e the a z i m u t h a l quantum numbers, m = - l , . . . , 0 , . . . / , a r e the a z i m u t h a l s p h e r i c a l harmonics i n d i c e s . The ( 2 / + 1 ) - f o l d degeneracy of / 27 i n m can be l i f t e d by the r o t a t i o n of the s t a r . A review of m - s p l i t t i n g has been given by Cox [1984a]. The c o n d i t i o n m>0 corresponds to waves t r a v e l l i n g i n the opp o s i t e d i r e c t i o n to the s t e l l a r r o t a t i o n while m<0 corresponds to waves t r a v e l l i n g i n the same d i r e c t i o n . The /=0 mode corresponds to the s p e c i a l case of r a d i a l p u l s a t i o n . The /=1 mode i s the case of n o n r a d i a l d i p o l e o s c i l l a t i o n while 1=2,3 correspond to quadrupole and octupole o s c i l l a t i o n s , r e s p e c t i v e l y . There are a l s o the r a d i a l quantum numbers k, which d i s t i n g u i s h the number of nodes i n the displacement from the c e n t r e of the s t a r . The k=0 mode corresponds to the fundamental p u l s a t i o n while £=1,2 are the f i r s t and second overtone modes, r e s p e c t i v e l y . T i d a l e f f e c t s on the v a r i a b l e i n b i n a r y o r b i t , e.g. 14 Aur, may a l s o cause h y p e r f i n e s p l i t t i n g s of the p u l s a t i o n modes ( F i t c h and Wisniewski [1979]). There are four types of s p h e r o i d a l n o n r a d i a l o s c i l l a t i o n s ( p , f , g + , g ~ ) . Each type of o s c i l l a t i o n corresponds to an i n f i n i t e d i s c r e t e spectrum of the eigenvalues a2. The p-modes are the pressure ( a c o u s t i c ) modes with the r e s t o r i n g f o r c e due to pre s s u r e v a r i a t i o n s . The propagation zones of the p-modes are g e n e r a l l y s i t u a t e d i n the envelope of the s t a r . The a2 v a l u e s of these modes inc r e a s e with i n c r e a s i n g k (the number of r a d i a l nodes) .and / v a l u e s . The p-modes u s u a l l y have mainly v e r t i c a l motions as w e l l as l a r g e pressure and d e n s i t y v a r i a t i o n s . The valu e s of a2 are g e n e r a l l y l a r g e f o r the p-modes i n comparison with the g-modes. The g or g r a v i t y modes have g r a v i t y (buoyancy) as the r e s t o r i n g f o r c e . The propagation zones of the g-modes 28 are g e n e r a l l y s i t u a t e d i n the i n t e r i o r of the s t a r . For a g i v e n / v a l u e , the a2 v a l u e s of the g-modes approach 0 as k i n c r e a s e s . The g-modes u s u a l l y have main ly h o r i z o n t a l motions as w e l l as s m a l l p r e s s u r e and d e n s i t y v a r i a t i o n s . For the g + -modes , a2 are p o s i t i v e w h i l e a2 are n e g a t i v e for the g~-modes. The c o n d i t i o n a2<0 i m p l i e s the modes are d y n a m i c a l l y u n s t a b l e . The e i g e n f u n c t i o n s of the g + -modes are o s c i l l a t o r y i n the r a d i a t i v e r e g i o n s of the s t a r w h i l e the g~-modes can be o s c i l l a t o r y o n l y i n the c o n v e c t i v e l y u n s t a b l e r e g i o n s . The f or K e l v i n modes e x i s t o n l y w i t h the lowest k v a l u e ( i . e . no r a d i a l node) and f o r l>2. The e f f e c t s of the f-modes on the p r e s s u r e and d e n s i t y v a r i a t i o n s as w e l l as t h e i r a2 v a l u e s are i n t e r m e d i a t e between those of the p - and g-modes. In s imple homogeneous s t e l l a r models , the f-modes can be c o n s i d e r e d as s u r f a c e waves on a f l u i d s p h e r e . T o r o i d a l modes may a l s o be p r e s e n t f o r a r o t a t i n g p u l s a t o r . P a p a l o i z o u and P r i n g l e [1978] have g i v e n the r-modes d e s i g n a t i o n to a c l a s s of t o r o i d a l modes i n s l o w l y r o t a t i n g s t a r s . They are R o s s b y - l i k e waves d r i v e n by C o r i o l i s f o r c e s . T h e i r motions are h o r i z o n t a l ( v o r t e x - l i k e ) and c a r r y no i n f o r m a t i o n from one a tmospher ic depth to another (Smith [1981] , S a i o [1982] ) . The r a d i a l d i s p l a c e m e n t %(r,t) due to s p h e r o i d a l n o n r a d i a l o s c i l l a t i o n s can be expres sed as : i(r ,t ) = ( lf, Jie, £ 0 ) e x p ( / t n ) ir = a ( r ) P^(cos0) exp(/m0) (1 .5) (1 .6) ir. = b ( r ) d(P' (cos0) )/dd exp(/m0) (1 .7) 29 = imb(r) P (cos0) expdmcj)) / sin0 (1.8) air) = <//(')/r2 (1.9) bir) = ( d x / d r ) / ( / ) r ) (1.10) In a r o t a t i n g s t a r with angular frequency fl, o = a 0 -mnc where a 0 i s the eigenfrequency i n the n o n - r o t a t i n g case. The constant C i s given i n Cox [1980] and i s a f u n c t i o n of both the s t e l l a r s t r u c t u r e and the p u l s a t i o n mode. The f u n c t i o n s air) and bir) are the r a d i a l p a r t of the displacement i n the r a d i a l and h o r i z o n t a l d i r e c t i o n , r e s p e c t i v e l y . The f u n c t i o n s i//(r) and x ( r ) are given in Ledoux and Walraven [1958]. For the p- and f-modes, they can be approximated by power s e r i e s e.g. \j/ir) = r l + ] L a k r 2 k and xir) = r l + ] L b k r 2 k (Clement [1984]). In most models with a t h i n p u l s a t i n g envelope e.g. Osaki [1971] and Kubiak [1978], they are t r e a t e d as c o n s t a n t s and with the r a t i o b ( r ) / a ( r ) = (Q/0.116) 2. The v e l o c i t y change i s then simply the time d e r i v a t i v e of Hr,t). Recent d i s c u s s i o n s on the t h e o r i e s of s t e l l a r p u l s a t i o n can be found i n Ledoux and Walraven [1958], C h r i s t y [1966], Cox [1976], P a p a l o i z o u and P r i n g l e [1978], T a s s o u l [1978], Dziewbowski [1979], Unno et a l . [1979], F i t c h and Wisniewski [1979], Cox [1980], S t o t h e r s [1981], Saio [1982], Barranco et a l . [1982], Perdang and B l a c h e r [1982], Buchler and G o u p i l [1984], Aikawa [1984], Clement [1984], and Cox [1984a]. 30 1.5.2 MODE IDENTIFICATION TECHNIQUES 1.5.2.1 P e r i o d r a t i o s Mode i d e n t i f i c a t i o n s f o r 5 S c u t i s t a r s can be accomplished by s e v e r a l methods. U s u a l l y , no s i n g l e method can unambiguously i d e n t i f y a l l the d i f f e r e n t p u l s a t i o n modes present i n the s t a r . The p u l s a t i o n modes of s e v e r a l s t a r s , however, have been determined using a combination of s e v e r a l methods. One of the sim p l e s t and most common methods i s l o o k i n g at p e r i o d r a t i o s . T h e o r e t i c a l p e r i o d r a t i o s have been computed f o r a v a r i e t y of s t e l l a r models and p u l s a t i o n modes (Peterson [1975], S t e l l i n g w e r f [1979], Cox and Hodson [1980], Andreasen et a l . [1980], F i t c h [1980]). Observed p e r i o d s of the v a r i a b l e can be checked f o r c e r t a i n "magic" p e r i o d r a t i o s e.g. P T / P O = 0.76 and P 2 / P 1 = 0.81 f o r r a d i a l p u l s a t i o n . Ambiguities may a r i s e f o r higher overtone r a d i a l p u l s a t i o n s . From t h e o r e t i c a l models, Andreasen et a l . [1983] have found that as the s t a r evolves with a downwards helium d e p l e t i o n from the s t e l l a r outer zones, the p e r i o d r a t i o between the fundamental and f i r s t overtone p u l s a t i o n i n c r e a s e s . Cox [1984b] has p o i n t e d out that the l a r g e observed p e r i o d r a t i o i n VZ Cnc c o u l d be the r e s u l t of such e v o l u t i o n a r y e f f e c t . No s p e c i f i c n o n r a d i a l p u l s a t i o n p e r i o d r a t i o has been determined. G e n e r a l l y , any observed p e r i o d r a t i o which does not agree with any of the s p e c i f i c r a d i a l mode p e r i o d r a t i o s would i n d i c a t e the presence of n o n r a d i a l modes. The ex i s t e n c e of three almost e q u a l l y - s p a c e d f r e q u e n c i e s i n V474 Mon has suggested the presence of n o n r a d i a l modes with 31 m - s p l i t t i n g due to s t e l l a r r o t a t i o n (Shobbrook and St o b i e [1979]). D e t a i l e d a n a l y s i s by Balona and S t o b i e [1980b] has i n d i c a t e d t hat of the three f r e q u e n c i e s , one i s an overtone r a d i a l mode while the other two are n o n r a d i a l d i p o l e modes with r o t a t i o n a l m - s p l i t t i n g . 1.5.2.2 Use of the p u l s a t i o n constant Q The observed p u l s a t i o n constant Q of a v a r i a b l e can be used to d i s t i n g u i s h between o s c i l l a t i o n modes. With the narrow band uvby0 photometry, and knowing the main p u l s a t i o n p e r i o d P, Q can be c a l c u l a t e d u s i n g the f o l l o w i n g formula from Breger and Bregman [1975]: l o g Q = -6.454 + l o g P + 0.5 l o g g +0.1 M b Q l + l o g T e f f (1.11) The q u a n t i t i e s M^ 0 j f T e f f ' a n c ^ ^ c a n ^ e c a l c u l a t e c ^ with the uvby0 i n d i c e s from the c a l i b r a t i o n s of H a r r i s [1963], A l l e n [1973], Crawford [1975,1979], Breger [1977], and P h i l i p and Relyea [1979]. A Q value of 0.033 day w i l l i n d i c a t e fundamental r a d i a l p u l s a t i o n . While i t i s not always the case, Q = 0.025 day may i n d i c a t e f i r s t overtone r a d i a l p u l s a t i o n and Q value s of l e s s than 0.025 day may imply n o n r a d i a l modes. The Q value s f o r v a r i o u s s t e l l a r models and p u l s a t i o n modes have been computed by S t e l l i n g w e r f [1979] and F i t c h [1981]. T h i s method cannot be a p p l i e d to the SX Phe subgroup s i n c e the uvbyj3 c a l i b r a t i o n s are v a l i d only f o r normal s t a r s . 32 1.5.2.3 L i n e p r o f i l e a n a l y s i s L i n e p r o f i l e a n a l y s i s p r o v i d e s means to examine the instantaneous d i s t r i b u t i o n of v e l o c i t i e s a c r o s s the p u l s a t i n g s t e l l a r s u r f a c e . A time s e r i e s of l i n e p r o f i l e s can be used to i d e n t i f y p u l s a t i o n modes e.g. r a d i a l p u l s a t i o n s cause l a r g e r a d i a l - v e l o c i t y v a r i a t i o n s but only small l i n e w i d t h v a r i a t i o n s while n o n r a d i a l p u l s a t i o n s tend to show the o p p o s i t e . The observed time s e r i e s of l i n e p r o f i l e s i s g e n e r a l l y f i t t e d by a time s e r i e s of t h e o r e t i c a l l y generated l i n e p r o f i l e s which have been d i s t o r t e d by a s p e c i f i e d p u l s a t i o n mode. T h i s i s more r e l i a b l e than j u s t f i t t i n g l i n e - p r o f i l e parameters l i k e r a d i a l v e l o c i t i e s , h a l f widths, and e q u i v a l e n t widths. The t h e o r e t i c a l s t e l l a r d i s k i s f i r s t d i v i d e d i n t o many g r i d p o i n t s . The e f f e c t of r a d i a l - v e l o c i t y changes due to p u l s a t i o n s i s then imposed upon the unperturbed i n t r i n s i c l i n e p r o f i l e a s s o c i a t e d with each g r i d p o i n t . The unperturbed p r o f i l e i n each square g r i d i s g e n e r a l l y computed from s t e l l a r model atmospheres. The r a d i a l - v e l o c i t y changes are c a l c u l a t e d from the time d e r i v a t i v e of £(r ,t) f o r a given set of the modal constants (I,m), frequency, and p u l s a t i o n amplitude. The e f f e c t of superimposed modes can be l i n e a r l y added. Each p r o f i l e at each g r i d p o i n t i s then c o r r e c t e d f o r the e f f e c t of s t e l l a r r o t a t i o n , limb darkening, and other effe c t s - . Summation of a l l the per t u r b e d p r o f i l e s from a l l the g r i d p o i n t s over the e n t i r e v i s i b l e s t e l l a r d i s k w i l l then p r o v i d e a t h e o r e t i c a l l i n e p r o f i l e . A time s e r i e s of l i n e p r o f i l e s 33 can be generated by v a r y i n g the <j> v a l u e s i n the e x p r e s s i o n f o r £ { r , t ) . The time parameter t i n t h i s case i s i r r e l e v a n t . T h i s method r e q u i r e s high s/n data as w e l l as long time coverage f o r the time s e r i e s . Due to the l a r g e number of a v a i l a b l e degrees of freedom i n the f i t , m u l t i p e r i o d i c v a r i a b l e s with two or more superimposed modes w i l l pose a problem f o r t h i s method, e s p e c i a l l y i f there are not enough o b s e r v a t i o n a l data. Campos and Smith [1980] and Smith [1982] used t h i s method to model the l i n e p r o f i l e s of the 5 S c u t i v a r i a b l e s . G e n e r a l l y , the i d e n t i f i c a t i o n of e i t h e r r a d i a l or n o n r a d i a l p u l s a t i o n f o r the dominant mode was accomplished. However, the i d e n t i f i c a t i o n of the modal c o n s t a n t s (/,m) as w e l l as the secondary superimposed modes were not c o n c l u s i v e . The method a l s o f a i l e d f o r the v a r i a b l e 20 CVn. In t h i s case, the e f f e c t of the Doppler imaging on the l i n e p r o f i l e i s reduced by the low p r o j e c t e d r o t a t i o n a l v e l o c i t y of the s t a r . 1.5.2.4 Use of simultaneously observed data Dziembqwski [1977], Balona and S t o b i e [1979a,1979b,1980a], Balona [1981], and Stamford and Watson [1981] have d e r i v e d l i n e a r i s e d e x p r e s s i o n s f o r the v a r i a t i o n s i n l i g h t , c o l o u r , and r a d i a l v e l o c i t y of a p u l s a t i n g s t a r . The major assumption i s that the s u r f a c e b r i g h t n e s s v a r i a t i o n s are p r o p o r t i o n a l to the c o l o u r v a r i a t i o n s . For r a d i a l p u l s a t i o n s , Balona and S t o b i e [1979b] obtained e x p r e s s i o n s f o r AL/L, AF/F, and AV r which are the b r i g h t n e s s , c o l o u r , 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 , 3 4 r e s p e c t i v e l y . The e x p r e s s i o n s a r e : AL/L = e|/(f 2 + 4fcos\//+4) c o s ( a * + T ? ) (1.12) AF/F = ef cos(a«+^) (1.13) AV r = 0.708eaR o c o s ( a f - 7 r / 2 ) (1.14) tanr? = f sini///( 2 + f cos^) (1.15) where a = frequency e = semi-amplitude ty = phasing between c o l o u r and v e l o c i t i e s TJ = phasing between l i g h t and v e l o c i t i e s R 0 = e q u i l i b r i u m r a d i u s of the s t a r f = s c a l i n g f a c t o r f o r the amplitude With simultaneous l i g h t , c o l o u r , and r a d i a l - v e l o c i t y data, one can s o l v e f o r the unknowns e, f, ty, R 0, and 77 i n the four equations above. Balona and S t o b i e [1980b] a p p l i e d the above equations to the data of V474 Mon which had been prewhitened to i s o l a t e only the r a d i a l mode i . e . removing the e f f e c t s of the two n o n r a d i a l modes from the d a t a . They were ab l e to o b t a i n a value f o r the r a d i u s of the s t a r . S i m i l a r l y , f o r n o n r a d i a l p u l s a t i o n , Balona and S t o b i e [1979a,1980a] and Balona [1981] have obtained e x p r e s s i o n s f o r AV, AC, and AV r which are the l i g h t , c o l o u r , 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 , r e s p e c t i v e l y . The e x p r e s s i o n s a r e : AV = 1.086e P ^ ( c o s i ) F v cosiat+ty^) (1.16) AC = (1 .086eP^(cosi)F c/A) cos(at+<t>c) (1.17) AV r = a R 0 e P ^ ( c o s i ) F R V cos(ar-ir/2) (1.18) 35 F„sinc6„ = - ( f b . s i n ^ ) (1.19) F„cos</>„ = -(fb,cosi//+(2 + / ) (/-1 )b, ) (1.20) F c s i n 0 c = -(fb^sin*//) F ccosc6 c = -( f b/cosi//+(2 + / ) (/-1 ) (bj-21 ,)) (1.21) (1.22) F RV (2-0)12+1-51 3 + 74.6Q 2(/ (/ + 1 ) / 2 ) ( ( 2 - 0 ) I 2 + /3I3) 0 .23) b (2-/3)1 , + 1 .5/51 2 (1.24) I n j l c o s n 0 P. (cos©) dcost? (1.25) where 0 = the limb-darkening c o e f f i c i e n t 1 = the i n c l i n a t i o n angle ty = phase l a g of the f l u x r e l a t i v e to the r a d i u s A = f a c t o r r e l a t i n g AC with s u r f a c e b r i g h t n e s s P^ = Legendre Polynomial of order / Simultaneous l i g h t , c o l o u r , and r a d i a l - v e l o c i t y data can prov i d e three amplitudes and two phase d i f f e r e n c e s i . e . f i v e q u a n t i t i e s f o r the above equations which have s i x unknowns i n /, f, ty, e, A, and R 0. Some c o n s t r a i n t s , however, can be p l a c e d on the value of / s i n c e i t i s an i n t e g e r . For ty-ti/2 (6 S c u t i s t a r s ) , one can i d e n t i f y a value f o r / from the p o s i t i o n of the data on a p l o t between (0y-0^) and (0 V-<6 R V), which are the phase d i f f e r e n c e between l i g h t and c o l o u r v a r i a t i o n s and the phase d i f f e r e n c e between 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 , r e s p e c t i v e l y . T h e o r e t i c a l l i n e s of constant f and ty can be generated on the p l o t f o r each d i s c r e t e v alue of / using 36 rough estimates of the other unknowns e.g. R 0. For odd values of /, however, the p r o j e c t e d area v a r i a t i o n of the p u l s a t o r i s z e r o . . The l i g h t v a r i a t i o n i s then determined only by the s u r f a c e b r i g h t n e s s v a r i a t i o n s i . e . 0V~<6C=O for a l l odd v a l u e s of /. N e v e r t h e l e s s , one can s t i l l get an estimate f o r / from the phase d i f f e r e n c e p l o t e.g. t/»v—0^,<O i m p l i e s / =0 and c6v~0c>O i m p l i e s I >2. For ty^n, I can be i d e n t i f i e d from the p o s i t i o n of the data on a p l o t between the q u a n t i t i e s F y / F c and F V / F R V which i s s i m i l a r to a p l o t between the c o r r e s p o n d i n g amplitude r a t i o s . T h i s i s a refinement of the l e s s r e l i a b l e method of examining the r a t i o of the v e l o c i t y and l i g h t amplitude f o r mode-typing. Balona et a l . [1980b,1981] have s u c c e s s f u l l y a p p l i e d t h i s e x t e n s i v e mode a n a l y s i s technique to the v a r i a b l e s V474 Mon and 5 S c u t i . Burki and Mayor [1981] a l s o a p p l i e d t h i s mode-typing method to the v a r i a b l e HD37819. One of the main disadvantages of t h i s method of mode-typing i s the l a r g e amount of r e q u i r e d d a t a . Simultaneous data of l i g h t , c o l o u r , and r a d i a l v e l o c i t i e s are normally d i f f i c u l t to o b t a i n . The method i s a l s o best a p p l i e d to only one p a r t i c u l a r p u l s a t i o n mode at a time, hence the data have to be prewhitened to i s o l a t e each mode. T h i s makes the data a n a l y s i s r a t h e r t e d i o u s . Stamford and Watson [1981] r e d e r i v e d a l l the above equations f o r n o n r a d i a l p u l s a t i o n but t a k i n g i n t o account a l s o the v a r i a t i o n s of the e f f e c t i v e g r a v i t y as w e l l as the wavelength dependence i n the limb-darkening f u n c t i o n . They found f o r n o n r a d i a l p u l s a t i o n s , the assumption that the 37 v a r i a t i o n s of s u r f a c e b r i g h t n e s s a r e p r o p o r t i o n a l to the v a r i a t i o n s of c o l o u r i s i n v a l i d . Ba lona [1981] s t u d i e d the e f f e c t of s u r f a c e g r a v i t y v a r i a t i o n s on h i s mode- typ ing method and c o n c l u d e d that the e f f e c t i s too s m a l l in most cases to a f f e c t the r e s u l t . N e v e r t h e l e s s , he generated c o r r e c t i o n f a c t o r s to be used i n c o n j u n c t i o n w i t h h i s mode- typ ing method. In the new d e r i v a t i o n by Stamford and Watson [1981] , one can s t i l l i d e n t i f y / from the (<j>^-<j>c) vs ( 0 V ~ 0 R V ) or the F v / F c vs F V / F R V p l o t , but the t h e o r e t i c a l l i n e s on the p l o t s are d i s t o r t e d from those i n Balona [1981] . Mode d i s c r i m i n a t i o n f o r odd v a l u e s of / i s a l s o p o s s i b l e i n the new d e r i v a t i o n . Stamford and Watson [1981] r e - a n a l y s e d the data on V474 Mon u s i n g the new method but o b t a i n e d a lmost the same r e s u l t as Ba lona and S t o b i e [1980b] . They p o i n t e d out t h a t , even w i t h the new d e r i v a t i o n , mode i d e n t i f i c a t i o n of />3 w i l l be r a t h e r i n s e c u r e when the s u r f a c e r a d i a l - v e l o c i t y v a r i a t i o n s cannot be i n f e r r e d s imply from the v a r i a t i o n s of the mean wavelengths of the l i n e p r o f i l e . The l i m b - d a r k e n i n g f u n c t i o n from c u r r e n t model atmospheres becomes inadequate for h i g h e r v a l u e s of / . They a l s o p o i n t e d out tha t the c o r r e c t i o n f a c t o r f o r the c e n t r e - t o - l i m b e f f e c t of 1.31 between the observed r a d i a l v e l o c i t y and p u l s a t i o n v e l o c i t y from Parsons [1972] can be used f o r /<2, p r o v i d e d tha t the r o t a t i o n a l v e l o c i t y f i e l d i s l e s s than the p u l s a t i o n a l v e l o c i t y f i e l d . 38 1.5.2.5 Use of p o l a r i s a t i o n measurements Mode-typing by measuring the p o l a r i s a t i o n v a r i a t i o n s has not yet been a p p l i e d to the 5 S c u t i s t a r s . Stamford and Watson [1980] have shown that at v i s i b l e wavelengths, the amplitude of the p o l a r i s a t i o n v a r i a t i o n s would be too small f o r d e t e c t i o n i n even most B s t a r s except f o r BW V u l which has been mode-typed using t h i s method by O d e l l and Tapia [1981]. P o l a r i s a t i o n v a r i a t i o n s have been r e p o r t e d f o r the p e c u l i a r large-amplitude 5 S c u t i v a r i a b l e SX Phe (Haefner et a l . [1976]).] A recent d i s c u s s i o n of the method i s given by Watson [ 1983]. 1.6 SPECTROSCOPIC OBSERVATIONS OF DELTA SCUTI STARS 1.6.1 DIFFICULTIES WITH LOW-AMPLITUDE VARIABLES Observation of the s p e c t r o s c o p i c v a r i a t i o n s of the low-amplitude 8 S c u t i v a r i a b l e s i s u s u a l l y d i f f i c u l t . The amplitude of the r a d i a l - v e l o c i t y v a r i a t i o n s , 2K, i s g e n e r a l l y l e s s than I0kms~ 1 and can be l e s s than 2kms~1 f o r many of the v a r i a b l e s . T h i s i m p l i e s the need of v e r y - h i g h - d i s p e r s i o n spectroscopy. The main p u l s a t i o n p e r i o d i s g e n e r a l l y very short and i s of the order of three hours while many are of the order of j u s t one hour. The exposure time i n each spectrum i s g e n e r a l l y l i m i t e d to at most 10% or 20% of the p u l s a t i o n p e r i o d i n order to a v o i d s i g n i f i c a n t phase-smearing e f f e c t s on both the l i n e p r o f i l e s and the r a d i a l - v e l o c i t y c u rve. The e f f e c t of exposure time upon the l i n e p r o f i l e has been s t u d i e d by Huang and Struve [1955]. 39 Except f o r )3 Cas and p Pup, however, a l l the other 8 S c u t i v a r i a b l e s are not b r i g h t e r than t h i r d magnitude. T h i s c e r t a i n l y imposes a very s t r i n g e n t l i m i t a t i o n on the a t t a i n a b l e s/n i n each spectrum. One method of improvement i s t o o b t a i n an average r a d i a l - v e l o c i t y curve over many p e r i o d s . Not t a k i n g i n t o account any systematic e f f e c t , the improvement i s only by a f a c t o r of /n where n i s the number of o b s e r v a t i o n s i n each phase b i n . T h i s i m p l i e s many o b s e r v a t i o n s are r e q u i r e d i f one wants the e r r o r i n each phase b i n of the mean r a d i a l - v e l o c i t y curve to be reduced f o r example from 1 kms"1 to 0.1 kms - 1. The a c t u a l improvement w i l l be worse than /n when other e f f e c t s are c o n s i d e r e d . The c o r r e c t phasing of the r a d i a l v e l o c i t i e s w i l l be very d i f f i c u l t without p r i o r knowledge of the p r e c i s e p e r i o d and almost impossible i f the v a r i a b l e i s m u l t i p e r i o d i c . C y c l e - t o - c y c l e v a r i a t i o n s may a l s o be pr e s e n t . The use of p h o t o e l e c t r i c r a d i a l - v e l o c i t y scanner-type systems has enabled the r a d i a l - v e l o c i t y v a r i a t i o n s to be measured f o r s e v e r a l narrow l i n e v a r i a b l e s . No proper s p e c t r a l "mask" f o r the v e l o c i t y scanner i s a v a i l a b l e i f the v a r i a b l e has a s p e c t r a l type e a r l i e r than FO. I t i s d i f f i c u l t to get a p e r f e c t match between the v a r i a b l e and the s p e c t r a l mask e s p e c i a l l y with the changing s p e c t r a l c h a r a c t e r i s t i c s of the v a r i a b l e d u r i n g the p u l s a t i o n c y c l e . D i f f i c u l t i e s w i l l a l s o be encountered i f the v a r i a b l e has broad l i n e s with v s i n i g r e a t e r than 30kms~ 1. In f a c t , no r a d i a l - v e l o c i t y " d i p " can be achieved with the DAO RVS 40 system on the b r o a d - l i n e v a r i a b l e j3 Cas (Ninkov [1982]). Furthermore, one cannot study l i n e - p r o f i l e v a r i a t i o n s with the scanner-type systems. Although some i n f o r m a t i o n on the asymmetry of the mean l i n e p r o f i l e can be i n f e r r e d from the shape of the c r o s s - c o r r e l a t i o n f u n c t i o n e.g. by Burki and Mayor [1981], one s t i l l cannot examine the i n d i v i d u a l l i n e p r o f i l e s . The v a r i a t i o n s i n both the p r o f i l e and the r a d i a l v e l o c i t y may be d i f f e r e n t f o r d i f f e r e n t l i n e s . G e n e r a l l y , the r a d i a l - v e l o c i t y scanner technique has been a p p l i e d only to the la r g e - a m p l i t u d e 8 S c u t i s t a r s . 1.6.2 RECENT OBSERVATIONS Recent photographic s p e c t r o s c o p i c o b s e r v a t i o n s i n c l u d e the simultaneous l i g h t and r a d i a l - v e l o c i t y measurements of 8 D e l , p Pup, and 6 Set by Kuhi and Danziger [1967]. Leung and Wehlau [1967] a l s o measured the r a d i a l - v e l o c i t y v a r i a t i o n s of 8 D e l . C h e v a l i e r et a l . [1968] obtained simultaneous l i g h t and r a d i a l - v e l o c i t y data on 14 Aur. B e s s e l l [1969] used a p h o t o e l e c t r i c spectrum scanner i n a d d i t i o n t o the Coude p l a t e s i n h i s study of the energy d i s t r i b u t i o n and l i n e p r o f i l e v a r i a t i o n s of the v a r i a b l e s p Pup, 8 D e l , and 8 Set. A curve-of-growth a n a l y s i s was a l s o a p p l i e d to the v a r i a b l e s . Simultaneous l i g h t and r a d i a l - v e l o c i t y data were obtained f o r the v a r i a b l e s 28 And, 20 CVn, and HR7331 by Nishimura [1969], P e n f o l d [1971], and Breger et a l . [1976], r e s p e c t i v e l y . Breger [1969] was able to show the r a d i a l - v e l o c i t y v a r i a t i o n s of e Cep us i n g the data of Abt [1965]. Dravins et a l . [1977] measured the 41 r a d i a l - v e l o c i t y v a r i a t i o n s of p Pup. They a l s o r e p o r t e d Ca II K l i n e emission at a c e r t a i n phase of the p u l s a t i o n . T h i s may be caused by shocks propagating through the atmosphere. Shock generated emission has a l s o been observed i n VZ Cnc by Garbuzov and M i t s k e v i c h [1984]. R a d i a l - v e l o c i t y v a r i a t i o n s of the v a r i a b l e s HR432, HR515, and HR8006 were measured by V a l t i e r et a l . [1979]. Auvergne et a l . [1979] observed the evolved 6 S c u t i s t a r y Boo. They found l i n e core s p l i t t i n g i n the Ca II and H I p r o f i l e s i n c o n j u n c t i o n with "bumps" i n the r a d i a l - v e l o c i t y c u r v e s . Meanwhile, Duncan and Preston [1979] measured the r a d i a l - v e l o c i t y v a r i a t i o n s i n both 5 Del A and 6 Del B with photographic p l a t e s . The suspected v a r i a b l e j3 A r i was a l s o observed with the photographic p l a t e s by Barbiano Di B e l g i o s o et a l . [1983]. Recent p h o t o e l e c t r i c r a d i a l - v e l o c i t y scanner o b s e r v a t i o n s i n c l u d e the measurement of the r a d i a l - v e l o c i t y v a r i a t i o n s of s e v e r a l RRs s t a r s by Van C i t t e r s [1976]. Imbert [1980] measured the p h o t o e l e c t r i c r a d i a l v e l o c i t y v a r i a t i o n s of HD200925. Simultaneous l i g h t and p h o t o e l e c t r i c r a d i a l - v e l o c i t y data on V474 Mon were obtained by Balona and St o b i e [1980b]. Simultaneously observed l i g h t and p h o t o e l e c t r i c r a d i a l - v e l o c i t y data were a l s o o b t a i n e d f o r the v a r i a b l e s p Pav, HD37819, and 5 Set by Ku r t z [1981], Burki and Mayor [1980,1981], and Balona et a l . [1981], r e s p e c t i v e l y . Campos and Smith [1980] and Smith [1982] used a Retico n d e t e c t o r t o measure both the r a d i a l - v e l o c i t y and the 42 l i n e - p r o f i l e v a r i a t i o n s of the v a r i a b l e s p Pup, 6 Set, 28 And, 44 Tau, 6 Del A, 6 Del B, V474 Mon, 28 A q l , 14 Aur A, and 20 CVn. Nishimura et a l . [1983] used an i n t e n s i f i e d Reticon d e t e c t o r to measure the r a d i a l - v e l o c i t y v a r i a t i o n s of 20 CVn. Simultaneous l i g h t v a r i a t i o n s were a l s o o b t a i n e d . Reay et a l . [1983a] used a s e r v o - c o n t r o l l e d Fabry-Perot r a d i a l - v e l o c i t y spectrometer to measure the r a d i a l - v e l o c i t y v a r i a t i o n s of p Pup. The I n t e r n a t i o n a l U l t r a v i o l e t E x p l o r e r (IUE) was used by F r a c a s s i n i and P a s i n e t t i [1982] and F r a c a s s i n i et a l . [1983] to observe the Mg II H and K l i n e emissions of s e v e r a l 5 S c u t i v a r i a b l e s . 1.6.3 USE OF PRECISION RADIAL-VELOCITY TECHNIQUES The small 2K v a l u e s of many of the 6 S c u t i v a r i a b l e s demand that r a d i a l - v e l o c i t y o b s e r v a t i o n s should have a p r e c i s i o n of at l e a s t ±0.5 kms" 1. Heacox [1980] has p o i n t e d out the advantages of a p p l y i n g p r e c i s i o n r a d i a l - v e l o c i t y techniques to the study of 6 S c u t i v a r i a b l e s . For many 5 S c u t i s t a r s , the r a t i o of v e l o c i t y to l i g h t amplitude i s about 92 kms" 1mag" 1 (Breger [1979]). Hence, i f one can achieve a r a d i a l - v e l o c i t y p r e c i s i o n of b e t t e r than ±0.1 kms - 1 i n the r a d i a l - v e l o c i t y c u r v e s , i t would be e q u i v a l e n t to a p r e c i s i o n of about ±0.001 mag i n the l i g h t c u r ve. Most modern p r e c i s i o n r a d i a l - v e l o c i t y techniques can c e r t a i n l y achieve t h i s ; i n f a c t , many of the methods can achieve almost ±0.01 kms"1 i n p r e c i s i o n which i s e q u i v a l e n t to having a p r e c i s i o n of ±0.0001 mag i n the photometry. T h i s would mean more than an order of magnitude improvement i n 43 the a b i l i t y t o study the 6 S c u t i p u l s a t i o n . The HF a b s o r p t i o n c e l l technique i s very s u i t a b l e i n i t s a p p l i c a t i o n to the study of 8 S c u t i v a r i a b l e s . I n d i v i d u a l l i n e p r o f i l e s can a l s o be obt a i n e d . T h i s a l l o w s the study of the i n d i v i d u a l l i n e p r o f i l e v a r i a t i o n s as w e l l as the d e t e r m i n a t i o n of p r e c i s e r a d i a l v e l o c i t i e s f o r i n d i v i d u a l s p e c t r a l l i n e s . B r o a d - l i n e s t a r s would not be a problem f o r the HF technique. A p r e c i s i o n of about ±50 ms"1 can be achieved with only a moderate s/n f o r the spectrum and with only a moderate amount of data r e d u c t i o n . One would be able to d e t e c t p r e v i o u s l y unseen low-amplitude p u l s a t i o n modes. The e f f e c t s of p u l s a t i o n on d i f f e r e n t s p e c t r a l l i n e s can a l s o be s t u d i e d . The d i f f e r e n t p u l s a t i o n amplitude of d i f f e r e n t s p e c t r a l l i n e s may h e l p one to study the s t r u c t u r e of the p u l s a t i n g s t e l l a r atmosphere. The o b s e r v a t i o n of the Van Hoof e f f e c t (van Hoof [1957]) which i s c h a r a c t e r i s e d by phase l a g s between the r a d i a l - v e l o c i t y curves of d i f f e r e n t s p e c t r a l l i n e s e.g. hydrogen and the m e t a l l i c l i n e s , c o u l d a l s o be used to probe the .structure of the p u l s a t i n g s t e l l a r atmosphere. P r e s e n t l y , most of the e x i s t i n g mode-typings of 5 S c u t i s t a r s are based on photometric s t u d i e s e.g. using the Q v a l u e s . S p e c t r o s c o p i c data can g i v e another .independent d e t e r m i n a t i o n of the p u l s a t i o n mode. One can use the observed 2K value of a 6 S c u t i s t a r to d i s c r i m i n a t e between r a d i a l and n o n r a d i a l p u l s a t i o n s . L i n e - p r o f i l e a n a l y s i s o f f e r s another independent method f o r mode d i s c r i m i n a t i o n . These w i l l enable one to study, u s i n g s p e c t r o s c o p i c data, 44 the s y s t e m a t i c s of the 6 S c u t i s t a r s at d i f f e r e n t p a r t s of the i n s t a b l i l i t y s t r i p . Many 8 S c u t i s t a r s can be s t u d i e d with the HF technique i f one uses a t e l e s c o p e with an a p e r t u r e s i m i l a r to that of the CFH 3.6m t e l e s c o p e . S e v e r a l 5 S c u t i v a r i a b l e s have been s t u d i e d using the HF technique. I t i s the o b j e c t i v e here to r e p o r t on the study. Chapter two of t h i s t h e s i s d i s c u s s e s the c r i t e r i a to choose the d e t e c t o r and the r e f e r e n c e a b s o r p t i o n gas. The R e t i c o n d e t e c t o r i s a l s o d i s c u s s e d with p a r t i c u l a r emphasis on noise r e d u c t i o n , incomplete readout, and the p e r s i s t e n c e phenomenon. Chapter two a l s o g i v e s a d e s c r i p t i o n of the a b s o r p t i o n c e l l and i t s o p e r a t i o n . Chapter three d e s c r i b e s the d e r i v a t i o n of new molecular c o n s t a n t s f o r HF and hence improved wavelengths for the (3-0) band. The l i n e s t r e n g t h s f o r the (3-0) band are a l s o c a l c u l a t e d . T h i s enables the d e r i v a t i o n of r e l a t i v e gas temperature and pressure d i r e c t l y from the observed s p e c t r a . The theory of molecular c o l l i s i o n s e l f - b r o a d e n i n g i s a l s o d e s c r i b e d i n Chapter t h r e e . T h i s i s f o l l o w e d by a t h e o r e t i c a l f o r m u l a t i o n and e v e n t u a l c a l c u l a t i o n of the HF l i n e w i d t h s and s h i f t s . Chapter four d e s c r i b e s the procedure to preprocess R e t i c o n data i n order to achieve optimum s/n f o r the s p e c t r a . The procedure to reduce HF data i s a l s o d e s c r i b e d . There i s a l s o d i s c u s s i o n on the Fahlman-Glaspey d i f f e r e n c e technique, b a r y c e n t r i c c o r r e c t i o n s , and v e l o c i t y accuracy. Chapter f i v e r e p o r t s on the a p p l i c a t i o n of the HF technique to the v a r i a b l e 20 CVn. Chapter s i x r e p o r t s on p Pup. Chapter seven r e p o r t s on o 1 E r i while Chapter e i g h t r e p o r t s on /3 Cas. Chapter 2 THE HF ABSORPTION CELL SYSTEM 2.1 INTRODUCTION One can e l i m i n a t e most of the problems a s s o c i a t e d with the use of t e l l u r i c l i n e s as wavelength r e f e r e n c e l i n e s . T h i s i s accomplished i f the r e f e r e n c e l i n e s are generated by an a b s o r p t i o n c e l l c o n t a i n i n g a s e l e c t e d gas. The a b s o r p t i o n c e l l would be p l a c e d i n the t e l e s c o p e beam before the s t a r l i g h t e n t e r s the spectrograph. Since the temperature and pres s u r e of the absorbing gas can be c o n t r o l l e d , the generated a b s o r p t i o n l i n e s w i l l be more s t a b l e than the t e l l u r i c l i n e s . Blending problems i n the t e l l u r i c - l i n e technique can be minimised with a c a r e f u l c h o i c e of the abso r b i n g gas. Moreover, as opposed to the case with the t e l l u r i c l i n e s , s t e l l a r s p e c t r a with or without the imposed a b s o r p t i o n l i n e s as w e l l as s p e c t r a with o n l y the imposed a b s o r p t i o n l i n e s can be e a s i l y generated. These s p e c t r a are very u s e f u l i n removing the l i n e - b l e n d i n g - i n d u c e d e r r o r s i n l i n e - p o s i t i o n measurements. 2.2 CHOOSING THE DETECTOR Measuring very a c c u r a t e l i n e p o s i t i o n s r e q u i r e s very h i g h s i g n a l - t o - n o i s e r a t i o s f o r the s p e c t r a . Photon noise i s the dominant source of noise i n hig h s/n s p e c t r a . T h e r e f o r e , i n order to achieve h i g h s/n, l a r g e numbers of photons must be d e t e c t e d . T h i s would imply the d e t e c t o r must have a l a r g e dynamic range. I t i s a l s o d e s i r a b l e f o r the d e t e c t o r to have 45 46 a l i n e a r response. High p o s i t i o n a l s t a b i l i t y i s important i n order that i n s t r u m e n t a l induced s p e c t r a l s h i f t s ( r a s t e r s h i f t s ) , as seen i n t e l e v i s i o n type d e t e c t o r s or image tubes, would not occur. The d e t e c t o r should have hi g h d e t e c t i v e quantum e f f i c i e n c y (DQE). The sensing area of the d e t e c t o r should a l s o be l a r g e and i n a c o n f i g u r a t i o n s u i t a b l e f o r spectroscopy. T h i s i s important i n the c o n s i d e r a t i o n to maximise the number of d e t e c t e d photons. The d e t e c t o r which s a t i s f i e s most of the above mentioned c r i t e r i a i s a l i n e a r R e t i c o n photodiode a r r a y . The DQE of the R e t i c o n can be as high as 75% i n the X8000 s p e c t r a l r e g i o n (Walker et a l . [1985]). T h i s i s higher than most c o n v e n t i o n a l d e t e c t o r s e.g. photographic p l a t e s and most CCDs. The Re t i c o n a l s o does not have the i n e f f i c i e n t charge t r a n s f e r problem as found i n many CCDs. The r e c t a n g u l a r shape ( l a r g e aspect r a t i o ) of the R e t i c o n sensing area (diode) i s more s u i t a b l e f o r spectroscopy than the square-shaped p i x e l of the CCD. The n e c e s s i t y to sum a column of CCD p i x e l s negates the very-low-readout-noise advantage of the CCD over the a v a i l a b l e low-readout-noise R e t i c o n (Walker et a l . [1985]). Johnson [1984a] has p o i n t e d out t h a t , i n the f u t u r e , the summation along a column of CCD p i x e l s can be accomplished w i t h i n the c h i p such that the readout n o i s e of the summed column i s the same as that of a s i n g l e CCD p i x e l . At the present, only three p i x e l s i n the same column can be added i n t h i s manner. In any case, readout n o i s e i s not the dominant source of noise i n h i g h s/n s p e c t r a . The R e t i c o n a r r a y a l s o has e x c e l l e n t 47 ge o m e t r i c a l s t a b i l i t y , a l a r g e charge storage c a p a c i t y i . e . l a r g e dynamic range, as w e l l as having no dead ( i n s e n s i t i v e ) space between diodes. 2.3 THE RETICON DETECTOR 2.3.1 INTRODUCTION Re t i c o n i s the commercial name of a group of s e l f - s c a n n e d or i n t e g r a t e d diode a r r a y s (IDA). The s i l i c o n diode a r r a y i s a l a r g e s c a l e i n t e g r a t e d c i r c u i t f a b r i c a t e d on a s i n g l e s i l i c o n c h i p . The c h i p i s mainly a 300jum t h i c k n-doped s i l i c o n s u b s t r a t e . I t has t h i n channels of p-doped s i l i c o n a t r e g u l a r i n t e r v a l s on the s u r f a c e and a common gol d s u b s t r a t e on the bottom as anode. The number of channels corresponds to the number of photodiodes i n the a r r a y . A t r a n s p a r e n t l a y e r of S i 0 2 covers the whole s u r f a c e of the c h i p to reduce the e f f e c t i v e r e f l e c t i v i t y of the a r r a y . When a reverse b i a s of 5 v o l t s i s a p p l i e d to the diodes, a space-charge l a y e r ( d e p l e t i o n region) i s c r e a t e d at the i n t e r f a c e between the n-doped and the p-doped s i l i c o n . A b s o r p t i o n of photons with s u f f i c i e n t energy (to r a i s e the e l e c t r o n s i n t o the conduction band of the semiconductor) w i t h i n the s i l i c o n w i l l form e l e c t r o n / h o l e p a i r s . I f these excess f r e e charge c a r r i e r s are formed i n the p-doped s i l i c o n , they w i l l d r i f t to the d e p l e t i o n r e g i o n and p a r t i a l l y d i s c harge the b i a s . And i f they are formed i n the n-doped s i l i c o n , they a l s o w i l l move by d i f f u s i o n to the d e p l e t i o n r e g i o n . T h i s r e s u l t s i n having e f f e c t i v e l y no dead 48 space between di o d e s . When the diode i s r e b i a s e d again a f t e r a set exposure time, the amount of charge r e q u i r e d to r e s t o r e the o r i g i n a l 5 v o l t s p o t e n t i a l w i l l g i v e a measure of the number of absorbed photons. In a readout ( r e b i a s i n g the d i o d e s ) , a s h i f t r e g i s t e r i s c l o c k e d to connect each diode s e q u e n t i a l l y to i t s a p p r o p r i a t e video output l i n e . The analog output s i g n a l from the video l i n e i s f i r s t a m p l i f i e d and then d i g i t i s e d f o r storage i n a computer. Higher energy (blue) photons tend to generate the e l e c t r o n / h o l e p a i r s more e a s i l y than the lower energy (red) photons. Hence the red photons have a longer p e n e t r a t i o n depth i n t o the s i l i c o n . I f the wavelength of the photons i s too long f o r e l e c t r o n / h o l e p a i r g e n e r a t i o n i . e . beyond the red c u t o f f , the photons may be r e f l e c t e d back from the bottom g o l d s u b s t r a t e of the c h i p and i n t e r f e r e with incoming photons to produce f r i n g i n g e f f e c t s . Geary [1976] has p o i n t e d out t h a t , f o r the t h i c k a r r a y s , such i n t e r f e r e n c e e f f e c t s are s m a l l . T h i s i s because the rear s u r f a c e of the c h i p would be r e q u i r e d to have a very f i n e f i n i s h i n order to produce c o l l i m a t e d r e f l e c t i o n s . In any case, proper f l a t - f i e l d c a l i b r a t i o n procedure can remove any small-amplitude f r i n g e p a t t e r n from the spectrum. The e l e c t r o n / h o l e p a i r s c r e a t e d by the a b s o r p t i o n of the photons can recombine before reaching the space-charge l a y e r . T h i s leads to a reduced resp o n s i v e quantum e f f i c i e n c y (RQE). The d e n s i t y of the recombination c e n t r e s i s higher near the s u r f a c e of the s i l i c o n (Geary [1976]). Consequently, the blue photons, which have shor t p e n e t r a t i o n 49 depths, are more s e r i o u s l y a f f e c t e d . Of course, the response i s s t i l l dominated by the r e f l e c t i v i t y of s i l i c o n . The response i n the f a r red w i l l a l s o be a f f e c t e d by the i n c r e a s e d recombination l o s s due to the longer d i s t a n c e the charges have to t r a v e l to reach the space-charge l a y e r . With the long d i s t a n c e of t r a v e l by the f r e e charges, there w i l l a l s o be a higher chance that the f r e e charges would d i f f u s e to adjacent p i x e l s r a t h e r than the one d i r e c t l y above. T h i s w i l l r e s u l t i n gradual l o s s of s p a t i a l r e s o l u t i o n with i n c r e a s i n g wavelength. In f a c t , at the s p e c t r a l r e g ion chosen f o r the HF program, the i n s t r u m e n t a l p o i n t - s p r e a d f u n c t i o n i s dominated by t h i s e f f e c t . The c r o s s t a l k between adjacent diodes due to charge s p i l l o v e r from s a t u r a t e d (the space-charge l a y e r completely discharged) diodes has been found to be very small (Talmi and Simpson [1980]). 2.3.2 DARK CURRENT Dark c u r r e n t (thermal leakage) e x i s t s when f r e e charges are t h e r m a l l y generated to d i s c h a r g e the space-charge l a y e r . I t reduces the e f f e c t i v e dynamic range of the a r r a y as w e l l as adding an e x t r a n o i s e component i n t o the system. Thermal dark c u r r e n t i s a l s o n o n l i n e a r with exposure time. T h i s i s because of the dependence of the leakage on the width of the d e p l e t i o n r e gion and hence the number of absorbed photons. Some of the diodes (hot diodes) can a l s o have a much higher amount of thermal leakage than the average diode i n the a r r a y . The amount'of dark c u r r e n t , however, can be reduced by c o o l i n g the diode a r r a y . At an o p e r a t i n g temperature of 50 -130°C with l i q u i d N 2 c o o l i n g , the dark c u r r e n t becomes almost n e g l i g i b l e (Vogt et a l . [1978], Vogt [1981]). The c o o l i n g of the a r r a y , however, w i l l i n c r e a s e the bandgap energy as w e l l as the p e n e t r a t i o n depth of the photons. Vogt et a l . [1978] rep o r t e d a marked drop i n the RQE i n the longer wavelengths (redward of X7000) with c o o l i n g . The peak RQE f o r R e t i c o n s cooled to 90K i s about 80% at about X7000 (Geary [1976]). To a v o i d l a r g e systematic readout e r r o r s from d i f f e r e n t i a l h e a t i n g (Campbell et a l . [1981]) as w e l l as dark s i g n a l from "charge pumping" e f f e c t s (Vogt [1981]), c l o c k p u l s e s to the a r r a y are turned o f f between readouts. Vacuum housing f o r the a r r a y p r o v i d e s temperature i n s u l a t i o n and prevents the formation of f r o s t on the c o o l e d a r r a y s u r f a c e . The UBC b u i l t a r r a y housing ( c o l d box) i s f i r s t rough pumped with a mechanical pump. The c r y o g e n i c pumping m a t e r i a l , z e o l i t e , w i l l then generate a good vacuum by absorbing the r e s i d u e gas when c o o l e d (Campbell et a l . [1981]). Temperature s t a b i l i t y of the UBC b u i l t R e t i c o n d e t e c t o r s i s maintained to ±0.2°C by a heater/diode-temperature-sensor loop (Campbell et a l . [1981]). 2.3.3 LINEARITY IN RESPONSE The l i n e a r i t y of the R e t i c o n response has been examined by Geary [1976], Campbell [1977], Vogt et a l . [1978], Talmi and Simpson [1980], and Vogt [1981]. E x i s t e n c e of the n o n l i n e a r dark c u r r e n t i n the s i g n a l w i l l c e r t a i n l y produce 51 a n o n l i n e a r response. N o n l i n e a r i t y can a l s o r e s u l t from imperfect removal of the a d d i t i v e f i x e d p a t t e r n s i g n a l ( b a s e l i n e ) from the data. Furthermore, the a m p l i f i e r s and the a n a l o g - t o - d i g i t a l c o n v e r t e r s of the a r r a y output can have n o n l i n e a r c h a r a c t e r i s t i c s . G e n e r a l l y , with the R e t i c o n p r o p e r l y c o o l e d , the s i g n a l i s l i n e a r over four orders of magnitude. Small n o n l i n e a r i t i e s i n the response can be removed with proper i n s t r u m e n t a l c a l i b r a t i o n and data r e d u c t i o n procedure. The dynamic range of the Reticon depends on the amount of charge s u f f i c i e n t to s a t u r a t e a s i l i c o n photodiode. The s a t u r a t i o n charge of a RL-1872F/30 Re t i c o n photodiode i s 3.2 pcoul or 2x10 7 e q u i v a l e n t photons. 2.3.4 FIXED LINE PATTERN The output s i g n a l c o n t a i n s an a d d i t i v e f i x e d p a t t e r n component. I t i s the r e s u l t of c a p a c i t i v e c o u p l i n g of the c l o c k - d r i v e r s i g n a l s onto the v i d e o output l i n e s . The amplitude of t h i s f i x e d p a t t e r n i s u s u a l l y l a r g e . The f i x e d p a t t e r n has a saw-tooth waveform with s p a t i a l p e r i o d s i n m u l t i p l e s of the number of output v i d e o l i n e s . I t , however, can be very constant over long p e r i o d s of time i f the c l o c k s i g n a l waveforms are s t a b l e and the temperature of the R e t i c o n i s kept c o n s t a n t . Consequently, i t can be e a s i l y removed by n u m e r i c a l l y s u b t r a c t i n g o f f a second readout o b t a i n e d s h o r t l y a f t e r the i n i t i a l readout of the data frame (Walker et a l . [1976]). Because of a r r a y s e l f - h e a t i n g and other phenomena, a small-amplitude r e s i d u a l f i x e d p a t t e r n can s t i l l remain a f t e r the s u b t r a c t i o n of the second readout 52 ( b a s e l i n e ) . T h i s can be removed with proper i n s t r u m e n t a l c a l i b r a t i o n and d a t a - r e d u c t i o n procedure. 2.3.5 REDUCTION OF READOUT NOISE The readout noise of a p r o p e r l y c o o l e d R e t i c o n i s dominated by the r e s e t (KTC) n o i s e . In r e b i a s i n g a diode, the Johnson noise i n the r e s i s t a n c e of the s h i f t - r e g i s t e r s witch causes an u n c e r t a i n t y i n the amount of r e s t o r i n g charges a p p l i e d to the diode. The rms n o i s e AQ i s given by Walker et a l . [1985]: AQ = 2.5x10 9 /(kTC) (2.1) Where AQ i s i n number of e q u i v a l e n t photons C i s the di.ode c a p a c i t a n c e i n pF k i s the Boltzmann's constant T i s the temperature i n k e l v i n The diode c a p a c i t a n c e i s about 0.6 pF. Hence, i f there i s no other s t r a y c a p a c i t a n c e , the r e s e t n o i s e would be l e s s than 300 e" at an o p e r a t i n g temperature of 150K. In c o n v e n t i o n a l readout schemes of the diode a r r a y , however, the video l i n e v o l t a g e i s monitored with r e s p e c t to ground both before and a f t e r r e b i a s i n g each diode. T h i s would s u b j e c t the measured charge of the diode to the KTC n o i s e a s s o c i a t e d with the shunt c a p a c i t a n c e of the video l i n e which i s t y p i c a l l y about 50 pF. T h i s problem has been minimised by Walker et a l . [1985] u s i n g the s i g n a l p r o c e s s i n g technique of c o r r e l a t e d double sampling (CDS) (White et a l . [1974], Geary [1979]). The CDS s i g n a l f o r the nth diode i s the d i f f e r e n c e between the measured s i g n a l a f t e r r e b i a s i n g the (n-1)th diode (with 53 the video l i n e shunt c a p a c i t o r s h o r t - c i r c u i t e d and the ( n - l ) t h diode connected to the video l i n e ) and the measured s i g n a l a f t e r connecting the nth diode to the video l i n e (with the ( n - l ) t h diode d i s c o n n e c t e d from the video l i n e ) . W i t h i n the short double sample i n t e r v a l , the noise from the v i d e o l i n e i s c o r r e l a t e d and i s removed i n the d i f f e r e n c e . Only the u n c o r r e l a t e d noise due to the d i f f e r e n c e between the ( n - l ) t h and the nth diode remains. T h i s i s the r e s e t n o i s e with a c h a r a c t e r i s t i c c a p a c i t a n c e of the same magnitude as that of a s i n g l e diode. The readout n o i s e i n a n a l o g - t o - d i g i t a l c o n v e r t o r u n i t s (adcu) i s e a s i l y o b tained from the v a r i a n c e of a short dark exposure. I t i s 2 adcu rms per diode per readout f o r the c u r r e n t UBC-built DAO 1872-Reticon d e t e c t o r o p e r a t i n g i n the h i g h gain mode of the a m p l i f i e r s . In order to make comparison with the number of d e t e c t e d photons, one needs to know the c o n v e r s i o n f a c t o r between adcu and the number of e q u i v a l e n t photons. T h i s can be accomplished i n two ways. Knowing both the number of adcu f o r a s a t u r a t e d diode and the manufacturer s p e c i f i e d s a t u r a t i o n charge, a c o n v e r s i o n f a c t o r of 185 e" per adcu was o b t a i n e d f o r the UBC-built DAO R e t i c o n d e t e c t o r (Walker et a l . [1985]). By computing the r e s i d u a l between two s p e c t r a of heavy modulation i . e . a l a r g e range of s i g n a l l e v e l s , one can p l o t the s i g n a l v a r i a n c e a g a i n s t the s i g n a l . The c o n v e r s i o n f a c t o r w i l l then be the r e c i p r o c a l of the s l o p e . Walker et a l . [1985] used s p e c t r a of Betelgeuse exposed to the same s i g n a l l e v e l f o r t h i s a l t e r n a t e c a l i b r a t i o n method. The l a r g e number of 54 TABLE 2.1 UBC-built 1872-Reticons ********************************************************** Telescope e"/adcu ( h i gain) e"/adcu ( l o gain) DAO 1,2m/l.8m 185 185x5 CFHT 3.6m 250 250x3.5 UBC 0.4m 150 150x3.5 ********************************************************** s t r o n g a b s o r p t i o n l i n e s i n the spectrum p r o v i d e d the range i n the s i g n a l l e v e l s . A c o n v e r s i o n f a c t o r of 180 e" per adcu was o b t a i n e d . T h i s agrees very w e l l with the p r e v i o u s c a l i b r a t i o n and i m p l i e s a nois e l e v e l of only 370 e q u i v a l e n t photons i n a s i n g l e readout. Recent improvements i n the e l e c t r o n i c s have f u r t h e r reduced the readout noise to be l e s s than 320 e q u i v a l e n t photons. The UBC-built CFHT R e t i c o n has s i m i l a r n o i s e c h a r a c t e r i s t i c s . T able 2.1 l i s t s the three c u r r e n t UBC-built R e t i c o n s and t h e i r e'/adcu c o n v e r s i o n f a c t o r s . The valu e s l i s t e d f o r the CFHT R e t i c o n are based on " r e a l " adc u n i t s . The CFHT R e t i c o n - c o n t r o l software uses a "pseudo" adc u n i t ( l o g i c a l l y s h i f t e d by one b i t ) which i s e q u i v a l e n t to two " r e a l " adc u n i t . Hence the corresponding c o n v e r s i o n f a c t o r s based on t h i s pseudo u n i t would be h a l f of those l i s t e d i n Table 2.1. 2.3.6 THE INCOMPLETE READOUT PHENOMENON I t has been r e p o r t e d that a r e s i d u a l charge can s t i l l remain a f t e r a readout ( L i v i n g s t o n et a l . [1976], Vogt et a l . [1978]). T h i s phenomenon of incomplete readout or readout l a g , however, was found to be n e g l i g i b l e by Geary 55 [1976] and Talmi and Simpson [1980]. If the diodes are not r e b i a s e d back to the o r i g i n a l o p e r a t i n g p o t e n t i a l (5 v o l t s ) a f t e r a readout, i t would have the same e f f e c t as a reduced response. Moreover, the r e s i d u a l charge w i l l contaminate the next exposure. The incomplete readout can be reduced or e l i m i n a t e d by having a s u f f i c i e n t l y low scan r a t e f o r the readout to allow f o r more complete r e b i a s i n g . M u l t i p l e e rasure readouts can a l s o be used to i n s u r e that a l l diodes are f u l l y charged before the s t a r t of an exposure (Vogt et a l . [1978]). The power d i s s i p a t i o n i n a quick sequence of readouts, however, w i l l cause s e l f - h e a t i n g of the a r r a y . T h i s w i l l r a i s e the l e v e l of the dark c u r r e n t as w e l l as i t would take time f o r the a r r a y to reach temperature e q u i l i b r i u m again. The e f f e c t of incomplete readout has been examined with the DAO R e t i c o n . T h i s i s accomplished by s e t t i n g a r e l a t i v e l y long exposure time f o r a Hg a r c spectrum. But before the end of the exposure d u r a t i o n , the arc i s switched o f f and the o p t i c a l path to the spectrograph blocked o f f i n f r o n t of the image s l i c e r . T h i s i s to e l i m i n a t e any a f t e r glow from the a r c . The R e t i c o n i s set to continue performing exposures ( r e a d o u t s ) . The f i r s t readout would be s a t u r a t e d at the p o s i t i o n s of the strong arc l i n e s and can be d i s c a r d e d . The second readout, however, should be a dark exposure. A f t e r s u b t r a c t i n g o f f the u n d e r l y i n g b a s e l i n e , any s i g n a l d e t e c t e d i n t h i s readout would then represent r e s i d u a l s l e f t behind by the incompleteness of the f i r s t readout. F i g u r e 2.1 shows t h i s remnant spectrum together F i g u r e 2.1 R e s i d u a l s from i n c o m p l e t e r e a d o u t . F i g u r e 2 . 2 The p e r s i s t e n c e phenomenon 57 R0B = 2. T = 37.5s R0H = 3. T=62.5s R0H = 4. T=87.5s R0H=5. T = 112.5s R08=6. T = 137.5s R0W = 7. T = 162.5s R0H=8. T=212.5s R0H=9. T=262.5s RO» = 10. r = 312.5s 'R0H = 11. T=362.5s R0H = I2. T=412.5s >i*m»m n»*wm>»»»1*4w*mnpQ^_)3 7 = 452 5s ^ ^ R 0 t t = !4. T=512.5s I^OOOe" per 100s H ^ ^ I ^ N M * ^ ! ' ^ n^m.!*^ PQĴ  = 1 5 y=562 5s _i i_ 100 500 900 1300 1700 P i x e l M F i g u r e 2.3 Time decay of the. p e r s i s t e n c e phenomenon. i s 1 « SQOI j a d . a 59 with an unsaturated Hg arc spectrum. The sharp weaker l i n e s in the remnant spectrum would correspond to l i n e s not s a t u r a t e d i n the f i r s t readout. The e q u a l l y strong l i n e s (each of about 1.8xl0 5e") would correspond to p o s i t i o n s of s a t u r a t e d l i n e s i n the f i r s t readout. Since the s a t u r a t i o n charge i s 2x10 7e~, t h i s would imply that almost 1% of the s i g n a l c o u l d be r e t a i n e d by the incompleteness of a s i n g l e readout. T h i s i s a r a t h e r l a r g e e f f e c t . Consequently, m u l t i p l e erasure readouts should normally be used. 2.3.7 THE PERSISTENCE PHENOMENON Fi g u r e 2.2 shows the r e s u l t of subsequent readouts a f t e r the incomplete readout t e s t . The readout number and the time elapsed s i n c e the i n i t i a l readout are a l s o shown fo r each spectrum. The i n i t i a l s a t u r a t e d readout would have a readout number RO#=0. The s p e c t r a are a l l s c a l e d to the same exposure time of 100 seconds. The apparent higher n o i s e in the f i r s t couple of s p e c t r a i s the a r t i f a c t of s c a l i n g from a s h o r t e r exposure time. One can observe from t h i s time s e r i e s i n F i g u r e 2.2 that the r e s i d u a l s at the p o s i t i o n s of the arc l i n e p e r s i s t f o r a long time and a f t e r many readouts. T h i s p e r s i s t e n c e phenomenon does not have the same c h a r a c t e r i s t i c s as the e f f e c t of incomplete readout d i s c u s s e d e a r l i e r . The e f f e c t cannot be reduced by performing m u l t i p l e readouts. I t would decay only as time e l a p s e s . The reason f o r t h i s p e r s i s t e n c e phenomenon i s s t i l l unknown. There has been su g g e s t i o n that i t r e s u l t s from the c r e a t i o n of s u r f a c e s t a t e s i n the s i l i c o n by the i n i t i a l 60 l i g h t exposure. The e l e c t r o n / h o l e p a i r s from these regions would d i f f u s e towards the d e p l e t i o n region only over a long time s c a l e . F i g u r e 2.3 shows the time decay of the phenomenon. In view of t h i s , one r e a l l y should delay new exposures f o r a c e r t a i n amount of time a f t e r a s t r o n g exposure. T h i s i s e s p e c i a l l y important i f the new exposure i s long and weak. 2.3.8 COSMIC-RAY EVENTS E l e c t r o n / h o l e p a i r s can be generated i n the s i l i c o n diode from the passage of a cosmic ray event. Such event w i l l produce a narrow emission spike of s e v e r a l p i x e l s i n width, superimposed on the spectrum (Vogt [1981]). Because the frequency of such occurrence i s about one i n two to four hours (Vogt et a l . [1978]), long exposures can be a f f e c t e d . Johnson [1984b], however, has p o i n t e d out t h a t sharp emission spik e s i n the s p e c t r a can a l s o be the r e s u l t of h i g h frequency (around the sampling frequency) n o i s e p i c k u p s . Cosmic-ray events have been observed with the U B C - b u i l t R e t i c o n s on long exposures or time s e r i e s . In f a c t , one can see a few of these s p i k e s on the time s e r i e s p l o t i n F i g u r e 2.2. The two most n o t i c e a b l e but t y p i c a l s p i k e s can be seen i n the s p e c t r a at T=263s and at T=463s. The cosmic-ray s p i k e s are g e n e r a l l y each of about 2000e~ i n h e i g h t . T h e i r occurrence i n R e t i c o n spectrum i s about ten per hour at DAO and about twice t h i s at CFHT. N a t u r a l l y , t h e i r p i x e l p o s i t i o n s on the spectrum are random. At times, t h e r e can al,so be s p i k e s caused by alpha p a r t i c l e s from the 61 •nuclear decay of r a d i o a c t i v e elements in the d e t e c t o r . These are very e n e r g e t i c events of s e v e r a l hundreds of thousands of e q u i v a l e n t photons. A more d e t a i l e d d i s c u s s i o n on the cosmic ray phenomenon can be found i n Ninkov [1985]. Reviews on R e t i c o n d e t e c t o r s can be found in L i v i n g s t o n et a l . [1976], Walker et a l . [1976], Geary [1976], L i v i n g s t o n [1976], Campbell [1977], Vogt et a l . [1978], Geary [1979], P e r c i v a l and Nordieck [1980], Talmi and Simpson [1980], Cochran and Barnes III [1981], Campbell et a l . [1981 ], Vogt [1981], Timothy [1983], and Walker et a l . [1985]. 2.4 THE GAS ABSORPTION SYSTEM 2.4.1 CHOOSING THE ABSORBING GAS A s u i t a b l e gas f o r the a b s o r p t i o n c e l l should have a b s o r p t i o n l i n e s i n the s p e c t r a l r e g i o n where the R e t i c o n has high d e t e c t i v e quantum e f f i c i e n c y . The r e f e r e n c e s p e c t r a l l i n e s should at l e a s t be as s t r o n g as the s t e l l a r l i n e s i n the r e g i o n . The l i n e s t r e n g t h , of course, w i l l be l i m i t e d by a f i x e d a b s o r p t i o n path l e n g t h . A convenient l e n g t h f o r the a b s o r p t i o n c e l l i n most t e l e s c o p e Coude i n s t a l l a t i o n s i s about one metre. I t i s d e s i r a b l e f o r the spectrum of the r e f e r e n c e a b s o r p t i o n l i n e s to be simple with very few weak l i n e s while the s t e l l a r spectrum has some f a i r l y s t r o n g l i n e s i n the same s p e c t r a l r e g i o n . The r e f e r e n c e a b s o r p t i o n l i n e s should p r e f e r a b l y be d i s t r i b u t e d evenly i n wavelength a c r o s s the coverage of about 150 angstroms by the R e t i c o n a r r a y at a r e c i p r o c a l 62 d i s p e r s i o n commonly between 2 A/mm and 10 A/mm. I t i s a l s o important f o r the number of r e f e r e n c e l i n e s i n the spectrum to be s u f f i c i e n t i n the c a l i b r a t i o n of any high or low frequency s t r u c t u r e i n the d i s p e r s i o n r e l a t i o n . But there should s t i l l be s u f f i c i e n t gaps between the r e f e r e n c e l i n e s where unblended s t e l l a r l i n e s can be found. Conversely, i t i s a l s o d e s i r a b l e f o r the r e f e r e n c e l i n e s not to be blended too h e a v i l y with strong s t e l l a r f e a t u r e s . The blending between the l i n e s i s , however, c o n s t r a i n e d by the range of the t o p o c e n t r i c v e l o c i t i e s of the s t a r s . A r o t a t i o n a l band s t r u c t u r e from a s e l e c t e d gas of diatomic or l i n e a r polyatomic molecules would s a t i s f y the above c r i t e r i a . In t h i s case, the r e f e r e n c e a b s o r p t i o n band of the molecule should not be contaminated by other s t a b l e i s o t o p i c bands of the molecule. Moreover the s p e c t r a l region i s to be chosen such that contamination by t e l l u r i c l i n e s would be minimal. T h i s i s important i n order to a v o i d the l i n e - b l e n d i n g problems caused by the v a r i a b l e t e l l u r i c l i n e s . Campbell and Walker [1979] s e l e c t e d hydrogen f l u o r i d e (HF) to be the absorbing gas as suggested by Gerhard Herzberg. The R-branch of the (3-0) v i b r a t i o n r o t a t i o n band of HF i n the X8700 region f u l f i l s most of the mentioned c r i t e r i a . For a given r a d i a l - v e l o c i t y change, the X8700 reg i o n a l s o g i v e s a f a c t o r of two l a r g e r Doppler s h i f t i n wavelength than in the c o n v e n t i o n a l blue s p e c t r a l r e g i o n . There are s e v e r a l weakly-blended st r o n g s t e l l a r l i n e s i n the s p e c t r a l r e g i o n around X8700 f o r ' s t a r s of s p e c t r a l type between F5 and K. F i g u r e 2.4 shows s e v e r a l t y p i c a l s p e c t r a 63 F i g u r e 2.4 S t e l l a r s p e c t r a i n the r e g i o n of X8700 25% I I I 8660 8695 8730 8765 W a v e l e n g t h 64 i n t h i s s p e c t r a l r e g i o n . The prominent s t e l l a r l i n e s i n the region are v a r i o u s Fe I l i n e s and the Ca II X8662 l i n e . For s t a r s of e a r l i e r s p e c t r a l type than F5, s e v e r a l Paschen and N I l i n e s are a l s o p r e s e n t . There are almost no t e l l u r i c l i n e s i n t h i s p a r t i c u l a r s p e c t r a l r e gion except towards the longer wavelength where the P-branch of the a b s o r p t i o n band i s l o c a t e d . F l u o r i n e has no other s t a b l e i s o t o p e while the molecular bands of DF are not i n the same s p e c t r a l r e g i o n . For s i m i l a r reasons, the HF spectrum i n the i n f r a r e d has been c o n s i d e r e d to be the most s u i t a b l e f o r the study of diatomic molecular l i n e shapes (Kuipers [1958]). Moreover, there i s an i n c r e a s e d i n t e r e s t i n the use of HF l a s e r s to d e l i v e r l a r g e s p e c i f i c power d e n s i t i e s (Hough [1977]). Consequently, HF has a l s o been one of the most completely s t u d i e d d i a t o m i c molecules i n c h e m i s t r y . Wavelengths and other parameters f o r the a b s o r p t i o n l i n e s i n the band are much b e t t e r known than f o r other molecules. Cochrane et a l . [1982], a f t e r an e x t e n s i v e search, have a l s o concluded that HF may be the most s u i t a b l e gas. There are some disadvantages i n u s i n g the (3-0) HF l i n e s . HF i s a dangerous gas and i s not the e a s i e s t to handle. Moreover, the a b s o r p t i o n c e l l must a l s o be heated to 100°C i n order to a v o i d p o l y m e r i s a t i o n of the HF molecules. The X8700 r e g i o n of the (3-0) band a l s o c o i n c i d e s with the low r e f l e c t i v i t y d i p of aluminum. For f r e s h l y d e p o s i t e d aluminum, the r e f l e c t a n c e at X8700 i s only about 89%. The r e f l e c t a n c e of overcoated aluminum i s very much lower. Because there are up to ten or more r e f l e c t i o n s i n a 65 Coude-Reticon system, the l o s s i n the thro.ugh-put becomes very h i g h . The use of s i l v e r high red r e f l e c t i n g s u r f a c e s to improve on the through-put i s almost mandatory. The r e f l e c t a n c e of s i l v e r at X8700 i s over 98%. The s i l v e r o p t i c s on the DAO 1.22m t e l e s c o p e , however, d e t e r i o r a t e badly w i t h i n a week a f t e r they are f r e s h l y r e - s u r f a c e d . T h i s i m p l i e s that the high red r e f l e c t a n c e Coude t r a i n would not be r e a d i l y a v a i l a b l e at a l l times. T h i s r a p i d d e t e r i o r a t i o n of the red o p t i c s has not been observed on the red Coude t r a i n at e i t h e r CFHT or UBC. T h i s may be helped by the much d r i e r c o n d i t i o n s at CFHT while an a i r c o n d i t i o n i n g system i s used at the UBC 0.4m t e l e s c o p e . Gold o p t i c s have now been used at the DAO 1.22m Coude t r a i n to improve the red through-put. One way to a l l e v i a t e the problem i s to use the (4-0) band at X6750 i n s t e a d of the (3-0) band. More (4-0) HF l i n e s can a l s o be observed over the same amount of s p e c t r a l coverage. T h i s may h e l p to improve the wavelength c a l i b r a t i o n . At the present time, however, the a v a i l a b l e a b s o r p t i o n c e l l s are too short such that the (4-0) HF l i n e s are only about 10-15% i n depth compared to about the 60-70% depth f o r the (3-0) l i n e s . To i n c r e a s e the HF l i n e s t r e n g t h , one has to i n c r e a s e the a b s o r p t i o n path l e n g t h and t h i s w i l l imply the use of m u l t i - p a s s a b s o r p t i o n c e l l . There are a l s o a few more t e l l u r i c l i n e s i n the region of X6750 to c o m p l i c a t e matters. 66 2.4.2 PHYSICAL AND CHEMICAL PROPERTIES OF HF Hydrogen f l u o r i d e (anhydrous h y d r o f l u o r i c a c i d ) has a chemical formula of simply HF. I t i s c o l o u r l e s s and has a b o i l i n g p o i n t of 19.51°C. Hence i t can be i n l i q u i d or gas phase over the normal range of observatory temperatures. I t i s a very strong d e h y d r a t i n g agent. In f a c t , because of i t s s t r o n g a f f i n i t y f o r water, a leak of HF gas i n t o moist a i r w i l l produce v i s i b l e fumes. There w i l l a l s o be a h i g h heat r e l e a s e when HF i s d i s s o l v e d i n water. HF i s very c o r r o s i v e and w i l l r a p i d l y a t t a c k q u a r t z , s i l i c a t e s , and g l a s s . S t e e l w i l l be r e a d i l y corroded by HF at temperatures above 65°C. Being an a c i d , HF w i l l r e a c t with oxides and hydroxides of metals to form water and f l u o r i d e s a l t s . I t i s a l s o a c a t a l y s t f o r organic r e a c t i o n s . The p h y s i c a l and chemical p r o p e r t i e s of HF "have been d e s c r i b e d i n Simons [1950,1964]. 2.4.3 SAFETY PRECAUTIONS ON WORKING WITH HF HF i s t o x i c . I n h a l a t i o n of HF vapour can cause coughing and a sense of hampered b r e a t h i n g . The sharp and p e n e t r a t i n g odour of the vapour, however, w i l l u s u a l l y l i m i t the amount of i n h a l a t i o n to below a s e r i o u s t o x i c l e v e l . In the event t h a t one cannot escape from the vapour, b r e a t h i n g through dry c l o t h i n g or c l o t h may save one's l i f e . I n h a l a t i o n of 100% oxygen must be a d m i n i s t e r e d to the v i c t i m immediately a f t e r the exposure to HF vapour. HF i n e i t h e r the vapour or l i q u i d phase i s very c o r r o s i v e and w i l l a t t a c k unprotected s k i n . One who has c o n t a c t with HF must be s u b j e c t e d immediately to a drenching shower of water. A p h y s i c i a n 67 should be c o n s u l t e d as soon as p o s s i b l e a f t e r any exposure, even i f the i n j u r y appears to be minor. Adequate s a f e t y p r e c a u t i o n s must be taken when and wherever HF i s i n v o l v e d . A v a i l a b i l i t y of adequate gas masks, HF r e s i s t a n t g l o v e s , goggles, and head c o v e r i n g s i s e s s e n t i a l . One should a l s o have a f u l l coverage of c l o t h i n g at a l l times when working with HF. Even with c a u t i o n , a c c i d e n t s can s t i l l happen e.g. the HF equipment may explode due to sudden p r e s s u r e b u i l d u p as a r e s u l t of r e a c t i o n with contaminants. A more d e t a i l e d d i s c u s s i o n on the s a f e t y p r e c a u t i o n s as w e l l as recommended procedure i n the case of a c c i d e n t a l exposure i s given i n the p u b l i s h e d data sheet on hydrogen f l u o r i d e by the Matheson Gas Products company. 2. 4."4 THE ABSORPTION CELL The HF system i s e s s e n t i a l l y a c l o s e d system c o n s i s t i n g of a gas a b s o r p t i o n c e l l which i s connected to a r e s e r v o i r (sample l i n e ) of l i q u i d HF. T h i s r e s e r v o i r of l i q u i d HF w i l l be maintained at a constant temperature to ensure a constant gas pressure (HF vapour pressure) f o r the c e l l . The a b s o r p t i o n c e l l i t s e l f w i l l a l s o be maintained at a constant o p e r a t i n g temperature. The temperature of the c e l l may be d i f f e r e n t than that of the HF sample l i n e . The gas h a n d l i n g system c o n s i s t s of v a r i o u s c o n t r o l v a l v e s , manometer, and connecting tubings f o r use i n the i n t r o d u c t i o n and removal of HF from the system. The whole system has to be leak p r o o f . 68 The a b s o r p t i o n c e l l i t s e l f i s b a s i c a l l y a metal tube with windows at both ends. The l e n g t h of the c e l l i s l i m i t e d by the amount of a v a i l a b l e space in f r o n t of the Coude entrance s l i t i n the p a r t i c u l a r o b s e r v a t o r y . T h i s i s g e n e r a l l y about one metre. The i n s i d e diameter of the c e l l should be l a r g e enough to accommodate the p a r t i c u l a r Coude l i g h t beam. T h i s i s to ensure that v i g n e t t i n g would not occur when the c e l l i s moved i n t o the beam. At CFHT, t h i s i s accomplished by a c e l l with a l a r g e diameter. The HF system used at the DAO 1.22m t e l e s c o p e , however, has a c e l l with two d i f f e r e n t i n s i d e diameters to match the s i z e of the converging Coude beam. The l a r g e i n s i d e diameter at one end of the c e l l i s about 4.0cm while the s m a l l e r diameter at the other end of c e l l , towards the spectrograph"entrance s l i t , i s about 1.0cm. The o u t s i d e diameter of the c e l l i s a l s o s m a l l e r at the small end to a v o i d g e t t i n g i n the way of the g u i d i n g beam. HF gas molecules have a tendency to form polymers at temperatures l e s s than 70°C (Meredith [1972b]). In a d d i t i o n to the monomers, dimers ( H F ) 2 , z i g - z a g shaped c h a i n tetramers (HF) f l, and the r i n g shaped c y c l i c hexamers ( H F ) 6 are a l s o present at lower temperatures. D i s c u s s i o n s on the p o l y m e r i s a t i o n of gaseous HF molecules can be found i n J a r r y and Davis [1953], Smith [1958,1959], Herget et a l . [1960], Huong and Couzi [1969], Janzen and B a r t e l l [1969], Himes and Wiggins [1971], and Hinchen and Hobbs [1979], The presence of these polymers would g r e a t l y reduce the number of monomeric HF molecules a v a i l a b l e f o r the formation of the 69 r e f e r e n c e a b s o r p t i o n l i n e s . Consequently, the HF a b s o r p t i o n c e l l has to be heated and maintained at a temperature g r e a t e r than 100°C i n order to minimise p o l y m e r i s a t i o n (Kuipers [1958], Herndon et a l . [1962], Meredith and Smith [1974]). Operation at the high temperature i s accomplished by h e a t i n g c o i l s wrapped around the c e l l and pa r t of the gas h a n d l i n g system. A thermocouple i s used to monitor the temperatures at both ends of the c e l l . A feed-back temperature c o n t r o l u n i t was o r i g i n a l l y used to maintain a s t a b l e temperature. I t was found that a s t a b l e v o l t a g e a p p l i e d to the h e a t i n g c o i l would be s u f f i c i e n t . I t takes s e v e r a l hours f o r the temperature of the c e l l to reach e q u i l i b r i u m a f t e r the s t a r t of the h e a t i n g p r o c e s s . The average temperature o b t a i n e d from the two ends of the c e l l may not be the same as that of the HF gas i n s i d e the c e l l . I t i s a l s o q u i t e c o n c e i v a b l e f o r the HF gas to have a r a t h e r non-uniform temperature p r o f i l e along any c r o s s - s e c t i o n of the c e l l . The heated a b s o r p t i o n c e l l i s t h e r m a l l y i n s u l a t e d from the surroundings i n order to minimise i t s e f f e c t on the seeing c o n d i t i o n of the s t e l l a r image. T h i s a l s o h e l p s to maintain a s t a b l e c e l l temperature. F i b r e g l a s s i s used to p r o v i d e the i n s u l a t i o n . An e a r l i e r v e r s i o n of the HF system even has the e n t i r e c e l l p l a c e d i n a p o l y e t h y l e n e c o v e r i n g . Continuous f o r c e d a i r c i r c u l a t i o n to the outdoor was then a p p l i e d t o the p o l y e t h y l e n e cover. T h i s a l s o h e l p s to minimise the danger of a p o s s i b l e HF gas leakage from the system. The system at CFHT uses a p l e x i g l a s s cover i n s t e a d 70 of the p o l y e t h y l e n e c o v e r i n g . At room temperature, s t e e l HF a b s o r p t i o n c e l l s had been used i n chemistry experiments (Naude and V e r l e g e r [1950], J a f f e et a l . [1965]). Some metals become HF r e s i s t a n t a f t e r a c o a t i n g of f l u o r i d e s has been formed on the s u r f a c e . N i c k e l i s one such metal and i t has been used f o r the c o n s t r u c t i o n of HF a b s o r p t i o n c e l l s or p a r t s of the a s s o c i a t e d gas handl i n g system (Kuipers et a l . [1956], Kuipers [1958], Smith [1958], R o t h s c h i l d [1964], Huong and Couzi [1969]). The f l u o r o c a r b o n polymer Kel-F (copolymers of v i n y l i d e n e f l u o r i d e and c h l o r o t r i f l u o r o e t h y l e n e ) has a l s o been used f o r the c o n s t r u c t i o n of HF c e l l s and gas ha n d l i n g systems (Hinchen [1974], Hinchen and Hobbs [1979], W i c k l i f f e and R o l l e f s o n [1979]). D i s c u s s i o n s on the p r o p e r t i e s of Kel-F can be found i n Bryce [1964]. The most common m a t e r i a l used f o r the c o n t r u c t i o n of HF c e l l s and gas ha n d l i n g systems i n chemistry experiments i s monel (Kuipers et a l . [1956], K u i p e r s [1958], Herndon et a l . [1962], Mason and N i e l s e n [1967], Janzen and B a r t e l l [1969], Huong and Couzi [1969], Wiggins et a l . [1970], Himes and Wiggins [1971], Meredith [1972b], S p e l l i c y et a l . [1972], G u e l a c h v i l i [1976]). Monel i s one of the most HF r e s i s t a n t metals at high temperatures. Consequently, i t has been chosen f o r the c o n s t r u c t i o n of the HF c e l l and the a s s o c i a t e d gas ha n d l i n g system f o r both HF systems at CFHT and DAO. I t i s advantageous to use the same metal f o r both the c e l l and the gas h a n d l i n g system. D i f f e r e n t i a l thermal expansion between d i f f e r e n t metals c o u l d cause gas leakage at the v a r i o u s 71 j o i n t s i n the system. C o n t r o l v a l v e s as w e l l as the p r e s s u r e gauge used i n the gas h a n d l i n g system are a l s o HF r e s i s t a n t . 2.4.5 THE CELL WINDOWS The t r a n s p a r e n t windows f o r the c e l l must a l s o be HF r e s i s t a n t at the high o p e r a t i n g temperatures. Many types of window, depending on the s p e c t r a l r e g i o n , have been used on HF c e l l s i n chemistry experiments. R o t h s c h i l d [1964] used p o l y e t h y l e n e windows f o r the c e l l . S i l v e r c h l o r i d e windows have a l s o been used (Adams and Katz [1956], Smith [1958], Mason and N i e l s e n [1967]). Kuipers et a l . [1956] and Kuipers [1958] used c a l c i u m f l u o r i d e windows. Huong and Couzi [1969], however, used a l l three types of window depending on the wavelength r e g i o n . P l e x i g l a s s windows have a l s o been used (Naude and V e r l e g e r [1950], Fishburne and Rao [1966]). P l e x i g l a s s windows were once used on the DAO system. T h e i r t r a n s m i s s i o n p r o p e r t y d e t e r i o r a t e d r a p i d l y a f t e r r e a c t i n g with HF at the high o p e r a t i n g temperatures. The most common types of window used f o r HF c e l l s i n chemistry experiments are made of sapphire (Herget et a l . [ i 9 6 0 ] , Mann et a l . [1961], Herndon et a l . [1962], Webb and Rao [1966], Wiggins et a l . [1970], Himes and Wiggins [1971], S p e l l i c y et a l . [1972], Meredith [1972b], Hinchen [1974], G u e l a c h v i l i [1976], Hinchen and Hobbs [1979]). Sapphire has e x c e l l e n t o p t i c a l q u a l i t y and i s very HF r e s i s t a n t at the h i g h o p e r a t i n g temperatures. Consequently, sapphire windows were chosen f o r the HF c e l l s at both CFHT and DAO. The sapphire windows are a l s o made with optimum t r a n s m i s s i o n at the X8700 72 r e g i o n . T e f l o n ( t e t r a f l u o r o e t h y l e n e polymers) r i n g s or gaskets are used to s e a l the windows onto the a b s o r p t i o n c e l l . T hin windows give higher t r a n s m i s s i o n and are more e a s i l y heated to the o p e r a t i n g temperature. They are, however, more f r a g i l e and have been known to crack under pre s s u r e d u r i n g the tightening-down pr o c e s s . Thick windows have the advantage of m i n i m i s i n g the e f f e c t of i n t e r f e r e n c e f r i n g e p a t t e r n generated from poor q u a l i t y t h i n windows. B a s i c a l l y , a t h i n window can a c t as a Fabry-Perot e t a l o n . F r i n g e p a t t e r n s had been observed with d e f e c t i v e t h i n windows. F i g u r e 2.5 shows the spectrum of a Lyrae taken with and without the HF a b s o r p t i o n c e l l c o n t a i n i n g the d e f e c t i v e window. F i g u r e 2.5a i s the spectrum taken without the c e l l i n the beam. Fi g u r e 2.5b i s that taken with an empty (no HF) a b s o r p t i o n c e l l i n the beam. F i g u r e s 2.5c and 2.5d a r e , r e s p e c t i v e l y , spectrum of a Lyrae and a f l a t - f i e l d lamp taken with the loaded ( c o n t a i n i n g HF) a b s o r p t i o n c e l l i n the beam. The observed f r i n g e p a t t e r n i s wavelength dependent and i s not p e r f e c t l y r e p r o d u c i b l e . F l a t - f i e l d i n g techniques are not adequate to reduce the r e s i d u a l f r i n g e p a t t e r n to much below one percent of the continuum. F i g u r e 2.6a shows the f r i n g e p a t t e r n a f t e r d i v i d i n g a lamp spectrum taken with an empty HF c e l l by another lamp spectrum taken with the c e l l out of the beam. F i g u r e 2.6b shows the p a t t e r n a f t e r d i v i d i n g the spectrum i n F i g u r e 2.5b by that i n F i g u r e 2.5a. F i g u r e 2.6c shows the r e s i d u a l of the f r i n g e p a t t e r n a f t e r d i v i d i n g the spectrum i n F i g u r e 2.5b by those i n F i g u r e s 73 74 75 2.5a and 2.6a. F i g u r e 2.6d shows the r e s u l t of d i v i d i n g the spectrum i n F i g u r e 2.5c by those in F i g u r e s 2.5a and 2.5d. 2.4.6 OPERATION OF THE GAS HANDLING SYSTEM The HF system has to be l e a k - p r o o f . A leak of HF gas i n t o the surroundings c o u l d be d i s a s t r o u s to the o p t i c s i n most o b s e r v a t o r i e s . A leak of a i r i n t o the HF system can a l s o be very u n d e s i r a b l e . Contaminants e.g. water vapour can re a c t with the HF and cause severe damage ( c o r r o s i o n ) to the e n t i r e HF system. The pressure gauge i s e s p e c i a l l y v u l n e r a b l e to t h i s type of c o r r o s i o n . C o l l i s i o n of the HF molecules with f o r e i g n gases w i l l both broaden and a l t e r the r e s t wavelengths of the HF r e f e r e n c e l i n e s . T h i s h i g h l y v a r i a b l e systematic e r r o r i s almost impossible to c o r r e c t at the time of data r e d u c t i o n . A rough search f o r p o s s i b l e l e a k s i n the system can be accomplished using the p r e s s u r e gauge and the v a r i o u s c o n t r o l v a l v e s to i s o l a t e , one at a time, s e l e c t e d s e c t i o n s of the system. A helium l e a k - t e s t e r i s r e q u i r e d f o r a more d e t a i l e d check on the system. I t i s d e s i r a b l e to mount the whole gas h a n d l i n g system on a s i n g l e metal p l a t e . T h i s w i l l h e l p to minimise v i b r a t i o n - i n d u c e d l e a k s when the system i s being t r a n s p o r t e d . J o i n t s have been known to become loose d u r i n g t r a n s p o r t a t i o n of the HF system. There are s e v e r a l o p e r a t i o n modes f o r the HF system. They are : (a) d i s t i l l a t i o n of s u f f i c i e n t HF from the l e c t u r e b o t t l e i n t o the HF r e s e r v o i r (sample l i n e ) . 76 (b) i n t r o d u c t i o n of the HF gas i n t o the heated a b s o r p t i o n c e l l and m a i n t a i n i n g a constant HF gas p r e s s u r e by using a m e l t i n g i c e bath around the sample l i n e . (c) condense a l l the HF gas from the a b s o r p t i o n c e l l back i n t o the sample l i n e . (d) removing a l l the HF gas from the system. Normal usage of the HF system i n v o l v e s only the o p e r a t i o n modes (b) and ( c ) . Modes (a) and (d) are used only very i n f r e q u e n t l y . Operations of the HF system should be performed near an exhaust fume-hood. T h i s i s e s p e c i a l l y recommended f o r the modes (a) and ( d ) . F i g u r e 2.7 shows the schematic of a g e n e r a l i s e d HF system. H a l f - i n c h monel t u b i n g s are used throughout the system. T h i s l a r g e diameter helps to speed up any HF d i s t i l l a t i o n p r o c e s s . The v a l v e V6 i s the c o n t r o l v a l v e a s s o c i a t e d with the commercial l e c t u r e b o t t l e of HF e.g. from Matheson Gas Products. I t i s c l o s e d f o r most of the times. For the o p e r a t i o n mode ( a ) , a l l the c o n t r o l v a l v e s i n the system are i n i t i a l l y c l o s e d a f t e r a c h i e v i n g a good vacuum throughout the e n t i r e the system. With the Kel-F tube immersed i n a methanol-dry-ice bath, the v a l v e s V6 and V5 are opened. The amount of HF d i s t i l l e d over i n t o the Kel-F tube can be measured by examining the l e v e l of the l i q u i d HF in t h i s t r a n s l u c e n t tube. A lamp i s u s u a l l y r e q u i r e d to h e l p observe the l i q u i d i n the tube. Pure l i q u i d HF should be c o l o u r l e s s . Any v i s i b l e c o l o u r seen on the l i q u i d i n the Kel-F tube would imply t h a t the HF i s probably contaminated gure 2.7 Schematic of a g e n e r a l i s e d HF system 78 and should be d i s c a r d e d . T h i s can be accomplished by f i r s t c l o s i n g v a l v e V 6 and immersing the c o l d t r a p i n a l i q u i d - n i t r o g e n bath. With the vacuum pump turned on, opening v a l v e s V 1 and V 4 w i l l remove a l l the HF from the system i n t o the c o l d t r a p . The vacuum pump and c o l d t r a p ( i n a methanol-dry-ice bath) can a l s o be used to speed up the d i s t i l l a t i o n of HF from the l e c t u r e b o t t l e i n t o the Kel-F tube. If the HF v i s i b l e i n the Ke l - F tube i s c l e a r , i t can be d i s t i l l e d i n t o the sample l i n e . With v a l v e V 6 c l o s e d and a methanol-dry-ice or l i q u i d - n i t r o g e n bath around the sample l i n e , opening v a l v e s V 4 and V 3 w i l l t r a n s f e r a l l the HF from the Kel-F tube i n t o the sample l i n e . The procedure can be repeated u n t i l an a p p r o p r i a t e amount of HF has been d i s t i l l e d i n t o the sample l i n e . T h i s i s about 6 cm 3 of l i q u i d HF f o r the HF system at DAO. The amount d i s t i l l e d i s monitored by the intermediate t r a n s f e r i n t o the Kel-F tube of known volume. There should be s u f f i c i e n t HF i n the sample l i n e such that some l i q u i d HF would s t i l l remain i n the sample l i n e a f t e r HF gas i s r e l e a s e d i n t o the a b s o r p t i o n c e l l . T h i s a l s o imposes a lower l i m i t on the volume of the sample l i n e . I n i t i a l l y , a d d i t i o n a l HF i s a l s o r e q u i r e d to form the f l u o r i d e c o a t i n g s on the inner s u r f a c e of the c e l l and gas h a n d l i n g system. A f t e r completing the t r a n s f e r , v a l v e V 3 can be c l o s e d and r e s i d u a l HF i n the r e s t of the system i s removed u s i n g the vacuum pump and c o l d t r a p . With v a l v e V 3 opened and a methanol-dry-ice bath around the sample l i n e , any f o r e i g n gas i n the system can a l s o be 79 removed u s i n g the pump and c o l d t r a p . T h i s should be a very b r i e f process s i n c e the vapour pressure of HF i s non-zero at d r y - i c e temperature. Using l i q u i d n i t r o g e n , however, would a l s o f r e e z e any trapped a i r . A more i n v o l v e d procedure to d i s t i l l higher p u r i t y HF from the standard commercial HF l e c t u r e b o t t l e i s d e s c r i b e d by W i c k l i f f e and R o l l e f s o n [1979]. For o p e r a t i o n mode (b), most of the n o n - e s s e n t i a l p a r t s of the system can be removed. These are the vacuum pump, c o l d t r a p , Kel-F tube, and HF l e c t u r e b o t t l e . The v a l v e s V1 and V4 are c l o s e d f o r the e n t i r e d u r a t i o n of the o p e r a t i o n . The HF a b s o r p t i o n c e l l must be heated to the o p e r a t i n g temperature of about 100°C p r i o r to the i n t r o d u c t i o n of the HF gas. Condensation of the HF gas can occur i f the c e l l i s not heated when the HF gas e n t e r s the c e l l . Brown c o l o u r e d l i q u i d condensations on the windows may be r e l a t e d to t h i s problem. Heating the windows with a 'heat-gun' may remove the condensation. With a heated c e l l , opening v a l v e s V2 and V3 w i l l i n t r o d u c e HF gas i n t o the c e l l . A m e l t i n g - i c e bath i s p l a c e d around the sample l i n e to ensure a constant gas pressure i n the system. T h i s w i l l be the vapour p r e s s u r e of HF at 0°C and i s about 360 mm Hg. The f a c t t h a t the pre s s u r e i s l e s s than the atmospheric p r e s s u r e has added an e x t r a margin of s a f e t y to the o p e r a t i o n . If a leak develops, a i r w i l l leak i n t o the system r a t h e r than having s u b s t a n t i a l amount of HF l e a k i n g out. Valve V7 can be opened t o allow the p r e s s u r e gauge to monitor the HF p r e s s u r e . The vapour pre s s u r e of HF i s given by J a r r y and Davis [1953] : 80 l o g 1 0 P = A - B / ( C + t ) (2.2) where P = HF vapour p r e s s u r e i n mm Hg A = 8.38036 ± 0.10896 B = 1952.55 ± 125.85° C = 335.52 ± 8.15° t = temperature i n °C In order to maintain a more c o n s i s t e n t temperature everytime, one uses d i s t i l l e d water and i c e formed from d i s t i l l e d water f o r the i c e bath. A more s t a b l e temperature can be maintained i f the i c e i s crushed i n t o s m a l l e r p i e c e s before being used i n the i c e bath. For o p e r a t i o n mode ( c ) , c o l l e c t i n g the HF back i n t o the sample l i n e can be accomplished by slowly f r e e z i n g the sample i i n e to l i q u i d n i t r o g e n temperature. Rapid f r e e z i n g may cause c l o t t i n g of the system by frozen HF. R e s i d u a l HF or f o r e i g n gases i n the system can again be removed with the vacuum pump and c o l d t r a p . S i m i l a r l y , f o r o p e r a t i o n mode (d) to remove a l l HF from the system, the vacuum pump and c o l d t r a p are used with v a l v e s V1, V2, V3, V4, V5, and V7 opened. Dry a i r or n i t r o g e n can a l s o be int r o d u c e d to f l u s h the system by opening v a l v e V8. In t h i s case, the vacuum pump and c o l d t r a p would be r e p l a c e d by a tubing to bubble the output gas through a sodium-hydroxide s o l u t i o n and i n t o an exhaust fume-hood. 2.4.7 PLACEMENT OF THE CELL Instead of the c o n v e n t i o n a l s l i t , an image s l i c e r i s g e n e r a l l y used to reduce the l o s s i n through-put caused by 81 poor seeing c o n d i t i o n s . The image s l i c e r would a l s o reduce the amount of g u i d i n g e r r o r . At both CFHT and DAO, the HF c e l l i s p l a c e d between the l a s t Coude f l a t m i r r o r and the image s l i c e r . The c e l l can a l s o be a c c u r a t e l y and e a s i l y moved i n or out of the Coude beam. The alignment of the c e l l with r e s p e c t to the beam has to be acc u r a t e i n order to a v o i d v i g n e t t i n g . G e n e r a l l y , the c e l l i s a l i g n e d a g a i n s t the i r i s diaphragm that i s i n p l a c e f o r use on the lamp exposures. T h i s i r i s diaphragm i s g e n e r a l l y w e l l a l i g n e d with r e s p e c t to the beam. At DAO, the alignment of the i r i s can be checked by viewing the primary m i r r o r through the i r i s from behind the image s l i c e r . T h i s i s accomplished by p o i n t i n g the t e l e s c o p e at the blue sky or r e f l e c t i n g l i g h t o f f the i n s i d e of the dome. The image s l i c e r may a l s o be r e p l a c e d by a small c i r c u l a r a p e r t u r e f o r t h i s purpose. To a l i g n the c e l l with the image s l i c e r i n p l a c e , one can shine l i g h t along the beam path from behind the image s l i c e r by way of the viewing o p t i c s . Alignment of the c e l l can then be accomplished by viewing the l i g h t rays coming through both the c e l l and the i r i s . Chapter 3 THE HF SPECTRUM 3.1 INTRODUCTION The chosen wavelength r e f e r e n c e l i n e s a r i s e from the v i b r a t i o n - r o t a t i o n t r a n s i t i o n s i n the ground e l e c t r o n i c s t a t e ( X 1 2 + ) of HF between the v i b r a t i o n a l l e v e l v=0 (ground l e v e l ) and u=3 i . e . they belong to the second overtone of the v i b r a t i o n - r o t a t i o n band, where v i s the v i b r a t i o n a l quantum number. The e a r l i e s t experimental study of the HF v i b r a t i o n - r o t a t i o n band (that of the (1-0) fundamental) was c a r r i e d out by Imes [1919]. The second overtone (3-0) band as w e l l as the (4-0) band were f i r s t photographed by K i r k p a t r i c k and Salant [1935]. Since then, the wavenumbers of the v a r i o u s v i b r a t i o n - r o t a t i o n bands have been measured. The (1-0) and the (2-0) bands were measured by T a l l e y et a l . [1950], Kuipers et a l . [1956], Herget et a l . [1962], Webb and Rao [1968], and G u e l a c h v i l i [1976], Naude and V e r l e g e r [1950] measured the (2-0), (3-0), and the (4-0) bands while Mann et a l . [1961] measured, from flame s p e c t r a , 23 d i f f e r e n t bands ranging from (1-0) through (9-4). Deutsch [1967a] measured the (1-0), (2-1), and (3-2) v i b r a t i o n - r o t a t i o n bands from HF l a s e r t r a n s i t i o n s while Kwok et a l . [1970] measured only the (1-0) and (2-1) bands. Sengupta et a l . [1979] measured the HF l a s e r t r a n s i t i o n s from the (1-0) through to the (6-5) band. The most recent measurement of the (3-0) band was performed by Fishburne and Rao [1966] who a l s o observed the (4-0) and (5-0) bands. Pure 82 83 r o t a t i o n a l t r a n s i t i o n s of HF have a l s o been observed by Kuipers et a l . [1956], R o t h s c h i l d [1964], Deutsch [1967b], Mason and N i e l s e n [1967], A k i t t and Yardley [1970], and Sengupta et a l . [1979]. Meanwhile, e l e c t r o n i c t r a n s i t i o n s have been observed by Safary et a l . [1951], Johns and Barrow [1959], Di Lonardo and Douglas [1973], and Douglas and Greening [1979]. Most of the above r e f e r e n c e s a l s o p r ovided e x p e r i m e n t a l l y d e r i v e d molecular constants f o r HF. The r o t a t i o n a l s e l e c t i o n r u l e f o r the v i b r a t i o n - r o t a t i o n bands i s AJ=±1 with J being the r o t a t i o n a l quantum number. The AJ=+1 t r a n s i t i o n s give r i s e to the l i n e s i n the R-branch of the band while AJ=-1 t r a n s i t i o n s form the P-branch. There i s no Q-branch (AJ=0 t r a n s i t i o n s ) f o r HF. The v a r i o u s t r a n s i t i o n s i n the (3-0) v i b r a t i o n - r o t a t i o n band are i l l u s t r a t e d i n F i g u r e 3.1. The running index number m i s equal to J^+1 (or or J') f o r the l i n e s i n the R-branch. And i t takes on the v a l u e s of - J ^ (or -J") f o r the l i n e s i n the P-branch. J ^ (or J") and J ^ (or J') a r e , r e s p e c t i v e l y , the J values i n the lower (u=0) and the upper (u=3) v i b r a t i o n a l l e v e l . 3.2 MOLECULAR CONSTANTS FOR HF 3.2.1 BASIC EQUATIONS AND CONSTANTS The wavenumber of a l i n e can be obtained d i r e c t l y from the d i f f e r e n c e between the p a r t i c u l a r upper and lower energy l e v e l s . The term value T(u,J) f o r the u - v i b r a t i o n and J - r o t a t i o n l e v e l i s the sum of a v i b r a t i o n term G(u) and a 8 4 F i g u r e 3 .1 T h e ( 3 - 0 ) v i b r a t i o n - r o t a t i o n b a n d o f HF — ° Ti« " I I II i ' IIII II I' II l : i : • * a to II 2 —v- r A . I l —v— O ~ ' l 85 r o t a t i o n term Fy(J) : T(w,J) = G(v) + F y ( J ) (3.1) G(i>) = co (u+1/2) • x (u+1/2) 2 e e e + co y (y+1/2) 3 - co z (u+1/2)" (3.2) e y e e e F (J) = B y j ( j + 1 ) - D y J 2 ( J + l ) 2 + H y J 3 ( j + 1 ) 3 - L^JMJ+1 ) 4 (3.3) The terms u> , co x , co y , and co z are some of the e e e e'e e e anharmonic c o n s t a n t s . The v a l u e s of these c o n s t a n t s are l i s t e d i n Table 3.1. Other anharmonic c o n s t a n t s can be found r e f e r e n c e d i n Huber and Herzberg [1979], The r o t a t i o n c onstants B , D , and H can be expressed i n power s e r i e s of (u+1/2) with the anharmonic c o n s t a n t s as c o e f f i c i e n t s (Herzberg [1950], Rao and Mantz [1972], Johns and Barrow [1959]). The anharmonic c o n s t a n t s themselves can be expressed i n terms of the Dunham c o e f f i c i e n t s . The correspondence between the anharmonic co n s t a n t s and Dunham c o e f f i c i e n t s , Dunham c o r r e c t i o n s , can be found i n Dunham [1932]. The term value T(u,J) as w e l l as G(u), B y, D , and H y can be expressed more e x p l i c i t l y u s i ng the Dunham c o e f f i c i e n t s : T(u,J) = I. Y. ( u + 1 / 2 ) i J m ( J + l ) m (3.4) G(u) = Z i Y i 0 ( u + l / 2 ) 1 (3.5) Bv = Z i Y i 1 ( u + 1 / 2 ) 1 ( 3- 6> Dv = - 2 i Y i 2 d ' + l / 2 ) 1 (3.7) Hv = Z i Y i 3 ( u + l / 2 ) 1 ( 3 , 8 ) 86 Lv = - ^ i Y ^ t o + l ^ ) 1 (3.9) The v a l u e s f o r most of the Dunham c o e f f i c i e n t s can be found in Webb and Rao [1968]. Values f o r Y 0 „ and Y 1 3 can be found i n Deutsch [1967b] and Mann et a l . [1961], r e s p e c t i v e l y . The Dunham anharmonic c o e f f i c i e n t s are u s u a l l y expressed i n terms of B^, u£, and the Dunham p o t e n t i a l c o n s t a n t s a,,..., a 6 . Formulae f o r the Dunham c o e f f i c i e n t s have been p u b l i s h e d by Dunham [1932], Sandeman [1940], Woolley [1962], Niay et a l . [1977], Bouanich [1978a], and O g i l v i e and T i p p i n g [1983], The co n v e r s i o n formulae between the p o t e n t i a l c o n s t a n t s a^ and c^ are given i n Sandeman [1940]. T h e o r e t i c a l Dunham c o e f f i c i e n t s have been c a l c u l a t e d with c o n t r i b u t i o n s up to the order of ( a , ) 6 . The v a l u e s f o r the p o t e n t i a l c onstants a^ used i n the c a l c u l a t i o n are taken from O g i l v i e and Koo [1976]. The formulae used f o r Y 0 6 , Y 1 5 , and Y 2 f t are taken from Woolley [1962] while those f o r Y 0 7 , Y 0 8 , and Y 1 6 are from O g i l v i e and T i p p i n g [1983]. Formulae f o r the other c o e f f i c i e n t s are taken from Bouanich [1978a]. These c a l c u l a t e d Dunham c o e f f i c i e n t s p r ovide a easy way to compute the v a r i o u s energy l e v e l s which w i l l be r e q u i r e d ' i n any l i n e - s t r e n g t h and l i n e w i d t h c a l c u l a t i o n s . 3.2.2 DERIVATIONS OF NEW CONSTANTS AND WAVELENGTHS Using Equation 3.3, one can d e r i v e the wavenumbers of l i n e s i n a (u-0) v i b r a t i o n - r o t a t i o n band. The wavenumbers are f u n c t i o n s of the v and J v a l u e s , the band c e n t r e v0 (the wavenumber corres p o n d i n g to which would be the pure 87 Table 3.1 Pu b l i s h e d molecular c o n s t a n t s f o r HF *************************************************** r e f e r e n c e co = 41 38.7666 cm" 1 e co x = 89.88 cm" e e co y = 0.90 cm e 7 e - i co z =-0.0110 cm' 1 e e B = 20.9561 cm'1 e a = 0.798 cm"1 e 1.27x10" 2 cm'1 D = 0.0021497 cm"1 e 0 = 6 . 1 3 3 X 1 0 " 5 cm" H = 1.6445x10" 7 cm"1 e r = 0.9168A e v0 (1-0) = 3 9 6 1 . 4 2 2 9 ± 0 . 0 0 0 2 5 cm"1 v0 (2-0) = 7 7 5 0 . 7 9 4 9 ± 0 . 0 0 1 5 cm"1 v0 (3-0) = 1 1 3 7 2 . 8 0 7 ± 0 . 0 0 7 cm"1 v0 (4-0) = 1 4 8 3 1 . 6 2 7 ± 0 . 0 0 7 cm"1 B 0 = 2 0 . 5 5 9 7 4 3 ± 0 . 0 0 0 0 1 4 cm'1 Webb and Rao [1968] Webb and Rao [1968] Webb and Rao [1968] Webb and Rao [1968] O g i l v i e and Koo [1976] Webb and Rao [1968] Webb and Rao [1968] O g i l v i e and Koo [1'976] O g i l v i e and Koo [1976] O g i l v i e and Koo [1976] O g i l v i e and Koo [1976] G u e l a c h v i l i [1976] G u e l a c h v i l i [1976] Fishburne and Rao [1966] Fishburne and Rao [1966] G u e l a c h v i l i [1976] D 0 = (2 . 1 2 0 4 8 0 ± 0 . 0 0 0 0 4 6 ) x l 0 ' 3 cm"'1 Sengupta et a l . [1979] H 0 = (1 .665310 .0064)X10" 7 cm"1 L 0 = (1 .8110 .12)x10- 11 cm"1 ' B, = 19.78747810.000014 cm"1 Sengupta et a l . [1979] Sengupta et a l . [1979] G u e l a c h v i l i [1976] D, = (2.063996+0.000056)X10" 3 cm 1 1 Sengupta et a l . [1979] H, = (1 .594210 .0068)X10" 7 cm"1 Sengupta et a l . [1979] 88 L, = (1.46±0.11)x10" 1 1 cm"1 Sengupta et a l . [1979] B 2 = 19.034931±0.000032 cm"1 G u e l a c h v i l i [1976] D 2 = (2.00958±0.00011)x10" 3 cm"1 Sengupta et a l . [1979] H 2 = (1.523±0.013)x10" 7 cm'1 Sengupta et a l . [1979] L 2 = (1,32±0.12)x10 _ 1 1 cm"1 Sengupta et a l . [1979] B 3 = 18.2995±0.0005 cm"1 Fishburne and Rao [1966] D 3 = (1.948±0.006)X10" 3 cm"1 Fishburne and Rao [1966] H 3 = (1.43±0.10)X10" 7 cm"1 Mann et a l . [1961] B„ = 17.5829±0.0007 cm"1 Fishburne and Rao [1966] D„ = (1.911±0.006)x10- 3 cm'1 Fishburne and Rao [1966] H, = (1.23±0.15)X10" 7 cm"1 Mann et a l . [1961] ************************************** v i b r a t i o n a l t r a n s i t i o n ) , and the r o t a t i o n c o n s t a n t s B 0, D 0, H 0, L 0 , B y, D y, H y, and L y . For example, Mann et a l . [1961] have given a power-series r e p r e s e n t a t i o n f o r the wavenumbers. The r e c e n t l y p u b l i s h e d v a l u e s of v a r i o u s v0, B y, D y, and H y f o r the (1-0), (2-0), (3-0), and (4-0) bands are l i s t e d i n Table 3.1. The most recent v a l u e s f o r the other r o t a t i o n c o n s t a n t s can be found i n Mann et a l . [1961], Fishburne and Rao [1966], and Di Lonardo and Douglas [1973], A set of c o n s i s t e n t molecular c o n s t a n t s i s r e q u i r e d f o r the c a l c u l a t i o n of the wavenumbers. The f a c t t h at the valu e s f o r B 0, D 0, H 0, and L 0 have been a c c u r a t e l y determined by Sengupta et a l . [1979], i m p l i e s t h a t only the f i v e molecular c o n s t a n t s v0, B 3, D 3, H 3, and L 3 remain to be determined from p u b l i s h e d experimental wavenumbers. The problem i s b a s i c a l l y a l e a s t - s q u a r e s problem to f i n d the f i v e c o n s t a n t s 89 which would minimise the sum of the squares of r e s i d u a l s between the experimental wavenumbers of the l i n e s and the c a l c u l a t e d v a l u e s . T h i s can be accomplished by o p t i m i s a t i o n methods. The experimental wavenumbers used f o r the (3-0) band are those from Fishburne and Rao [1966], A " s t a t e of the a r t " o p t i m i s a t i o n program has been used to determine optimal v a l u e s f o r the c o n s t a n t s . The theory of the o p t i m i s a t i o n method as w e l l as the program i s d e s c r i b e d i n Moore [1984]. A review on u s i n g the l e a s t - s q u a r e s method to estimate molecular c o n s t a n t s i s given by A l b r i t t o n et a l . [1976], As expected from the accuracy of the experimental data, the procedure g i v e s reasonable values f o r v0, B 3, and D 3 but not f o r H 3 and L 3 . A b e t t e r approach i s to d e r i v e the v a l u e s f o r H 3 and L 3 independently of the data and s o l v e f o r only vQ, B 3, and D 3 i n the o p t i m i s a t i o n p r o c e s s . With the improved v a l u e s of H 0, H,, and H 2 from Sengupta et a l . [1979], one can s o l v e f o r Y 0 3 , Y 1 3 , and Y 2 3 i n Equation 3.8 with terms up to i = 2. The terms H y f o r t>>3 can then be e v a l u a t e d u s i n g Equation 3.8. T h i s i s e s s e n t i a l l y s o l v i n g f o r H y as a f u n c t i o n of u, H 0, H 1 f and H 2. A s i m i l a r approximation formula f o r H 3 i n terms of only H 0 and H, has been given by Rank et a l . [1965]. A value of 1 . 4523x 1 0" 7cm." 1 i s subsequently d e r i v e d f o r H 3. S i m i l a r l y , a value of 1.39x10" 1 1cm _ 1 i s c a l c u l a t e d f o r L 3 u s i n g the improved molecular c o n s t a n t s from Sengupta et a l . [1979]. Most of the d e r i v e d r o t a t i o n c o n s t a n t s are w i t h i n the u n c e r t a i n t i e s of t h e i r experimental v a l u e s . T h i s i s a l s o the f i r s t time that L 3 i s determined. The method i s l i m i t e d by 90 having a c c u r a t e r o t a t i o n c o n s t a n t s only f o r u = 0, 1, and 2. Hence, t r u n c a t i o n e r r o r s are i n t r o d u c e d by using only up to the i=2 term i n the e q u a t i o n s . A f t e r adopting the d e r i v e d values f o r H 3 and L 3 , the o p t i m i s a t i o n procedure i s used to d e r i v e the other three c o n s t a n t s from the data. T h i s proves very s u c c e s s f u l . The f i n a l r e s u l t s f o r the three c o n s t a n t s are a l l compatible with t h e i r experimental v a l u e s given i n Table 3.1. The mean r e s i d u a l between the experimental wavenumbers and the c a l c u l a t e d v a l u e s , however, i s reduced from ±0.02 cm - 1 i n u s i n g the o l d molecular c o n s t a n t s to ±0.005 cm - 1 with the new molecular c o n s t a n t s . In f a c t , t h i s i s even s l i g h t l y s m a l l e r than the r e s i d u a l o b t a i n e d i n a f o r c e d f o u r t h order polynomial f i t of the data. A small improvement may be obtained i f one i n c l u d e s the experimental data from Sengupta et a l . [1979] on the (3-2) band i n the o p t i m i s a t i o n p r o c e s s . T h i s , however, w i l l add more unknowns i n t o the problem e.g. the (3-2) band c e n t r e . The f i n a l adopted v a l u e s f o r the molecular c o n s t a n t s are l i s t e d i n Table 3.2. Table 3.3 l i s t s both the experimental vacuum wavenumbers and the wavenumbers c a l c u l a t e d with the adopted c o n s t a n t s . The c o r r e s p o n d i n g J ^ , J ^ , and m v a l u e s f o r each l i n e are a l s o l i s t e d . The vacuum wavenumbers can be converted i n t o a i r wavelengths by knowing the r e f r a c t i v e index of the a i r . The a i r r e f r a c t i v e index, however, i s a f u n c t i o n of the wavelength, temperature, a i r p r e s s u r e , and water-vapour p r e s s u r e . The r e l e v a n t formulae are given i n Edlen [1966]. Table 3.4 l i s t s the s t a n d a r d - a i r wavelengths f o r both the 91 Table 3.2 Adopted molecular c o n s t a n t s f o r the (3-0) band ****************************************** = 1 1372.81 08 cm-1 B 0 = 20.559743 cm"1 D 0 = 2.12048xl0" 3 cm' H 0 =• 1 .6653x1 0- 7 c n r 1 L 0 = 1 .81x10" 1 1 cm'1 B 3 = 18.30016 cm"1 D 3 = 1 . 9 5 5 1 X 1 0 " 3 cm"1 H 3 - 1.4523x10- 7cm" 1 L 3 = 1 . 39x10" 1 1 cm"1 ************************************************************ Table 3.3 Vacuum wavenumbers f o r the (3-0) band of HF ************************************************************ J i J f m experimental c a l c u l a t e d 3 2 -3 1 1236. 129 cm"1 1 1236.1296 cm" 2 1 -2 11286.120 cm" 1~ 1 1 286.1211 cm" 1 0 -1 11331.707 cm" 1 1 1331.6998 cm" 0 1 1 11409.413 cm" 1 1 1409.4033 cm" 1 2 2 11441.428 cm"1 1 1441.4304 cm" 2 3 3 11468.841 cm" 1 1 1468.8493 cm" 3 4 4 11491.616 cm"1 1 1491.621 3 cm" 4 5 5 11509.714 cm"1 11509.71 1 9 cm" 5 6 6 11523.092 cm" 1 11523.0911 cm" 6 7 7 11531.735 cm' 1 11531.7330 cm" 7 8 8 11535.614 cm"1 11535.6160 cm" ************************************************************ 92 experimental and c a l c u l a t e d v a l u e s as w e l l as the c o r r e s p o n d i n g a i r r e f r a c t i v e i n d i c e s . The "standard a i r " or the s p e c t r o s c o p i c standard temperature and pressure (SSTP) i s at 760 t o r r s , 15°C, and zero water-vapour pressure (Edlen [1966]). Before the r a d i a l v e l o c i t y of a p a r t i c u l a r s t e l l a r l i n e can be obtained, one has to convert the measured p o s i t i o n of the s t e l l a r l i n e on the spectrum i n t o a wavelength measurement. The r e l a t i o n s h i p between p o s i t i o n s on the spectrum and wavelengths i s the d i s p e r s i o n r e l a t i o n . The HF l i n e s are the w a v e l e n g t h - c a l i b r a t i o n marks on the spectrum. Each HF l i n e p r o v i d e s a correspondence between one p o s i t i o n measurement and an a b s o l u t e wavelength. One of the s i m p l e s t ways of e x p r e s s i n g the d i s p e r s i o n r e l a t i o n i s i n terms of a polynomial f i t between the p o s i t i o n s and wavelengths of the HF l i n e s . The accuracy of a r a d i a l - v e l o c i t y d e t e r m i n a t i o n depends c r i t i c a l l y on how w e l l t h i s polynomial r e p r e s e n t s the d i s p e r s i o n r e l a t i o n . There are many causes that can a f f e c t the accuracy of the polynomial r e p r e s e n t a t i o n . Two of the most simple causes are the accuracy of the HF l i n e - p o s i t i o n measurements and the accuracy of t h e i r c o r r e s p o n d i n g a b s o l u t e wavelengths. The l a t t e r , of course, depends somewhat on the c r i t e r i a of how the l i n e p o s i t i o n s are determined. T h i s would be a systematic e f f e c t . Systematic z e r o - p o i n t e r r o r s may a l s o e x i s t i n the experimental v a l u e s from Fishburne and Rao [1966]. Meanwhile, random experimental e r r o r s i n these wavelength measurements would a l s o be i n t r o d u c e d i n t o the polynomial 9 3 Table 3.4 SSTP wavelengths f o r the (3-0) band of HF ******************************************* m exper imental c a l c u l a t e d r e f r a c t i v e index 8 8 6 6 6 . 4 2 5 6 A 8 6 6 6 . 4 2 4 1 A 1 . 0 0 0 2 7 4 6 7 7 8 6 6 9.3408A 8 6 6 9.3423A 1 . 0 0 0 2 7 4 6 7 6 8 6 7 5.8433A 8 6 7 5.8440A 1 . 0 0 0 2 7 4 6 7 5 8 6 8 5 . 9 2 7 5 A 8 6 8 5 . 9 2 9 1 A 1.00027466 4 8 6 9 9 . 6 0 6 9 A 8 6 9 9 . 6 0 2 9 A 1.00027466 3 8 7 1 6 . 8 8 2 8 A 8 7 1 6 . 8 7 6 5 A 1 . 0 0 0 2 7 4 6 5 2 8 7 3 7 . 7 6 8 0 A 8 7 3 7 . 7 6 6 2 A 1 . 0 0 0 2 7 4 6 4 1 8 7 6 2 . 2 8 6 4 A 8 7 6 2 . 2 9 3 9 A 1 . 0 0 0 2 7 4 6 3 - 1 8 8 2 2 . 3 7 3 1 A 8 8 2 2 . 3 7 8 8 A 1 . 0 0 0 2 7 4 6 0 - 2 8 8 5 8 . 0 0 8 7 A 8 8 5 8 . 0 0 7 8 A 1 . 0 0 0 2 7 4 5 8 - 3 8 8 9 7 . 4 1 9 3 A 8 8 9 7 . 4 1 8 9 A 1 . 0 0 0 2 7 4 5 7 ************************************************************ d i s p e r s i o n f i t . T h i s would be a random e f f e c t . The experimental values from Fishburne and Rao [1966] might a l s o c o n t a i n l i n e - t o - l i n e wavelength v a r i a t i o n s caused by pr e s s u r e - i n d u c e d d i f f e r e n t i a l s h i f t s . The amount of these s h i f t s i s not a p p r o p r i a t e f o r the p a r t i c u l a r o p e r a t i o n c o n d i t i o n of the HF c e l l . I t i s probably more s a t i s f a c t o r y to adopt the wavelengths c a l c u l a t e d from molecular c o n s t a n t s to be the a b s o l u t e r e f e r e n c e wavelengths. A p p r o p r i a t e s h i f t c o r r e c t i o n s can then be a p p l i e d to them. A very small z e r o - p o i n t e r r o r may s t i l l e x i s t i n the adopted r e f e r e n c e wavelengths. T h i s would be caused by z e r o - p o i n t e r r o r i n the experimental data of Fishburne and Rao [1966]. For most a p p l i c a t i o n s and e s p e c i a l l y i n r e l a t i v e r a d i a l - v e l o c i t y 94 works, however, the adopted val u e s should be adequate. F u r t h e r improvement on the r e f e r e n c e wavelengths can be achieved when l i n e - t o - l i n e p r e s s u r e - i n d u c e d l i n e s h i f t s are taken i n t o account. T h i s i s e s p e c i a l l y the case f o r a p p l i c a t i o n at the 360 t o r r s p r e s s u r e of the HF c e l l . Small l i n e s h i f t s caused by s l i g h t temperature and pressure d e v i a t i o n s from the mean o p e r a t i o n c o n d i t i o n of the c e l l can a l s o be c o r r e c t e d . T h i s p o i n t w i l l be d i s c u s s e d i n d e t a i l l a t e r i n the ch a p t e r . A t h i r d order polynomial d i s p e r s i o n f i t was a p p l i e d to a t y p i c a l HF spectrum taken at the CFHT Coude. The r e c i p r o c a l d i s p e r s i o n was 4.8A/mm or 0 . 0 7 1 3 A / p i x e l . A rms r e s i d u a l of ± 0 . 0 0 2 2 4 A was o b t a i n e d f o r the f i t when the experimental v a l u e s from Table 3.4 were used as the re f e r e n c e wavelengths. An improved rms r e s i d u a l of ± 0 . 0 0 1 5 5 A was obtained when the c a l c u l a t e d v a l u e s from Table 3.4 were used as the r e f e r e n c e wavelengths. The improved f i t may imply an improved r e p r e s e n t a t i o n of the d i s p e r s i o n r e l a t i o n . 3 . 3 THE TEMPERATURE AND PRESSURE OF THE HF GAS 3.3.1 INTRODUCTION One of the t e c h n i c a l d i f f i c u l t i e s with the HF technique i s the i n a b i l i t y to measure the temperature and pre s s u r e of the HF gas a c c u r a t e l y i n s i d e the c e l l . The temperature measured with the thermocouple i s only the mean o u t s i d e s u r f a c e temperature of the c e l l . The temperature of the HF gas i n s i d e the c e l l can be q u i t e d i f f e r e n t . Moreover, there 95 may not be a uniform temperature p r o f i l e on the c r o s s - s e c t i o n of the c e l l . T h i s c r o s s - s e c t i o n a l temperature p r o f i l e may a l s o be v a r i a b l e along the length of the c e l l . T h i s i s e s p e c i a l l y the case near the windows as w e l l as the l o c a t i o n where c o l d e r HF molecules are int r o d u c e d i n t o the heated c e l l . Hence, depending on the p a r t i c u l a r path of the s t e l l a r beam through the c e l l , the e f f e c t i v e temperature of the HF gas forming the HF r e f e r e n c e l i n e s can be d i f f e r e n t . T h i s d i f f e r e n c e may depend on many f a c t o r s such as alignment of the c e l l with re s p e c t to the s t e l l a r beam, the atmosphere and Coude seei n g , temperature s t a b i l i t y of the c e l l , as w e l l as the focus of the beam. The measured pressure i s s u b j e c t e d to z e r o - p o i n t e r r o r of the manometer as w e l l as temperature v a r i a t i o n s of the i c e bath around the sample l i n e . I t i s important to maintain a constant temperature f o r the HF gas throughout any long-term HF p r o j e c t . Assuming the usual temperature-dependent Boltzmann d i s t r i b u t i o n f o r the p o p u l a t i o n at the v a r i o u s energy l e v e l s of HF, a change in temperature would a f f e c t the l i n e s t r e n g t h of the HF l i n e s . Many numerical techniques to reduce the o b s e r v a t i o n a l data e.g. to determine r e l a t i v e HF l i n e p o s i t i o n s , would be a f f e c t e d by having t o compare HF l i n e s which are d i f f e r e n t i n s t r e n g t h and shape. The p r e s s u r e - i n d u c e d l i n e s h i f t s of the HF l i n e s are a l s o s e n s i t i v e t o temperature v a r i a t i o n s . T h i s can be e x p l i c i t l y due to the v a r i a t i o n s i n the Maxwellian v e l o c i t y d i s t r i b u t i o n as w e l l as i m p l i c i t l y from the v a r i a t i o n s i n the energy l e v e l p o p u l a t i o n . The temperature, of course, would a l s o a f f e c t the r e l a t i v e 96 number of monomers (HF), and hexamers ( H F ) 6 . T h i s a f f e c t s both the l i n e s t r e n g t h as w e l l as the p r e s s u r e s h i f t s . A l l these problems can be minimised i f one can determine the temperature and pressure of the HF gas d i r e c t l y from the observed spectrum. With known temperature and p r e s s u r e , l i n e - s h i f t c o r r e c t i o n s can be made at the time of the data reduct i o n . 3.3.2 HF LINE STRENGTH 3.3.2.1 Basi c equations The l i n e s t r e n g t h of a v i b r a t i o n - r o t a t i o n l i n e per unit-atmosphere gas pressure and i n u n i t s of (atm.cm 2)" 1 has been given by Meredith [1972b] and Pugh and Rao [1976] as: S(m) = { ( 1 .01 325X1 0 6 ) 8 i r 3 | m | y N / (3hcpZ) } { exp( - E ( u i , J i ) / k T ) |<u. J . |/z(r) | u f J f>| 2 } (3.10) c = speed of l i g h t h = Planck constant k = Boltzmann's constant T = temperature i n k e l v i n v = wavenumber of l i n e m = the running index of the l i n e N = number of absorbing molecules per cm 3 P = gas pressure i n atmospheres Z = t o t a l p a r t i t i o n f u n c t i o n E(u.,J.) = energy l e v e l of the lower s t a t e 97 u(r) = e l e c t r i c d i p o l e moment f u n c t i o n Equation 3.10 uses the convention of s u b s c r i p t i (or double primes) and s u b s c r i p t f (or s i n g l e prime) to i n d i c a t e the lower and upper energy l e v e l , r e s p e c t i v e l y . The D i r a c bra < | and ket | > r e p r e s e n t a t i o n s are a l s o used to express the matrix element of the e l e c t r i c d i p o l e moment. Equation 3.10 does not take i n t o account the e f f e c t s of induced emission by not i n c l u d i n g the f a c t o r {1-exp(-hev/kT)}. T h i s f a c t o r , however, i s very c l o s e to 1 f o r v i b r a t i o n - r o t a t i o n t r a n s i t i o n s i n the i n f r a r e d and at moderate temperatures (Pugh and Rao [1976]). 3.3.2.2 The Herman-Wallis f a c t o r s Herman and W a l l i s [1955] have p o i n t e d out that the e l e c t r i c d i p o l e matrix elements |<v^J^ | u ( r ) | v > | 2 can be w r i t t e n as the product of a v i b r a t i o n a l f a c t o r and a v i b r a t i o n - r o t a t i o n i n t e r a c t i o n f u n c t i o n F(v^,v^,m): \<viJi\n(r)\vfJf>\2 = | < u i | M ( r ) | y f > | 2 F ( o . , » f , m ) (3.11) The term F(v^,v^,m) i s the Herman-Wallis f a c t o r or F f a c t o r . I t can be c a l c u l a t e d d i r e c t l y from the r a t i o between the t h e o r e t i c a l v i b r a t i o n - r o t a t i o n and r o t a t i o n e l e c t r i c d i p o l e matrix elements. T h i s method has been d i s c u s s e d by Bouanich [1977,1978b] who has a l s o given t h e o r e t i c a l e x p r e s s i o n s f o r the matrix elements. D i r e c t formulae f o r the f a c t o r s were f i r s t g iven by Herman et a l . [1958] f o r the (1-0), (2-0), (3-0), and (2-1) 98 Table 3.5 The a.'s and the M i's f o r HF **************************************** a, = = -2.2538 a 2 = = 3.4882 a 3 = = -4.4986 a„ = = 4.704 a 5 = = -2.91 a 6 = = -1.76 M 0 = = 1.80306 M, = = 1.39366 M 2 = = -0.0583 M 3 = = -0.8861 M« = = -0.599 M 5 = = -0.931 ************************************************************ t r a n s i t i o n s . Improvements on these formulae have been given by Toth et a l . [1969,1970], T i p p i n g and Herman [1970a], T i p p i n g and Forbes [1971], Meredith [1972b], T i p p i n g [1976], O g i l v i e et a l . [1980], T i p p i n g and O g i l v i e [1982], and O g i l v i e and T i p p i n g [1983]. S e v e r a l of these recent p u b l i c a t i o n s a l s o give formulae f o r the higher-overtone t r a n s i t i o n s . The f a c t o r can be expressed in terms of a power s e r i e s expansion i n m : F ( u i f u f , m ) = 1 + C ( u i , u f ) m + D ( u i f u f ) m 2 (3.12) The f u n c t i o n s C(v^.v^) and B(v^.v^) are expressed i n terms of the e q u i l i b r i u m i n t e r n u c l e a r s e p a r a t i o n r g , the Dunham p o t e n t i a l c o n s t a n t s a^'s, <Uj | n( r) | u ̂>, B^, u>e , and the 99 M^'s. The M^'s are the c o e f f i c i e n t s i n the power s e r i e s expansion of the e l e c t r i c d i p o l e moment: n(r) = I. M i [ ( r - r e ) / r g ] i (3.13) The v a l u e s f o r these 's have been ev a l u a t e d by S p e l l i c y et a l . [1972], Meredith and Smith [1973], L i e [1974], S i l e o and Cool [1976], and O g i l v i e et a l . [1980]. Meanwhile, the a^'s have been e v a l u a t e d by Mann et a l . [1961], Webb and Rao [1968], and O g i l v i e and Koo [1976]. Table 3.5 l i s t s the value s of the M i's from O g i l v i e et a l . [1980] as w e l l as the va l u e s of the a^'s from O g i l v i e and Koo [1976]. The v i b r a t i o n a l f a c t o r <v^\n(r) |i>f> can a l s o be expressed i n terms of these c o n s t a n t s : <vL\u(r) | i>f> = Li M i<u i | ( r / r e ) 1 | u f> (3.14) Formulae f o r the e x p e c t a t i o n v a l u e s <v^ | ( r / r ? ) 1 | u ^ > i n terms of B g, cjg, the a^'s, and M^'s are given i n T i p p i n g [1973]. More general e x p r e s s i o n s can be found i n Niay et a l . [1979]. The Herman-Wallis f a c t o r s have not been e x p l i c i t l y p u b l i s h e d f o r the (3-0) band of HF. One, however, can d e r i v e t h e i r v a l u e s from the r a t i o s between the experimental <v^\n(r)\v^> and < v ^ \ u ( r ) \ v ^ > values measured by S p e l l i c y et a l . [1972]. These are l i s t e d i n Table 3.5. Appl y i n g a l e a s t - s q u a r e s o p t i m i s a t i o n procedure on the experimental data, o p t i m a l v a l u e s of -1 .0788xl0- 2 and 6.3528x10-* are determined f o r C(i > i f u f ) and D(u^,y^), r e s p e c t i v e l y . The l e a s t - s q u a r e s f i t t e d f a c t o r s are a l s o l i s t e d i n Table 3.5. T h e o r e t i c a l values 100 Table 3.6 Herman-Wall i s f a c t o r s f o r the (3-0) band of HF ********************************************** m experimental l e a s t - s q u a r e s f i t t e d -4 1 .0676 1.0533 -3 1 .0301 1.0381 -2 1 .0086 1.0241 -1 1.0068 1.0114 1 0.9969 0.9899 2 0.9908 0.9810 3 0.9601 0.9734 4 0.9920 0.9670 5 0.9386 0.9619 6 0.9668 0.9581 7 0.9546 0.9556 ************************************************************ f o r the Herman-Wallis f a c t o r s have been c a l c u l a t e d u sing the formulae given independently i n T i p p i n g and O g i l v i e [1982], T i p p i n g and Forbes [1971], and Toth et a l . [1969]. The agreement between the t h e o r e t i c a l and experimental v a l u e s i s not very s a t i s f a c t o r y . T h i s may be caused by the inadequacies of the t h e o r i e s . The experimental values may a l s o c a r r y s i z e a b l e e r r o r s . In view of these, choosing the l e a s t - s q u a r e s f i t t e d f a c t o r s f o r the c a l c u l a t i o n of gas temperature may be more a p p r o p r i a t e . 101 3.3.2.3 D e r i v a t i o n of gas temperature For two l i n e s , m, and m2, i n the same v i b r a t i o n - r o t a t i o n band, the r a t i o between t h e i r l i n e s t r e n g t h as given by Equation 3.10 w i l l simply be: Sdn,) / S(m 2) = A B exp( [E 2 - E , ] / k T ) (3.15) A = {|m,|p.} / {|m2|p2] (3.16) B = F ( u i f u f f m , ) /»F( v i, o f,m 2) (3.17) The m v a l u e s , Herman-Wallis f a c t o r s , and the wavenumbers v are known f o r the (3-0) band of HF. The d i f f e r e n c e i n energy, E 2 - E , , between the l e v e l of the two J s t a t e s i n the ground v i b r a t i o n a l l e v e l i s simply the d i f f e r e n c e between the corresponding v a l u e s of i n Equation 3.3 m u l t i p l i e d by he. Values f o r B 0, D 0, H 0, and L 0 are a c c u r a t e l y known. Hence, i f the two l i n e s have d i f f e r e n t v a l u e s of J ^ , the temperature T can be c a l c u l a t e d from Equation 3.15. The Beer-Lambert law g i v e s the r e l a t i o n which d e s c r i b e s the t r a n s m i s s i o n of r a d i a t i o n through a homogeneous gas as: I(p) = I 0 exp( -K(p)pL ) (3.18) The terms I 0 and l(i>) are the continuum and l i n e i n t e n s i t y , r e s p e c t i v e l y . The f u n c t i o n K(P) i s the ab s o r p t i o n c o e f f i c i e n t . The term L i s the l e n g t h of the ab s o r p t i o n path, and p i s the pres s u r e of the absorbing gas. The l i n e shape depends on the form of K(P). E x p r e s s i o n s f o r K(P) i n the cases of Gaussian, L o r e n t z i a n , and V o i g t p r o f i l e s have been given by Mandin et a l . [1980]. The l i n e s t r e n g t h S i s simply the 1 02 i n t e g r a t e d a b s o r p t i o n c o e f f i c i e n t (Pugh and Rao [1976], Lowry and F i s h e r [1982], Overend [1982]): S = ; l i n e K(U) d» (3.19) The r e l a t i o n s h i p between l i n e s t r e n g t h and e q u i v a l e n t width i s given i n Overend [1982] and Korb et a l . [1968]. By c o n s i d e r i n g the d i s p e r s i o n of the spectrum, the i n t e g r a l i n Equation 3.19 can be performed over the l i n e l i m i t s i n p i x e l number. The d i s p e r s i o n i s simply the - d e r i v a t i v e of the polynomial f i t between the wavenumbers of the l i n e s and the c o r r e s p o n d i n g l i n e p o s i t i o n s . T h i s together with Equations 3.18 and 3.19 g i v e s : S = (-1/pL) J l i n e L n ( l ( x ) / I 0 ) |df/dx| dx (3.20) The f u n c t i o n I ( x ) / l 0 i s simply the r e c t i f i e d spectrum. Equation 3.20 can be used "with Equation 3.15 to determine the temperature of the gas. The c o n s t a n t s p and L are c a n c e l l e d from the equation when, i n the temperature d e t e r m i n a t i o n , one c o n s i d e r s only the r a t i o s between the l i n e s t r e n g t h s . Average temperatures of 360.9±1.4K and 368.2±4.3K have been c a l c u l a t e d f o r the two s e t s of l i n e s t r e n g t h s p u b l i s h e d i n S p e l l i c y et a l . [1972]. These agree with the assumed temperature of about 100°C. The procedure has a l s o been a p p l i e d to a t y p i c a l HF spectrum o b t a i n e d from the CFH Coude. The l i n e l i m i t s f o r a l l the HF l i n e s have been set at p o s i t i o n s of only 0.6% a b s o r p t i o n with r e s p e c t to the continuum. A temperature of 373.7±0.7K i s o b t a i n e d . T h i s agrees very w e l l with the expected temperature of 100°C. The 1 03 accuracy of the procedure depends c r i t i c a l l y on the accuracy of the measured l i n e s t r e n g t h s which are e s s e n t i a l l y the e q u i v a l e n t widths of the l o g a r i t h m of the l i n e s . The main source of e r r o r i s i n the placement of the continuum. T h i s i s hampered mainly by the c o l l i s i o n a l l y broadened l i n e p r o f i l e s . Blendings at the f a r wings of the l i n e s may become s i g n i f i c a n t and would r e s t r i c t the c h o i c e of l i n e l i m i t s . C o r r e c t i o n curves f o r the e f f e c t s on measured l i n e s t r e n g t h s and widths from i n s t r u m e n t a l broadening have been generated by Meredith [1972]. An i t e r a t i v e method to c o r r e c t f o r the l o s s of the f a r wings of the l i n e i n the measurement of l i n e s t r e n g t h and e q u i v a l e n t width has been d e s c r i b e d i n Korb et a l . [1968] and G i v e r et a l . [1982]. Pugh and Rao [1976] and Overend [1982] have suggested the use of curve-of-growth procedure to measure the l i n e s t r e n g t h s . N e v e r t h e l e s s , s i n c e only the r a t i o s between l i n e s t r e n g t h s are used by the present procedure, any m u l t i p l i c a t i v e systematic e r r o r i n the e q u i v a l e n t - w i d t h measurements would have no e f f e c t on the r e s u l t . Meanwhile, the use of other Herman-Wallis f a c t o r s e.g. those t h e o r e t i c a l v a l u e s from O g i l v i e and T i p p i n g [1982], c o u l d i n c r e a s e the mean temperature by as much as 10K. In s p i t e of t h i s , to monitor the r e l a t i v e changes i n the temperature, the present procedure i s probably adequate. The d i f f e r e n c e between the e q u i v a l e n t width of l i n e s can be more e a s i l y measured from a d i f f e r e n c e 104 spectrum (Campbel l [1984a] ) . C o n s e q u e n t l y , s t r e n g t h s of HF l i n e s in a p a r t i c u l a r spectrum can be measured by examining the d i f f e r e n c e spectrum produced by s u b t r a c t i n g a "s tandard" HF spec trum. The HF l i n e s i n the "s tandard" spectrum would have known e q u i v a l e n t widths and l i n e s t r e n g t h s . R e l a t i v e l i n e s t r e n g t h s can then be c a l c u l a t e d w i t h r e s p e c t to these "s tandard" l i n e s . These can then be used to determine a temperature r e l a t i v e to that c a l c u l a t e d for the "s tandard" HF s p e c t r u m . 3 . 3 . 2 . 4 D e r i v a t i o n of the gas p r e s s u r e The HF gas p r e s s u r e p i n E q u a t i o n 3.20 can be e v a l u a t e d i f one can c a l c u l a t e the t h e o r e t i c a l l i n e s t r e n g t h per u n i t - a t m o s p h e r e u s i n g E q u a t i o n s 3.10 and 3 . 1 1 . By assuming i d e a l gas law for the HF gas , the term N / p i n E q u a t i o n 3.10 would become 1 / k T . The t o t a l p a r t i t i o n f u n c t i o n Z i n E q u a t i o n 3.10 can be expres sed a s : Z(T) = LvLj (2J+1) exp( - E ( o , J ) / k T ) (3 .21) The energy l e v e l E ( u , J ) can be c a l c u l a t e d from the term v a l u e o b t a i n e d through E q u a t i o n 3.4 u s i n g the c a l c u l a t e d Dunham c o e f f i c i e n t s . In E q u a t i o n 3 . 1 1 , the term <t>^|M (r ) |v j> can be c a l c u l a t e d from E q u a t i o n 3.14 t o g e t h e r wi th formulae for <v^ \(r/rg)1|v^> which are g i v e n i n T i p p i n g [1973] . A v a l u e of 1 . 6 0 9 5 x l 0 ~ 2 1 esu.cm (1 debye i s 1 0 " 1 8 esu.cm) i s o b t a i n e d when the v a r i o u s c o n s t a n t s l i s t e d i n T a b l e 3 .5 are used i n the 105 Table 3.7 L i n e s t r e n g t h s of the (3-0) band of HF ***************************************** m experimental. c a l c u l a t e d -3 0.0235 cm" 2atm" 1 0.02285 cm" 2atm" -2 0.0245 cm" 2atm" 1 0.02467 cm" 2atm" - 1 0.0164 cm" 2atm" 1 0.01697 cm" 2atm" 1 0.0193 cm" 2atm" 1 0.01970 cm- 2atrrr 2 0.0327 cm" 2atm" 1 0.03324 cm" 2atm" 3 0.0353 cm" 2atm" 1 0.03574 cm" 2atm" 4 0.0287 cm- 2atm" 1 0.02903 cm" 2atm" 5 0.0188 cm" 2atm" 1 0.01881 cm" 2atm" 6 0.0099 cm" 2atm" 1 0.00996 cm" 2atm" 7 0.0039 cm" 2atm" 1 0.00437 cm" 2atm" ************************************************************ c a l c u l a t i o n . T h i s agrees with the experimental value of 1.628x10" 2 1 esu.cm measured by S p e l l i c y et a l . [1972]. Other experimental and t h e o r e t i c a l v a l u e s f o r t h i s matrix element can be found r e f e r e n c e d i n Werner and Rosmus [1980]. The minute e f f e c t of induced emission on the l i n e s t r e n g t h can a l s o be taken i n t o account by i n c l u d i n g the f a c t o r {1-exp(-hc?/kT)} i n t o Equation 3.10. With a l l these terms known, t h e o r e t i c a l l i n e s t r e n g t h s can then be c a l c u l a t e d as a f u n c t i o n of temperature. Table 3.7 l i s t s both the experimental and c a l c u l a t e d l i n e s t r e n g t h s per u n i t atmosphere f o r the (3-0) band of HF. The experimental v a l u e s are those given i n S p e l l i c y et a l . [1972] corresponding to the 0.467atm d a t a . The c a l c u l a t e d v a l u e s used the 106 temperature of 360.9K obtained from the temperature-determination method d e s c r i b e d e a r l i e r . The agreement between the two s e t s of values i n Table 3.7 i s very good. Pres s u r e of the HF gas can be determined by comparing the measured l i n e s t r e n g t h a g a i n s t the c o r r e s p o n d i n g t h e o r e t i c a l unit-atmosphere l i n e s t r e n g t h . The procedure has been a p p l i e d to the "standard" CFHT HF spectrum used e a r l i e r i n the temperature d e t e r m i n a t i o n . In f a c t , t h a t p a r t i c u l a r determined temperature i s used i n t h i s c a l c u l a t i o n . A value of 300.4±0.7 t o r r s i s o b t a i n e d f o r the gas p r e s s u r e . T h i s i s about 17% lower than the expected value of 360 t o r r s . The small s t a t i s t i c a l standard e r r o r a s s o c i a t e d with the determined value as w e l l as the e x c e l l e n t r e s u l t o b t a i n e d e a r l i e r with the r a t i o s of the l i n e s t r e n g t h s i n the temperature d e t e r m i n a t i o n , would s t r o n g l y suggest that the problem here i s the e x i s t e n c e of a m u l t i p l i c a t i v e s ystematic e r r o r f a c t o r . The f a c t t h a t t h e r e i s a good agreement between the t h e o r e t i c a l c a l c u l a t i o n s and the r e s u l t of S p e l l i c y et a l . [1972], i m p l i e s the source of e r r o r i s probably i n the measured l i n e s t r e n g t h s . The r e l a t i v e e r r o r i n the e q u i v a l e n t - w i d t h measurements caused by s c a t t e r e d l i g h t i s of the order of only 3% f o r the CFH Coude sp e c t r o g r a p h . Systematic e r r o r s i n the placement of continuum f o r the spectrum c o u l d not be the s o l e cause f o r the l a r g e d i s c r e p a n c y . The e r r o r caused by the f a c t 107 that the chosen l i n e l i m i t s are not at the true continuum should be s m a l l . Furthermore, s i n c e the l o g a r i t h m of the spectrum i s used in the computation of the l i n e s t r e n g t h s , the small m u l t i p l i c a t i v e e r r o r f a c t o r i n the continuum placement would become an a d d i t i v e f a c t o r . One of the m u l t i p l i c a t i v e f a c t o r s used i n the c a l c u l a t i o n of the l i n e s t r e n g t h s i s the d i s p e r s i o n of the spectrum. However, i t i s d i f f i c u l t to understand the e x i s t e n c e of any l a r g e e r r o r i n t h i s parameter. Another m u l t i p l i c a t i v e f a c t o r used i n the c a l c u l a t i o n i s the o p t i c a l a b s o r p t i o n path l e n g t h of the c e l l . The chosen value of 100 cm i s the measured l e n g t h of the a b s o r p t i o n c e l l . T h i s assumption that the e f f e c t i v e a b s o r p t i o n path l e n g t h i s the same as the l e n g t h of the c e l l , however, may not be q u i t e c o r r e c t . The c o l d e r HF molecules near the windows as w e l l as the ones f r e s h l y i n t r o d u c e d i n t o the heated c e l l from the sample l i n e would be (HF) 6 hexamers r a t h e r than monomers. They would then not c o n t r i b u t e towards the observed a b s o r p t i o n l i n e s . I t i s c o n c e i v a b l e that they may be i n s i g n i f i c a n t numbers i n the c e l l near the windows and the l o c a t i o n where the sample l i n e i s connected. T h i s has the e f f e c t of reducing the e f f e c t i v e a b s o r p t i o n path l e n g t h , and hence reducing the observed l i n e s t r e n g t h s . Although a b s o l u t e pressure d e t e r m i n a t i o n using t h i s l i n e s t r e n g t h method i s not p o s s i b l e , r e l a t i v e p r essure d e t e r m i n a t i o n s should be p o s s i b l e . One can use the  109 method to monitor the v a r i a t i o n s i n the gas pressure with re s p e c t to that of a "standand" HF spectrum. The r e l a t i v e temperature dependence of the l i n e s t r e n g t h s f o r the v a r i o u s HF l i n e s can be s t u d i e d with the t h e o r e t i c a l l i n e - s t r e n g t h c a l c u l a t i o n s . F i g u r e 3.2' shows the l i n e - s t r e n g t h v a r i a t i o n s with temperatures f o r the (3-0) band of HF. At the c e l l o p e r a t i o n temperature of about 100°C, the HF l i n e s of |m|<5 would decrease i n s t r e n g t h with temperature while l i n e s of m>6 would i n c r e a s e i n s t r e n g t h with temperature. Meanwhile, the p r e s s u r e of the gas i s d i r e c t l y p r o p o r t i o n a l to the measured l i n e s t r e n g t h s . For each i n d i v i d u a l l i n e , the r a t i o between the measured l i n e s t r e n g t h and p r e s s u r e would be equal to the corresponding t h e o r e t i c a l unit-atmosphere l i n e s t r e n g t h . T h i s value i s constant f o r a given temperature. 3.4 THE COLLISIONALLY BROADENED LINEWIDTHS 3.4.1 INTRODUCTION The p r e s s u r e of the HF gas can a l s o be determined from the l i n e w i d t h s of the HF l i n e s . T h i s i n v o l v e s c a l c u l a t i n g the t h e o r e t i c a l c o l l i s i o n a l l y broadened l i n e w i d t h s . The broadening i s a f u n c t i o n of both the temperature and p r e s s u r e of the gas. The amount of broadening is- d i f f e r e n t f o r each HF l i n e . A very much r e l a t e d phenomenon i s the c o l l i s i o n a l l y induced l i n e s h i f t s . One needs to know these s h i f t s as f u n c t i o n s of both temperature and p r e s s u r e i n 1 10 order to make c o r r e c t i o n s to the ref e r e n c e wavelengths. The s h i f t s , to a c e r t a i n e x t e n t , determine the u l t i m a t e accuracy of the r e f e r e n c e wavelengths. They are d i f f e r e n t i n both d i r e c t i o n and magnitude f o r each r e f e r e n c e HF l i n e . In chemistry, the study of l i n e w i d t h s and s h i f t s can be used to determine the i n t e r m o l e c u l a r f o r c e s between the c o n s t i t u e n t s of the gas e.g. the e l e c t r i c quadrupole and oct o p o l e moments of many molecules have been determined i n t h i s manner. The a p p l i c a t i o n of t h e o r e t i c a l broadened l i n e w i d t h s of molecules i s numerous. The CO c o n c e n t r a t i o n and temperature i n combustion exhaust can be determined from the broadened l i n e w i d t h s (Lowry and F i s h e r [1982]). I t i s a l s o important to the design and o p e r a t i o n of molecular l a s e r s , the study of p o l l u t a n t s i n the atmosphere, as w e l l as the study of r a d i a t i v e t r a n s f e r i n t e r r e s t r i a l and p l a n e t a r y atmospheres. Line-broadening theory has been a p p l i e d to determine the ozone d e n s i t y d i s t r i b u t i o n i n the atmosphere (Monnanteuil and Colmont [1983]). Abundances of helium i n J u p i t e r can be determined from the helium c o l l i s i o n a l l y broadened l i n e w i d t h s of CH 3 l i n e s (Varanasi [1971]). The p r e s s u r e and temperature i n the J o v i a n atmosphere can a l s o be determined from the broadening of the methane l i n e s ( M a r g o l i s [1971]). Meanwhile, c o l l i s i o n a l l y broadened HC1 and HF l i n e w i d t h s i n C0 2 atmospheres have been s t u d i e d f o r a p p l i c a t i o n on Venus (Shaw and L o v e l l [1969], V a r a n a s i et a l . [1971]) . 111 3.4.2 THEORIES ON COLLISIONAL LINE BROADENING The shape of s p e c t r a l l i n e s can be a l t e r e d by the i n t e r a c t i o n of each i n d i v i d u a l r a d i a t i n g molecule with other surrounding molecules. Line-broadening e f f e c t s have g e n e r a l l y been d i v i d e d i n t o two l i m i t i n g c a ses: the low-pressure impact ( i n t e r r u p t i o n ) broadening and the hi g h - p r e s s u r e s t a t i s t i c a l broadening. S t a t i s t i c a l broadening theory was f i r s t given by Margenau [1935]. Impact broadening, however, was f i r s t c o n s i d e r e d by Michelson [1895] and Lorentz [1906]. E a r l y impact t h e o r i e s are summarised and expanded in F o l e y [1946] and Mizushima [1951], These e a r l y impact t h e o r i e s assume the r a d i a t i n g molecule to be a c l a s s i c a l o s c i l l a t o r and that the c o l l i s i o n s are strong enough that the o s c i l l a t i o n process i s i n t e r r u p t e d . Formulations i n formal quantum-mechanical framework have been given by Baranger [I958ab], Kolb and Griem [1958], and Fano [1963]. Recent f o r m u l a t i o n s using the b i n a r y - c o l l i s i o n approximation have been given by Ross [1966], Gordon [1966], B e z z e r i d e s [I967ab], Murphy and Boggs [1967ab,1968], Z a i d i [1968], Di Giacomo and T a r r i n i [1970], Smith et a l . [I971ab], N i e l s e n and Gordon [1973], Smith et a l . [1976], Lam [1977], Boulet and Robert [1978], and Clough et a l . [1983]. More e l a b o r a t e f o r m u l a t i o n s i n v o l v i n g the use of s t a t i s t i c a l mechanics and quantum many-body techniques have been given by Z a i d i [1972], Davies [1975], Roney [I975ab], and Davies and O l i [1978], Recent reviews on l i n e broadening t h e o r i e s i n c l u d e Boggs [1972], R a b i t z [1974], L e a v i t t [ 1980], Breene [1981], L e a v i t t and K o r f f [1981 ], and 1 1 2 Bu f f a and T a r r i n i [1983]. Anderson [1949] a p p l i e d p e r t u r b a t i o n methods to a s e m i c l a s s i c a l impact theory and produced computational e x p r e s s i o n s f o r general pressure broadening. T h i s was l a t e r extended by Tsao and Curnutte [1962], Robert et a l . [1969], L e a v i t t [1980], and L e a v i t t and K o r f f [1981]. Hewitt [1976], F r o s t [1976], and Boulet et a l . [1977] extended the Anderson theory to permit c a l c u l a t i o n of l i n e s h i f t s . E a r l y extensions of the impact theory to c a l c u l a t e noble-gas p r e s s u r e - i n d u c e d l i n e w i d t h s and s h i f t s i n c l u d e Ben-Reuven et a l . [1963], Herman [1963], T i p p i n g and Herman [1970b], Herman and T i p p i n g [1970], Levy et a l . [1973], Boulet et a l . [1973], and J a r e c k i and Herman [1975]. The Anderson-Tsao-Curnutte (ATC) theory i s the most widely used f o r the computation of c o l l i s i o n a l l y broadened l i n e w i d t h s . The MB method proposed by Murphy and Boggs [1967a] produces l i n e w i d t h s which are i n b e t t e r agreement with experimental r e s u l t s than the e a r l y ATC formalism. The l a t e r m o d i f i e d ATC t h e o r i e s ( F r o s t and M a c G i l l i v r a y [1977], L e a v i t t and K o r f f [1981]), however, can produce as good i f not b e t t e r agreement. The ATC theory i s a l s o g e n e r a l l y c o n s i d e r e d to be founded on more a c c e p t a b l e p h y s i c a l premises than the MB theory. The Quantum F o u r i e r Transform (QFT) method proposed by Davies [1975] has not been as widely used as the other two methods. Mandin et a l . [1980] have a p p l i e d a l l three methods to the c a l c u l a t i o n of se l f - b r o a d e n e d H 20 l i n e s . They found that there i s not much d i f f e r e n c e between the ATC and QFT methods, while the 1 1 3 r e s u l t s from the MB method are s y s t e m a t i c a l l y too low. F r o s t and M a c G i l l i v r a y [1977] have a l s o found that the MB method g i v e s poorer c a l c u l a t e d l i n e s h i f t s . The m o d i f i e d MB method ( C a t t a n i [1972], F r o s t and M a c G i l l i v r a y [1977]), however, i s comparable to the m o d i f i e d ATC method i n accuracy. L e a v i t t and K o r f f [ 1981 ] have shown that e a r l y p h a s e - s h i f t t h e o r i e s (Van Vleck and Weisskopf [1945]), the s e m i c l a s s i c a l theory of Gordon [1966], and the MB t h e o r i e s are a l l l i m i t i n g cases of the m o d i f i e d ATC theory. 3.4.3 THE ANDERSON-TSAO-CURNUTTE (ATC) THEORY 3.4.3.1 Basic approach The ATC theory uses a s e m i c l a s s i c a l p e r t u r b a t i o n approach with the binary-impact approximation. In the impact approximation, the d u r a t i o n of the c o l l i s i o n s which a f f e c t the s t a t e s of the absorbing (or r a d i a t i n g ) molecules by s h i f t i n g t h e i r phases or causing t r a n s i t i o n s i s assumed to be short i n comparision with the time, i n t e r v a l between s u c c e s s i v e c o l l i s i o n s . T h i s enables one to t r e a t the i n d i v i d u a l molecular c o l l i s i o n s as u n c o r r e l a t e d with the r a d i a t i v e p r o c e s s , s i n c e they occur on d i f f e r e n t time s c a l e s . The molecular t r a n s l a t i o n would then be decoupled from the i n t e r n a l degrees of freedom of the molecules. Quantum mechanics can then be used to t r e a t the i n t e r n a l degrees of freedom while the t r a n s l a t i o n a l motion i s t r e a t e d c l a s s i c a l l y . A s t r a i g h t - l i n e c l a s s i c a l t r a j e c t o r y i s g e n e r a l l y used f o r the c o l l i s i o n . T h i s procedure i s 1 1 4 q u i t e common f o r most s e m i c l a s s i c a l t h e o r i e s . Smith et a l . [1976], however, have proposed the use of a curved c l a s s i c a l t r a j e c t o r y . The t i m e - e v o l u t i o n operator ( c o l l i s i o n matrix) which d e s c r i b e s the c o l l i s i o n dynamics i s expanded by p e r t u r b a t i o n theory i n the ATC approach to o b t a i n a simple and p r a c t i c a l computational procedure. The phenomenon of resonance broadening i s co n s i d e r e d i n the ATC theory. Normally, the p e r t u r b e r molecules are d i s t i n g u i s h a b l e from the r a d i a t o r molecules as i n the case of f o r e i g n - g a s broadening. In the case of s e l f - b r o a d e n i n g , however, e x c i t a t i o n can be t r a n s f e r r e d from one molecule to another and one of the molecules can s t i l l be c o n s i d e r e d as among the r a d i a t o r s . In t h i s case, e x c i t a t i o n can be t r a n s f e r r e d with a zero net change i n the i n t e r n a l energy of the two-molecule system. The d i s t i n c t i o n between the pe r t u r b e r and r a d i a t o r would become meaningless. T h i s has the e f f e c t of s h o r t e n i n g the l i f e t i m e of the e x c i t e d s t a t e or broadening the upper s t a t e . T h i s would i n c r e a s e the amount of l i n e broadening r e l a t i v e to the case of d i s t i n c t p e r t u r b e r - r a d i a t o r i n t e r a c t i o n . Both e l a s t i c and i n e l a s t i c c o l l i s i o n e f f e c t s are c o n s i d e r e d by the ATC theory. The c o l l i s i o n e f f e c t of r e o r i e n t a t i o n (Gordon [1966], Sharma [1971]), however, i s not co n s i d e r e d by most e a r l y ATC t h e o r i e s . T h i s i s the e f f e c t i n which the d i r e c t i o n of angular momentum of the r a d i a t o r i s changed i n the c o l l i s i o n while the i n t e r n a l energy i s u n a l t e r e d . 1 1 5 3.4.3.2 C o l l i s i o n c r o s s s e c t i o n In the ATC theory, the l i n e w i d t h 7 and l i n e s h i f t 8 per unit-atmosphere gas pr e s s u r e of the p e r t u r b e r and in u n i t s of cm - 1 are r e l a t e d to the complex c o l l i s i o n c r o s s s e c t i o n a(v) : 7 = ( 1 .01 325x1 0 6 )N/(27rcp) fg v F ( v ) a R ( v ) dv (3.22) 8 = ( 1 .0l325xl0 6)N/(27rcp) SQ v F ( v ) a : ( v ) dv (3.23) a(v) = a R ( v ) + / a j ( v ) (3.24) The f u n c t i o n F(v) i s the Maxwellian p r o b a b i l i t y d i s t r i b u t i o n f o r the r e l a t i v e v e l o c i t y v. The f u n c t i o n o R ( v ) i s the r e a l p a r t of a(v) while Ojiv) ' i s the imaginary p a r t . The f u n c t i o n o(v) i s a complex q u a n t i t y because of the noncommutative c h a r a c t e r i s t i c of the c o l l i s i o n matrix (Herman [1963]). The i n t e g r a t i o n over the v e l o c i t y d i s t r i b u t i o n can be removed i f one uses the approximation: /Q v F ( v ) o ( v ) dv =* <v> a(<v>) (3.25) <v> = •( 8kT / (TTM) ) (3.26) The term <v> i s the mean c o l l i s i o n v e l o c i t y and M i s the reduced mass of the molecule. T h i s i s a very commonly used approximation and may a f f e c t the c a l c u l a t e d l i n e w i d t h by only a few p e r c e n t . T h i s p o i n t has been i n v e s t i g a t e d by many r e s e a r c h e r s . Rabitz [1974] has p o i n t e d out that the small e r r o r i s the r e s u l t of the f a c t t h a t , at room temperature, the c r o s s s e c t i o n tends to peak near the peak of the v e l o c i t y d i s t r i b u t i o n f u n c t i o n . T h i s c o n d i t i o n may not be v a l i d i n a l l 1 16 a p p l i c a t i o n s . Buffa and T a r r i n i [1983] have a l s o p o i n t e d out that even i f the approximation i s good f o r the c a l c u l a t i o n of l i n e w i d t h s , a l a r g e e r r o r can be i n t r o d u c e d i n t o the l i n e - s h i f t c a l c u l a t i o n s . Meanwhile, Hewitt [1976] found no s i g n i f i c a n t d i f f e r e n c e between the use of <v> and e x p l i c i t numerical i n t e g r a t i o n over the v e l o c i t y d i s t r i b u t i o n . 3.4.3.3 The c o l l i s i o n e f f i c i e n c y f u n c t i o n The c o l l i s i o n c r o s s s e c t i o n a(v) i s r e l a t e d to the c o l l i s i o n e f f i c i e n c y f u n c t i o n S(b,v), where b i s the impact parameter. In the ATC theory, S(b,v) i s p e r t u r b a t i o n expanded i n powers of the i n t e r a c t i o n p o t e n t i a l between the p e r t u r b e r and r a d i a t o r : S(b,v) = S 0 + S,(b,v) + S 2(b,v) +... (3.27) The sum i s u s u a l l y t r u n c a t e d a f t e r the second order term. The z e r o t h order term S 0 i s zero. The f i r s t order term S,(b,v) i s imaginary and would be a c o n t r i b u t o r to only the l i n e s h i f t s . F r o s t [1976] has shown that S ^ b j v ) can v a n i s h i n some a p p l i c a t i o n s . Boulet et a l . [1977] have remarked that S,(b,v) would v a n i s h u n l e s s one c o n s i d e r s the c o n t r i b u t i o n s from the i n t e r n a l v i b r a t i o n a l degrees of freedom. C o n t r i b u t i o n to the l i n e w i d t h comes from the r e a l p a r t of the second order term S 2(b,v) while those f o r the l i n e s h i f t are from S,(b,v) and the imaginary p a r t of S 2 ( b , v ) . The f u n c t i o n S 2(b,v) has a l s o 'been r e f e r r e d to as" the i n t e r r u p t i o n f u n c t i o n . The f u n c t i o n s a R and can then be w r i t t e n as: 1 1 7 V V ) = Lv2J2 pv2J2 ° R ( v ^ 2 , J 2 ) ( 3 . 2 8 ) a I ( v ) = Lv7J7 PV9J, ° i ^ > v 2 > J 2 ) ( 3 . 2 9 ) o R ( v , u 2 , J 2 ) = JQ 27rb R e { S 2 (b,v, v2 , J 2 ) } db ( 3 . 3 0 ) a I ( v , u 2 , J 2 ) = /Q 2nb [ S 1 ( b , v , » 2 , J 2 ) + I m { S 2 ( b , v , u 2 , J 2 ) } ] db ( 3 . 3 1 ) P„ -r = ( 2 J 2 + 1) exp( - E ( u 2 , J 2 ) / k T ) / Z(T) ( 3 . 3 2 ) U 2 J 2 The v a r i a b l e s v2 and J 2 r e f e r to the v i b r a t i o n a l and r o t a t i o n a l quantum number of the p e r t u r b e r , r e s p e c t i v e l y . The f u n c t i o n Z(T) i s simply the t o t a l p a r t i t i o n f u n c t i o n from Equation 3 . 2 1 . 3. 4 . 3 . 4 C u t o f f procedure f o r smal l impact parameter The f u n c t i o n S 2 ( b , v ) produced by the ATC theory, however, w i l l d i v e r g e f o r small values of b. In e a r l y ATC t h e o r i e s , S 2 ( b , v ) i s r e a l and a c u t o f f procedure of assuming S 2 ( b , v ) = 1 f o r b^b 0 i s used. Equation 3.30 would then become: a R ( v , y 2 , J 2 ) = *bg + f£ o 27rb R e { S 2 ( b , v , u 2 , J 2 ) } db ( 3 . 3 3 ) The term b 0 i s the c u t o f f impact parameter and i s d e f i n e d by the c o n d i t i o n that S 2 ( b 0 , v ) = 1 . I f one c o n s i d e r s the c o n t r i b u t i o n of the imaginary part of S 2 ( b , v ) as w e l l , the c o n d i t i o n d e f i n i n g b 0 would be one of the f o l l o w i n g s : R e { S 2 ( b 0 , v ) } = 1 ( 3 . 3 4 ) ( R e { S 2 ( b 0 , v ) } ) 2 + ( l m { S 2 ( b 0 , v ) } ) 2 = 1 ( 3 . 3 5 ) 118 Re{S 2(b 0,v)} + |lm{S 2(b 0,v)}| = 1 (3.36) Equation 3.34 was used by Sharma and Caledonia [1971] while Equation 3.36 was recommended by T i p p i n g and Herman [1970b]. Equation 3.35, however, i s more n a t u r a l and has been more widely used. Other c u t o f f c r i t e r i a are summarised i n R a b i t z [1974]. F r o s t and M a c G i l l i v r a y [1977] have suggested that a c u t o f f procedure would not be necessary i f S 2(b,v) i s r e p l a c e d by a new convergent c o l l i s i o n e f f i c i e n c y f u n c t i o n X(b,v): X(b,v) = 1 - exp( -S 2(b,v) ) (3.37) Re{X(b,v)} = 1 - cos(Im{S 2(b,v)})exp(-Re{S 2(b,v)}) (3.38) Im{X(b,v)} = sin(Im{S 2(b,v)})exp(-Re{S 2(b,v)}) (3.39) The f u n c t i o n X(b,v) would approach 1 f o r small b and S 2(b,v) f o r l a r g e b. 3.4.3.5 The c u t o f f - f r e e theory The most s o p h i s t i c a t e d f o r m u l a t i o n i n the ATC framework i s given by L e a v i t t [1980] and L e a v i t t and K o r f f [1981]. A S(b,v) which i s bound and i n t e g r a b l e over a l l values of b has been d e r i v e d f o r the c u t o f f - f r e e theory.. The e f f e c t of anharmonicity i n the i n t e r n a l v i b r a t i o n a l motion of the molecule (Giraud et a l . [1973]) has a l s o been i n c l u d e d i n the f o r m u l a t i o n . The c o l l i s i o n e f f i c i e n c y f u n c t i o n i s given as: S(b,v) = 1 - e x p ( - / A i + / A f - S ° ^ e r - S ^ e r - S m i d d l e ) (3.40) 119 The s u b s c r i p t s i and f r e f e r to the i n i t i a l and f i n a l o p t i c a l s t a t e of the p a r t i c u l a r s p e c t r a l l i n e t r a n s i t i o n , r e s p e c t i v e l y . The most d e t a i l e d e x p r e s s i o n s f o r the c o n t r i b u t o r s A, S ° U t e r , and s m i d d l e are given i n L e a v i t t [1980]". They are f u n c t i o n s of the b-dependent i n t e r m o l e c u l a r p o t e n t i a l between the c o l l i d i n g molecules e.g. e l e c t r i c d i p o l e - d i p o l e , d i p o l e - q u a d r u p o l e , and quadrupole-quadrupole i n t e r a c t i o n s . These are f u n c t i o n s of the v a r i o u s molecular c o n s t a n t s e.g. p o l a r i z a b i l i t y , e l e c t r i c d i p o l e , quadrupole, and o c t o p o l e moments. Higher-order i n t e r a c t i o n s are summarised i n Townes and Schawlow [1975] and Buckingham [1967], The c o n t r i b u t o r s to S(b,v) a l s o depend on the Clebsch-Gordan c o e f f i c i e n t s c o r r e s p o n d i n g to each p o s s i b l e c o l l i s i o n - i n d u c e d t r a n s i t i o n . Some higher order terms a l s o depend on the Racah c o e f f i c i e n t s . There i s a l s o a dependence on the v a r i o u s "resonance f u n c t i o n s " fn(k), gn(k), If (k), and I g n ( & ) . The v a r i a b l e k i s the resonance parameter and i s a measure how c l o s e the v a r i o u s s t a t e s are to exact resonance: k = 27rb/(hv) ( E A - E? + E A - E B ) (3.41a) or k = 27rb/(hv) ( E A - E B + E A - Ef ) (3.41b) The s u p e r s c r i p t s A and B i n d i c a t e the s t a t e s before and a f t e r the c o l l i s i o n , r e s p e c t i v e l y . The s u b s c r i p t s i and f r e f e r to the upper and lower s t a t e s of the a c t i v e molecule, r e s p e c t i v e l y . The s u b s c r i p t 2 i n d i c a t e s the s t a t e of the p e r t u r b e r . The resonance f u n c t i o n s 120 f n(/c) and 9 n ^ ) a r e even f u n c t i o n s while I f (A:) and Ig n(A;) are odd. E x p l i c i t e x p r e s s i o n s f o r a l l the resonance f u n c t i o n s i n terms of the m o d i f i e d B e s s e l f u n c t i o n s are given i n L e a v i t t [1980] , Power- s e r i e s approximations for some of the resonance f u n c t i o n s are given i n F r o s t [1976] and Messer et a l . [1982] . 3 . 4 . 3 . 6 B a s i c f o r m u l a t i o n s of s i m p l i e d theory Less e l a b o r a t e e x p r e s s i o n s f o r S(b,v) have been used i n the c a l c u l a t i o n s of broadened (3-0) HF l i n e w i d t h s . S p e l l i c y et a l . [1972] c o n s i d e r e d only the r e a l p a r t of S(b,v) together with the c u t o f f procedure i n Equation 3 . 33 . A more s o p h i s t i c a t e d but s t i l l r a t h e r simple e x p r e s s i o n f o r S(b,v) i n the case of se l f - b r o a d e n i n g can be formulated as: Re{S(b,v)} = Z . f [ Z j b - ' c ^ A,(v) f , U ) + Z_j b - 6 c 2 : j A 2 ( v ) f 2 U ) ~+ Z j b - 8 c 3 j A 3 ( v ) f 3 U ) ] (3.42) Im{S(b,v)} = - Z i f [ Z j b - ' c ^ A,(v) I f , U ) + Z j b - 6 c 2 : j A 2 ( v ) It2(k) + Z j b - 8 c 3 : j A 3 ( v ) I f 3 ( * ) ] (3 .43) A , ( v ) = (4/9) M" [27r/(hv)] 2 (3 .44) A 2 ( v ) = (4/45) u2 62 [27r/(hv)] 2 (3 .45) A 3 ( v ) = (1/25) 6* [27r/(hv)] 2 (3 .46) T h i s s i m p l i f i e d S(b,v) i n c l u d e s c o n t r i b u t i o n s from only 121 the i n t e r m o l e c u l a r e l e c t r i c d i p o l e - d i p o l e , d i p o l e - q u a d r u p o l e , and quadrupole-quadrupole i n t e r a c t i o n s . The c o n t r i b u t i o n s are summed over the upper and lower s t a t e s of the a c t i v e molecule to take i n t o account c o l l i s i o n - i n d u c e d t r a n s i t i o n s from those s t a t e s . The f u n c t i o n s A , ( v ) , A 2 ( v ) , and A 3 ( v ) are taken from Tsao and Curnutte [1962]. The parameters ju and 6 i n the above formulae are the e l e c t r i c d i p o l e and quadrupole moments, r e s p e c t i v e l y . The c n j ' s are Clebsch-Gordan c o e f f i c i e n t s and are given i n Benedict and Herman [1963]. Equations 3.42 through 3.46 can be used with Equations 3.37 through 3.39 i n a c u t o f f - f r e e a p p l i c a t i o n of the ATC theory. 3.4.4 SURVEY OF EXPERIMENTS AND CALCULATIONS C o l l i s i o n a l l y broadened l i n e w i d t h s and s h i f t s of HF l i n e s have been measured by many r e s e a r c h e r s . Kuipers [1958] and Herget et a l . [1962] measured both the se l f - b r o a d e n e d l i n e w i d t h s and s h i f t s of HF l i n e s belonging t o the (1-0) band. L o v e l l and Herget [1962] and Hinchen and Hobbs [1979] a l s o measured the l i n e w i d t h s of se l f - b r o a d e n e d (1-0) band HF l i n e s . M eredith [1972] and S p e l l i c y et a l . [1972] have measured the se l f - b r o a d e n e d l i n e w i d t h s of l i n e s i n the (2-0) and (3-0) bands, r e s p e c t i v e l y . S e l f - i n d u c e d l i n e s h i f t s i n the (2-0) band have been measured by G u e l a c h v i l i and Smith [1978]. Meanwhile, Campbell and Walker [1979] and Campbell [1983] have measured the l i n e s h i f t s f o r the l i n e s i n the (3-0) band. J a f f e et a l . [1956], Wiggins et a l . [1970], and 1 22 Pine [1980] measured the rar e - g a s - i n d u c e d l i n e w i d t h s and s h i f t s of the HF l i n e s i n the (1-0) and (2-0) bands. Shaw and L o v e l l [1969] and Varanasi et a l . [1972] measured the C0 2 induced l i n e w i d t h s and s h i f t s of the l i n e s i n the (1-0) band. Meredith and Smith [1973] measured the N 2-, H 2-, and D 2-induced l i n e w i d t h s of the (1-0) HF l i n e s . G u e l a c h v i l i and Smith [1978] have measured the ra r e - g a s - , C0 2-, N 2-, and HCl-induced s h i f t s f o r l i n e s i n the (1-0) and (2-0) bands. C a l c u l a t i o n s of se l f - b r o a d e n e d l i n e w i d t h s f o r the (1-0) HF l i n e s i n c l u d e those by Benedict and Herman [1963] and Hough [1977]. Meredith [1972] and S p e l l i c y et a l . [1972] c a l c u l a t e d the self-broadened l i n e w i d t h s f o r the l i n e s i n the (2-0) and (3-0) bands, r e s p e c t i v e l y . Meanwhile, Boulet et a l . [1977] c a l c u l a t e d the l i n e w i d t h s and s h i f t s f o r the l i n e s i n the (0-0), (1-0), (2-0), and (3-0) bands. C a l c u l a t i o n s of rare-gas-induced l i n e w i d t h s and s h i f t s f o r l i n e s i n the (0-0), (1-0), and (2-0) bands were made by J a r e c k i and Herman [1975], J a r e c k i [1977], and Pine [1980]. Smith and Meredith [1974], meanwhile, a l s o c a l c u l a t e d DF-induced l i n e w i d t h s f o r l i n e s i n the (1-0) band. 3.4.5 THE CALCULATION OF LINEWIDTHS AND LINE SHIFTS The c u t o f f - f r e e c o l l i s i o n e f f i c i e n c y f u n c t i o n d e s c r i b e d i n Equations 3.38 and 3.39 has been used to c a l c u l a t e s e l f - i n d u c e d l i n e w i d t h s and s h i f t s f o r the (3-0) HF l i n e s . C o n t r i b u t i o n s from i n t e r m o l e c u l a r i n t e r a c t i o n s to the c o l l i s i o n e f f i c i e n c y f u n c t i o n were taken to be those given i n Equations 3.42 through 3.46. The d i p o l e moment f o r HF was 123 taken to be 1 .80306x10- 1 8 esu.cm ( O g i l v i e et a l . [1980]). T h i s i s the same as the c o e f f i c i e n t M 0 l i s t e d i n Table 3.5. The quadrupole moment f o r HF was taken to be the observed value of 2 . 2 1 X 1 0 - 2 6 esu.cm 2 (De Leeuw and Dymanus [1973]). Other v a l u e s f o r the c o n s t a n t s are given i n Werner and Ramus [1980], S i l e o and Cool [1976], L i e [1974], Meredith and Smith [1973], and Muenter and Klemperer [1970]. They are a l s o r e f e r e n c e d i n the c a l c u l a t i o n s by Boulet et a l . [1976] and Benedict and Herman [1963]. A 32-point Gauss-Laguerre-quadrature numerical i n t e g r a t i o n was used f o r the s e m i - i n f i n i t e range i n t e g r a t i o n of Equations 3.43 and 3.44. The same method can a l s o be a p p l i e d to the i n t e g r a t i o n over the v e l o c i t y d i s t r i b u t i o n i n Equations 3.42 and 3.36. In f a c t , a 7-point Gauss-Laguerre quadrature was used by Boulet et a l . [1976]. In the a c t u a l c a l c u l a t i o n , however, because of l i m i t e d computing r e s o u r c e s , the approximation i n Equations 3.25 and 3.26 was used i n s t e a d of the e x p l i c i t numerical i n t e g r a t i o n over v e l o c i t i e s . The c a l c u l a t i o n c o n s i d e r s a l l c o n t r i b u t i o n s which have Boltzmann f a c t o r s g r e a t e r than that of u=0 and J=23. T h i s should be s u f f i c i e n t to account f o r most of the e n e r g y - l e v e l p o p u l a t i o n at the low temperature of 100°C f o r HF. In f a c t , Benedict and Herman [1963] have p o i n t e d out t h a t even at 1200K, l e s s than 0.3% of the molecules are i n r o t a t i o n a l s t a t e s J>20. The method r e q u i r e s many e v a l u a t i o n s of the v a r i o u s energy l e v e l s . The amount of computation can be g r e a t l y reduced i f one e s t a b l i s h e s a common database on the energy l e v e l s . T h i s can then be r e f e r e n c e d by the v a r i o u s p a r t s of the computer 1 24 program and hence would e l i m i n a t e any repeated e n e r g y - l e v e l c a l c u l a t i o n . The c a l c u l a t e d 7 and 5 are the unit-atmosphere l i n e w i d t h and s h i f t , r e s p e c t i v e l y . The l i n e w i d t h i s d e f i n e d as the h a l f w i d t h of the l i n e at 50% of the maximum l i n e absorbance. From the d e f i n i t i o n of the l i n e absorbance f u n c t i o n K(V) i n Equation 3.18, the l i n e w i d t h would be the h a l f w i d t h at h a l f maximum of the f u n c t i o n - L n ( I ( v ) / l 0 ) . The f u n c t i o n s l ( p ) and I 0 are the l i n e and continuum i n t e n s i t y , r e s p e c t i v e l y . Reviews on the d e f i n i t i o n s of v a r i o u s l i n e - s h a p e parameters are given i n Meredith [1972] and Mandin et a l . [1980]. 3.4.6 CALCULATED LINEWIDTHS FOR THE HF LINES To t e s t the theory and the computational procedure, l i n e w i d t h s f o r the l i n e s i n the (1-0) and (2-0) bands are c a l c u l a t e d . They are then compared a g a i n s t the experimental v a l u e s . A gas temperature of 373.15K has been used i n the c a l c u l a t i o n s . In g e n e r a l , the c a l c u l a t e d v a l u e s are c o n s i s t e n t l y lower than the experimental v a l u e s by s e v e r a l percent f o r l i n e s i n the P-branch. The agreement f o r l i n e s i n the R-branch i s b e t t e r . Table 3.8 l i s t s the l i n e w i d t h s f o r the l i n e s i n the (1-0) band. The experimental v a l u e s are taken from L o v e l l and Herget [1962]. The agreement between the c a l c u l a t e d and experimental l i n e w i d t h s i s q u i t e good e s p e c i a l l y f o r the R-branch l i n e s with m<6. For the l i n e s i n the (2-0) band, the agreement i s e x c e l l e n t . Table 3.9 l i s t s the c a l c u l a t e d and experimental v a l u e s f o r the l i n e s i n the 125 T a b l e 3.8 L i n e w i d t h s of the (1-0) band of HF * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * m c a l c . e x p t . -1 0.4336 c r r r 1 a t n r 1 0.462 cm" 'atm" 1 0.4310 cm" 1 atm" 1 0.458 c n r ' a t n r 2 0.501 6 c r r r ' a t m - 1 0.501 c n r ' a t m - 3 0.5133 c m - ' a t m - ' 0.563 c n r 1 a t n r 4 0.4555 c r r r 1 a t r r r 1 0.454 c n r 1 a t r r r 5 0.3636 c m - 1 a t n r 1 0.345 cm" 1 atm" 6 0.2821 c m - ' a t m - 1 0.259 cm-'atm" 7 0.2058 c m - ' a t m ' 1 0.173 cm" 1 atm" ************************************************************ T a b l e 3.9 L i n e w i d t h s of the (2-0) band of HF * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * m c a l c . e x p t . -1 0.4263 c r r r 1 a t m - 1 0.459 cm" 1 atm" 1 1 0.4090 cm- 'a tm"' 0.406 c m - ' a t m " 1 2 0.4586 c m - ' a t m " 1 0.464 cm- 'a tm"' 3 0.4696 c m - ' a t m " 1 0.471 c m ^ a t m " 1 4 0.4268 c m - ' a t m " 1 " 0.419 cm" 1 atm" 1 5 0.3493 c m - ' a t m " 1 0.347 c n r ' a t m - 1 6 0.2738 cm"'atm" 1 7 0.2064 c m - ' a t m " 1 0.184 cm" 1 atm" 1 8 0.1536 cm" 1 atm" 1 0.140 c m - 1 a t m " 1 ************************************************************ 1 26 Table 3.10 Linewidths of the (3-0) band of HF ***************************************** m c a l c expt. -2 0.4492 cm- 1atm" 1 0.506 cm" 1 a t m" -1 0.4136 cm" 1atm" 1 0.447 cm" 'atnr 1 0.3975 cm" 1atm" 1 0.397 cm" 1atm" 2 0.4268 cm" 1atm" 1 0.443 cm" 1atm" 3 0.4473 cm" 1atm" 1 0.453 cm" 1 atm" 4 0.4201 cm" 1atm" 1 0.412 cm" ̂ tm" 5 0.3514 cm" ̂ t i r r 1 0.331 cm" 1atm" 6 0.2820 cm" 1atm" 1 0.254 cm" ̂ tm" 7 0.2187 cm" 1atm" 1 0. 1 64 cm" 1atm" ************************************************************ (2-0) band. The experimental l i n e w i d t h s , i n t h i s case, are taken from Meredith [1972]. The r e s u l t f o r the (3-0) band i s given i n Table 3.10. The experimental values are taken from S p e l l i c y et a l . [1972]. Again, the agreement i s q u i t e good fo r the R-branch l i n e s with m<6. Of course, the agreement i s much b e t t e r than the r e s u l t from the l e s s s o p h i s t i c a t e d c a l c u l a t i o n by S p e l l i c y et a l . [1972], F i g u r e 3.3 shows the l i n e w i d t h v a r i a t i o n s with temperature f o r the (3-0) band. The r e s u l t i s not unexpected. I t can be seen from Equation 3.22 that 7 i s p r o p o r t i o n a l to the number d e n s i t y of the p e r t u r b e r n, and the gas v e l o c i t y v. For constant gas p r e s s u r e , n i s invers-ely p r o p o r t i o n a l to temperature T while v i s p r o p o r t i o n a l to /T. Hence 7 would be i n v e r s e l y p r o p o r t i o n a l to v/T. The temperature dependence of the c o l l i s i o n c r o s s 1 27 s e c t i o n i s small f o r the very small change i n temperature c o n s i d e r e d here. Meanwhile, the observed l i n e w i d t h s should be d i r e c t l y p r o p o r t i o n a l to the p r e s s u r e . Linewidths are known to have a n o n l i n e a r pressure dependence (Meredith and Smith [1974], Smith and Meredith [1974]). But the n o n l i n e a r i t y i s very small over the pressure range of i n t e r e s t here. 3.4.7 CALCULATED LINE SHIFTS FOR THE HF LINES In comparing the c a l c u l a t e d l i n e s h i f t s f o r the (1-0) band a g a i n s t the experimental v a l u e s from Herget et a l . [1962], the agreement i s poor. G e n e r a l l y , there i s agreement only to the same order of magnitude. The c a l c u l a t e d s h i f t s tend to be p r o g r e s s i v e l y too p o s i t i v e f o r l i n e s with m>4. The s i t u a t i o n i s the same when comparing the c a l c u l a t e d l i n e s h i f t s f o r the (2-0) band a g a i n s t the experimental v a l u e s from G u e l a c h v i l i and Smith [1978] or the c a l c u l a t e d v a l u e s from Boulet et a l . [1976]. Table 3.11 l i s t s the c a l c u l a t e d l i n e s h i f t s f o r the (3-0) band. The experimental v a l u e s i n the t a b l e are taken from Campbell and Walker [1979], There i s s l i g h t agreement between the c a l c u l a t e d and experimental v a l u e s f o r l i n e s with m<5. Although the c a l c u l a t i o n s f o r the l i n e w i d t h s and s h i f t s are based on the same theory, the c a l c u l a t e d l i n e w i d t h s appear t o be s i g n i f i c a n t l y more ac c u r a t e that those of the l i n e s h i f t s . T h i s phenomenon has been commented on by Buf f a and T a r r i n i [1983] t o be a consequence "of u s i n g the approximation i n Equation 3.25 i n s t e a d of performing an gure 3.3 The temperature dependence of the l i n e w i d t h s 1 29 Table 3.11 L i n e s h i f t s of the (3-0) band of HF ********************************************* m c a l c . expt. -2 +0.0288 cm - 1atm" 1 -1 +0.0763 cm" 'atm" 1 1 +0.0763 cm" 1atm" 1 +0.0445 cm" 1 atm" 2 +0.0061 cm" 1atm" 1 +0.0070 cm" 'atm" 3 -0.0237 cm" 1atm" 1 -0.0140 cm" 1 atm" 4 -0.0046 cm" 'atm - 1 -0.0171 cm" 1 atm" 5 +0.0281 cm" 1atm" 1 -0.0201 cm" 1 atm" 6 +0.0541 cm" 1atm" 1 -0.0101 cm" 1 atm" 7 +0.0758 cm" 1atm" 1 -0.0121 cm" 1atm" ************************************************************ e x p l i c i t v e l o c i t y i n t e g r a t i o n . The s i m p l i f i e d theory a l s o ignores the terms / A ^ and /A ^ from Equation 3.40. These terms are pure imaginary q u a n t i t i e s and hence w i l l c o n t r i b u t e towards the l i n e s h i f t s and not the l i n e w i d t h s . Boulet et a l . [1976], u s i n g a more s o p h i s t i c a t e d theory together with e x p l i c i t v e l o c i t y i n t e g r a t i o n , have produced c a l c u l a t e d l i n e s h i f t s f o r the (1-0) and (2-0) bands which are i n b e t t e r agreement with the experimental v a l u e s . In f a c t , f o r l i n e s with m>4, Boulet et a l . [1976] have produced c a l c u l a t e d l i n e s h i f t s i n the (2-0) band which agree i n the s h i f t d i r e c t i o n with those of the (3-0) band measured by Campbell and Walker [1979]. The use of a more s o p h i s t i c a t e d theory l i k e t h a t by L e a v i t t [1980] to improve the c a l c u l a t e d r e s u l t i s beyond the scope of t h i s i n i t i a l study. In f a c t , i t w i l l s i g n i f i c a n t l y d e p l e t e the a v a i l a b l e computing 1 30 resources which would, at the present time, be more f r u i t f u l l y u t i l i s e d on s p e c t r a l data r e d u c t i o n . 3.4.8 SHIFT-CORRECTED REFERENCE WAVELENGTHS Table 3.12 l i s t s the vacuum wavenumbers and s t a n d a r d - a i r wavelengths of the (3-0) band l i n e s . These have been c o r r e c t e d f o r s h i f t s at a gas temperature of 100°C and pressure of 360 t o r r s . T h i s i s accomplished by a p p l y i n g the experimental s h i f t s of Campbell and Walker [1979] to the adopted wavenumbers l i s t e d i n Table 3.6. L i s t e d i n Table 3.13 are the unit-atmosphere s h i f t s as given by Campbell and Walker [1979]. The temperature dependence of the unit-atmosphere s h i f t s , 35/9T, are a l s o l i s t e d i n Table 3.13. These are the experimental v a l u e s taken from Campbell [1983]. Again, the s h i f t s are l i n e a r i n p r e s s u r e over the small range c o n s i d e r e d here. T h i s has been demonstrated by Campbell and Walker [1979]. The use of the s h i f t - c o r r e c t e d wavelengths from Table 3.12 has f u r t h e r reduced the r e s i d u a l in the e a r l i e r d i s p e r s i o n f i t of the standard HF spectrum. The rms r e s i d u a l i s reduced from ±0.001 55A" down to ±0.00128A. T h i s corresponds to an improvement from ±53ms~ 1 to ±44ms" 1 i n v e l o c i t y . 3.4.9 DOPPLER AND CELL-WALL BROADENING There are other l i n e - b r o a d e n i n g mechanisms besides molecular c o l l i s i o n broadening. These are summarised i n Townes and Schawlow [1975]. The two more s i g n i f i c a n t ones are the Doppler broadening and the c o l l i s i o n broadening with 131 Table 3.12 S h i f t - c o r r e c t e d wavenumbers and wavelengths *************************************** m V X 1 114 0 9 .4244 cm - 1 8762.2777A 2 11441.4337 cm" 1 8737 . 7 6 3 7 A 3 11468.8426 cm - 1 8716.8815A 4 11491.6132 cm - 1 8 6 9 9 . 6 0 9 0 A 5 115 0 9.7024 cm - 1 8 6 8 5 . 9 3 6 3 A 6 11523 . 0 8 6 3 cm" 1 8 6 7 5 . 8 4 7 6 A 7 11531.7272 cm' 1 8 6 6 9 . 3 4 6 6 A ************************************************************ Table 3.13 Line s h i f t s f o r the R-branch of the (3-0) band ************************************************************ m 5 95/3T 1 -0.0342 A.atnr 1 -0.115 mA.K" 1atnr 2 -0.0053 A.atnr 1 +0.013 mA.K" 1 atnr 3 +0.0106 A.atnr 1 +0.036 mA.K" 1atnr 4 +0.0129 A.atnr 1 -0.006 mA. K" 1 atnr 5 +0.0152 A.atnr 1 -0.044 mA.K" 1atm" 6 +0.0076 A.atnr 1 -0.084 mA.K- 'atnr 7 +0.0091 A.atnr 1 -0.082 mA.K" 1 atnr ************************************************************ the a b s o r p t i o n c e l l w a l l . Doppler broadening i s caused by the motion of the molecule i n the d i r e c t i o n p a r a l l e l to that of the r a d i a t i o n being absorbed. The Doppler h a l f w i d t h 7 D has been given i n Townes and Schawlow [1975] as: 1 32 7 D = (v/c) v/( (2kT/M) ln2 ) (3.47) The term M i n the above equation i s the molecular mass. At the o p e r a t i o n temperature of 100°C of the HF c e l l , 7 D i s of the order of 0.02cm"1 f o r the (3-0) l i n e s . The l i n e s are a l s o broadened by the molecules c o l l i d i n g a g a i n s t the a b s o r p t i o n c e l l w a l l . The c o l l i s i o n would i n t e r r u p t the a b s o r p t i o n process and hence cause l i n e broadening. A r i g o r o u s treatment to c a l c u l a t e the w a l l - c o l l i s i o n h a l f w i d t h 7 w has been given by Danos and Geschwind [1953]. A simpler but adequate e x p r e s s i o n has been given i n Townes and Schawlow [1977] as: 7 w = (A/V) •( (RT) / (8ir 3M) ) (3.48) In the above equation, A i s the t o t a l i n s i d e s u r f a c e area of the c e l l , V i s the volume of the c e l l , R i s the gas co n s t a n t , and M i.s the molecular mass. The value of 7 i s w g e n e r a l l y much l e s s than the pressure-broadened width. Chapter 4 HF DATA REDUCTION 4. 1 INTRODUCTION The most time-consuming p a r t of the HF program i s the r e d u c t i o n of the observed s p e c t r a to o b t a i n r e l a t i v e r a d i a l v e l o c i t i e s . The very high p r e c i s i o n r e q u i r e d f o r the r a d i a l v e l o c i t i e s causes the d a t a - r e d u c t i o n procedure to be more complex and d e t a i l e d than i n c o n v e n t i o n a l techniques. C o n s i d e r a t i o n s have to be made f o r e f f e c t s too minute to be important i n c o n v e n t i o n a l data r e d u c t i o n but which would a f f e c t the r e s u l t i n the h i g h - p r e c i s i o n case. The d a t a - r e d u c t i o n procedure can be roughly d i v i d e d i n t o two p a r t s . The f i r s t p art i n v o l v e s p r e p r o c e s s i n g the R e t i c o n data i n t o c o n t i n u u m - r e c t i f i e d s p e c t r a while the second p a r t i n v o l v e s deducing r a d i a l - v e l o c i t y i n f o r m a t i o n from these s p e c t r a . The two p a r t s are not mutually e x c l u s i v e s i n c e r a d i a l - v e l o c i t y i n f o r m a t i o n i s needed in the c o n t i n u u m - r e c t i f i c a t i o n procedure. The data r e d u c t i o n of the s p e c t r a i s accomplished through the use of the d a t a - r e d u c t i o n program R e t i c e n t (Yang [1980], P r i t c h e t et a l . [1982]). 4.2 PREPROCESSING OF RETICON SPECTRA 4.2.1 BASELINE SUBTRACTION As mentioned p r e v i o u s l y , an a d d i t i v e f i x e d l i n e p a t t e r n has to be removed from each raw R e t i c o n spectrum. T h i s can 133 1 3 4 be accomplished by s u b t r a c t i n g a dark ( b a s e l i n e ) exposure taken s h o r t l y a f t e r the data exposure. In many a p p l i c a t i o n s , t h i s dark exposure can simply be a short exposure with s t a r l i g h t s t i l l i n c i d e n t on the R e t i c o n . The l i n e a r response of the R e t i c o n means that any photon s i g n a l d e t e c t e d i n t h i s short, exposure would not a f f e c t the s p e c t r a l c h a r a c t e r i s t i c s of the longer data exposure a f t e r the s u b t r a c t i o n . T h i s i s e s p e c i a l l y the case f o r f a i n t e r o b j e c t s . A dark exposure, however, would be more d e s i r a b l e . A s h u t t e r i s now a v a i l a b l e on the CFH Retico n system. I t w i l l c l o s e and block o f f a l l l i g h t from the d e t e c t o r d u r i n g the b a s e l i n e exposure. There c o u l d be z e r o - p o i n t d i f f e r e n c e s between s p e c t r a taken with d i f f e r e n t exposure times. The amounts of the z e r o - p o i n t d r i f t s are a l s o d i f f e r e n t between the output video l i n e s . The d i f f e r e n c e s are g e n e r a l l y of the order of s e v e r a l adc u n i t s depending on the exposure time. I t would be the best i f the dark b a s e l i n e exposure has the same exposure time as the data spectrum. T h i s i s u s u a l l y d i f f i c u l t t o achieve f o r the longer exposures. Besides lowering the e f f i c i e n c y i n the use of t e l e s c o p e time, the p h y s i c a l s t a t e of the Re t i c o n may not remain the same long enough f o r both the dark b a s e l i n e and data exposures. Moreover, one would want to use the mean of many dark b a s e l i n e exposures f o r the s u b t r a c t i o n i n order to reduce the no i s e c o n t r i b u t i o n as w e l l as the e f f e c t of cosmic-ray events. G e n e r a l l y , depending on R e t i c o n s , a s h o r t e r dark exposure would be s u f f i c i e n t . The c h a r a c t e r i s t i c s of the a m p l i f i e r e l e c t r o n i c s become q u i t e constant f o r exposures 135 longer than s e v e r a l hundred seconds. Dark b a s e l i n e exposures of s i m i l a r exposure times, however, are s t i l l d e s i r a b l e f o r the s h o r t e r exposures such as those f o r the incandescent c a l i b r a t i o n lamps. 4.2.2 USE OF "EXTRA-READOUT" POINTS The e f f e c t of the z e r o - p o i n t d r i f t s can a l s o be minimised or c o r r e c t e d by the use of "extra-readout" p o i n t s (Walker et a l . [1983]). B a s i c a l l y , these readout p o i n t s are the r e s u l t of performing readout sequences on the Reticon but without a d d r e s s i n g or r e b i a s i n g the i n d i v i d u a l photodiodes. I t i s e s s e n t i a l l y sampling the v o l t a g e s from the v a r i o u s output v i d e o l i n e s . These "extra-readout" p o i n t s are o b t a i n e d both before and a f t e r the readouts of the 1872 photodiodes. There are about 28 u s e f u l "extra-readout" p o i n t s f o r each output video l i n e . By comparing the r e s i d u a l s i n these "extra-readout" p o i n t s a f t e r s u b t r a c t i n g the c o r r e s p o n d i n g ones from the dark b a s e l i n e exposure, minor z e r o - p o i n t d r i f t s can be n u m e r i c a l l y c o r r e c t e d f o r each output video l i n e . The "extra-readout" p o i n t s are not p e r f e c t . I t has been found that i n incandescent c a l i b r a t i o n lamp exposures, photon s i g n a l i s a l s o d e t e c t e d i n these "extra-readout" p o i n t s . T h i s i s p o s s i b l y caused by the f a c t t h a t the s h i f t r e g i s t e r s on the R e t i c o n are a l s o l i g h t s e n s i t i v e and would d e t e c t photons i f the l i g h t i n t e n s i t y i s h i g h . G e n e r a l l y , t h i s e f f e c t i s small i n comparison with the amount of z e r o - p o i n t d r i f t s . Some Retic o n s are a l s o known to c o n t a i n l i g h t s i g n a l s i n the second set of the 1 36 "extra-readout" p o i n t s . These are the p o i n t s obtained a f t e r the readouts of the 1872 photodiodes. The amount of s i g n a l i n these p o i n t s i s found to be a f u n c t i o n of the s i g n a l accumulated i n the 1872 photodiodes. T h i s behaves as i f r e s i d u a l charges are l e f t i n the output video l i n e s a f t e r the readouts of the photodiodes. Consequently, the second set of the "extra-readout" p o i n t s may be u n s u i t a b l e f o r use i n z e r o - p o i n t - d r i f t c o r r e c t i o n s . 4.2.3 RELATIVE GAIN CORRECTION 4.2.3.1 L i n e - n o r m a l i s a t i o n procedure The r e l a t i v e g ains of the output a m p l i f i e r s are u s u a l l y a d j u s t e d only to w i t h i n 5%. T h i s i m p l i e s that the gain i s s l i g h t l y d i f f e r e n t f o r each output v i d e o l i n e . The r e l a t i v e g ains e f f e c t i s a m u l t i p l i c a t i v e e f f e c t and a d i f f e r e n t m u l t i p l i c a t i v e n o r m a l i s a t i o n constant has to be a p p l i e d n u m e r i c a l l y to the output from each video l i n e . T h i s i s e s s e n t i a l l y the simple m u l t i p l i c a t i v e l i n e - n o r m a l i s a t i o n process where each n o r m a l i s a t i o n constant i s taken to be the r a t i o between the mean s i g n a l of the p a r t i c u l a r output l i n e and the mean s i g n a l over a l l output l i n e s . The m u l t i p l i c a t i v e e f f e c t , however, would be c o m p l i c a t e d by any r e s i d u a l z e r o - p o i n t d r i f t which i s an a d d i t i v e e f f e c t . One cannot use a simple numerical a d d i t i v e or m u l t i p l i c a t i v e l i n e - n o r m a l i s a t i o n process to p e r f e c t l y decouple and c o r r e c t the two e f f e c t s . To c o m p l i c a t e matters, the gain of each output l i n e a l s o has a small dependence on the 1 37 accumulated s i g n a l l e v e l . T h i s e f f e c t i s a small n o n l i n e a r i t y . The change i n gain i s t y p i c a l l y about 5x10" 8 per adc u n i t over the s i g n a l range i n the high gain mode of the R e t i c o n . I t becomes g r e a t e r at higher s i g n a l l e v e l s where the accumulated s i g n a l i s a s i g n i f i c a n t f r a c t i o n of s a t u r a t i o n . In c o n v e n t i o n a l techniques, the r e l a t i v e gains e f f e c t i s c o r r e c t e d by d i v i d i n g the data spectrum by a lamp spectrum exposed to the same continuum s i g n a l l e v e l . A f o u r - or e i g h t - l i n e m u l t i p l i c a t i v e or a d d i t i v e n o r m a l i s a t i o n process i s then a p p l i e d . S e v e r a l problems may e x i s t i n t h i s procedure. The f a c t that the data spectrum c o n t a i n s s p e c t r a l l i n e s which w i l l have s i g n a l l e v e l s q u i t e d i f f e r e n t from the continuum s i g n a l l e v e l , d i v i s i o n by a continuous lamp spectrum would cause i n c o r r e c t gain c o r r e c t i o n s t o be a p p l i e d t o the s p e c t r a l l i n e s . R e s i d u a l a d d i t i v e p a t t e r n from the b a s e l i n e s u b t r a c t i o n would a l s o a f f e c t the accuracy of a simple l i n e - n o r m a l i s a t i o n p r o c e s s . 4.2.3.2 Use of step lamps One method to c a l i b r a t e the r e l a t i v e gains e f f e c t and the small n o n l i n e a r i t y i s through the use of ste p incandescent lamp s p e c t r a . B a s i c a l l y , one o b t a i n s a s e r i e s of continuous lamp s p e c t r a with each of them exposed to a d i f f e r e n t s i g n a l l e v e l . The s e r i e s of lamp s p e c t r a should cover the e n t i r e range of s i g n a l l e v e l s . One can c a l c u l a t e the m u l t i p l i c a t i v e l i n e - n o r m a l i s a t i o n f a c t o r s f o r each spectrum and hence f o r each d i f f e r e n t 1 3 8 s i g n a l l e v e l . A polynomial can then be used to f i t the n o r m a l i s a t i o n f a c t o r s as a f u n c t i o n of s i g n a l l e v e l s in each separate video l i n e . These polynomials can then be used to c o r r e c t the r e l a t i v e gains and n o n l i n e a r i t y e f f e c t s f o r each i n d i v i d u a l p o i n t i n the data spectrum. F i r s t - o r d e r polynomials are g e n e r a l l y adequate to prov i d e a good f i t over the e n t i r e s i g n a l l e v e l range in the' high-gain mode of the R e t i c o n . Higher-order polynomials are needed i n order to cover the e n t i r e range of s i g n a l l e v e l up to s a t u r a t i o n . U s u a l l y , ten or more s p e c t r a are r e q u i r e d f o r the f i t . The goodness of the polynomial f i t s depends on the q u a l i t y of the lamp s p e c t r a . A p p r o p r i a t e dark b a s e l i n e s should be used f o r these lamp s p e c t r a . And any z e r o - p o i n t d r i f t as i n d i c a t e d by the e x t r a readout p o i n t s should a l s o be c o r r e c t e d . The re g i o n i n the continuous lamp s p e c t r a where the n o r m a l i s a t i o n f a c t o r s are c a l c u l a t e d should be r e l a t i v e l y f l a t i . e . have a small range i n s i g n a l l e v e l s . T h i s i s u s u a l l y the case when one i s working i n the near i n f r a r e d and at high d i s p e r s i o n s . Otherwise, f i l t e r s have to be used to produce a f l a t r e g i o n i n the lamp s p e c t r a . 4.2.4 FLAT-FIELDING A f t e r dark b a s e l i n e s u b t r a c t i o n and a p p l i c a t i o n of the v a r i o u s l i n e - n o r m a l i s a t i o n procedures, the data s t i l l have to be f l a t - f i e l d e d . T h i s i s to remove di o d e - t o - d i o d e s e n s i t i v i t y v a r i a t i o n s as w e l l as n o n - u n i f o r m i t i e s i n the 1 39 Ret i c o n and spectrograph s p e c t r a l responses. Dust p a r t i c l e s on the R e t i c o n s u r f a c e would a l s o a l t e r the u n i f o r m i t y of the response. T h i s i s e s p e c i a l l y r e l e v a n t to the Reticon d e t e c t o r at CFHT. Removal of n o n - u n i f o r m i t i e s i n the response a c r o s s the R e t i c o n spectrum would a l s o ease the a p p l i c a t i o n of continuum r e c t i f i c a t i o n . F l a t - f i e l d i n g can be accomplished by n u m e r i c a l l y d i v i d i n g the data spectrum by an incandescent lamp spectrum. Dark b a s e l i n e s u b t r a c t i o n as w e l l as a l l the necessary l i n e - n o r m a l i s a t i o n p r e p r o c e s s i n g should have been a p p l i e d t o t h i s lamp spectrum before the d i v i s i o n . Care should be taken to ensure the l i g h t beams from both the lamp and the s t a r i l l u m i n a t e a l l the o p t i c s i d e n t i c a l l y . I t i s e s p e c i a l l y important f o r the lamp beam to i l l u m i n a t e e x a c t l y the same part of the R e t i c o n a r r a y as the s t e l l a r beam. T h i s has been achieved to a c e r t a i n extent by the use of an image s l i c e r which c r e a t e s a one dimensional e q u i v a l e n t of a Fabry-image (Campbell et a l . [1981]). However, small e f f e c t s may s t i l l e x i s t i f the lamp and s t e l l a r beams are not i d e n t i c a l l y matched. One method p r e v i o u s l y used at DAO to match the lamp and s t e l l a r beams was to, use the lamp l i g h t r e f l e c t e d o f f the i n s i d e of the dome. The lamps were mounted on the top end of the t e l e s c o p e . Each lamp spectrum was a l s o taken at the same t e l e s c o p e o r i e n t a t i o n as the p a r t i c u l a r s t a r . E x c e l l e n t f l a t - f i e l d lamp s p e c t r a were obtained. T h i s c l a i m i s based on how w e l l the lamp spectrum would f l a t - f i e l d the response caused by a dust p a r t i c l e on the Re t i c o n a r r a y . One of the major disadvantages of t h i s method to o b t a i n matched 1 40 f l a t - f i e l d lamp s p e c t r a i s that very long exposures are g e n e r a l l y needed. T h i s i s e s p e c i a l l y bad when one wants to use the mean of many of these s p e c t r a to reduce noise c o n t r i b u t i o n . The b r i g h t lamps would a l s o degrade the seeing by warming up the dome as w e l l as a t t r a c t i n g moths to the t e l e s c o p e . Another method i s adopted at the present time at both CFHT and DAO. An i r i s diaphragm i s used to i s o l a t e the t e l e s c o p e e x i t p u p i l formed by the converging l e n s which i s the l a s t element i n the Coude t r a i n . A lamp i l l u m i n a t i n g t h i s p u p i l would produce a beam which w i l l match the s t e l l a r beam. The p o s i t i o n of the diaphragm needs to be a d j u s t e d f o r each s t e l l a r o b j e c t because of s l i g h t m i s - c o l l i m a t i o n of the Coude t r a i n . The lamp exposures are g e n e r a l l y s h o r t . I f the lamp spectrum has a s i g n i f i c a n t s l o p e , as i s the case i n the blue s p e c t r a l r e g i o n , a low order polynomial f i t of the lamp spectrum can be d i v i d e d out from the lamp spectrum. Otherwise, d i v i d i n g each p o i n t i n the lamp spectrum by the mean va l u e would be s u f f i c i e n t to produce a normalised f l a t - f i e l d . A f t e r d i v i d i n g the s t e l l a r spectrum by the f l a t - f i e l d lamp, a four-channels a d d i t i v e l i n e - n o r m a l i s a t i o n procedure can be a p p l i e d to remove any smal l r e s i d u a l l i n e p a t t e r n . T h i s g e n e r a l l y i s a very small c o r r e c t i o n and i s of the order of an adc u n i t or l e s s . I f the lamp spectrum was, however, not taken at the same p h y s i c a l s t a t e of the Reticon as the s t e l l a r spectrum, the four-channels l i n e - n o r m a l i s a t i o n would not reduce the r e s i d u a l l i n e 141 p a t t e r n . T h i s can occur when the lamp or s t e l l a r spectrum was taken when the Re t i c o n was i n the process of reaching temperature e q u i l i b r i u m a f t e r a l i q u i d - N 2 r e f i l l . An e i g h t - c h a n n e l s l i n e n o r m a l i s a t i o n may be u s e f u l i n t h i s case. I f the lamp spectrum i s of poor q u a l i t y , i t should be smoothed to about 5% of the Nyquist frequency to avoid degrading the s t e l l a r spectrum. The a d d i t i v e l i n e - n o r m a l i s a t i o n process should not be a p p l i e d to arc s p e c t r a . 4 . 3 REDUCTION OF HF DATA 4 . 3 . 1 CONTINUUM RECTIFICATION There are four types of s p e c t r a one has to o b t a i n i n an obse r v i n g run i n order to produce a r e l a t i v e r a d i a l - v e l o c i t y measurement. These are the s t e l l a r s p e c t r a with imposed HF l i n e s , the s t e l l a r s p e c t r a with no imposed HF l i n e s , the continuous lamp s p e c t r a with HF l i n e s , and the continuous lamp s p e c t r a without the imposed HF l i n e s . A l l these s p e c t r a are used i n the d a t a - r e d u c t i o n procedure to measure r a d i a l v e l o c i t i e s f o r the s t e l l a r l i n e s i n the spectrum with the imposed HF l i n e s . For g r e a t e r c o n s i s t e n c y , the i d e n t i c a l s p e c t r a l r e g i o n i s used at a l l times i n the HF programs. In f a c t , the same coverage i s used at both DAO and CFHT where the r e c i p r o c a l d i s p e r s i o n i s the same 4 .8A/mm. The i d e n t i c a l s p e c t r a l coverage i s achieved by a l i g n i n g the Retico n and spectrograph such that the arc l i n e Ar I X 8 6 6 7 would always 142 l o c a t e at the same p i x e l p o s i t i o n of 351 on the Reticon a r r a y . Furthermore, to av o i d any i n c o n s i s t e n c y i n the focus of the s p e c t r a due to the curved f o c a l plane, the same c r i t e r i a are used to focus the R e t i c o n . The Hartmann mask i s always used to o b t a i n optimum focus on the same arc l i n e of Fe I X8689. T h i s i s e s s e n t i a l l y to monitor the p i x e l p o s i t i o n of the arc l i n e when d i f f e r e n t p a r t s of the beam are masked o f f . In most HF programs, a s t e l l a r spectrum i s g e n e r a l l y used as a r a d i a l - v e l o c i t y standard (or r e f e r e n c e ) spectrum f o r t h at same s t a r . S i m i l a r l y , an HF spectrum i s used as a re f e r e n c e spectrum f o r HF l i n e s i n a l l s p e c t r a . These standard s p e c t r a can a l s o be used as the s p e c t r a l standards f o r pseudo-continuum r e c t i f i c a t i o n of the data s t e l l a r s p e c t r a . The standard s t e l l a r s p e c t r a are g e n e r a l l y taken without the imposed HF l i n e s . Pseudo-continuum r e c t i f i c a t i o n of the standard s p e c t r a i s the same as i n c o n v e n t i o n a l t e c h n i q u e s . T h i s i s achieved by d i v i d i n g the spectrum by a polynomial f i t to the continuum p o i n t s i n the spectrum. Each continuum p o i n t i n the f i t i s g e n e r a l l y taken to have the mean s i g n a l of s e v e r a l neighbouring p o i n t s . Due to the p a u c i t y of good a v a i l a b l e continuum p o i n t s , a low-order polynomial f i t i s g e n e r a l l y used. T h i s i s e s p e c i a l l y the case f o r the s p e c t r a with strong and broad l i n e s e.g. those of a L y r . A s i m i l a r procedure can be a p p l i e d to the standard HF spectrum which i s a continuous lamp spectrum imposed with HF l i n e s . More continuum p o i n t s are a v a i l a b l e i n t h i s p a r t i c u l a r a p p l i c a t i o n and a h i g h e r - o r d e r polynomial f i t can 1 43 be used. Again, the continuum p o i n t s are chosen to a v o i d the broad wings of the HF l i n e s . A g e n e r a l way to continuum r e c t i f y a data spectrum i s f i r s t to d i v i d e the data spectrum by the r e c t i f i e d standard s t e l l a r spectrum. D i v i s i o n by the r e c t i f i e d HF standard spectrum i s a l s o performed i f the data spectrum c o n t a i n s the imposed HF l i n e s . The r e s i d u a l r a t i o spectrum i s then f i t t e d by an a p p r o p r i a t e f u n c t i o n (polynomial or o t h e r s ) . A r e c t i f i e d data spectrum i s then o b t a i n e d by d i v i d i n g the o r i g i n a l data spectrum by t h i s f u n c t i o n f i t . The r e s i d u a l r a t i o spectrum g e n e r a l l y c o n t a i n s besides the general shape of the s p e c t r a l response, many f e a t u r e s such as l o w - s p a t i a l - f r e q u e n c y v a r i a t i o n s and even f r i n g e p a t t e r n s . G e n e r a l l y , these p a t t e r n s appear when one d i v i d e s the data spectrum by a standard spectrum taken on another o b s e r v i n g date or run. The p a t t e r n s are much l e s s v i s i b l e when the data spectrum i s compared to a standard spectrum taken on the same n i g h t . In most a p p l i c a t i o n s , one can a c t u a l l y ignore a l l these small f e a t u r e s . I t i s only when one i s going f o r very h i g h p r e c i s i o n i . e . b e t t e r than ±20ms" 1 that they may become important. A s i n g l e polynomial f i t to a l l the f e a t u r e s i n the complicated spectrum i s u s u a l l y i m p o s s i b l e . One can, however, s u b d i v i d e the spectrum i n t o s m a l l e r s e c t i o n s and apply polynomial or s i n u s o i d a l f u n c t i o n f i t s to each i n d i v i d u a l s e c t i o n . The d i s c o n t i n u o u s j o i n t s between the polynomial f i t s from d i f f e r e n t s e c t i o n s can be n u m e r i c a l l y smoothed. A simpler method i s to use a c u b i c - s p l i n e f i t r a t h e r than the pseudo-piecewise-polynomial 1 44 f i t . The goodness of f i t to the r e s i d u a l spectrum i s very dependent on how w e l l the standard spectrum matches the p a r t i c u l a r data spectrum. T h i s i s one of the main reasons why the standard spectrum i s u s u a l l y t h a t of the same s t a r . Other f a c t o r s that would a f f e c t the match i n c l u d e d i f f e r e n c e s between the i n s t r u m e n t a l p r o f i l e , focus, r a d i a l v e l o c i t y , and d i s p e r s i o n . The former are g e n e r a l l y kept to a minimum by c a r e f u l alignment of the equipments while the l a t t e r can be c o r r e c t e d by numerical methods. However, one w i l l have d i f f i c u l t i e s i f there are i n t r i n s i c l i n e - p r o f i l e v a r i a t i o n s . R a d i a l - v e l o c i t y and d i s p e r s i o n c o r r e c t i o n s can be a p p l i e d to the standard spectrum to produce a b e t t e r matching spectrum. T h i s , however, r e q u i r e s the knowledge of at l e a s t rough r a d i a l - v e l o c i t y and d i s p e r s i o n "information. A b e t t e r method would be the use of a polynomial f i t between the p o s i t i o n s of corresponding s p e c t r a l l i n e s i n the data and standard s p e c t r a . Based on the polynomial f i t , an i n t e r p o l a t i o n procedure can then be a p p l i e d to the standard spectrum to produce a b e t t e r matching spectrum. However, t h i s method would become r a t h e r complicated f o r v a r i a b l e - s t a r s p e c t r a where d i f f e r e n t l i n e s may have d i f f e r e n t v e l o c i t i e s . I t i s a l s o d i f f i c u l t f o r a s i n g l e p o l y nomial to represent a l l the h i g h e r - f r e q u e n c y d i s p e r s i o n v a r i a t i o n s . A more s u i t a b l e method i s again the use of a c u b i c - s p l i n e f i t between the two s e t s of l i n e p o s i t i o n s . A s i n c - i n t e r p o l a t i o n method i s g e n e r a l l y used with the f i t to produce the matching spectrum. The same procedure i s a l s o 1 45 a p p l i e d to the continuum r e c t i f i c a t i o n of the HF s p e c t r a . The f i n a l r e s i d u a l spectrum w i l l probably s t i l l c o n t a i n e f f e c t s of poor l i n e c a n c e l l a t i o n s . These can be "overlooked" i n the continuum f i t by c a r e f u l c h o i c e of the continuum p o i n t s . Imperfect c a n c e l l a t i o n between l i n e s ' w i l l be d i s c u s s e d i n more d e t a i l l a t e r when i t s e f f e c t on l i n e - p o s i t i o n d e t e r m i n a t i o n s i s examined. 4.3.2 LINE-POSITION DETERMINATION 4.3.2.1 Line c a n c e l l a t i o n s The data s t e l l a r s p e c t r a have both HF and s t e l l a r l i n e s . Blendings between the two s e t s of l i n e s would s e r i o u s l y a f f e c t any d i r e c t l i n e - p o s i t i o n measurement from the s p e c t r a . D i v i d i n g the data spectrum by the matched standard HF system w i l l g i v e a s t e l l a r spectrum e s s e n t i a l l y f r e e of the contaminations by the HF l i n e s . Any small r e s i d u a l due to imperfect l i n e . c a n c e l l a t i o n s would not be a s e r i o u s problem. One can simply ignore the few a f f e c t e d s t e l l a r l i n e s and there would s t i l l be p l e n t y of u n a f f e c t e d s t e l l a r l i n e s l e f t . The s t e l l a r l i n e p o s i t i o n s can be d e r i v e d from the r e l a t i v e l i n e p o s i t i o n s obtained by comparing t h i s HF-free s t e l l a r spectrum with the standard s t e l l a r spectrum. T h i s i s accomplished f o r each i n d i v i d u a l l i n e by means of the d i f f e r e n c e technique d e s c r i b e d by Fahlman and Glaspey [1973], Since one i s comparing the corresponding i n d i v i d u a l l i n e p r o f i l e s , i t i s not necessary to use the matched spectrum as the r e f e r e n c e spectrum. The o r i g i n a l 1 4 6 standard spectrum i s s u f f i c i e n t f o r the purpose. Any d i f f e r e n c e i n the d i s p e r s i o n between the s p e c t r a over the narrow width of the i n d i v i d u a l s t e l l a r l i n e p r o f i l e i s n e g l i g i b l e . In f a c t , one should use the same standard spectrum as the comparison spectrum f o r a l l other data s p e c t r a of the same s t a r . T h i s w i l l ensure that the same c r i t e r i a are used to determine each r e l a t i v e . l i n e p o s i t i o n . R e l a t i v e r a d i a l v e l o c i t i e s determined f o r each i n d i v i d u a l l i n e w i l l be i n t e r n a l l y c o n s i s t e n t f o r a l l the s p e c t r a of the same s t a r . One can then make d i r e c t comparison between these v e l o c i t i e s without the need of any a d d i t i o n a l c o r r e c t i o n . The HF l i n e p o s i t i o n s can a l s o be measured with the d i f f e r e n c e technique by comparing the HF l i n e s i n the data spectrum a g a i n s t those i n the standard HF spectrum. D i v i d i n g the data spectrum by the matched s t e l l a r spectrum w i l l g i v e an HF spectrum almost f r e e of the contaminations by the s t e l l a r l i n e s . The contaminations by the s t e l l a r l i n e s can a l s o be minimised by d i v i d i n g the data spectrum by a s t e l l a r spectrum which has been n u m e r i c a l l y s h i f t e d i n p i x e l p o s i t i o n . The amount of s h i f t used i s such that i t would produce minimal r e s i d u a l i n the neighbourhood of the HF l i n e s a f t e r the spectrum d i v i s i o n . A rough l i n e c a n c e l l a t i o n can be achieved i f a s i n g l e mean s h i f t i s used. For more r e f i n e d work, d i f f e r e n t optimal values of s h i f t can be used f o r d i f f e r e n t HF l i n e s . However, t h i s would imply a s i g n i f i c a n t i n c r e a s e i n the amount of r e q u i r e d 1 47 computation when i n d i v i d u a l spectrum c a n c e l l a t i o n i s used f o r each HF l i n e - p o s i t i o n d e t e r m i n a t i o n . Moreover, d i f f e r e n t d i f f e r e n c e - t e c h n i q u e c a l c u l a t i o n has to be a p p l i e d to each l i n e . The same technique can a l s o be used to n u m e r i c a l l y minimise the contaminations on the s t e l l a r l i n e s by the HF l i n e s . No matter which l i n e - c a n c e l l a t i o n technique i s used, there w i l l always be small r e s i d u a l s due to imperfect l i n e c a n c e l l a t i o n s . These r e s i d u a l s are caused by a v a r i e t y of small e f f e c t s . These i n c l u d e i n t r i n s i c l i n e - p r o f i l e v a r i a t i o n s , imperfect p r e p r o c e s s i n g of the Ret i c o n s p e c t r a , and imperfect d i s p e r s i o n match. One of the most s e r i o u s l i m i t a t i o n s i s an e f f e c t caused by the nonzero-width of the i n s t r u m e n t a l p r o f i l e and l i n e b l e n d i n g s . Campbell and Walker [1985] have p o i n t e d out that the observed spectrum i s a c t u a l l y the c o n v o l u t i o n between the i n t r i n s i c spectrum with the in s t r u m e n t a l p r o f i l e I ( X ) . For the i n t r i n s i c s t e l l a r spectrum S(X) and the HF spectrum H(X), the observed s p e c t r a are S(X ) * I(X) and H(X)*I(X), r e s p e c t i v e l y . The o p e r a t i o n symbol * denotes c o n v o l u t i o n . S i m i l a r l y , the observed s t e l l a r spectrum with imposed HF l i n e s would then be [S ( X ) H ( X ) ] * I ( X ) . Consequently, l i n e c a n c e l l a t i o n by f l a t - f i e l d i n g - t y p e d i v i s i o n technique i s not q u i t e c o r r e c t i . e . {[S ( X ) H ( X ) ] * I ( X ) } / [ H ( X ) * I ( X ) ] * S(X ) * I(X) (4.1) { [ S ( X ) H ( X ) ] * l ( X ) } / [ S ( X ) * l ( X ) ] * H(X)*I(X) (4.2) 1 48 E q u a l i t y i n Equations 4.1 and 4.2 would h o l d only i f there i s no b l e n d i n g between the s t e l l a r and HF l i n e s or the i n s t r u m e n t a l p r o f i l e i s very narrow. However, e i t h e r of these c o n d i t i o n s i s almost impossible to a t t a i n . There w i l l always be s t e l l a r l i n e s blended with the HF l i n e s e s p e c i a l l y f o r the l a t e r - s p e c t r a l - t y p e s t a r s . The i n s t r u m e n t a l p r o f i l e i s f i x e d by the p r o j e c t e d s l i t width of the image s l i c e r and by the red r e s o l u t i o n l o s s of the R e t i c o n . I t i s the l i n e b l e n d i n g s that n u l l i f y the e q u a l i t y i n Equations 4.1 and 4.2 and hence c o n t r i b u t e to the r e s i d u a l i n the l i n e c a n c e l l a t i o n s . The r e s i d u a l i s a f u n c t i o n of the r e l a t i v e l i n e s t r e n g t h s between the s t e l l a r and HF l i n e s , the l i n e w i d t h s , the r e l a t i v e l i n e p o s i t i o n s , and the c r i t e r i a used to determine l i n e p o s i t i o n s . The l i n e b l e n d i n g s a l s o change with the apparent r a d i a l - v e l o c i t y v a r i a t i o n s of the s t a r . D i f f e r e n t s t e l l a r l i n e s w i l l be blended with the HF l i n e s as the apparent r a d i a l v e l o c i t y of the s t a r changes with the b a r y c e n t r i c E a r t h ' s o r b i t a l motion, E a r t h ' s r o t a t i o n , and any i n t r i n s i c r a d i a l - v e l o c i t y v a r i a t i o n s of the s t a r . These cause the r e s i d u a l from imperfect l i n e c a n c e l l a t i o n to be r a d i a l v e l o c i t y and s p e c t r a l type dependent. The e f f e c t of contamination by HF l i n e r e s i d u a l s on s t e l l a r l i n e s i s not c r i t i c a l s i n c e one can simply ignore the more s e r i o u s l y a f f e c t e d s t e l l a r l i n e s . In the case of the contamination by s t e l l a r l i n e r e s i d u a l s on HF l i n e s , the problem cannot be ignored. There are only 149 a few HF l i n e s i n the spectrum. Campbell and Walker [1985] have s t u d i e d the problem i n d e t a i l . They adopted a more p a s s i v e and n u m e r i c a l l y s t a b l e method to c o r r e c t the e f f e c t than the obvious method of d i r e c t i n s t r u m e n t a l p r o f i l e d e c o n v o l u t i o n . A p p r o p r i a t e a r t i f i c i a l S(X), H(X), and I(X) are f i r s t generated i n order to compute the a r t i f i c i a l s p e c t r a S ( X ) * I ( X ) , H ( X ) * I ( X ) , and [ S ( X ) H ( X ) ] * I ( X ) . The r a t i o spectrum { [ S ( X ) H ( X ) ] * I ( X ) } / [ S ( X ) * I ( X ) ] can then be c o n s t r u c t e d fo r d i f f e r e n t r e l a t i v e s t e l l a r - H F l i n e p o s i t i o n s . I f the blen d i n g i s not severe, c o r r e c t i o n to any measured HF l i n e p o s i t i o n can be made by comparing the observed and a r t i f i c i a l l y generated r e s i d u a l s . G e n e r a l l y , these c o r r e c t i o n s are necessary only i f one wants to achieve a v e l o c i t y p r e c i s i o n of b e t t e r than ±20ms" 1. Young [1978] has a l s o s t u d i e d the e f f e c t of l i n e s h i f t s due to l i n e b l e n d i n g . When one i s comparing s p e c t r a taken from d i f f e r e n t o b s e r v i n g runs, imperfect c a n c e l l a t i o n s of the s t e l l a r l i n e s can a l s o be caused by changes i n I(X) between the runs. I t has been n o t i c e d i n s p e c t r a of the same s t a r t h a t the s t e l l a r l i n e s may appear more asymmetric i n some observing runs. C o r r e c t i o n f o r t h i s e f f e c t i s g e n e r a l l y d i f f i c u l t and may have to r e l y on the comparison between the standard s p e c t r a (both HF and s t e l l a r ) taken i n the d i f f e r e n t o b s e r v i n g runs. 1 50 4.3.2.2 L i n e p o s i t i o n i n standard s p e c t r a The l i n e p o s i t i o n s i n the standard s p e c t r a are g e n e r a l l y determined by the c r i t e r i o n of i n t e n s i t y - w e i g h t e d c e n t r e - o f - g r a v i t y : x = ( I. I n x. ) / ( L. I? ) (4.3) The measured l i n e p o s i t i o n i s x, and. 1^ i s the l i n e i n t e n s i t y corresponding to the p o i n t or p i x e l x^. The power n would be equal to one f o r the usual c e n t r e - o f - g r a v i t y c r i t e r i o n . The c o n d i t i o n n=2 c o u l d r e s u l t i n a b e t t e r d i s p e r s i o n f i t to arc l i n e s . The summation i s c a r r i e d over the whole l i n e p r o f i l e between a short-wavelength l i n e l i m i t and a long-wavelength l i n e l i m i t . I t would be optimal i f these l i m i t s c o i n c i d e with the p o s i t i o n s of zero-percent l i n e a b s o r p t i o n . To use the l i n e - p o s i t i o n d e f i n i t i o n as s p e c i f i e d i n Equation 4.3, the spectrum has to be continuum r e c t i f i e d and i n v e r t e d such that a b s o r p t i o n l i n e s would appear as emission l i n e s . The advantage of the i n t e n s i t y - w e i g h t e d c e n t r e - o f - g r a v i t y c r i t e r i o n i s that i t i s a nonparametric e s t i m a t i o n of the l i n e p o s i t i o n . I t can be c o n s i d e r e d simply as the a r i t h m e t i c mean p o s i t i o n of the l i n e p r o f i l e with each p o i n t weighted by i t s i n t e n s i t y . No theory of the l i n e shape i s u t i l i s e d as compared to most l i n e - p r o f i l e - f i t t i n g t e c h niques. T h i s i s e s p e c i a l l y important f o r v a r i a b l e s t a r s where l i n e - p r o f i l e v a r i a t i o n s may occur. I f one i s , however, only i n t e r e s t e d i n the r e l a t i v e r a d i a l v e l o c i t i e s of the s t a r , the c r i t e r i o n of how one would determine the l i n e 151 p o s i t i o n s i n the standard spectrum i s not important. 4.3.2.3 Use of the d e r i v a t i v e of the l i n e p r o f i l e The main weakness of the i n t e n s i t y - w e i g h t e d c e n t r e - o f - g r a v i t y c r i t e r i o n i s the requirement of l i n e l i m i t s . T h i s i s not a s e r i o u s problem f o r i s o l a t e d l i n e s as i n the HF spectrum where p o s i t i o n s of n e a r l y zero-percent a b s o r p t i o n can be found. L i n e b l e n d i n g s , however, would l i m i t the a b i l i t y to pl a c e l i m i t s f o r the s t e l l a r l i n e s . One should use a c o n s i s t e n t c r i t e r i o n to l o c a t e l i n e l i m i t s . C r i t e r i a such as using the p o s i t i o n s of 50% l i n e a b s o r p t i o n e t c . are good except that they cannot be e a s i l y and unambiguously measured from the d i g i t a l data. A more easy and p r a c t i c a l approach i s to choose the p o s i t i o n s c o i n c i d i n g with the maximum and minimum of the f i r s t d e r i v a t i v e of the l i n e p r o f i l e . F i g u r e s 4.1, 4.2, and 4.3 show the f i r s t d e r i v a t i v e of the HF spectrum, the p Pup spectrum, and the 38 E r i spectrum, r e s p e c t i v e l y . I t can be seen from the f i g u r e s t h a t the p o s i t i o n s c o r r e s p o n d i n g to the maximum and minimum i n the d e r i v a t i v e of the l i n e p r o f i l e s are sh a r p l y d e f i n e d . The z e r o - c r o s s i n g s i n the d e r i v a t i v e spectrum correspond to the p o s i t i o n s of maximum l i n e a b s o r p t i o n i . e . "peak" l i n e p o s i t i o n s . The p i x e l p o s i t i o n s f o r the two l i m i t s are obtained by simply examining the d e r i v a t i v e spectrum. For a symmetric l i n e p r o f i l e , the two l i m i t s w i l l a l s o have the same percentage of l i n e a b s o r p t i o n . In f a c t f o r a Gaussian l i n e p r o f i l e , the two l i m i t s correspond to the p o s i t i o n s 1 52 Fi g u r e 4.1 The f i r s t d e r i v a t i v e of the HF spectrum 15  1 55 d e f i n i n g the a h a l f w i d t h of the l i n e p r o f i l e , where a i s the s t a n d a r d - d e v i a t i o n parameter f o r the Gaussian l i n e p r o f i l e . Of course, i n order to i n c l u d e more of the l i n e p r o f i l e , one can choose l i m i t s corresponding to ma, where m i s g r e a t e r than one e.g. m=v/(21n2) would be e q u i v a l e n t to the h a l f w i d t h at h a l f the l i n e depth. The e f f e c t of l i n e b l e n d i n g s w i l l , however, become p r o p o r t i o n a l l y g r e a t e r . If one wants to l o c a t e the p o s i t i o n s of the two l i m i t s more p r e c i s e l y than to the nearest p i x e l , one can n u m e r i c a l l y f i n d the c o r r e s p o n d i n g z e r o - c r o s s i n g s i n the second d e r i v a t i v e spectrum. The d e r i v a t i v e of a spectrum can be e a s i l y computed n u m e r i c a l l y u s i n g the F o u r i e r Transform method. For a spectrum S(X) and i t s F o u r i e r Transform T(a>), the f i r s t d e r i v a t i v e i s simply the i n v e r s e F o u r i e r Transform of /wT(a>). S i m i l a r l y , the second d e r i v a t i v e i s simply the i n v e r s e F o u r i e r Transform of -C J 2 T ( C J ) . For low s/n data, the d e r i v a t i v e spectrum may have to be smoothed before i t can be used. The method of i n t e n s i t y - w e i g h t e d c e n t r e - o f - g r a v i t y i s not the most i d e a l f o r very h i g h p r e c i s i o n d e t e r m i n a t i o n of r e l a t i v e l i n e p o s i t i o n s . I t i s q u i t e s e n s i t i v e to the c h o i c e of l i n e l i m i t s . Fahlman [1982] has shown that e r r o r terms w i l l be i n t r o d u c e d i n t o the r e s u l t i f the l i m i t s are not chosen to be at the p o s i t i o n s of zero-percent l i n e a b s o r p t i o n . The e r r o r terms w i l l a l s o be c a n c e l l e d i f both l i m i t s are chosen at the p o s i t i o n s which have the same percentage of l i n e 1 56 a b s o r p t i o n . None of these c o n d i t i o n s can be e a s i l y a c h i e v e d . The n e c e s s i t y to sum between no n - i n t e g e r - v a l u e d l i n e l i m i t s i s not an easy computational procedure, even i f the l i m i t s can be d e f i n e d with s u f f i c i e n t accuracy. The presence of nonsymmetrical l i n e p r o f i l e s w i l l f u r t h e r complicate the problem. I t i s more d e s i r a b l e to use a method f o r the det e r m i n a t i o n of r e l a t i v e l i n e p o s i t i o n s which i s l e s s s e n s i t i v e to the accuracy i n the placement of l i n e l i m i t s but would s t i l l p r e s erve the nonparametric e s t i m a t i o n p r o p e r t y of the c e n t r e - o f - g r a v i t y c r i t e r i o n . 4.3.2.4 The Fahlman-Glaspey d i f f e r e n c e technique The most g e n e r a l and nonparametric technique to determine the r e l a t i v e p o s i t i o n between two s p e c t r a l l i n e s i s probably the technique of c r o s s - c o r r e l a t i o n . The maximum of the c r o s s - c o r r e l a t i o n f u n c t i o n between two l i n e p r o f i l e s w i l l occur a t - the p o s i t i o n c o r r e s p o n d i n g t o the r e l a t i v e s h i f t between them. A s i m i l a r method i s the d i f f e r e n c e - f u n c t i o n technique d e s c r i b e d by Fahlman and Glaspey [1973]. In f a c t , f i n d i n g the minimum of the d i f f e r e n c e f u n c t i o n i s mathematically e q u i v a l e n t to f i n d i n g the maximum of the c r o s s - c o r r e l a t i o n f u n c t i o n (Fahlman [1984]). The d i f f e r e n c e - f u n c t i o n technique i s , however, more i n t u i t i v e l y a t t r a c t i v e as i t can be expressed i n terms of l e a s t - s q u a r e s m i n i m i s a t i o n . I t can a l s o p rovide an e r r o r estimate f o r the c a l c u l a t e d r e l a t i v e l i n e s h i f t . S i m i l a r to the c e n t r e - o f - g r a v i t y method, the accuracy of 157 a l i n e - s h i f t measurement with the d i f f e r e n c e - f u n c t i o n technique depends on the e q u i v a l e n t width of the l i n e i . e . the amount of i n f o r m a t i o n i n the l i n e . T h i s c o n c l u s i o n , however, does not i n c l u d e any c o n s i d e r a t i o n of the n o i s e . T h i s p o i n t w i l l be d i s c u s s e d l a t e r i n the chapter. B a s i c a l l y , the data spectrum B(X) i s n u m e r i c a l l y s h i f t e d ( d i s p l a c e d i n p i x e l p o s i t i o n ) by v a r i o u s amounts and s u b t r a c t e d from the standard spectrum A(X). The sum of squares of the r e s i d u a l s i n the d i f f e r e n c e spectrum over the p a r t i c u l a r l i n e p r o f i l e i s computed f o r each d i f f e r e n t s h i f t of the data spectrum. One can then formulate a d i f f e r e n c e f u n c t i o n D(s) which i s a f u n c t i o n of the s h i f t s, and takes on the value of the c orresponding sum of squares of the r e s i d u a l s : D(s) = I. [ A(X-) - B(X i+s) ] 2 (4.4) The term A(X^) i s the i n t e n s i t y at p i x e l i i n the spectrum A(X). The minimum of the d i f f e r e n c e f u n c t i o n would give the optimal s h i f t which would produce the minimum sum of the squares of the r e s i d u a l s . As i n the case of the c r o s s - c o r r e l a t i o n technique, non-integer v a l u e s f o r s are g e n e r a l l y r e q u i r e d f o r h i g h p r e c i s i o n r e s u l t s . I n t e r p o l a t i o n has to be used to produce a spectrum s h i f t e d by a non-integer amount. T h i s can be e a s i l y accomplished by the F o u r i e r S h i f t theorem. The s h i f t e d spectrum B(X+s) i s simply the i n v e r s e F o u r i e r Transform of exp( io)s )T(w), where T(o>) i s the F o u r i e r Transform of B(X). T h i s procedure i s e q u i v a l e n t to the 1 58 sin e i n t e r p o l a t i o n . An assumption of i d e n t i c a l l i n e shape f o r both A(X) and B(X) would mean B(X) = A ( X - s 0 ) , where s0 i s the i n t r i n s i c r e l a t i v e l i n e s h i f t . Fahlman and Glaspey [1973] have shown that under t h i s assumption, D(s) i n Equation 4.4 i s a parabola when h(\+s-s0) i s T a y l o r expanded to the f i r s t order of s - s 0 . T h i s means a second-order polynomial f i t to the d i f f e r e n c e f u n c t i o n i s optimal and the minimum of the parabol a corresponds to the optimal s h i f t . The T a y l o r expansion c a r r i e s the a d d i t i o n a l assumption that s~s0 i s s m a l l . T h i s assumption i s almost always true s i n c e any i n t e g e r p o r t i o n of the s h i f t s can be e x p l i c i t l y c o r r e c t e d before a p p l i c a t i o n of the method. G e n e r a l l y , f o r the high s/n HF data, s i x t h - o r d e r polynomials are found to provide b e t t e r f i t s to the d i f f e r e n c e f u n c t i o n s . T h i s i s using about 14 d i f f e r e n t values, of s i n steps of 0.05 p i x e l about the d i f f e r e n c e f u n c t i o n minimum. The parabola f i t s , however, are s t i l l very good approximations near the minimum of the d i f f e r e n c e f u n c t i o n s . The p o s i t i o n of the minimum d i f f e r e n c e i s found by n u m e r i c a l l y s o l v i n g f o r the r o o t s to the d e r i v a t i v e of the polynomial f i t . D i s c r i m i m a t i n g the r o o t s i s not a problem. Conceivably, with i n c r e a s e d computation, one can dispense with the polynomial f i t and apply o p t i m i s a t i o n techniques to f i n d the minimum of the d i f f e r e n c e f u n c t i o n d i r e c t l y . G e n e r a l l y , the d i f f e r e n c e f u n c t i o n technique i s a p p l i e d to i n d i v i d u a l l i n e s . The summation i n Equation 1 5 9 4.4 i s best to be performed over the whole l i n e p r o f i l e . I f the l i n e l i m i t s f o r the summation are d e f i n e d over only a p a r t i a l l i n e p r o f i l e , i t w i l l r e s u l t i n the c a l c u l a t i o n of a d i f f e r e n c e f u n c t i o n d i f f e r e n t than that summed over the whole p r o f i l e . But i f the r e s i d u a l s are the r e s u l t of pure r e l a t i v e l i n e - p o s i t i o n s h i f t , t h i s d i f f e r e n c e f u n c t i o n should a l s o be minimal at the same s h i f t p o s i t i o n as that summed over the whole p r o f i l e or any other p a r t of the p r o f i l e . T h e r e f o r e , the c a l c u l a t e d r e l a t i v e l i n e s h i f t i s not very s e n s i t i v e to the accuracy i n the placement of the l i n e l i m i t s . In cases of s m a l l i n t r i n s i c s h i f t s , r e s i d u a l s i n the d i f f e r e n c e spectrum at the f a r wings of the l i n e are u s u a l l y small compared to that from the l i n e c o r e . The squaring of the r e s i d u a l s i n Equation 4.4 f u r t h e r decreases the r e l a t i v e c o n t r i b u t i o n from the wings to the d i f f e r e n c e f u n c t i o n . Consequently, f o r r e l a t i v e r a d i a l - v e l o c i t y work, a c o n s i s t e n t c r i t e r i o n t o d e f i n e l i n e l i m i t s c o v e r i n g the whole l i n e core would be adequate. Of course, the more of the p r o f i l e i s u t i l i s e d , the more v e l o c i t y i n f o r m a t i o n i s used, and hence a more ac c u r a t e d e t e r m i n a t i o n w i l l r e s u l t . However, the wider the l i n e l i m i t s , the more acute w i l l be the imperfect l i n e - c a n c e l l a t i o n e f f e c t s caused by i n c r e a s e d s t e l l a r - H F l i n e b l e n d i n g s . In f a c t , t h i s i s one of the major l i m i t a t i o n s on the accuracy of the measured HF l i n e p o s i t i o n s and hence the accuracy of the measured v e l o c i t i e s . Consequently, the l i n e l i m i t s f o r the HF 160 l i n e s are u s u a l l y more r e s t r i c t e d than the corresponding ones used f o r the c e n t r e - o f - g r a v i t y l i n e - p o s i t i o n measurements i n the standard spectrum. A set of l i n e l i m i t s corresponding to twice that i n d i c a t e d by the p o s i t i o n s of maximum and minimum i n the d e r i v a t i v e i s g e n e r a l l y used. Less r e s t r i c t i v e l i m i t s may be used f o r the imposed HF l i n e s from b r o a d - s t e l l a r - l i n e s p e c t r a . In t h i s case, the imperfect l i n e - c a n c e l l a t i o n e f f e c t i s l e s s severe. For the s t e l l a r l i n e s , the same l i n e l i m i t s used f o r the c e n t r e - o f - g r a v i t y l i n e - p o s i t i o n measurements i n the standard spectrum are a l s o used f o r the d i f f e r e n c e - f u n c t i o n technique. 4.3.2.5 L i n e - p r o f i l e v a r i a t i o n s In the case of l i n e - p r o f i l e v a r i a t i o n s , a c o n s i s t e n t c r i t e r i o n to d e f i n e the l i n e l i m i t s becomes important. L i n e - p r o f i l e v a r i a t i o n s can be q u a n t i f i e d i n terms of r e l a t i v e r a d i a l - v e l o c i t y v a r i a t i o n s . The d i f f e r e n c e f u n c t i o n technique can a l s o be a p p l i e d to measure these pseudo r a d i a l - v e l o c i t y v a r i a t i o n s . In t h i s case, the r e s i d u a l s i n Equation 4.4 are produced by p r o f i l e v a r i a t i o n s r a t h e r than by pure displacement of the l i n e p r o f i l e . The r e s i d u a l s w i l l be d i f f e r e n t at d i f f e r e n t p a r t s of the p r o f i l e and w i l l a l s o be a f u n c t i o n of the type of p r o f i l e v a r i a t i o n s . Hence the c r i t e r i o n to choose the l i n e l i m i t s f o r the d i f f e r e n c e - f u n c t i o n technique i s e s s e n t i a l l y the c r i t e r i o n to sample a p a r t i c u l a r p a r t of the p r o f i l e v a r i a t i o n s . The l i n e l i m i t s a l s o determine the r e l a t i v e 161 c o n t r i b u t i o n s toward the measured pseudo r a d i a l v e l o c i t y from d i f f e r e n t p a r t s of the p r o f i l e . As f o r example, i n most types of p r o f i l e v a r i a t i o n s , the v a r i a t i o n s near the l i n e core are g r e a t e r than those at the l i n e wings. The measured r a d i a l - v e l o c i t y amplitude f o r the v a r i a t i o n s would then be p r o p o r t i o n a l to the l i n e l i m i t s ' r e l a t i v e coverage between the core and the wings. S i m i l a r l y , the e f f e c t s on the shape and phase of the r a d i a l - v e l o c i t y curve i n cases where the l i n e - p r o f i l e v a r i a t i o n s are c h a r a c t e r i s e d by " f e a t u r e s " moving ac r o s s the l i n e , would be even more complicated. To make a comparison between r e s u l t s from d i f f e r e n t s t a r s or with model c a l c u l a t i o n s , i t i s important to have a c o n s i s t e n t c r i t e r i o n to choose the l i n e l i m i t s . In g e n e r a l , f o r na r r o w - l i n e s p e c t r a with or without p r o f i l e v a r i a t i o n s , the. adopted c r i t e r i o n i s to choose the l i n e l i m i t s as the p o s i t i o n s c o i n c i d i n g with the maximum and minimum i n the f i r s t d e r i v a t i v e of the l i n e p r o f i l e . For b r o a d - l i n e s p e c t r a , broader l i n e l i m i t s may be p o s s i b l e depending on the s e v e r i t y of the l i n e b l e n d i n g s . The l i n e l i m i t s are always chosen to be those on the standard spectrum. T h i s i s a computational d e t a i l to ensure that the same l i n e l i m i t s and c r i t e r i a are used f o r the same s t e l l a r l i n e i n a l l of the s p e c t r a . And hence i s the main reason why i n Equation 4.4 that the s h i f t s are a p p l i e d to the data s p e c t r a B(X) and not to the standard spectrum A(X). 1 62 4.3.2.6 O p t i m i s i n g d i f f e r e n c e f u n c t i o n Besides l i n e - p r o f i l e v a r i a t i o n s and r a d i a l - v e l o c i t y s h i f t s , there are other systematic e f f e c t s which c o u l d c o n t r i b u t e to the r e s i d u a l s and the d i f f e r e n c e f u n c t i o n . They are the very small e f f e c t s caused by imperfect data p r e p r o c e s s i n g , presence of dark c u r r e n t or p e r s i s t e n c e phenomenon induced non-zero o f f s e t s i n the d e t e c t o r ' s response, imperfect continuum r e c t i f i c a t i o n , and d i f f e r e n c e s between the in s t r u m e n t a l p r o f i l e s . These e f f e c t s should be removed or c o r r e c t e d from the d i f f e r e n c e f u n c t i o n such that the accuracy of the f u n c t i o n to measure s p e c t r a l s h i f t s i s l i m i t e d by only the f i n i t e s/n of the s p e c t r a . T h i s can be accomplished to a f i r s t approximation by n o r m a l i s i n g the two s p e c t r a A(X) and B(X) to have the same l i n e depth or e q u i v a l e n t width e t c . One of the most e f f e c t i v e yet s t i l l very simple procedures i s to formulate a m o d i f i e d d i f f e r e n c e f u n c t i o n E(s) : E(s) = S. [ ( A U j J - a ) - 0(B(X i + s ) - i ) ] 2 (4.5) The term a i s the mean of the spectrum A(X) over the l i n e l i m i t s , and b i s the mean of the spectrum B(X+s') over the l i n e l i m i t s . The term s' i s the i n i t i a l e stimate f o r the i n t r i n s i c r e l a t i v e l i n e s h i f t s0. I t i s c a l c u l a t e d as the p o s i t i o n of D(s)'s minimum. The term 0 i s a s c a l e f a c t o r that would minimise E ( s ' ) . An a n a l y t i c e x p r e s s i o n f o r 0 can be d e r i v e d from the c o n d i t i o n 9E(s,/3)/3/J = 0 at s = s' . One o b t a i n s : 0 = [ I . A U . j B U . + s ' ) - nab ] / 1 6 3 [ Z ^ B U . + s ' )2 - nb2 ] ( 4 . 6 ) E(s) L i A ( X i ) 2 - 2 ( a - / 3 6 ) L . A ( X i ) - 2 / 3 Z i A ( X i ) B ( X i + 5 ) + 0 2 I . B ( X . + s ) 2 - 2 / 3 ( f l - / 3 i ) E i B ( X i + 5 ) + n(a-pb)2 ( 4 . 7 ) D ( j ) L . A ( X . ) 2 - 2 Z i A ( X i ) B ( X i + ^ ) + Z i B ( X i + ^ ) 2 ( 4 . 8 ) T h e t e r m n i s t h e n u m b e r o f p o i n t s o v e r t h e l i n e l i m i t s . E q u a t i o n s 4 . 7 a n d 4 . 8 a r e m a t h e m a t i c a l l y e q u i v a l e n t t o E q u a t i o n s 4 . 6 a n d 4 . 5 , r e s p e c t i v e l y . T h e f u n c t i o n E ( s ) i s r e w r i t t e n i n a m o r e e a s y c o m p u t a t i o n a l f o r m s u c h t h a t i t c a n b e c o n s t r u c t e d f r o m t h e s a m e b a s i c i n f o r m a t i o n c o m p u t e d i n o b t a i n i n g s ' f r o m D ( s ) . T h i s a v o i d s t h e n e e d t o a p p l y a d d i t i o n a l s h i f t s t o B ( X ) a n d c a l c u l a t i n g m o r e r e s i d u a l s . T h e m o d i f i e d d i f f e r e n c e f u n c t i o n E(s) i s b a s i c a l l y t h e r e s u l t o f a d j u s t i n g A ( X ) a n d B ( X + s ) t o t h e s a m e m e a n v a l u e o r o f f s e t a n d t h e n a p p l y i n g a s c a l e f a c t o r t o m o d i f y t h e l i n e d e p t h o f B ( X + s ) . T h i s p r o c e d u r e r e s u l t s i n m i n i m i s i n g t h e d i f f e r e n c e f u n c t i o n . T h e p o s i t i o n o f m i n i m a l E ( s ) , s", w o u l d b e a n i m p r o v e d e s t i m a t e f o r t h e o p t i m a l s h i f t o v e r 5 / . I n f a c t , E ( s " ) i s a l w a y s l e s s t h a n D ( s ' ) . T h e d i f f e r e n c e s b e t w e e n t h e t w o v a l u e s a r e u s u a l l y s m a l l e x c e p t i n c a s e s w h e r e t h e r e q u i r e d c o r r e c t i o n s a r e l a r g e i . e . a b e i n g q u i t e d i f f e r e n t t h a n b, o r /3 n o t c l o s e t o o n e . O f c o u r s e , o n e s h o u l d b e a w a r e o f t h e c o n s e q u e n c e s i n a p p l y i n g E(s) r a t h e r t h a n D{s) f o r l i n e s w i t h i n t r i n s i c p r o f i l e 1 6 4 v a r i a t i o n s . 4 . 3 . 2 . 7 E r r o r e s t i m a t i o n The minimum of the d i f f e r e n c e f u n c t i o n r e p r e s e n t s the minimal mean square r e s i d u a l p o s s i b l e under the e f f e c t of noise present i n the s p e c t r a . In the case of n o i s e - f r e e data, the minimum of the f u n c t i o n would be zer o . T h i s w i l l a l s o happen i f the noise i s c o r r e l a t e d between A(X) and B(X) such that the r e s i d u a l goes to zero at s = s". The minimum value of the d i f f e r e n c e f u n c t i o n can provide a measure of the accuracy i n the measured r e l a t i v e s h i f t . Fahlman [ 1 9 8 4 ] has p o i n t e d out that the minimum of the d i f f e r e n c e f u n c t i o n f o l l o w s a x 2 v d i s t r i b u t i o n with v .= 2 ( « ~ 1 ) degrees of freedom. In the case of E ( s ) or other m o d i f i e d d i f f e r e n c e f u n c t i o n s , the number of degrees of freedom would be decreased by the number of a d d i t i o n a l estimated parameters i n the formula e.g. a, b, and /3. Since s" i s only an estimate f o r the tr u e s h i f t s0, E(s") i s a l s o an estimate f o r the true minimal d i f f e r e n c e . With a confidence i n t e r v a l , e.g. 6 8 . 3 % f o r equivalence to that one standard d e v i a t i o n , x 2 can be evaluated and one can then get an estimate f o r the standard d e v i a t i o n a of E ( s " ) : a = v Ejis") / x 2 ( 4 . 9 ) T h i s i m p l i e s that there i s a 6 8 . 3 % chance that the true minimal d i f f e r e n c e would l i e w i t h i n a of E ( s " ) . T h i s standard d e v i a t i o n can then be t r a n s l a t e d d i r e c t l y from the shape of the d i f f e r e n c e f u n c t i o n i n t o a standard 1 65 d e v i a t i o n f o r the measured optimal s h i f t s " , and hence, a standard d e v i a t i o n f o r the measured r e l a t i v e r a d i a l v e l o c i t y . The r e c i p r o c a l of the d i f f e r e n c e f u n c t i o n minimum should a l s o p r o v i d e an estimate f o r the e f f e c t i v e s/n of the s p e c t r a . I f the n o i s e i n A ( X ) i s very s m a l l , then the s/n of B ( X ) i s e s s e n t i a l l y b/ySEis"). 4.3.2.8 D i s p e r s i o n r e l a t i o n The a b s o l u t e p o s i t i o n of any l i n e can be d e r i v e d from i t s r e l a t i v e l i n e s h i f t with r e s p e c t to the a b s o l u t e l i n e p o s i t i o n on the standard spectrum. The p o s i t i o n i s then c o r r e c t e d f o r the e f f e c t of imperfect l i n e c a n c e l l a t i o n s caused by l i n e b l e n d i n g s and c o n v o l u t i o n with the i n s t r u m e n t a l p r o f i l e . To t r a n s l a t e the l i n e p o s i t i o n i n t o a wavelength measurement, the d i s p e r s i o n r e l a t i o n has to be c a l c u l a t e d . A polynomial f i t t o the HF l i n e p o s i t i o n s a g a i n s t t h e i r wavelengths would p r o v i d e the d i s p e r s i o n c a l i b r a t i o n . The HF wavelengths used i n the d i s p e r s i o n f i t would be the adopted a b s o l u t e wavelengths c o r r e c t e d f o r any a b s o l u t e or d i f f e r e n t i a l l i n e s h i f t s induced by the temperatureand pressure-dependent molecular c o l l i s i o n s . G e n e r a l l y , t h i r d o r lower-order polynomials are used t o f i t the HF l i n e s . T h i s i s imposed by the f a c t that a h i g h e r - order polynomial f i t i s not f e a s i b l e f o r the small number of HF l i n e s i n the spectrum. Each l i n e i s weighted by the square of i t s l i n e depth i n the d i s p e r s i o n f i t . For c o n s i s t e n c y , the same weight i s used 1 66 f o r the same l i n e i n a l l of the s p e c t r a . The p a r t i c u l a r c h o i c e of the weighting f u n c t i o n has been shown by Fahlman [1984] to be a d i r e c t consequence of minimising the d i f f e r e n c e f u n c t i o n f o r the m u l t i - l i n e case. Without the d e t a i l e d l i n e - p o s i t i o n or reference-wavelength s h i f t c o r r e c t i o n s , a standard d e v i a t i o n of the order of 0.0015A can be achieved in a d i s p e r s i o n f i t of the HF spectrum from the CFHT Coude. T h i s corresponds to a standard d e v i a t i o n of 50ms"1 f o r the v e l o c i t i e s . The d i s p e r s i o n f i t i s b a s i c a l l y the mean d i s p e r s i o n r e l a t i o n f o r the spectrum and i t can be e v a l u a t e d at any s t e l l a r l i n e p o s i t i o n to give the observed wavelength of that l i n e . The HF l i n e s i n the spectrum are separated from each other by about 10 to 25 angstroms. Hence only the low-frequency components of the d i s p e r s i o n r e l a t i o n are c a l i b r a t e d by the HF l i n e s . High-frequency s t r u c t u r e may s t i l l be present in the d i s p e r s i o n r e l a t i o n . T h i s s t r u c t u r e c o u l d be the r e s u l t of o p t i c a l e f f e c t s w i t h i n the spectrograph as w e l l as i r r e g u l a r spacing between the R e t i c o n photodiodes. D i f f e r e n c e s i n the shape of the photometric response a c r o s s each i n d i v i d u a l photodiode would be manifested as d i f f e r e n c e s i n the spacings between each photodiode's o p t i c a l c e n t r e s . For a v e l o c i t y p r e c i s i o n of ±l0ms" 1, the wavelengths have to be measured to an accuracy of ±3x10""A. Hence i f the high-frequency s t r u c t u r e can cause a d i f f e r e n c e from the mean disp e r s i o n ' r e l a t i o n of more than ±3x10" "A, v e r y - h i g h - p r e c i s i o n r e s u l t s can be a f f e c t e d . 1 6 7 To decrease the e f f e c t s of the high-frequency s t r u c t u r e i n r e l a t i v e r a d i a l - v e l o c i t y work, the Re t i c o n i s always a l i g n e d such that the HF l i n e s are at the same p o s i t i o n s on the Re t i c o n a r r a y i . e . having the same i d e n t i c a l s p e c t r a l r e gion at a l l times. The mean d i s p e r s i o n r e l a t i o n can be improved by i n c l u d i n g the s t e l l a r l i n e s i n t o the d i s p e r s i o n f i t . In t h i s case, the s t e l l a r wavelengths are the apparent wavelengths with the same v e l o c i t y s h i f t f o r a l l l i n e s . Hence an i t e r a t i v e procedure has to be used between the d i s p e r s i o n r e l a t i o n and the d e r i v e d v e l o c i t y i n order to achieve i n t e r n a l c o n s i s t e n c y . Of course, t h i s cannot be a p p l i e d to v a r i a b l e s t a r s where d i f f e r e n t l i n e s can have d i f f e r e n t v e l o c i t i e s . One simple ad hoc way to take i n t o account the int e r m e d i a t e - f r e q u e n c y s t r u c t u r e i n the wavelength c a l c u l a t i o n of a s t e l l a r l i n e i s t o weight each HF l i n e i n v e r s e l y with i t s s e p a r a t i o n from the p a r t i c u l a r s t e l l a r l i n e . A d i f f e r e n t d i s p e r s i o n r e l a t i o n can be generated f o r each HF l i n e such that the wavelength r e s i d u a l a t the p o s i t i o n of the HF l i n e i s zero. T h i s i s accomplished by r e t a i n i n g a l l the high-order terms from the mean d i s p e r s i o n f i t s but c o r r e c t i n g the z e r o t h - o r d e r term to give zero r e s i d u a l at the p a r t i c u l a r HF l i n e p o s i t i o n . Hence f o r N HF l i n e s i n the spectrum, there would be N d i f f e r e n t d i s p e r s i o n r e l a t i o n s . And N d i f f e r e n t wavelength measurements can then be eva l u a t e d f o r a p a r t i c u l a r s t e l l a r l i n e . The observed s t e l l a r l i n e wavelength would then be the 1 68 weighted average oth these N wavelengths. Each weight is inversely proportional to the separation between the particular HF line and the stellar line as well as directly proportional to the square of the HF line depth. Generally, the result of this weighted wavelength is different by about several metres per second in velocity from those obtained directly through the mean dispersion relation. Campbell et a l . [1981] have studied in detail the high-frequency structure of the dispersion relation. A Fabry-Perot etalon was used to impose a fringe pattern on incandescent lamp spectra. The fringes are about one angstrom apart and their relative wavelengths can be derived from their order numbers. These can then be compared to those calculated from the mean dispersion relation which was derived from the HF lines. By ti lt ing the Fabry-Perot etalon, the fringes can be made to move along the Reticon array. This enables the relative wavelengths at different pixel positions to be calculated. The high-frequency structure in the dispersion relation can be examined by calculating the residuals AX between the mean dispersion relation wavelengths and the Fabry-Perot wavelengths. Campbell et a l . [1981] found high-frequency structure with AX ranging between ±0.001$ and ±0.002A>. This is larger than the point-to-point scatter for AX of about ±0.00012A which could be caused by the errors in the fringe wavelengths. A single structure is generally greater 169 than 100 p i x e l s i n width. I t s shape a l s o appears to be constant w i t h i n the same observing night but v a r i e s between n i g h t s . Consequently, one can use the Fabry-Perot f r i n g e p a t t e r n s to c a l i b r a t e n i g h t l y the high-frequency components of the d i s p e r s i o n r e l a t i o n . T h i s would provide c o r r e c t i o n terms AX to the wavelengths d e r i v e d from the mean d i s p e r s i o n r e l a t i o n . These c o r r e c t i o n s are g e n e r a l l y not necessary unless one i s aiming at an accuracy of b e t t e r than 120ms" 1. In f a c t , Campbell [1984] has found that f o r low-amplitude r e l a t i v e r a d i a l - v e l o c i t y work, the n i g h t l y c o r r e c t i o n s can o n l y improve the v e l o c i t y p r e c i s i o n by about 1ms"1. 4.3.3 EFFECTIVE REST WAVELENGTHS To convert the measured wavelengths of the s t e l l a r l i n e s i n t o v e l o c i t i e s , one has to know the e f f e c t i v e r e s t wavelengths of these l i n e s . For s o l a r - t y p e s t e l l a r s p e c t r a , the most r e c e n t l y measured r e s t wavelengths are those on the s o l a r l i n e l i s t by P i e r c e and B r e c k i n r i d g e [1973]. I d e n t i f i c a t i o n s f o r these l i n e s can be found i n e a r l i e r s o l a r l i n e l i s t s by Moore et a l . [1966] and S t . John et a l . [1928]. Most wavelengths f o r other l i n e s are r e f e r e n c e d i n the second M.I.T. wavelength catalogue by Phelps [1982]. I t i s adequate to use e x c l u s i v e l y wavelengths f o r HF l i n e s and s t e l l a r l i n e s which are d e f i n e d with the r e f r a c t i v e index of standard a i r i . e . at a temperature of 15°C and an a i r p r e s s u r e of 760 t o r r s . The smooth change i n the a i r r e f r a c t i v e index over temperature and a i r pressure are 1 70 i m p l i c i t l y c o r r e c t e d i n the d i s p e r s i o n f i t when s t a n d a r d - a i r wavelengths are used c o n s i s t e n t l y . B a s i c a l l y , an almost constant r e f r a c t i v e index f a c t o r i s i n c l u d e d i n a l l the c o e f f i c i e n t s of the f i t t e d d i s p e r s i o n r e l a t i o n . Hence one can use s t a n d a r d - a i r wavelengths i n almost a l l a p p l i c a t i o n s even when the c o n d i t i o n s at the o b s e r v a t o r i e s are f a r from standard e.g. a i r pressure at CFHT i s always about 60% l e s s than one standard atmosphere. For a given l i n e , the e f f e c t i v e wavelength that one should adopt depends on s e v e r a l f a c t o r s . These i n c l u d e the p a r t i c u l a r c r i t e r i a used to measure l i n e p o s i t i o n s and ble n d i n g s with weaker l i n e s . Hence, the e f f e c t i v e wavelength i s a l s o s p e c t r a l type, r e s o l u t i o n , and d i s p e r s i o n dependent. One can perform a d i s p e r s i o n f i t t o the l i n e p o s i t i o n s on the standard spectrum using the catalogue r e s t wavelengths and then adopt the l e a s t - s q u a r e s - f i t t e d wavelengths as the e f f e c t i v e r e s t wavelengths f o r th a t s t a r . As p o i n t e d out i n Chapter one, the d e r i v e d a b s o l u t e v e l o c i t i e s cannot be ac c u r a t e down to the p r e c i s i o n p o s s i b l e i n the HF technique. Moreover, i t may not be p o s s i b l e t o p l a c e these v e l o c i t i e s onto a s t e l l a r r a d i a l - v e l o c i t y system which i s c a l i b r a t e d by r a d i a l - v e l o c i t y standard s t a r s . With the high p r e c i s i o n , even IAU r a d i a l - v e l o c i t y standard s t a r s may show v e l o c i t y v a r i a t i o n s . In almost a l l a p p l i c a t i o n s , however, one i s only i n t e r e s t e d i n the r e l a t i v e r a d i a l v e l o c i t i e s . The accuracy of the d e r i v e d r e l a t i v e v e l o c i t i e s i s not dependent on the e f f e c t i v e wavelengths as long as the same c o n s i s t e n t v a l u e s are used f o r the same s t a r . In some s t e l l a r s p e c t r a , 171 d i f f e r e n t s t e l l a r l i n e s may have d i f f e r e n t v e l o c i t i e s . In t h i s case, one i s i n t e r e s t e d i n the r e l a t i v e r a d i a l - v e l o c i t y v a r i a t i o n s i n the i n d i v i d u a l s t e l l a r l i n e s . 4.3.4 BARYCENTRIC CORRECTIONS Before one can make use of the observed v e l o c i t i e s i n any a p p l i c a t i o n , one has to convert them i n t o observed b a r y c e n t r i c v e l o c i t i e s . T h i s i n v o l v e s c o r r e c t i n g f o r the obs e r v e r ' s motion with respect to the ba r y c e n t r e of the s o l a r system. S i m i l a r l y , one should convert the t o p o c e n t r i c times of the observed v e l o c i t i e s i n t o b a r y c e n t r i c date i . e . the time that the s t a r l i g h t would have reached the bar y c e n t r e of the the s o l a r system. T h i s i n v o l v e s c o r r e c t i n g the o b s e r v e r 's p o s i t i o n with r e s p e c t to the ba r y c e n t r e of the s o l a r system. Conventional techniques only apply the c o r r e c t i o n s with r e s p e c t to the Sun as the v e l o c i t y frame of r e f e r e n c e i . e . to o b t a i n h e l i o c e n t r i c v e l o c i t i e s and h e l i o c e n t r i c d ates. A computer program to c a l c u l a t e h e l i o c e n t r i c v e l o c i t y c o r r e c t i o n s has been given by Gordon [1976]. I t has an ab s o l u t e accuracy of about 20ms - 1. With the h i g h p r e c i s i o n of the HF technique, the p e r t u r b a t i o n s by the p l a n e t s become important e s p e c i a l l y f o r long-term p r o j e c t s . The p l a n e t a r y p e r t u r b a t i o n s can cause a v e l o c i t y v a r i a t i o n of about 13ms - 1 (Gordon [1976]). I t i s convenient to perform most of the v e l o c i t y and p o s i t i o n c a l c u l a t i o n s i n the e q u a t o r i a l r e c t a n g u l a r c o o r d i n a t e system (or d i r e c t i o n c o s i n e s ) . In t h i s c o o r d i n a t e system, a s t a r with r i g h t a s c e n s i o n a and d e c l i n a t i o n 5 w i l l 1 72 have the d i r e c t i o n c o s i n e s (u,vtw): u = cosa cos5 (4.10) v = s i n a cos5 (4.11) w = sin6 (4.12) If the observer's b a r y c e n t r i c c o o r d i n a t e and v e l o c i t y are (x,y,z) and ( V X ' V ^ ' V Z K r e s p e c t i v e l y , the s t a r w i l l have a r a d i a l - v e l o c i t y c o n t r i b u t i o n V and a t a n g e n t i a l - v e l o c i t y component V f c: V„ = - ( uV + vV + >vV ) (4.13) r x y z V Q = -Mx s i n a + cos5 (4.14) Vfi = -V cosa sin6 - V s i n a sin5 + V cos6 (4.15) o x y z Vfc = •( V 2 + v 2 ) (4.16) A s p e c t r a l l i n e of wavelength X w i l l be Doppler s h i f t e d by the v e l o c i t i e s to X': (X/X') = •( 1 - V 2 / c 2 ) / ( 1 + V^/c ) (4.17) V 2 = V 2 + V 2 + V 2 (4.18) x y z V ' = - ( u ' V + v ' V + w ' V ) (4.19) r x y z i i ' = u + ( 1/c) (V x+uV r) + ( 1 /2c 2 ) ( K«+V J CV r ) (4.20) v ' = v + (1/c)(V +vV r) + ( 1/2C 2 ) (icv+VVr) (4.21) w' = w + (1/c)(V z+wV r) + ( 1 / 2 c 2 ) (K w + V z V r ) (4.22) K — 2V 2 — V 2 (4.23) Equations 4.17 through 4.23 are taken from Stumpff [1979]. The term V̂ . i n Equations 4.17 and 4.19 i s the apparent r a d i a l v e l o c i t y and i n c l u d e s the e f f e c t of a b e r r a t i o n . The use of V r r a t h e r than V£, i n Equation 4.17 would i n t r o d u c e an 1 73 e r r o r i n AX/X of up to 10~ 8. The second-order Doppler c o n t r i b u t i o n from Vfc i s a l s o i n c l u d e d i n Equation 4.17. Stumpff [1979] has p o i n t e d out that t h i s c o n t r i b u t i o n i s s i g n i f i c a n t i n h i g h - p r e c i s i o n measurements. The observer's b a r y c e n t r i c v e l o c i t y i s composed of both the b a r y c e n t r i c v e l o c i t y of the E a r t h and the observer's v e l o c i t y due to E a r t h ' s r o t a t i o n : ( V W - ( V V V V V V ( 4 - 2 4 ) The v e c t o r ( E x , E ,E z) i s the E a r t h ' s b a r y c e n t r i c v e l o c i t y and (T^,T ,T^) i s the d i u r n a l r o t a t i o n v e l o c i t y of the t e l e s c o p e . The d a i l y v a l u e s of ( E ^ E ,E z) are given i n the A s t r o n o m i c a l Almanac. They are the r e s u l t s generated by the r e l a t i v i s t i c ephemeris program JPL DE200. I t performs simultaneous i n t e g r a t i o n of the equations of motion of. the Sun and Moon as w e l l as of the p r i n c i p a l and minor p l a n e t s . An a l g o r i t h m to c a l c u l a t e (VX>V y>V z) a n d (x,y,z) has been given by Stumpff [1977,1979,1980]. T h i s method has been compared a g a i n s t an e a r l i e r ephemeris program JPL DE96. The maximum e r r o r s are of the order of 42cms" 1 f o r the v e l o c i t i e s and 4.6X10" 5AU f o r the b a r y c e n t r i c c o o r d i n a t e s . H e l i o c e n t r i c v e l o c i t i e s and c o o r d i n a t e s are a l s o c a l c u l a t e d by the program given i n Stumpff [1980]. In order to reduce the time of r e c e p t i o n of the s t a r l i g h t to a common o r i g i n at the b a r y c e n t r e , a time c o r r e c t i o n has to be added to the observed time. T h i s c o r r e c t i o n At i s simply: At = ( 1/c ) ( ux + vy + wz ) (4.25) The d i u r n a l r o t a t i o n a l v e l o c i t y of the t e l e s c o p e can be 174 c a l c u l a t e d from the f i g u r e of the E a r t h : C = 1 / •( c o s 2 ^ + ( 1 - f ) 2 sin24> ) (4.26) T = ( 27r/p ) ( aC + h ) costf> (4.27) T x -T sinty (4.28) T y T COSty (4.29) T z 0 (4.30) where a = e q u a t o r i a l r a d i u s of the E a r t h f = f l a t t e n i n g f a c t o r f o r the E a r t h p = s i d e r e a l r o t a t i o n p e r i o d of the E a r t h <p = geodetic l a t i t u d e of the obser v a t o r y h = e l e v a t i o n of the observatory ty = l o c a l mean s i d e r e a l time Equation 4.26 i s taken from Gurnette and Woolley [1974], I t enables the use of geodetic l a t i t u d e r a t h e r than g e o c e n t r i c l a t i t u d e i n Equation 4.27. The l a t e s t formulae t o c a l c u l a t e p and ty can be found i n the Ast r o n o m i c a l Almanac by Vohden and Boksenberg [1984]. I t a l s o c o n t a i n s the I.A.U. newly adopted v a l u e s f o r a l l the r e l e v a n t a s t r o n o m i c a l c o n s t a n t s e.g. a and f . Equations 4.10 through 4.30 can a l s o be used to o b t a i n h e l i o c e n t i c r a d i a l v e l o c i t i e s and time c o r r e c t i o n s i f the co r r e s p o n d i n g h e l i o c e n t r i c v e l o c i t i e s and p o s i t i o n s are used i n the eq u a t i o n s . Although the p r e c i s i o n i n the c a l c u l a t e d (E^,E^,E z) from Stumpff [1980] i s high, the f i n a l p r e c i s i o n i n the d e r i v e d v e l o c i t y c o r r e c t i o n s depends very much on the use of a p p r o p r i a t e s t e l l a r c o o r d i n a t e s (a,5). Stumpff [1980] has poi n t e d out t h a t the E-terms of a b e r r a t i o n must f i r s t be 1 75 removed from the p o s i t i o n . The E-terms c o r r e c t the e f f e c t on the a b e r r a t i o n caused by the e l l i p t i c i t y of the E a r t h ' s o r b i t (Vohden and Boksenberg [1984]). These e l l i p t i c a l a b e r r a t i o n terms are always present i n the fundamental s t e l l a r p o s i t i o n s which are i n the FK4 c a t a l o g u e . However, they w i l l be absent from the new equinox 2000 FK5 c a t a l o g u e . The formula to remove the E-terms i s given i n Vohden and Boksenberg [1984]. The e f f e c t of proper motion should a l s o be i n c l u d e d in (a,6). I f the a p p l i e d proper motions are based on the o l d p r e c e s s i o n c o n s t a n t s , any subsequent p r e c e s s i o n a p p l i e d to the p o s i t i o n s must a l s o be based on the o l d p r e c e s s i o n c o n s t a n t s . C o r r e c t i o n s to the proper motions must be made i f the new equinox 2000 p r e c e s s i o n c o n s t a n t s are used. Formulae to convert the o l d 1950 p o s i t i o n s and proper motions to the new system can be found i n Vohden and Boksenberg [1984]. Of course, c o r r e c t i o n s w i l l not be necessary i f the p o s i t i o n s and proper motions are taken from the new FK5 c a t a l o g u e . Stumpff [1979] has p o i n t e d out another p o s s i b l e source of e r r o r i n the c h o i c e of (a,5). The p o s i t i o n s should be r e f e r e n c e d to the same equator and equinox as the c a l c u l a t e d (Ex,E ,Ez). If (a,6) i s c o r r e c t e d f o r p r e c e s s i o n , the v e l o c i t y components should a l s o be p recessed a c c o r d i n g l y . S i m i l a r l y , i f n u t a t i o n c o r r e c t i o n s have been a p p l i e d to produce true equator and equinox p o s i t i o n s , the same c o r r e c t i o n s must a l s o be a p p l i e d to the v e l o c i t y components. 1 76 4.4. SIMULATION STUDIES 4.4.1 BASIC APPROACH The p r e c i s i o n of s t e l l a r l i n e - p o s i t i o n measurement depends on many f a c t o r s . These i n c l u d e the s/n of the spectrum, the depth of the l i n e , the width of the l i n e (when noise i s i n c l u d e d ) , l i n e - b l e n d i n g e f f e c t s , and the e f f e c t of cosmic-ray events. One of the s i m p l e s t ways to study the v a r i o u s e f f e c t s i s through the use of numerical s i m u l a t i o n . A r t i f i c i a l a b s o r p t i o n l i n e s p e c t r a can be n u m e r i c a l l y generated with the v a r i o u s e f f e c t s i n c l u d e d or excluded. One can then apply the v a r i o u s d a t a - r e d u c t i o n procedures to measure the l i n e p o s i t i o n on these s p e c t r a . Since the i n t r i n s i c p o s i t i o n of the a r t i f i c i a l l i n e i s known, the accuracy of the measurent can be assessed. 4.4.2 NOISE GENERATION The two types of noise to be i n c l u d e d i n the a r t i f i c i a l s p e c t r a are the readout n o i s e of the R e t i c o n and the photon shot n o i s e . The readout n o i s e would a f f e c t e q u a l l y a l l the p i x e l s i n the spectrum. The amount of readout n o i s e f o r each p i x e l can be o b t a i n e d through a random-number generator with a Standard Normal p r o b a b i l i t y d i s t r i b u t i o n . The mean value of the d i s t r i b u t i o n i s taken to be the mean readout noise of 350e". The photon noise should f o l l o w a Poisson d i s t r i b u t i o n . However, as a computational convenience, a Standard Normal d i s t r i b u t i o n random-number generator i s a l s o used. The mean value of t h i s d i s t r i b u t i o n i s the square root 1 77 of the amount of d e t e c t e d photons at the p a r t i c u l a r p i x e l . T h i s i s a c t u a l l y a very good assumption c o n s i d e r i n g the l a r g e number of d e t e c t e d photons. The C e n t r a l L i m i t theorem s t a t e s that the Poisson (or any) p r o b a b i l i t y d i s t r i b u t i o n should approach the Normal d i s t r i b u t i o n f o r l a r g e numbers. A Gaussian l i n e shape has been used for the s i m u l a t i o n . For s i m p l i c i t y , only an i s o l a t e d a b s o r p t i o n l i n e i s c o n s i d e r e d i n t h i s i n i t i a l study. A r t i f i c i a l f l a t - f i e l d lamp s p e c t r a are a l s o generated i n the same manner f o r d i v i s i o n i n t o the a b s o r p t i o n l i n e spectrum. The lamp s p e c t r a have the same s i g n a l l e v e l as the continuum of the s t e l l a r s p e c t r a . Each generated l i n e spectrum c o n t a i n s the e f f e c t of n o i s e r e d u c t i o n by using the mean of e i g h t b a s e l i n e s . The mean of four generated f l a t - f i e l d s p e c t r a i s used i n a l l cases. For s i m p l i c i t y , each i n d i v i d u a l p i x e l has been assumed to have zero width and hence a d e l t a - f u n c t i o n - t y p e response. For a more r e a l i s t i c s i m u l a t i o n of R e t i c o n s p e c t r a , the non-zero width of each p i x e l can be taken i n t o account by i n t e g r a t i n g over the l i n e p r o f i l e with the o v e r l a p p i n g t r a p e z o i d a l response f u n c t i o n of the i n d i v i d u a l p i x e l . 4 . 4 . 3 REDUCTION OF THE ARTIFICIAL SPECTRA The method which i s a p p l i e d to the r e a l s p e c t r a i s used to measure the l i n e p o s i t i o n s on these a r t i f i c i a l s p e c t r a . T h i s i n v o l v e s the use of the Fahlman-Glaspey d i f f e r e n c e technique. A d i f f e r e n t a r t i f i c i a l standard spectrum of the same s i g n a l l e v e l and other a t t r i b u t e s i s generated f o r each a r t i f i c i a l data spectrum. The standard spectrum, however, 178 has the l i n e c e n t r e d at p i x e l 150 while the l i n e p o s i t i o n i n the data spectrum i s p i x e l 151.3535. The two s p e c t r a are continuum r e c t i f i e d before . the unoptimised d i f f e r e n c e f u n c t i o n i n Equation 4.4 i s a p p l i e d to measure the r e l a t i v e s h i f t between them. The l i n e l i m i t s used are the ones co r r e s p o n d i n g to the maximum and minimum i n the f i r s t d e r i v a t i v e of the standard l i n e p r o f i l e . In t h i s case, the l i n e l i m i t s a l s o correspond to ±5, where 6 i s the s t a n d a r d - d e v i a t i o n parameter of the Gaussian l i n e p r o f i l e . At l e a s t f i v e p a i r s of these s p e c t r a are used to produce f i v e independent measurements of the s h i f t i n each e s t i m a t i o n of the accuracy. The standard d e v i a t i o n of these v a l u e s from the i n t r i n s i c value of -1.3535 would g i v e 'a measure of the accuracy. The s i m u l a t i o n t r i a l s can a l s o be used to study the e r r o r estimate d e r i v e d by the Fahlman-Glaspey technique. Out of 636 independent a p p l i c a t i o n s of the Fahlman-Glaspey method on the a r t i f i c i a l s p e c t r a , only i n 27 of these cases d i d the i n t r i n s i c s h i f t s l i e o u t s i d e the e r r o r bounds p r e d i c t e d by Equation 4.9. In another 24 cases, the i n t r i n s i c s h i f t s l i e almost e x a c t l y at the boundary of the p r e d i c t e d e r r o r bounds. T h e r e f o r e , the e r r o r estimate given by Equation 4.9 i s good on over 92% of a l l a p p l i c a t i o n s . T h i s i m p l i e s that the e r r o r estimate g i v e s a 2a c o n f i d e n c e l e v e l , r a t h e r than the usual one-o l e v e l . 179 4.4.4 THE EFFECT OF S/N The e f f e c t of s/n on the accuracy can be examined by v a r y i n g the s i g n a l l e v e l of the s p e c t r a while keeping a l l other parameters c o n s t a n t . The s/n of the generated s p e c t r a can be measured d i r e c t l y from the standard d e v i a t i o n i n t h e i r c o n t i n u a . F i g u r e 4.4 shows a" 1 p l o t t e d a g a i n s t s/n. The e r r o r a i s i n u n i t of p i x e l s and i s the standard d e v i a t i o n c a l c u l a t e d from s i x independent l i n e - s h i f t measurements. The l i n e used i n the s i m u l a t i o n has a depth of 0.4 with respect to a continuum of 1. The s t a n d a r d - d e v i a t i o n - h a l f w i d t h 5 of the l i n e i s 4 p i x e l s . The s t r a i g h t - l i n e f i t of the p l o t g i v e s : a" 1 = 0.1306 (s/n) -1.71 (4.31) The l i n e a r r e l a t i o n i s not unexpected. Campbell and Walker [1979] have given a r e l a t i o n which s t a t e s that the v e l o c i t y e r r o r i s i n v e r s e l y p r o p o r t i o n a l to the s/n. The v e l o c i t y e r r o r here i s simply ( a x c x d i s p e r s i o n ) / X. . The z e r o t h - o r d e r term i n Equation 4.31 i s n e a r l y zero as one would expect. 4.4.5 THE EFFECT OF LINE DEPTH The e f f e c t of l i n e depth on the accuracy i s examined i n F i g u r e 4.5 where a" 1 i s p l o t t e d a g a i n s t l i n e depth. The l i n e depths are i n f r a c t i o n s of the continuum. The l i n e has a s t a n d a r d - d e v i a t i o n - h a l f w i d t h of 4 p i x e l s and a s/n of about 1350. Each p o i n t on the graph i s the r e s u l t of f i v e independent measurements of the r e l a t i v e l i n e s h i f t . A l i n e a r f i t of the p l o t g i v e s : 180 F i g u r e 4.4 The e f f e c t of s /n on a c c u r a c y O CD CD CD Ul CD C\J —. (,I3XTd) tD gure 4.5 The e f f e c t of l i n e depth on a c c u r a c y ( t.I3Xjd) xD 1 82 a" 1 = 457.7 ( l i n e depth) - 10.1 (4.32) T h i s r e l a t i o n i s again not unexpected. Campbell and Walker [1979] have given an i n v e r s e r e l a t i o n s h i p between the v e l o c i t y e r r o r and l i n e depth. 4.4.6 THE EFFECT OF LINEWIDTH The e f f e c t of l i n e w i d t h on the accuracy i s examined i n F i g u r e 4.6 where a i s p l o t t e d a g a i n s t the s t a n d a r d - d e v i a t i o n - h a l f w i d t h 6 of the l i n e . F i g u r e 4.6 shows the e f f e c t of changing the l i n e w i d t h but keeping the e q u i v a l e n t width c o n s t a n t . The l i n e depth i n each spectrum of d i f f e r e n t l i n e w i d t h i s a d j u s t e d such that the e q u i v a l e n t width of the l i n e i s the same as the case with 5 being 4 p i x e l s and l i n e depth being 0.4. The s/n i n the s p e c t r a i s about 1350. The l i n e l i m i t s (±6) i n each case are v a r i e d to p r e s e r v e the same c r i t e r i o n of choosing the p o s i t i o n s at - extremal f i r s t d e r i v a t i v e . Each p o i n t i n F i g u r e 4.6 i s the r e s u l t of f i v e independent l i n e - s h i f t measurements. The f i t of the p o i n t s g i v e s : a = 0.000386 2 - 0.00185 + 0.0091 (4.33) The t r e n d between a and 8 i s not unexpected. One would expect the v e l o c i t y e r r o r to i n c r e a s e when more p i x e l s are i n c l u d e d w i t h i n the l i n e l i m i t s . T h i s i s because of the f a c t t h a t each a d d i t i o n a l p i x e l would add more noise (readout and photon) to the l i n e measurement while the e q u i v a l e n t width ( i n f o r m a t i o n content) of the l i n e has been kept c o n s t a n t . The amount of readout n o i s e would i n c r e a s e l i n e a r l y with the number of p i x e l s . The photon n o i s e , however, would i n c r e a s e 183 F i g u r e 4 . 6 The e f f e c t of l i n e w i d t h on a c c u r a c y (|3XTd) D 184 f a s t e r than l i n e a r l y with the number of p i x e l s . T h i s i s mainly because of the f a c t that the l i n e becomes shallower as the l i n e w i d t h i n c r e a s e s hence the photon noise at each p i x e l a l s o i n c r e a s e s . F i g u r e 4.7 shows the c o n v e r s i o n between s i g n a l ( i n adcu) and s/n. A readout noise of 350e~ i s used i n the c a l c u l a t i o n . Moreover, F i g u r e 4.7 a l s o assumes that e i g h t b a s e l i n e s and four f l a t - f i e l d lamps are used i n the p r e p r o c e s s i n g of the data. A c o n v e r s i o n f a c t o r of 250 e q u i v a l e n t photons per adcu has been used. T h i s f a c t o r i s a p p r o p r i a t e to the h i g h - g a i n mode of the CFHT 1872-Reticon. 4.4.7 THEORETICAL LINE-POSITION ACCURACY Walker [1982] has d e r i v e d a s i m p l i f i e d e x p r e s s i o n f o r the accuracy of a l i n e - p o s i t i o n measurement. One p o s s i b l e d e f i n i t i o n of a l i n e p o s i t i o n i s the p o s i t i o n of the l i n e b i s e c t o r which d i v i d e s the l i n e p r o f i l e i n t o two h a l v e s of equal number of d e t e c t e d photons. How w e l l t h i s b i s e c t i o n can be performed i s a f f e c t e d by the amount of n o i s e i n each h a l f of the l i n e p r o f i l e . The b i s e c t i o n i s e s s e n t i a l l y to determine a value f o r the h a l f w i d t h of the l i n e p r o f i l e at the continuum. Let t h i s h a l f w i d t h i n p i x e l s be h. For s i m p l i c i t y , one can c o n s i d e r a t r i a n g u l a r a b s o r p t i o n l i n e shape with a l i n e depth d. If. N i s the number of d e t e c t e d photons per p i x e l at the continuum, and S i s the number of d e t e c t e d photons i n each h a l f of the l i n e , then one o b t a i n s : S = N h ( 1 - d/2 ) (4.34) 5S = N ( 1 - d/2 ) 8h (4.35) adcu (hi gain) 186 U S ) 2 = S (4.36) (6/0 2 = h / ( N ( 1 - d/2 ) ) (4.37) o = 6h = |/(w/(2tf(l-d/2))) (4.38) Equation 4.36 c o n s i d e r s only photon n o i s e . T h i s i s a good assumption f o r high s/n data. The term CJ i n Equation 4.38 i s the f u l l width of the l i n e p r o f i l e at the continuum and would be equal to 2h i n t h i s simple symmetric l i n e p r o f i l e . Equation 4.38 s t a t e s the f a c t t h a t f o r a t r i a n g u l a r l i n e p r o f i l e , a, which i s the u n c e r t a i n t y i n the l i n e p o s i t i o n measurement i s the same as bh which i s the u n c e r t a i n t y i n the h a l f w i d t h o f the l i n e at the continuum l e v e l . A s i m i l a r method can be used to study the e f f e c t of a cosmic-ray event. In t h i s case, 5S i n Equation 4.36 would become the amount of e q u i v a l e n t photons i n a cosmic-ray s p i k e . Equation 4.38 g i v e s the accuracy estimate of an a b s o l u t e l i n e - p o s i t i o n measurement under a s p e c i f i c l i n e - p o s i t i o n c r i t e r i o n while the s i m u l a t i o n s t u d i e s g i v e o n l y the a c c u r a c i e s of r e l a t i v e l i n e - p o s i t i o n measurements. In s p i t e of t h i s and approximating the l i n e shape by a t r i a n g u l a r f u n c t i o n , Equation 4.38 should s t i l l g i ve at l e a s t an order-of-magnitude accuracy e s t i m a t i o n f o r comparison a g a i n s t the s i m u l a t i o n r e s u l t s . Consider a high s/n case i n which N i s 16000x250 de t e c t e d photons. F i g u r e 4.7 would then imply a s/n of about 1700. With rf=0.4 and OJ=16, Equation 4.38 would then imply a l i n e - p o s i t i o n a ccuracy of about 1.5x10" 3 p i x e l . The s i m u l a t i o n study i n F i g u r e 4.4, however, g i v e s an accuracy of 4 x l 0 " 3 p i x e l f o r a s/n of about 1700. I t i s not unexpected that the s i m u l a t i o n 187 study should give a lower value f o r the p r e c i s i o n , s i n c e r e a l i s t i c n o i s e c o n t r i b u t i o n s have been i n c l u d e d i n the s i m u l a t i o n study e.g. readout noise and noise c o n t r i b u t i o n from the f l a t - f i e l d spectrum. The s i m u l a t i o n study i s based upon the use of a standard spectrum with the same s/n. Hence, i f a much higher s/n standard spectrum i s a v a i l a b l e , the p r e c i s i o n should i n c r e a s e . Chapter 5 THE DELTA SCUTI VARIABLE 20 CVN 5.1 INTRODUCTION The s t a r 20 Canum Venaticorum ( AO CVn, HR5017 ) i s a 8 S c u t i v a r i a b l e with 8 D e l p h i n i - t y p e anomalous abundances. Parameters f o r 20 CVn are summarised in Table 5.1. Other parameters can be found i n Tsvetkov [1982b], Breger [1979], B a g l i n et a l . [1972], Gupta [1978], Breger and Bregman [1975], Leung [1970], and H o f f l e i t and Jaschek [1982]. Morgan and Abt [ 1972] have p o i n t e d out that the Ca 11 l i n e s i n 20 CVn are weaker than those i n other s t a r s of s i m i l a r s p e c t r a l type and l u m i n o s i t y c l a s s . Meanwhile, the Mg II X4481 l i n e has been commented on by H o f f l e i t and Jaschek [1982] to be anomalously s t r o n g i n 20 CVn. Abundance an a l y s e s of 20 CVn have been performed by Dickens et a l . [1971], Ishikawa [1975], Kurtz [1976], and Hauk et a l . [1985]. The value of 0.80 f o r [Fe/H] determined by Kurtz [1976] i s higher than the more t y p i c a l value of 0.44 f o r the Hyades of which 20 CVn may be a member. Leung [1970] has p o i n t e d out that there i s a d i s c r e p a n c y between the v a r i o u s r e p o r t e d v a l u e s f o r the a b s o l u t e v i s u a l magnitude M v of 20 CVn. Membership i n the Hyades would imply a value of about 0.45 f o r M v while Stromgrens photometry g i v e s a value of about 1.27. 188 189 5.2 VARIABILITIES OF 20 CVN The s t a r 20 CVn was f i r s t r e p o r t e d to be a v a r i a b l e by Wehlau et a l . [1966] and Danziger and Dickens [1966]. Other l i g h t curves f o r 20 CVn have been given by Danziger and Dickens [1967], Breger [1969], Shaw [1976], Pena and Gonzalez [1981], Chun et a l . [1983], and B o s s i et a l . [1983]. R a d i a l - v e l o c i t y v a r i a t i o n s have been measured by Smith [1982b]. Simultaneous r a d i a l - v e l o c i t y and photometric t i m e - s e r i e s o b s e r v a t i o n s have been performed by P e n f o l d [1971] and Nishimura et a l . [1983]. E a r l y photometric data of 20 CVn suggested a p u l s a t i o n p e r i o d between 0.13 and 0.14 day together with an amplitude Amv of about 0.03 m. V a l t i e r [1972] gave a r e c a l c u l a t e d p e r i o d of 0.135 day and an amplitude of 0.035 m. Shaw [1976] determined a p e r i o d of 0.12168 day or a frequency of 8.2183 day" 1 f o r the photometric v a r i a t i o n s . An amplitude of 0.0292 m was a l s o measured f o r the b r i g h t n e s s v a r i a t i o n s i n the blue while the amplitude i n yellow l i g h t was measured to be 0 . 0 l 9 6 m . Smith [1982b] adopted a value of 0.031 m f o r Am v« T h i s l i g h t amplitude has been s t a n d a r d i s e d to a c o l o u r of (b-v) = 0.20 f o r the s t a r . An amplitude of 0.022 m has been measured o f f the V data of Pena and Gonzalez [1981] by Bossi et a l . [1983]. Chun et a l . [1983] have observed amplitudes between 0.02 m and 0.04 m. Bossi et a l . [1983] have determined a frequency of 8.21 day" 1 and a V amplitude of 0.0174 m. Hence both the p e r i o d and the photometric amplitude appear to' be s t a b l e over the 14 years span of o b s e r v a t i o n s . P e n f o l d [1971] obtained a r a d i a l - v e l o c i t y amplitude 2K of 1.5km"1. 190 T a b l e 5.1 Parameters f o r 20 CVn * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * Reference HD number : 115604 SAO number : 44549 DM number : +41 2380 R . A . (1950) : 13 h 15 m 1 8 . 1 5 1 s Dec. (1950) : 4 0 ° 50' 7.483" Annual p a r a l l a x : +0.19" Proper motion in R . A . : - 0 . 1 2 7 " / y r Proper mot ion in Dec . : + 0 . 0 l 7 " / y r : 1 0 2 . 7 3 ° b1T : 7 5 . 5 2 ° S p e c t r a l type : F3 I I I - I V ' R a d i a l v e l o c i t y = 7.4 kms" 1 V s i n i = <5 k m s - 1 M y = +0.5 T e f f = 7500K l o g g = 3 .7 (cgs) Broad band photometry Pena et a l . [1981] Smith [1982b] Smith [1982b] Eggen [1979] K u r t z [1976] K u r t z [1976] I r i a t e et a l [1965] V = 4. 7 3 m B - V = +0. 30 m U - V = +0. 50 m V - R = +0. 2 5 m V - I = +0. 40 m I n f a r e d photometry : Verma et a l . [1983] 191 H = 4.17 m K = 3.95 m Intermediate band photometry : Kurtz [1976] 0 = 2.780 m b - y = 0.180 m m, = 0.231 m c, = 0.913 m 5m, = -0.035 m ***************************************** Smith [1982b] measured a r a d i a l v e l o c i t y amplitude of 1.3kms~1 and a p e r i o d which i s the same as that given by Shaw [1976]. Nishimura et a l . [1983] measured a p e r i o d of 0.127 day from t h e i r s p e c t r o s c o p i c data. One can a l s o determine a 2K value of 1.7kms"1 from the d a t a . From one n i g h t of simultaneous r a d i a l - v e l o c i t y and photometric o b s e r v a t i o n s by P e n f o l d [1971], there appears to be a phase s h i f t of 0.37 p e r i o d between the r a d i a l - v e l o c i t y and l i g h t c u r v e s . T h i s phase s h i f t i s d e f i n e d as the f r a c t i o n of the p e r i o d t h a t the r a d i a l - v e l o c i t y minimum l a g s behind the l i g h t maximum. The simultaneously observed s p e c t r o s c o p i c and photometric data by Nishimura et a l . [1983] i n d i c a t e t h a t the l i g h t maximum occurs at about 0.1 p e r i o d before the minimum r a d i a l v e l o c i t y . T h i s i s a t y p i c a l p h a s e - s h i f t value f o r 6 S c u t i s t a r s (Breger et a l . [1976]). 192 Table 5.2 Mid-exposure times f o r the 20 CVn s p e c t r a ************************************** # 24 Jan 83 UT+ B a r y c e n t r i c JD hour angle 0 ( 1 3 : 1 1 :30 .20) 2445359. 0520520 2h 1 4m 00s East 1 ( 1 3 :33 :22 .89) 2445359. 0672461 1h 52m 04s East 2 ( 1 3 :50 :38 .61 ) 2445359. 0792343 1h 34m 45s East 3 (1 4 :07 :49 .78) 2445359. 0911698 1h 1 7m 31s East 4 (1 4 :25 :07 .75) 2445359. 1031841 1h 00m 1 0s East 5 (14 :42 : 16 .58) 2445359. 1150925 Oh 42m 59s East 6 ( 1 4 :59 :25 .59) 2445359. 1270030 Oh 25m 47s East 7 ( 1 5 : 1 6 :31 .50) 2445359. 1388777 Oh 08m 38s East 8 (15 :33 :42 • 47) 2445359. 1.508109 Oh 08m 36s West 9 (1 5 :50 :56 .96) 2445359. 1627849 Oh 25m 53s West 10 (1 6 :08 :07 .57) 2445359. 1747139 Oh 43m 06s West ************************************************************ 5.3 THE OBSERVATIONS The s t a r 20 CVn was observed s p e c t r o s c o p i c a l l y with the HF a b s o r p t i o n c e l l at the Canada-France-Hawaii 3.6m t e l e s c o p e on the 24th of January 1983 UT. The time s e r i e s of s p e c t r a was ob t a i n e d with the red Coude t r a i n and image s l i c e r . The spectrograph used was the f/7.4 Coude fo u r - g r a t i n g - m o s a i c spectrograph (Brealey et a l . [1980]). The CFHT RL1872F/30 R e t i c o n d e t e c t o r was used u n i n t e n s i f i e d at the focus of the spectrograph. The g r a t i n g i s b l a z e d at X8000 and g i v e s a r e c i p r o c a l d i s p e r s i o n i n the f i r s t order of 4.8A/mm at X8700. T h i s corresponds to a d i s p e r s i o n of about 0 . 0 7 l A / p i x e l on the R e t i c o n a r r a y . The image s l i c e r g i v e s a p r o j e c t e d s l i t width of about 33a (Campbell et a l . 193 [1985] ) . The 15M width of a Re t i c o n p i x e l would then imply a r e s o l u t i o n of about 0.16A. The s p e c t r a l coverage i n each spectrum i s about 130A. The exposure time f o r each spectrum was 1000 seconds. T h i s corresponds to about one-tenth the "c y c l e - c o u n t " p e r i o d of 0.12168 day (2.92 h o u r s ) . The time s e r i e s of ten s p e c t r a covered one p e r i o d . S t e l l a r and lamp s p e c t r a without the imposed HF l i n e s were a l s o o b t a i n e d . These are shown i n F i g u r e 5.1. The mean s t e l l a r + H F spectrum with e i t h e r the s t e l l a r or HF l i n e s n u m e r i c a l l y removed are a l s o shown. Each s t e l l a r + H F spectrum has a mean s/n of about 266 at each p o i n t on the continuum. The mid-exposure times of the s p e c t r a are summarised i n Table 5 . 2 . Spectrum #0 i n Table 5.2 i s the 20 CVn spectrum without the imposed HF l i n e s . Approximate r a d i a l v e l o c i t i e s can s t i l l be measured from spectrum #0 by usi n g the d i s p e r s i o n r e l a t i o n determined f o r spectrum #1. For t h i s set of o b s e r v a t i o n s , the b a r y c e n t r i c times l a g the h e l i o c e n t r i c times by about 2.0 seconds. 5 .4 THE DATA REDUCTION The data were processed and reduced u s i n g the procedure d e s c r i b e d i n the p r e v i o u s c h a p t e r . A mean s t e l l a r spectrum i s c a l c u l a t e d f o r the time s e r i e s of s p e c t r a . T h i s spectrum with the HF l i n e s n u m e r i c a l l y removed has been chosen to be the r a d i a l - v e l o c i t y standard spectrum. I t p r o v i d e s both the l i n e shape and p o s i t i o n r e f e r e n c e s i n the a p p l i c a t i o n of the Fahlman-Glaspey d i f f e r e n c e technique i n determining the r e l a t i v e s h i f t s of i n d i v i d u a l s p e c t r a l l i n e s . The ch o i c e of F i g u r e 5.1 The 20 CVn spectrum 195 using a mean spectrum would imply an in c r e a s e d s/n spectrum i s used as the standard spectrum. The phase-smearing e f f e c t would broaden the s p e c t r a l l i n e s in the mean spectrum r e l a t i v e to those i n the i n d i v i d u a l spectrum. T h i s e f f e c t i s very sm a l l i n t h i s case because of the very small v e l o c i t y amplitude. Moreover, i n a second "pass" or i t e r a t i o n of the the d a t a - r e d u c t i o n procedure, the r e l a t i v e v e l o c i t y s h i f t s between i n d i v i d u a l s p e c t r a can be c o r r e c t e d before forming the mean spectrum. The chosen l i n e l i m i t s f o r the s t e l l a r l i n e s are those corresponding to the maximum and minimum values i n the f i r s t d e r i v a t i v e of the l i n e p r o f i l e s . In the de t e r m i n a t i o n of the HF l i n e p o s i t i o n s , the numerical c a n c e l l a t i o n s of the s t e l l a r l i n e s are achieved by d i v i d i n g the data spectrum by an a p p r o p r i a t e l y s h i f t e d spectrum #0. The amount of s h i f t a p p l i e d to spectrum #0 before each spectrum d i v i s i o n i s chosen such that i t would minimise the t o t a l r e s i d u a l over the p o s i t i o n s of a l l the s t e l l a r l i n e s i n the spectrum. T h i s i s e s s e n t i a l l y a mean value of the s h i f t over a l l the s t e l l a r l i n e s f o r the p a r t i c u l a r data spectrum. A s i m i l a r procedure of d i v i d i n g the data spectrum by an a p p r o p r i a t e l y s h i f t e d HF standard spectrum i s used to achieve numerical HF l i n e c a n c e l l a t i o n s . The a p p l i c a t i o n of an optimal s h i f t f o r each i n d i v i d u a l l i n e should r e s u l t i n b e t t e r l i n e c a n c e l l a t i o n s . However, t h i s would i n c r e a s e the amount of numerical computation i . e . computation cost by more than one order of magnitude. The unadjusted d i f f e r e n c e f u n c t i o n from Equation 4.4 has been used f o r the s t e l l a r l i n e s i n the a p p l i c a t i o n of 1 96 the Fahlman-Glaspey d i f f e r e n c e technique. T h i s i s to ensure that the measurement of any v e l o c i t y s h i f t caused by l i n e - p r o f i l e v a r i a t i o n s would not be a f f e c t e d by a r t i f a c t s of m i n i m i s i n g the d i f f e r e n c e f u n c t i o n . In any case, the d i f f e r e n c e s between the v e l o c i t i e s d e r i v e d u s i n g the unmodified and the m o d i f i e d d i f f e r e n c e f u n c t i o n were found to be very s m a l l . T h i s may be an i n d i c a t i o n of very l i t t l e p r o f i l e v a r i a t i o n s happening i n the l i n e s . The l i n e l i m i t s f o r the s t e l l a r l i n e s are chosen to be those corresponding to the maximum and minimum i n the f i r s t d e r i v a t i v e of the l i n e p r o f i l e . A t y p i c a l set of l i m i t s i s about 7 p i x e l s i n width. The l i n e l i m i t s f o r the HF l i n e s are chosen to be twice as broad as those c o r r e s p o n d i n g to the maximum and minimum v a l u e s i n the f i r s t d e r i v a t i v e of the l i n e p r o f i l e . A t y p i c a l set of l i m i t s i s about 16 "pixels i n width. The r e l a t i v e HF l i n e s h i f t s are measured u s i n g the m o d i f i e d d i f f e r e n c e f u n c t i o n from Equation 4.5. The same order of polynomial has been used to f i t the d i s p e r s i o n r e l a t i o n s of a l l the s p e c t r a . T h i s should minimise the p o s s i b i l i t y t h a t any s m a l l v a r i a t i o n i n the measured v e l o c i t i e s c o u l d be caused by i n c o n s i s t e n c i e s between the d i s p e r s i o n f i t s . I n d i v i d u a l r a d i a l - v e l o c i t y curves have been obtained f o r the s t e l l a r l i n e s Ca II X8662, Fe I X8675, X8689, X8710, X8763, S i I X8728, X8742, X8752, S I X8680, X8695, Mg I X8718, X8776, A l I X8773, X8774, and N I X8683. The s/n of the s p e c t r a are too low f o r a d e t e r m i n a t i o n of the H I X8750 v e l o c i t y c u r v e . A mean systemic v e l o c i t y of 9.532kms"1 i s measured over the observed c y c l e . T h i s was obtained by 1 97 averaging the v e l o c i t i e s of a l l the l i n e s from a l l the s p e c t r a . I t i s d i f f e r e n t from the value of 7.4kms"1 measured by Smith [1982b], G e n e r a l l y , one i s only i n t e r e s t e d i n the v e l o c i t y v a r i a t i o n s of the s t a r . Hence r e l a t i v e v e l o c i t i e s r a t h e r than a b s o l u t e v e l o c i t i e s are s u f f i c i e n t f o r the purpose. A r e l a t i v e r a d i a l - v e l o c i t y curve i s obtained by s u b t r a c t i n g a mean v e l o c i t y f o r the p a r t i c u l a r l i n e . 5.5 THE RADIAL VELOCITIES The r e l a t i v e r a d i a l - v e l o c i t y curve based only on the Ca II X8662 l i n e i s shown i n F i g u r e 5.2. The mean one - s t a n d a r d - d e v i a t i o n u n c e r t a i n t y i n each Ca II l i n e p o s i t i o n measurement i s about ± 0 . 0 9 3 k m s _ 1 . T h i s e r r o r estimate i s d e r i v e d from the formal e r r o r estimate o b t a i n e d with the Fahlman-Glaspey method. The t h e o r e t i c a l f o r m u l a t i o n s to c a l c u l a t e t h i s e f f e c t i v e e r r o r estimate have been d e s c r i b e d i n the p r e v i o u s chapter. The r a d i a l - v e l o c i t y curve based on the weaker l i n e Fe I X8689, i s shown i n F i g u r e 5.3. The mean on e - s t a n d a r d - d e v i a t i o n u n c e r t a i n t y i n each of these l i n e - p o s i t i o n measurements i s about ±0.l04kms~ 1. The o n e - s t a n d a r d - d e v i a t i o n u n c e r t a i n t i e s f o r the other even weaker l i n e s are mostly between ±0.1 and ±0.3kms~ 1. In order to i n c r e a s e the p r e c i s i o n , a weighted average v e l o c i t y curve i s c a l c u l a t e d f o r the weak l i n e s i . e . a l l l i n e s except the Ca II l i n e . The chosen weight f o r each i n d i v i d u a l v e l o c i t y curve i s p r o p o r t i o n a l to the square of the l i n e depth and i n v e r s e l y p r o p o r t i o n a l to the square of the v e l o c i t y p r e c i s i o n . The mean on e - s t a n d a r d - d e v i a t i o n 1 98 F i g u r e 5.2 The Ca II X8662 v e l o c i t y c u r v e of 20 CVn CO LO CO LO ^ * CNJ Q ~~) CD C D ,.sui>| Ad a A j i e j a d 199 F i g u r e 5 . 3 T h e F e I X 8 6 8 9 v e l o c i t y c u r v e o f 20 C V n C O C O C D 00 QJ (_> C D C M + C J ) L O C D L O C M Q ~i C O SUJ>I Ad a A j i e j a y 200 F i g u r e 5.4 The mean v e l o c i t y c u r v e of 20 CVn from weak l i n e s + C D L O C O L O C \ l Q ~> C O C D 201 u n c e r t a i n t y f o r t h i s mean curve has a c o n t r i b u t i o n of 0.083kms _ 1 per v e l o c i t y p o i n t from the s t e l l a r l i n e - p o s i t i o n measurements. T h i s mean curve i s shown i n F i g u r e 5.4. D i s p e r s i o n f i t s to the HF l i n e s i n the s p e c t r a would c o n t r i b u t e about ±60kms" 1 to the u n c e r t a i n t i e s of a l l v e l o c i t y p o i n t s . T h i s i s the mean standard e r r o r i n the d i s p e r s i o n f i t s and i s caused mainly by the u n c e r t a i n t i e s i n the HF l i n e - p o s i t i o n measurements. A s i m i l a r value i s obtained i f the standard e r r o r i s c a l c u l a t e d from the u n c e r t a i n t y estimates of the HF l i n e - p o s i t i o n measurements. Taking t h i s u n c e r t a i n t y of the d i s p e r s i o n r e l a t i o n i n t o account, the mean curve would have an u n c e r t a i n t y of about ±0.102kms" 1 per v e l o c i t y p o i n t while the corresponding value f o r the Ca II curve would be ± 0 . 1 1 1 k m s _ 1 . Table 5.3 l i s t s the v e l o c i t i e s shown i n F i g u r e s 5.2 through 5.4. The e r r o r s i n the mid-exposure times due to g u i d i n g i n c o n s i s t e n c i e s are very s m a l l . Large g u i d i n g e r r o r s would have caused l a r g e s h i f t s i n the HF l i n e p o s i t i o n s between the s p e c t r a . T h i s was not observed. In f a c t , the rms s c a t t e r of the HF l i n e p o s i t i o n s ( t h a t of the X8717 l i n e ) i s only about ±0.042 p i x e l over the e n t i r e time s e r i e s . T h i s small g u i d i n g e r r o r i s r e a l i s e d through the use of the image s l i c e r . The moderate exposure time of 1000 seconds would a l s o h e l p to smooth out any short-term e f f e c t s . In any case, the exposure-meter output was r a t h e r uniform f o r each exposure. Of course, i t would be b e t t e r to use a weighted mean time f o r each exposure. In t h i s case, the weight c o u l d be the i n t e n s i t y output from the exposure meter. 202 Table 5.3 R e l a t i v e r a d i a l v e l o c i t i e s of 20 CVn **************************************** # Ca II X8662 • . Fe I X8689 mean weak l i n e s 0 -0.188 kms"1 -0.302 kms"1 -0.344 kms"1 1 -0.519 kms"1 -0.605 kms"1 -0.547 kms"1 2 -0.934 kms-1 . -0.478 kms - 1 -0.568 kms"1 3 -0.327 kms"1 -0.309 kms"1 -0.249 kms"1 4 +0.071 kms - 1 -0.078 kms - 1 +0.001 kms - 1 5 +0.484 kms - 1 +0.414 kms - 1 +0.376 kms - 1 6 +0.677 kms"1 +0.525 kms - 1 +0.541 kms - 1 7 +0.630 kms - 1 +0.660 kms"1 +0.661 kms - 1 8 +0.450 kms'1 +0.437 kms"1 +0.446 kms"1 9 +0.011 kms"1 +0.022 kms"1 +0.027 kms"1 10 -0.354 kms - 1 -0.347 kms - 1 -0.345 kms'1 ************************************************************ 5.6 DISCUSSION The Ca II v e l o c i t y curve i n F i g u r e 5.2 has a 2K amplitude of 1.4kms"1 while the mean v e l o c i t y curve i n F i g u r e 5.4 has an amplitude of 1.2kms"1. These agree w e l l with the value of 1.3kms"1 d e r i v e d by Smith [1982b] from the Fe I X4476 and Fe II X4508 l i n e s . As i n Smith [1982b], no c o r r e c t i o n i s a p p l i e d to remove the phase-smearing e f f e c t caused by the l e n g t h of each exposure. In any case, the r e q u i r e d c o r r e c t i o n i s probably very s m a l l . The use of 10% of the dominant p e r i o d as the exposure time should cause l e s s severe phase smear than i n the case i n Smith [1982b] which used between 15% and 20% of the p e r i o d as the l e n g t h of exposure. Smith [1982b] has estimated t h a t , even i n that 203 case, the amount of r e d u c t i o n i n the 2K v a l u e s caused by phase smear would be l e s s than 10%. F i g u r e 5.5 g i v e s the v e l o c i t y d i f f e r e n c e between the Ca 11 X8662 v e l o c i t y curve and that from the mean of weak l i n e s . There appears to be a marked d e v i a t i o n at BJD2445359.08. The Ca II l i n e appears to be s h i f t e d shortward from the other l i n e s by about 0.37kms~ 1. In f a c t , a small gradual t r e n d towards s h o r t e r wavelengths can be seen from the neighbouring v e l o c i t y p o i n t s . The r e s u l t at BJD2445359.08 does not appear to be the e f f e c t of a s i n g l e aberrant v e l o c i t y p o i n t . In order to ensure that one i s sampling the i d e n t i c a l p u l s a t i o n phenomenon from each l i n e , the same c r i t e r i a have been used to determine a l l the s t e l l a r l i n e l i m i t s . T h i s c o u l d not be a c h i e v e d p e r f e c t l y because of the round-off e r r o r imposed by the i n t e g e r - v a l u e d l i n e l i m i t s . However, the use of the Fahlman-Glaspey technique to determine r e l a t i v e l i n e p o s i t i o n s i s not very s e n s i t i v e to the c h o i c e of l i n e l i m i t s . Moreover, t h i s d i s c r e t i s a t i o n e r r o r i s l e s s severe f o r the broad Ca II l i n e and i t i s minimal f o r a mean v e l o c i t y curve. The observed d e v i a t i o n s may s t i l l be caused by the f a c t t h a t a d i f f e r e n t p a r t of the p u l s a t i o n was sampled by the Ca II v e l o c i t i e s . The Ca II l i n e has a s l i g h t l y d i f f e r e n t l i n e shape than the other l i n e s ; t h e r e f o r e , the l i n e l i m i t s determined by the extrema of the p r o f i l e d e r i v a t i v e may have a d i f f e r e n t meaning than i n the case of the other l i n e s . Furthermore, the Ca II l i n e has an e x c i t a t i o n p o t e n t i a l q u i t e d i f f e r e n t from those of the other l i n e s and may have been formed at a 204 F i g u r e 5 . 5 V e l o c i t y d i f f e r e n c e ( Ca I I c u r v e - mean c u r v e ) cn cu a ra QJ ZX a ra cu I CM CO CD CO CO > C_J CD CN C O o CM O OO o o CD CD - 9 1 o 1 - - I - - / 1, - - V / \ &> \ - \ \ 1 / — CD O o C\J o CD O CD CD CD CD CD C D CD LO CO LO C\l Q ~o CO CD LD CD LO CD LD O CM in O O O CD CD (,.SUJ>|) 33U3J9J JTQ AlJ30J3/\ 205 F i g u r e 5.6 V e l o c i t y d i f f e r e n c e ( mean curve - Fe I curve ) CO CD O (XI CO o o oo CD CD CO CD - 9 1 T A / ? 1 o \ &> / - / f 1 Q - \ \ b / - \ i CO o .—• CD (\l <•—i CD CD CD OO CD O CO CD CD LO CO LO CNJ Q ~1 CD CD LD CD LO CD LD (XI CD (NJ LO CD CD CD CD CD (,.SUI>|) 3JU3J3JJTQ AlT30T3/\ 2 0 6 F i g u r e 5.7 V e l o c i t y d i f f e r e n c e ( Fe I curve - S i I curve ) c n c u c r CO o c r ro QJ c n cu t r QJ Li- ra QJ c r > CJ C D CM 4- CO LO CM Q -) CO o LO CO Ul CD LO C\J CD CM Ul O O CD O CD (,.sw>|) 3 J U 3 J 3 J J T Q A I T : J O T 3 / \ 207 F i g u r e 5 . 8 V e l o c i t y d i f f e r e n c e ( Fe I curve - S I curve ) C D C \ J QJ • t= I C D 1 cn QJ tz C D .—I r H C D QJ U_ u _ O O o C D an  C D QJ EH • • > C O C D C D I 1 \ © - / T 1 1 - - 1 \ - / Q 1 1 4 / 1 C O C D C D C M C D C D C O C D C D C O C D C D + C D L O C D L O "3- C M C D ~ D C O C D LD C D LD C D LD CM C D CM LD C D C D C D C D C D (,.SUI>J) 3 D U 3 J 3 J J T Q A J T D Q J 3 / \ 208 d i f f e r e n t r e g ion of the p u l s a t i n g s t e l l a r atmosphere. On the other hand, the propagation of shocks i n the p u l s a t i n g atmosphere c o u l d a l t e r the Ca II l i n e p r o f i l e and hence the deduced v e l o c i t y displacement. In f a c t , shock-generated s h o r t w a r d - s h i f t e d emission i n the Ca II K l i n e has been r e p o r t e d i n another 5 Del type 6 S c u t i v a r i a b l e , p Pup, by Dra v i n s et a l . [1977]. Shock-generated emission has a l s o been observed i n the large-amplitude 8 Set v a r i a b l e , VZ Cnc (Garbuzov and M i t s k e v i c h [1984]). The Ca II l i n e s i n 20 CVn have been found to be anomalous i n s t r e n g t h by Morgan and Abt [1972]. At the present, no concre t e e x p l a n a t i o n can be o f f e r e d f o r the observed e f f e c t . In f a c t , one cannot even begin to spe c u l a t e on whether i t i s r e l a t e d to the m e t a l l i c i s m or s o l e l y to the 6 S c u t i p u l s a t i o n of the s t a r . F i g u r e 5.6 g i v e s the v e l o c i t y d i f f e r e n c e between the mean curve and that d e r i v e d from the mean of the Fe I l i n e s . F i g u r e 5.7 shows the v e l o c i t y d i f f e r e n c e between the mean Fe I curve and the the mean S i I curve while the d i f f e r e n c e between the mean Fe I curve and the mean S I curve i s shown i n F i g u r e 5.8. There appears to be no systematic d e v i a t i o n i n any of these three c u r v e s . In f a c t , the s c a t t e r i n each curve i s probably a reasonable i n d i c a t i o n of the i n d i v i d u a l v e l o c i t y p r e c i s i o n . F i g u r e 5.6 has a standard d e v i a t i o n of ± 3 6 m s _ 1 . F i g u r e 5.7 has a standard d e v i a t i o n of ±l09ms~ 1 while F i g u r e 5.8 has a standard d e v i a t i o n of ±70ms~ 1. The sm a l l e r s c a t t e r i n F i g u r e 5.6 i s probably caused p a r t i a l l y by the f a c t that the Fe I l i n e s do c o n t r i b u t e s i g n i f i c a n t l y towards the mean curve because of t h e i r l a r g e number. 209 The short d u r a t i o n of the observed time s e r i e s w i l l cause l a r g e u n c e r t a i n t y i n any p u l s a t i o n p e r i o d d e r i v e d from the data. N e v e r t h e l e s s , a pseudo " c y c l e - c o u n t " p e r i o d can be i n f e r r e d from the observed change of almost one c y c l i c v a r i a t i o n i n the v e l o c i t i e s . The periodogram and Maximum L i k e l i h o o d power-spectrum a n a l y s i s of the Ca II v e l o c i t y curve i n F i g u r e 5.2 g i v e s a p e r i o d of 0.1266 . S i m i l a r l y , an a n a l y s i s of the mean v e l o c i t y curve in F i g u r e 5.4 g i v e s a p e r i o d of 0.1336 . Averaging the two gives a mean p e r i o d of 0.130 or a mean frequency of 7.69 d a y - 1 . These r e s u l t s agree very w e l l with the s p e c t r o s c o p i c r e s u l t of Nishimura et a l . [1983] and are not too d i f f e r e n t from the values given by Shaw [1976], Smith [1982b], and B o s s i et a l . [1983]. However, any agreement achieved here may be p u r e l y a c c i d e n t a l . The d e r i v e d p u l s a t i o n p e r i o d c a r r i e s a l a r g e u n c e r t a i n t y because of the l i m i t e d time coverage. Moreover, Smith [1982b] has r e p o r t e d t h a t the r a d i a l - v e l o c i t y v a r i a t i o n s of 20 CVn can be q u i t e d i f f e r e n t on d i f f e r e n t n i g h t s . The v e l o c i t i e s were found to be s i n u s o i d a l on one n i g h t while on the n i g h t b e f o r e , they were m o n o t o n i c a l l y d e c r e a s i n g . Smith [1982b] a t t r i b u t e d t h i s to the b e a t i n g e f f e c t caused by a second p e r i o d modulating the primary p e r i o d of 0.122 . B o s s i et a l . [1983] have r e p o r t e d modulation i n the photometric amplitude which may be caused by a beat between the main p e r i o d and a second p e r i o d at 0.143 d. The amplitudes of the two p e r i o d s are 0.0200 m and 0.0054 m, r e s p e c t i v e l y . The p e r i o d r a t i o , however, would d i s a g r e e with the i d e n t i f i c a t i o n of the second overtone 210 r a d i a l p u l s a t i o n as the main o s c i l l a t i o n by Pena and Gonzalez [1981]. T h i s mode t y p i n g was based on t h e i r d e r i v e d value of 0.02 day f o r Q, the p u l s a t i o n c o n s t a n t . Tsvetkov [1982b,1984], on the other hand, has adopted a Q value of 0.041 day f o r 20 CVn. Appl y i n g the parameters i n Table 5.1 to Equation 1.11, one o b t a i n s a Q value of 0.025 day f o r 20 CVn. T h i s value i m p l i e s f i r s t overtone r a d i a l p u l s a t i o n or even n o n r a d i a l p u l s a t i o n f o r 20 CVn. Tsvetkov [1984] has pl a c e d 20 CVn i n a l i s t of nine p e c u l i a r 6 S c u t i v a r i a b l e s . These p e c u l i a r s t a r s a l l have l a r g e d i s c r e p a n c i e s between t h e i r c a l c u l a t e d e v o l u t i o n a r y masses and p u l s a t i o n masses. Smith [1982b] has suggested a beat p e r i o d of 2 days and hence excluded the common r a d i a l and n o n r a d i a l p e r i o d r a t i o s . Furthermore, Smith [1982b] has found a sharp s e p a r a t i o n between the 2K/Amv v a l u e s f o r the r a d i a l l y and n o n r a d i a l l y p u l s a t i n g 5 S c u t i s t a r s . Based on a low value of 42kms _ 1mag" 1 adopted f o r 20 CVn, Smith [1982b] has suggested that the dominant mode i n 20 CVn i s probably n o n r a d i a l . T h i s i s i n s p i t e of the f a c t that the sharpness of the l i n e s i n 20 CVn has prevented Smith [1982b] from mode t y p i n g u s i n g l i n e - p r o f i l e a n a l y s i s . The nature of 20 CVn's p u l s a t i o n i s s t i l l u n clear and the s t a r deserves more e x t e n s i v e o b s e r v a t i o n s . The sky c o n d i t i o n was poor when t h i s set of data was taken. In f a c t , t here was a strong 60-knot wind which was a l s o the main reason f o r the short time coverage. The noise c h a r a c t e r i s t i c s of the Reticon at CFH have s i n c e been improved. T h i s together with the improved through-put of the 21 1 CFH red Coude t r a i n i m p l i e s that higher s/n data are now p o s s i b l e with the same in s t r u m e n t a l setup at CFH. Chapter 6 THE DELTA SCUTI VARIABLE p PUP 6.1 INTRODUCTION The s t a r p Puppis ( HR3185 ) has a l s o been c a l l e d c Argus by Reese [1903]. I t i s a 5 S c u t i v a r i a b l e with 6 D e l p h i n i - t y p e anomalous abundances. Parameters f o r p Pup are summarised i n Table 6.1. A d d i t i o n a l parameters are summarised i n Tsvetkov [1982b], H a l p r i n and Moon [1983], Gupta [1978], Breger and Bregman [1975], Leung [1970], A n t o n e l l o et a l . [1981 ], Breger [1979], B a g l i n et a l . [1972], and H o f f l e i t and Jaschek [1982]. Abundance a n a l y s i s of p Pup has been performed by G r e e n s t e i n [1948], B e s s e l l [1969], Breger [1970], and Kurtz [1976]. A s l i g h t l y enhanced value of 0.54 has been determined f o r [Fe/H]. Kurtz [1976] a l s o found among the l i g h t e r elements that the Ca I, Sc I I , T i I I , and V II abundances appear to be s l i g h t l y d e f i c i e n t . Bidelman [1951] has given p Pup a MK c l a s s i f i c a t i o n of F 6 I I . However, Bidelman [1951], McNamara and Augason [1962], and Morgan and Abt [1972] have p o i n t e d out that Ca II i s weak i n p Pup when compared to s t a r s of s i m i l a r s p e c t r a l type. 6.2 VARIABILITIES OF p PUP The s t a r p Pup was f i r s t r e p o r t e d to e x i b i t v a r i a b l e r a d i a l v e l o c i t i e s by Reese [1903] and Campbell and Moore [1928]. In f a c t , Spencer Jones [1928] has d e r i v e d a v e l o c i t y range of 11.6 kms"1 f o r p Pup. Photometric v a r i a t i o n s were f i r s t r e p o r t e d by Cousins [1951] and Eggen [1956,1957]. 212 213 Table 6.1 Parameters f o r p Pup *************************************** HD number : 67523 SAO number : 175217 DM number : -23 6828 R.A. (1950) : 8 h 5 m 24.798 s Dec. (1950) : -24° 9' 32.312" Annual p a r a l l a x : +0.031" Proper motion i n R.A. : -0.088"/yr Proper motion i n Dec. : +0.048"/yr / : i : 243.15° blz : 4.40° S p e c t r a l type : F5 H p R a d i a l v e l o c i t y = 46 kms V s i n i = 14 kms'1 M v = +1.7 T e f f = 7100K l o g g = 3.25 (cgs) Broad band photometry : V = = 2. 8 2 m B -- V = +0. 44 m u -- V = +0. 63 m v -- R = +0. 3 7m v -- I = +0. 5 8 m Reference Morgan and Abt [1972] Eggen [1979] Kurtz [1976] Eggen [1979] Kurtz [1976] Kurtz [1976] I r i a t e et a l . [ 1 965] I n f r a r e d photometry : Verma et a l . [1983] 214 H = 2.02 K = 1.99 m Intermediate band photometry : Eggen [1979] 0 = 2.715 m b - y = 0.260 m m, = 0.215 m c, = 0.730 m 8m, = -0.040 m *************************************** L i g h t curves f o r p Pup have a l s o been given by Ponsen [1963], T h u l s a s s i Doss [1969], and Trodahl et a l . [1973]. R a d i a l - v e l o c i t y curves have been given by Struve et a l . [1956], Buscombe [1957], Campos and Smith [1980], and F r a c a s s i n i et a l . [1983]. The v e l o c i t y o s c i l l a t i o n has been measured with a s e r v o - c o n t r o l l e d Fabry-Perot r a d i a l - v e l o c i t y spectrometer by Reay et a l . [1983a]. Simultaneous r a d i a l - v e l o c i t y and photometric t i m e - s e r i e s o b s e r v a t i o n s have been performed by Danziger and Kuhi [1966], Dravins et a l . [1977], and Balona and S t o b i e [1983]. B e s s e l l [1969] performed simultaneous r a d i a l - v e l o c i t y and p h o t o e l e c t i c spectrum-scanner o b s e r v a t i o n of the continuum. Emission l i n e s have been observed at the Mg II H and K l i n e s by Weiler and Oegerle [1979] with the Copernicus s a t e l l i t e . IUE o b s e r v a t i o n s of the emission l i n e s have been given by F r a c a s s i n i and P a s i n e t t i [1981] and F r a c a s s i n i et a l . [1983]. 215 The e a r l y r a d i a l - v e l o c i t y data from the L i c k Observatory (Reese [1903], Campbell and Moore [1928]) suggested a r a d i a l - v e l o c i t y amplitude 2K of 8.4kms~ 1. The data from Struve et a l . [1956] has an amplitude of 9.7kms~ 1. Struve et a l . [1956] measured a p e r i o d of 0.1409 and a l s o suggested a value of 0.14 as the amount i n phase by which the r a d i a l - v e l o c i t y minimum l a g s behind the l i g h t maximum. The photometric o b s e r v a t i o n s by Cousins [1951] show an amplitude of 0.146 m. Eggen [1956,1957] measured an amplitude of 0 . l 0 7 m and a p e r i o d of 0.14089 d. Ponsen [1963] has r e a n a l y s e d a l l the e a r l y data together with h i s photometric data and d e r i v e d a p e r i o d of 0.14088141 d (3.3712 hours) and a b l u e - l i g h t amplitude of 0.127 m. The l i g h t curve i s found to be n e a r l y s i n u s o i d a l with the maximum s l i g h t l y sharper than the minimum. Both the r i s i n g and the descending branches of the curve are almost e q u a l l y steep. Bappu [1959] has measured a l i g h t amplitude of 0.16 m at X4050 together with a 300K v a r i a t i o n i n the c o l o u r temperature. The simultaneous photometric and s p e c t r o s c o p i c o b s e r v a t i o n s by Danziger and Kuhi [1966] g i v e a 2K value of 11kms" 1, a l i g h t amplitude of 0.15 m at X4566, and an e f f e c t i v e - t e m p e r a t u r e v a r i a t i o n of 280K. Minimum l i g h t was found to occur at about 0.08 of a p e r i o d before maximum r a d i a l v e l o c i t y . B e s s e l l [1969] measured a l i g h t amplitude of 0.130 m at X4255, and a 2K value of 9.5kms~ 1. One can see from h i s data that the r a d i a l - v e l o c i t y minimum l a g s the maximum of the l i g h t curve by 0.075 i n phase. B e s s e l l [1969] has r e p o r t e d a temperature v a r i a t i o n of 130K and a r a d i u s v a r i a t i o n of 1.6xl0 f lkm. 216 T h u l a s i Doss [1969] has observed the photometric v a r i a t i o n s in the narrow pass bands at X3858, X4310, X4720, and X5875. The c o r r e s p o n d i n g amplitudes were given as 0 . l 7 m , 0.14 m, 0 . 1 2 m r and 0.09111, r e s p e c t i v e l y . The e f f e c t i v e - t e m p e r a t u r e curve was given with a f u l l amplitude of 320K. A p e r i o d of 0.14088067 was a l s o d e r i v e d f o r the p u l s a t i o n . Trodahl and S u l l i v a n [1977] measured a l i g h t amplitude of 0.105 m at X4850 and an e f f e c t i v e - t e m p e r a t u r e v a r i a t i o n of 175K. A corresponding amplitude of about 0.033 was given f o r the r e l a t i v e r a d i u s v a r i a t i o n s AR/R. Dravins et a l . [1977] measured a phase l a g of 0.06 between the r a d i a l - v e l o c i t y minimum and the v l i g h t maximum. A temperature v a r i a t i o n of 160K was measured from the loop t r a v e l l e d by p Pup i n the (c!,b-y)-diagram d u r i n g the p u l s a t i o n c y c l e . Dravins et a l . [1977] r e p o r t e d emission i n the blue wing of the Ca II K l i n e p r o f i l e at about 0.28 i n phase before maximum l i g h t . I t has been suggested that the emission i s caused by a shock wave propagating through the s t e l l a r atmosphere (Dravins et a l . [1977], H i l l [1977], Smith [1982]). With a R e t i c o n d e t e c t o r , Campos and Smith [1980] measured both the r a d i a l - v e l o c i t y and l i n e - p r o f i l e v a r i a t i o n s of the l i n e s Fe I X4476 and Fe II X4508. A 2K value of 11.5kms"1 was determined. Campos and Smith [1980] d i d not d e t e c t any l i n e w i d t h or l i n e - p r o f i l e v a r i a t i o n g r e a t e r than 5% of the continuum. Smith [1982b] has adopted a l i g h t amplitude Amy of 0 . l 0 2 m and a phase d i f f e r e n c e of 0.08 between the l i g h t maximum and r a d i a l - v e l o c i t y minimum. T h i s l i g h t amplitude has been s t a n d a r d i s e d to a c o l o u r of (b-v) = 0.20. Reay et 217 a l . [1983a] used a s e r v o - c o n t r o l l e d Fabry-Perot r a d i a l - v e l o c i t y spectrometer to measure the r a d i a l - v e l o c i t y v a r i a t i o n s of p Pup. From 226 minutes (3.76 hours) of o b s e r v a t i o n s , there was no evidence of p e r i o d i c i t i e s other than the p r i n c i p a l p e r i o d of 0.141 . T h i s i s to a l i m i t of 20ms - 1. F r a c a s s i n i et a l . [1983] have r e p o r t e d a time s e r i e s of h i g h - d i s p e r s i o n o b s e r v a t i o n s of the Mg II H and K l i n e s . The time s e r i e s covers one complete p u l s a t i o n c y c l e ; the exposure f o r each spectrogram was about 10 minutes. The Mg II emission was found to be present d u r i n g the e n t i r e p u l s a t i o n c y c l e and i t i n c r e a s e d with i n c r e a s i n g l u m i n o s i t y . 6.3 THE OBSERVATIONS The s t a r p Pup was observed s p e c t r o s c o p i c a l l y with the HF a b s o r p t i o n c e l l at ' the Canada-France-Hawaii 3.6m t e l e s c o p e on the 22nd, 24th, and 25th of January 1983 UT. The time s e r i e s of s p e c t r a were obtained under the same c o n d i t i o n s and with the same i n s t r u m e n t a l setups as f o r the time s e r i e s on 20 CVn d e s c r i b e d e a r l i e r . S i m i l a r o b s e r v i n g procedures were used. S t e l l a r and lamp s p e c t r a with and without the imposed HF l i n e s were obt a i n e d . These are shown in F i g u r e 6.1. The mean s t e l l a r + H F spectrum with e i t h e r the s t e l l a r or HF l i n e s n u m e r i c a l l y removed are a l s o shown. The l e n g t h of each exposure i s e i t h e r 600, 750, or 900 seconds. These correspond, r e s p e c t i v e l y , to 0.05, 0.062, and 0.074 of the " c y c l e - c o u n t " p e r i o d of 0.141 day (3.37 h o u r s ) . The time coverage i n "the three n i g h t s i s , r e s p e c t i v e l y , 83%, 112%, and 69% of t h i s p e r i o d . Each spectrum has f o r each p o i n t at 218 F i g u r e 6 . 1 T h e p P u p s p e c t r u m 219 Table 6.2 Mid-exposure times and exposures f o r p Pup ******************************************* # B a r y c e n t r i c JD exposure hour angle 22 Jan 83 UT+ 1 (09 :31 :43 .26) 2445356. 9011222 600s Oh 52m 20s E 2 (09 :42 : 1 1 .42) 2445356. 9083927 600s Oh 41m 50s E 3 (09 :52 :37 .84) 2445356. 9156430 600s Oh 31m 22s E 4 (10 :03 :04 .25) 2445356. 9228932 600s Oh 20m 53s E 5 (10 : 13 :36 .83) 2445356. 9302148 600s Oh 1 Om 1 9s E 6 ( 1 0 :24 :04 .30) 2445356. 9374773 600s Oh 00m 1 Os W 7 ( 1 0 :34 :31 .05) 2445356. 9447314 600s Oh 1 Om 39s W 8 (10 :44 :56 .65) 2445356. 9519722 600s Oh 21m 06s W 9 (10 :55 :22 .31 ) 2445356. 9592138 600s Oh 31m 33s W 1 0 (1 1 :05 :48 .43) 2445356. 9664606 600s Oh 42m 01 s W 1 1 ( 1 1 : 16 : 15 .70) 2445356. 9737208 600s Oh 52m 30s W 12 ( 1 1 :26 :41 .86) 2445356. 9809681 600s 1h 02m 58s W 1 3 ( 1 1 :37 :07 .68) 2445356. 9882115 600s 1h 1 3m 26s W 1 4 ( 1 1 :47 :33 .18) 2445356. 9954511 600s 1h 23m 52s W 1 5 (1 1 :57 :33 .18) 2445357. 0027053 600s 1h 34m 21 s W 16 (12 :08 :26 .45) 2445357. 0099568 600s 1h 44m 49s W 1 7 (12 :20 :39 .13) 2445357. 0184369 600s 1h 57m 04s W 24 Jan 83 UT+ 18 (08 :05 :52 .14) 2445358. 8415243 900s 2h 1 Om 32s E 19 (08 :21 :37 .01 ) 2445358. 8524604 900s 1h 54m 44s E 20 (08 :43 :45 .99) 2445358. 8678423 900s 1h 32m 32s E 21 (08 :59 :22 .30) 2445358. 8786793 900s 1h 1 6m 53s E 22 (09 : 14 :58 .28) • 2445358. 8895125 900s 1h 01m 14s E 23 (09 :30 :35 .19) 2445358. 9003565 900s Oh 45m 35s E 220 24 (09 :44: 54. 45) 2445358. 9103017 750s Oh 31m 1 3s E 25 (09 :57: 52. 83) 2445358. 9193108 750s Oh 1 8m 1 3s E 26 (10 : 1 2: 1 1 . 03) 2445358. 9292438 900s Oh 03m 52s E 27 (10 :27: 52. 65) 2445358. 9401423 900s Oh 1 1m 52s W 28 (1 0 :43: 28. 79) 2445358. 9509774 900s Oh 27m 31 s W 29 (10 :58: 55. 78) 2445358. 9617065 900s Oh 43m 00s W 30 ( 1 1 : 1 4: 23. 19) 2445358. 9724405 900s Oh 58m 30s W 31 (1 1 :29: 58. 27) 2445358. 9832633 900s 1h 1 4m 08s W 32 (1 1 :45: 26. 52) 2445358. 9940071 900s 1h 29m 39s W 33 ( 1 2 :00: 53. 85) 2445359. 0047401 900s in 45m 09s W 34 (1 2 : 1 6: 21 . 09) 2445359. 0154722 900s 2h 00m 39s W 35 ( 1 2 :31 : 48. 55) 2445359. 0262068 900s 2h 1 6m 09s W 25 Jan 83 UT+ 36 (08 :24: 20. 19) 2445359. 8543583 900s 1h 48m 04s E 37 (08 :39: 45. 69) 2445359. 8650702 900s 1h 32m 36s E 38 (08 :55: 12. 65) 2445359. 8757990 900s 1h 1 7m 06s E 39 (09 : 1 0: 39. 09) 2445359. 8865218 900s 1h 01m 37s E 40 (09 :26: 18. 42) 2445359. 8973938 900s Oh 45m 56s E 41 (09 :41 : 48. 19) 2445359. 9081551 900s Oh 30m 23s E 42 (09 :57: 19. 29) 2445359. 9189318 900s Oh 1 4m 50s E 43 (10 : 12: 48. 85) 2445359. 9296907 900s Oh 00m 43s W 44 (10 :28: 19. 31 ) 2445359. 9404600 900s Oh 1 6m 1 6s W 45 (10 :43: 45. 62) 2445359. 9511813 900s Oh 31m 44s W *********************************************** 221 the continuum, a mean s/n of 690, 800, and 400 f o r the three n i g h t s , r e s p e c t i v e l y . The mid-exposure time as w e l l as the leng t h of exposure f o r each spectrum are summarised i n Table 6.2. Spectrum #17, #18, and #19 are the s p e c t r a without the imposed HF l i n e s . Again as i n the case of 20 CVn, approximate r a d i a l v e l o c i t i e s can be measured from these s p e c t r a u s i n g the d i s p e r s i o n r e l a t i o n s determined f o r the other s p e c t r a . For the e n t i r e p Pup data s e t , the h e l i o c e n t r i c times l a g the b a r y c e n t r i c times by about 0.9 second. 6.4 THE DATA REDUCTION The data were processed and reduced using the same procedures which were a p p l i e d to the 20 CVn data. Spectrum #17 has been chosen to be the r a d i a l - v e l o c i t y standard spectrum f o r a l l the p Pup d a t a . I t p r o v i d e s both the li n e - s h a p e and l i n e - p o s i t i o n r e f e r e n c e s used i n determining the r e l a t i v e s h i f t s of i n d i v i d u a l s p e c t r a l l i n e s . A lamp+HF spectrum taken on the same n i g h t , the 24th, was used as the HF standard spectrum. These two standard s p e c t r a were used to reduce a l l three time s e r i e s . The use of i d e n t i c a l standard s p e c t r a ensures i n t e r n a l c o n s i s t e n c y f o r the measured v e l o c i t i e s between the three time s e r i e s . One can compare the measured v e l o c i t i e s of a p a r t i c u l a r l i n e d i r e c t l y between the three time s e r i e s without any a d d i t i o n a l c o r r e c t i o n . The same c r i t e r i a which were used f o r the 20 CVn r e d u c t i o n were used to choose the l i n e l i m i t s f o r both the 222 s t e l l a r and HF l i n e s . A t y p i c a l set of s t e l l a r l i n e l i m i t s i s about 10 p i x e l s i n width. The unmodified d i f f e r e n c e f u n c t i o n from Equation 4.4 was i n i t i a l l y used f o r the s t e l l a r l i n e s i n the a p p l i c a t i o n of the Fahlman-Glaspey technique. In the case of p Pup, the s t e l l a r l i n e s are broader than those i n 20 CVn. Because of the e f f e c t of Doppler imaging, i t may be e a s i e r to d e t e c t l i n e - p r o f i l e v a r i a t i o n s i f they are p r e s e n t . Hence i n i t i a l l y , the use of an o p t i m i s i n g d i f f e r e n c e f u n c t i o n i s l e s s d e s i r a b l e than i n the case of 20 CVn. However, the m o d i f i e d d i f f e r e n c e f u n c t i o n from Equation 4.5 was used f o r the the HF l i n e s . I n d i v i d u a l r e l a t i v e r a d i a l - v e l o c i t y curves were subsequently obtained f o r the s t e l l a r l i n e s Ca II X8662, H I X8750, Mg I X8719, Fe I X8689, X8710, X8713, X8757, X8764, S I X8671, X8680, X8695, S i I X8686, X8728, X8742, and X8752. The r e l a t i v e r a d i a l v e l o c i t y f o r any p a r t i c u l a r l i n e was obtained by s u b t r a c t i n g a mean v e l o c i t y c a l c u l a t e d over an observed p e r i o d . In order to preserve the i n t e r n a l c o n s i s t e n c y between data from d i f f e r e n t n i g h t s , the same value of mean v e l o c i t y was used f o r a l l three time s e r i e s . 6.5 THE LINE-PROFILE VARIATIONS F i g u r e s 6.2 and 6.3 show the v e l o c i t y curve measured on the n i g h t of the 24th f o r the Ca II X8662 and the Fe I X8689 l i n e s , r e s p e c t i v e l y . The Ca II. curve has an average o n e - s t a n d a r d - d e v i a t i o n u n c e r t a i n t y of about ±0.17kms" 1 i n each s t e l l a r l i n e - p o s i t i o n measurement. The u n c e r t a i n t y value i s the formal e r r o r estimate d e r i v e d from the 223 F i g u r e 6.2 The u n o p t i m i s e d Ca II X8662 v e l o c i t y c u r v e + C O L D CO L O "3" -d- C M Q "~) C O siu>| AcJ 3AT;ej3d 224 F i g u r e 6.3 The u n o p t i m i s e d Fe I X8689 v e l o c i t y c u r v e + 00 LO CO LO CN o -) CO S U I > J 3 A T i e | 3 y 225 Fahlman-Glaspey technique. I t i s a very l a r g e value i n comparison to the cor r e s p o n d i n g value of about ±0.09kms~ 1 o b t a i n e d f o r the much lower s/n 20 CVn da t a . The f a c t that the p Pup l i n e s are broader than the 20 CVn l i n e s can.not be the major reason f o r the d i s c r e p a n c y . S i m i l a r l y , the Fe I curve i n F i g u r e 6.3 has an on e - s t a n d a r d - d e v i a t i o n u n c e r t a i n t y of about ±0.31kms~ 1 i n each s t e l l a r l i n e - p o s i t i o n measurement. T h i s i s much l a r g e r than the c o r r r e s p o n d i n g value of about ±0.1kms~ 1 o b t a i n e d f o r the 20 CVn data. F i g u r e s 6.4 and 6.5 show the i n d i v i d u a l l i n e - p o s i t i o n measurement u n c e r t a i n t i e s f o r the Ca II and Fe I curves, r e s p e c t i v e l y . Each of these curves has been c o r r e c t e d f o r the e f f e c t caused by d i f f e r e n c e s i n the s/n between the s p e c t r a . T h i s i s accomplished by m u l t i p l y i n g each u n c e r t a i n t y value by the r a t i o between the spectrum's s/n and a mean s/n. The assumption t h a t the v e l o c i t y u n c e r t a i n t y i s i n v e r s e l y p r o p o r t i o n a l to the spectrum's s/n (Campbell and Walker [1979]) has been used. Comparing F i g u r e s 6.3 and 6.5, one sees that the u n c e r t a i n t i e s peak near the middle of the r i s i n g and d e c r e a s i n g branches of the v e l o c i t y curve. The minima of the v e l o c i t y u n c e r t a i n t y appear t o occur near the minima or maxima of the v e l o c i t y curve. T h i s e f f e c t can be e x p l a i n e d i f there i s a c o r r e l a t i o n between the v e l o c i t y u n c e r t a i n t i e s and the l i g h t curve which leads the v e l o c i t y curve by about 0.08 i n phase. F i g u r e 6.6 shows the Fe I X8757 v e l o c i t y c urve. Superimposed on the p l o t are the cor r e s p o n d i n g l i n e - p o s i t i o n u n c e r t a i n t i e s . I t i s evident 226 Fi g u r e 6.4 U n c e r t a i n t i e s i n the Ca II X8662 l i n e p o s i t i o n s cz o -—1 •4—1 . — » (_> CD c cu CJ cz QJ (_ CU r-U - CD u _ —H TD CD c n CD (_ CO + - » CD cn J C CD +-• —1 ' — ' CD CD CD CO 00 r < \—1 CD h-1 CD O L D -• CO CL CD ZZ) C L a + 00 L D CO L D C \ J a —> CO (t_sm>j) A;ujeiJ33un AJTDOJ3A F i g u r e 6.5 U n c e r t a i n t i e s in the Fe I X8689 l i n e p o s i t i o n s a o • M C D QJ (_) C QJ _ QJ CD u_ ^ CD O l — I _ ro CD l _ CO CO r - » CO CO CO oo C D CO CD QJ LO .. oo C X C D C L ex 0 \ \ 9 CD LD in C M oo CO CO CO oo in oo CD CD CD o C D (,.sui>i) A i u T e u a D u n AJTDOJ3A BJD2445358+ 229 from the p l o t that the u n c e r t a i n t i e s and the v e l o c i t i e s are c o r r e l a t e d . The u n c e r t a i n t y curve i s measured to be l e a d i n g the v e l o c i t y curve by about 0.07 i n phase. Hence the u n c e r t a i n t y curve would be i n phase or 180° out of phase with the l i g h t curve. The u n c e r t a i n t y peak near BJD2445358.9 c o i n c i d e s with the l i g h t minimum. The d e r i v e d u n c e r t a i n t i e s are measures of the r e s i d u a l s a f t e r d i f f e r e n c i n g between the standard and the i n d i v i d u a l l i n e p r o f i l e s . In a d d i t i o n to the i n t r i n s i c s/n l i m i t e d l i n e - p o s i t i o n e r r o r , l i n e - p r o f i l e v a r i a t i o n s w i l l c o n t r i b u t e towards t h i s formal e r r o r estimate d e r i v e d by the Fahlman-Glaspey technique. The systematic v a r i a t i o n s of the u n c e r t a i n t i e s s t r o n g l y suggest t h a t most of them are caused by systematic l i n e - p r o f i l e v a r i a t i o n s . A v e l o c i t y curve with simple l i n e - p r o f i l e v a r i a t i o n s w i l l appear more p r e c i s e or 'smoother' than a c o r r e s p o n d i n g curve with the same estimated formal e r r o r but with random nois e as the s o l e source of the u n c e r t a i n t y . T h i s i s p a r t l y because of the f a c t t h a t , i n the case of simple p r o f i l e v a r i a t i o n s , the r e s i d u a l s i n the d i f f e r e n c e d s p e c t r a would have a systematic or smooth t r e n d . In the technique's process of minimising the r e s i d u a l s , t h i s systematic t r e n d w i l l be r e f l e c t e d i n the measured l i n e p o s i t i o n s . In the case of the l i m i t e d s/n, however, the r e s i d u a l s are random. The formal e r r o r estimate from the technique i s d e r i v e d from the sum of the squares of these r e s i d u a l s and hence would not d i s t i n g u i s h between the two c a s e s . T h i s i s probably the e x p l a n a t i o n f o r F i g u r e 6.6 i n which the v e l o c i t y curve appears more p r e c i s e than i t s 230 formal e r r o r estimates imply. The s e n s i t i v i t y of the technique to l i n e - p r o f i l e v a r i a t i o n s depends very much on the type of v a r i a t i o n s . Simple v a r i a t i o n s l i k e l i n e - d e p t h and l i n e w i d t h v a r i a t i o n s would not s e r i o u s l y a f f e c t the d e r i v e d v e l o c i t y curve. Other more complicated l i n e shape v a r i a t i o n s would a f f e c t the d e r i v e d v e l o c i t y curve more s e r i o u s l y . Of course, i n those cases, i t may be the o b j e c t i v e to use the v e l o c i t y curve to parametrise the l i n e - p r o f i l e v a r i a t i o n s . Systematic l i n e - p r o f i l e v a r i a t i o n s may produce systematic pseudo v e l o c i t y v a r i a t i o n s . The standard p r o f i l e was taken at BJD2445358.85 which i s near a v e l o c i t y minimum. I t i s expected that t h i s p r o f i l e w i l l be s i m i l a r i n l i n e shape to those s p e c t r a taken near the other v e l o c i t y minimum and at a s i m i l a r photometric phase. T h i s would e x p l a i n why the u n c e r t a i n t y peak at BJD2445358.9 (near v e l o c i t y maximum) i s stronger than t h a t at BJD2445358.97 (near v e l o c i t y minimum). The f a c t that the v e l o c i t y u n c e r t a i n t i e s c o r r e l a t e with the l i g h t curve r a t h e r than with the v e l o c i t y curve • would suggest t h a t the l i n e - p r o f i l e v a r i a t i o n s may not be the r e s u l t of p u l s a t i o n - g e n e r a t e d s u r f a c e - v e l o c i t y - f i e l d v a r i a t i o n s . The p r o f i l e v a r i a t i o n s may be r e l a t e d to the temperature and s p e c t r a l - t y p e v a r i a t i o n s over the p u l s a t i o n c y c l e . L i n e - p r o f i l e v a r i a t i o n s can best be examined from a time s e r i e s d i f f e r e n c e - s p e c t r u m p l o t i . e . a stacked p l o t of the r e s i d u a l s a f t e r s u b t r a c t i n g a mean l i n e p r o f i l e from each spectrum. F i g u r e 6.7a shows the region of the Ca II X8662 l i n e f o r each spectrum a f t e r c o r r e c t i n g f o r the 2 3 1 6.7 The Ca II X8662 l i n e p r o f i l e s and t h e i r r e s i d u a l s (b) (a) 8 6 6 2 6 6 6 3 0.5X 0 . 8 6 7 8 4 0 . 8 7 8 6 8 0 .88951 0 .90036 0.91030 0.91931 0 . 9 2 9 2 4 0.94014 0 .95098 0.96171 0 . 9 7 2 4 4 0 . 9 8 3 2 6 0.99401 1.00474 1.01547 '1.02621 8662 8663 2 3 2 F i g u r e 6.8 The Fe I X8689 l i n e p r o f i l e s and t h e i r r e s i d u a l s (b) (a) 0.86784 0 . 8 7 8 6 8 0 .88951 0 . 9 0 0 3 6 0.91030 0.91931 0 . 9 2 9 2 4 0 .94014 0 . 9 5 0 9 8 0.96171 0 . 9 7 2 4 4 0 . 9 8 3 2 6 0 .99401 1.00474 1.01547 1.02621 Mean 8 6 8 7 8 6 8 9 1.01547 1.02621 8 6 8 7 8 6 8 9 gure 6.9 The S i I X8752 and Fe I X8757 l i n e p r o f i l e s '0.90112 '0.90839 '0.91564 0.92289 '0.93021 "0.93748 '0.94473 '0.9S197 '0.95921 "0.96646 '0.97372 '0.98097 "0.98821 '0.9954S "1.00271 '1.00996 "Heart 10X 8748 8750 8752 8754 8756 8758 gure 6.10 The S i I X8752 and Fe I X8757 r e s i d u a l s 8748 8750 8752 8754 8756 8758 gure 6.11 The S i I and Fe I r e s i d u a l s from BJD2445 356 8748 8750 8752 8754 8756 8758 236 measured v e l o c i t y s h i f t s . The s p e c t r a have been smoothed by a Gaussian t r a n s f e r f u n c t i o n which has a a of ±0.075&. The mid-exposure time i n f r a c t i o n of days from BJD2445358 i s i n d i c a t e d f o r each spectrum. F i g u r e 6.7b shows the r e s i d u a l s a f t e r s u b t r a c t i n g a mean l i n e p r o f i l e from each spectrum. The mean p r o f i l e i s chosen to be the average of the f i f t e e n l i n e p r o f i l e s from spectrum #21 through #35. I t re p r e s e n t s approximately the mean l i n e p r o f i l e averaged over one p u l s a t i o n c y c l e . F i g u r e s 6.8a and 6.8b show the time s e r i e s of Fe I X8689 l i n e p r o f i l e s and t h e i r r e s i d u a l s , r e s p e c t i v e l y . F i g u r e 6.9 shows the r e g i o n of the S i I X8752 and Fe I X8757 l i n e s i n each spectrum. The c o r r e s p o n d i n g r e s i d u a l p l o t i s shown i n F i g u r e 6.10. F i g u r e 6.11 shows the r e s i d u a l p l o t f o r the same s p e c t r a l l i n e s but the data are taken from the time s e r i e s observed on the night of the 22nd of January. The same mean l i n e p r o f i l e s were used to produce the r e s i d u a l s i n both F i g u r e s 6.10 and 6.11. The mid-exposure times i n f r a c t i o n of days from BJD2445356 are i n d i c a t e d i n F i g u r e 6.11. I t can be seen from F i g u r e s 6.8b, 6.10, and 6.11 that the l i n e - p r o f i l e v a r i a t i o n s can be c h a r a c t e r i s e d as sys t e m a t i c v a r i a t i o n s i n the l i n e depth or e q u i v a l e n t width of the l i n e s . Near the l i g h t maxima at both BJD2445358.90 and BJD2445359.03, a l l the s t e l l a r a b s o r p t i o n l i n e s are stronger than t h e i r mean l i n e p r o f i l e . But near the l i g h t minimum at BJD2445358.97, the s t e l l a r l i n e s are weaker than t h e i r mean l i n e p r o f i l e s . Near the nodes of the l i g h t curve, the v a r i a b i l i t i e s of the l i n e p r o f i l e s from t h e i r mean 237 p r o f i l e are minimal. The amplitudes of the v a r i a t i o n s are d i f f e r e n t between the v a r i o u s s t e l l a r l i n e s . The v a r i a t i o n s i n the l i n e depth of the Ca II X8662 l i n e are only at a l e v e l of 0.5% of the continuum. The strong Fe I X8689 l i n e shows v a r i a t i o n s at the 1% continuum l e v e l . The stacked r e s i d u a l p l o t of the l i n e in F i g u r e 6.8b shows that the v a r i a t i o n s are more complicated than simple l i n e - d e p t h or l i n e - i n t e n s i t y v a r i a t i o n s . The inner core of the s p e c t r a l l i n e appears to vary i n the reverse manner with r e s p e c t to the r e s t of the l i n e . I t i s shallower than the r e s t of the l i n e near l i g h t maxima but deeper near the l i g h t minimum. The amplitude of t h i s small r e v e r s a l i s l e s s than 0.3% of the continuum. The S i I X8752 and Fe I X8757 l i n e s show s i m i l a r l i n e - i n t e n s i t y v a r i a t i o n s as the Fe I X8689 l i n e . However the amplitudes are only about 0.8% of the continuum. The small r e v e r s e v a r i a t i o n s at the inner l i n e core can a l s o be seen i n these two weaker l i n e s but at a s m a l l e r amplitude. S i m i l a r but l a r g e r amplitude v a r i a t i o n s have been r e p o r t e d i n the p e c u l i a r l a rge-amplitude 8 Set v a r i a b l e SX Phe (Stock and Tapia [1971], Haefner et a l . [1976]). The a b s o r p t i o n - l i n e i n t e n s i t i e s were a l s o found to be minimal at minimum l i g h t while the maxima of the l i n e i n t e n s i t i e s occured near l i g h t maxima. T h i s same sequence of l i n e - i n t e n s i t y v a r i a t i o n s over a p u l s a t i o n c y c l e i s a l s o w e l l known as the s p e c t r a l - t y p e v a r i a t i o n i n Cepheid v a r i a b l e s . The s p e c t r a l type of l o n g - p e r i o d Cepheids can vary between F5 and K1 over a p u l s a t i o n c y c l e . N a t u r a l l y , 238 the c o r r e s p o n d i n g l i n e - i n t e n s i t y v a r i a t i o n s are very much l a r g e r than i n t h i s case of p Pup. The l a r g e r e f f e c t i n Cepheids i s e s s e n t i a l l y caused by t h e i r much l a r g e r change in the e f f e c t i v e temperature over the p u l s a t i o n c y c l e . In p Pup, the f u l l e f f e c t i v e - t e m p e r a t u r e amplitude i s only about 280K. The observed l i n e s of Ca I I , Fe I, S i I, and S I are known to have stronger i n t e n s i t i e s i n s t a r s of s p e c t r a l type s l i g h t l y l a t e r (or c o o l e r ) than p Pup. These l i n e s are a l s o weaker i n s t a r s of e a r l i e r (or h o t t e r ) s p e c t r a l type. T h e r e f o r e , i f the observed l i n e - i n t e n s i t y v a r i a t i o n s were caused mainly by the temperature v a r i a t i o n s over the p u l s a t i o n c y c l e , the l i n e s should then be stronger at temperature minima (almost c o i n c i d i n g with l i g h t maxima) and weaker at temperature maxima (almost c o i n c i d i n g with l i g h t minima). T h i s i s e x a c t l y the sequence of l i n e - i n t e n s i t y v a r i a t i o n s that i s observed i n p Pup. Temperature v a r i a t i o n s a f f e c t an observed l i n e depth through the temperature dependence of both the s t e l l a r continuum l e v e l and the e n e r g y - l e v e l p o p u l a t i o n of the p a r t i c u l a r s t e l l a r l i n e t r a n s i t i o n . C o n s i d e r i n g only weak s t e l l a r l i n e s and smal l temperature v a r i a t i o n s , one can d e r i v e a very simple l i n e a r theory to account f o r the main e f f e c t of the observed s p e c t r a l - t y p e v a r i a t i o n s . Assuming the continuum o p a c i t y KC i s much l a r g e r than the l i n e o p a c i t y K,  one o b t a i n s : d = Kl / ( Kl + Kc ) (6.1) Ad/d = (1-d) ( A K / / K / - A K C A C ) (6.2) 239 KL = A exp( -E L/kT ) (6.3) L \ K 1 / K 1 = (E L/kT) (AT/T) (6.4) A K C / K c = An H./n H_ (6.5) K H = n H + n e / nH (6.6) KH_ = n H n e / n H_ (6.7) n H_ = ( v/(n 3K H.) ) / K H. (6.8) An H . / n H . = (1/2) (AK^/K^) - (AK H_/K H-) (6.9) l o g ( K H _ ) = -(xu-)d - (3/2) logfl + ... (6.10) l o g ( K H ) = ~(xH)e - (3/2) logS + ... (6.11) AK H_/K H =2.3 (AT/T) [ CxH-)0 + 0.65 ] (6.12) AK H/K H = 2.3 (AT/T) [ (x H)0 + 0.65 ] (6.13) where 6 = 5040 / T n = n H + h e n e = e l e c t r o n number d e n s i t y n H = number d e n s i t y of H n H_ = number d e n s i t y of H" n H + = number d e n s i t y of H + d = l i n e depth T = temperature i n k e l v i n k = Boltzmann's constant = low energy l e v e l of the l i n e X H- = 0.75 ev X u = 13.6 ev 240 Equation 6.2 i s obtained by c o n s i d e r i n g the d e r i v a t i v e of Equation 6.1. B a s i c a l l y , Equation 6.3 i s simply s t a t i n g that the l i n e o p a c i t y i s p r o p o r t i o n a l to the number of atoms at the p a r t i c u l a r energy l e v e l . The term A i n Equation 6.3 i s independent of temperature (Johnson et a l . [1972]). The e f f e c t of induced emission has been n e g l e c t e d i n the f o r m u l a t i o n . Equation 6.4 i s obtained by c o n s i d e r i n g the d e r i v a t i v e of Equation 6.3. Equation 6.5 i s e s s e n t i a l l y s t a t i n g the assumption that the continuum o p a c i t y i s p r o p o r t i o n a l to the number d e n s i t y of the H" i o n s . T h i s i s a reasonable assumption f o r an F6 s t a r where the s t e l l a r continuum i s the r e s u l t of mainly f r e e - f r e e and bound-free t r a n s i t i o n s of the H~ i o n s . Equations 6.6 and 6.7 are, r e s p e c t i v e l y , the d e f i n i t i o n s of the d i s s o c i a t i o n c o n s t a n t s , K H and K H-. Equation 6.8 i s d e r i v e d by m a n i p u l a t i n g Equations 6.7 and 6.8 together with the assumption that n>>n e < Equation 6.9 i s obtained by c o n s i d e r i n g the d e r i v a t i v e of Equation 6.8. Equations 6.10 and 6.11 are Saha equations i n the form given by A l l e n [1973]. Equations 6.12 and 6.13 are the d e r i v a t i v e s of Equations 6.10 and 6.11, r e s p e c t i v e l y . The r e l a t i v e change i n the l i n e depth Ad/d as a f u n c t i o n of the r e l a t i v e temperature change AT/T can then be e v a l u a t e d u s i n g Equations 6.2, 6.4, 6.5, 6.9, 6.12, and 6.13 f o r any of the observed weak l i n e s . For p Pup, AT i s about ±140K. T h i s i s the observed value given by Danziger and Kuhi [1966] and i t agrees with most other d e t e r m i n a t i o n s . The most recent and probably the best d e t e r m i n a t i o n of T i s 7100K by Kurtz [1976]. T h e r e f o r e , 241 knowing E L and d f o r a l i n e , the corresponding Ad/d can be c a l c u l a t e d . The maximum l i n e - d e p t h change observed i n the Fe I X8757 l i n e i s about 0.8% of the continuum. Since the l i n e depth i s about 10% of the continuum, the observed Ad/d values are then +0.08 at temperature minimum and -0.08 at temperature maximum. The E L value f o r t h i s l i n e i s 2.83ev. The simple theory would then g i v e Ad/d a value of -0.081 f o r AT=+140K and a value of +0.081 f o r AT=-140K. These agree very w e l l with the observed v a l u e s . The agreement i s not as good when one c o n s i d e r s the stro n g Fe I X8689 l i n e . The c a l c u l a t e d Ad/d f o r t h i s l i n e i s ±0.08 while the observed value i s ±0.03. The d i s c r e p a n c y between the c a l c u l a t e d and the observed values becomes even l a r g e r when one c o n s i d e r s the Ca II X8662 l i n e . T h i s i s not unexpected s i n c e the simple theory i s only v a l i d f o r the weak l i n e s . The theory a l s o cannot be a p p l i e d to the S i I X8752 l i n e . T h i s l i n e i s superimposed on the broad Paschen 12 l i n e ; t h e r e f o r e , the H I X8750 l i n e o p a c i t y at the wavelength of the S i I l i n e a l s o has to be taken i n t o account i n Equation 6.1. N e v e r t h e l e s s , the theory i s s t i l l a b le to p r e d i c t the d i r e c t i o n of the l i n e - d e p t h v a r i a t i o n s f o r these l i n e s . Examining Equations 6.2 and 6.4, one can observe that f o r a l i n e with a s u f f i c i e n t l y l a r g e E^ value , the e f f e c t of AK^/K^ would be l a r g e r than that of A K C / K c « T h i s w i l l cause a d i r e c t i o n change f o r Ad/d. The l i n e would then vary i n the op p o s i t e d i r e c t i o n with r e s p e c t to the other l i n e s . I t would 242 become s t r o n g e s t at temperature maxima and weakest at temperature minima. The c o n d i t i o n i s true f o r the Paschen X8750 l i n e which has a value of 12.04ev. T h i s i s not unexpected s i n c e i t i s w e l l known that the hydrogen l i n e s become p r o g r e s s i v e l y weaker in s t a r s of s p e c t r a l types l a t e r (or c o o l e r ) than p Pup. Moreover, they are p r o g r e s s i v e l y stronger i n s t a r s of s l i g h t l y e a r l i e r (or h o t t e r ) s p e c t r a l types. Examining the stacked r e s i d u a l p l o t s i n F i g u r e 6.10 and 6.11, one can observe that at the p o s i t i o n of the Paschen l i n e , the v a r i a t i o n s do go i n the opposite d i r e c t i o n with r e s p e c t to the other l i n e s . In f a c t , the H I X8750 r e s i d u a l s make q u i t e a c o n t r a s t with the r e s i d u a l s of the S i I X8752 l i n e which v a r i e d i n the other d i r e c t i o n . In s p i t e of the f a c t t h at the simple theory p r e d i c t e d the observed v a r i a t i o n s , i t i s unable to account f o r the observed amplitude of the v a r i a t i o n s . The e f f e c t s of i o n i s a t i o n and hence d e p l e t i o n of the hydrogen atoms would have to be taken i n t o account. 6.6 THE RADIAL VELOCITIES The r a d i a l - v e l o c i t y curves shown i n F i g u r e s 6.2 and 6.3 are measured using the unmodified d i f f e r e n c e f u n c t i o n from Equation 4.4. The l i n e - d e p t h v a r i a t i o n s would a f f e c t the p r e c i s i o n of t h i s method to measure r e l a t i v e l i n e s h i f t s . The l i n e - d e p t h v a r i a t i o n s would cause d i f f e r e n c e s between the i n d i v i d u a l l i n e p r o f i l e and the standard l i n e p r o f i l e . In f a c t , the r e s i d u a l s have manifested i n t o the l a r g e formal e r r o r e s t i m a t e s . Since simple l i n e - d e p t h v a r i a t i o n s should 243 not a l t e r the l i n e p o s i t i o n , the d i f f e r e n c e f u n c t i o n should s t i l l be minimal at the same s h i f t p o s i t i o n as i n the case where there i s a p e r f e c t match between the p r o f i l e s . The d i f f e r e n c e f u n c t i o n , however, would be shallower than i n the case with the p e r f e c t l i n e - p r o f i l e match. T h i s would then make i t more d i f f i c u l t and l e s s p r e c i s e i n measuring the minimum p o s i t i o n of the d i f f e r e n c e f u n c t i o n . One way to minimise, to the f i r s t order, the e f f e c t of l i n e - i n t e n s i t y v a r i a t i o n s i s to use the m o d i f i e d d i f f e r e n c e f u n c t i o n from Equation 4.5 to measure the r e l a t i v e l i n e s h i f t s . T h i s d i f f e r e n c e f u n c t i o n w i l l s c a l e the i n d i v i d u a l l i n e p r o f i l e l i n e a r l y such that the r e s i d u a l from the standard l i n e p r o f i l e would be minimal. Consequently, the •modified d i f f e r e n c e f u n c t i o n has been a p p l i e d to re-measure a l l the s t e l l a r l i n e p o s i t i o n s i n the p Pup s p e c t r a . F i g u r e s 6.12 and 6.13 show the subsequently d e r i v e d formal e r r o r estimates f o r the Ca II X8662 and Fe I X8757 l i n e p o s i t i o n s , r e s p e c t i v e l y . Comparing these two f i g u r e s a g a i n s t the c orresponding unoptimised ones in F i g u r e s 6.4 and 6.6, one can observe that the p e r i o d i c trends caused by the l i n e - i n t e n s i t y v a r i a t i o n s have l a r g e l y disappeared i n F i g u r e s 6.12 and 6.13. Moreover, the formal e r r o r estimates have a l s o become much sm a l l e r and are i n agreement with those from the 20 CVn data s e t . T h e r e f o r e , i t appears that the a p p l i c a t i o n of the m o d i f i e d d i f f e r e n c e f u n c t i o n to the s t e l l a r l i n e s has been a s u c c e s s . F i g u r e s 6.14, 6.15, and 6.16 show the measured r e l a t i v e v e l o c i t y curves f o r the Ca II X8662, H I X8750, and Fe I 244 Figu r e 6.12 The optimised Ca II X8662 u n c e r t a i n t i e s LD CD C D LO O + CO LO CO LO C\l Q —) CO CD CD C D (,.sui>i) A i u j e u a j u n A I T 3 0 J 3 A 245 F i g u r e 6.13 The o p t i m i s e d Fe I X8757 u n c e r t a i n t i e s s—, cr o — i •M . — i CJ CD cr U- QJ CJ cr QJ l _ CU lo- CO ta-TP CD cu CO — I pz CO -•H 4—' CO CL . a CD JZ +-> — 1 - — CO LO CO • CO CD CD 0 0 LO CM CD CM LO CD LO CD O (,.su>|) A j U j e i J 3 3 U H AJT30J3A p Pup : Ca I I X 8 6 6 2 (with o p t i m i s e d d i f f e r e n c e f u n c t i o n ) 0 . 8 5 0 . 8 9 0 . 9 3 0 . 9 7 1 . 0 1 0 . 8 5 0 . 8 9 0 . 9 3 0 . 9 7 1 . 0 1 B J D 2 4 4 5 3 5 8 + p Pup : H I X 8 7 5 0 (with o p t i m i s e d d i f f e r e n c e f u n c t i o n ) 0 - 8 5 0 . 8 9 0 . 9 3 0 . 9 7 1 . 0 1 0 . 8 5 0 . 8 9 0 . 9 3 0 . 9 7 1 . 0 1 B J D 2 4 4 5 3 5 8 + 248 F i g u r e 6.16 The o p t i m i s e d Fe I X8689 v e l o c i t y curve rr o —i +-• '—i u CD £Z • ZD U - QJ U CZ QJ l_ QJ ij_ CO U _ —1 TD CD TD QJ cn • • H E= CD • M CD CL a CD _ c —1 ' — ' cn CD CO OO CD • CO CD r< i ( i i QJ Li_ LD « • OO CL CD ZD Q_ 00 LD 00 LD CM a ~o CQ (,.sui>|) Ad 3 A j i e T 3 c j 2 4 9 F i g u r e 6.17 The mean optimised Fe I v e l o c i t y curve + CO LO 0 0 LO CM a —> m (,.sui>|) Ad 3 A T ; e j 3 ^ 250 F i g u r e 6 . 1 8 The mean o p t i m i s e d S i I v e l o c i t y curve + CO LO CO LD C\l Q —> CO (,.sui>l) Ad 3AT +ej3y 251 F i g u r e 6.19 The mean o p t i m i s e d S I v e l o c i t y curve + CO LO CO LO Q —) CO (,.SUI>|) Ac! 3AT + BT3cJ 252 F i g u r e 6.20 The o p t i m i s e d Ca II v e l o c i t i e s from the 22nd + C D L O C O L O C M a CO. (,.sui>|) Ad 3 A T ; G T 3 ^ p Pup : Ca I I X 8 6 6 2 (with o p t i m i s e d d i f f e r e n c e f u n c t i o n ) 0 . 8 2 0 . 8 6 0 . 9 0 0 . 9 4 0 . 9 8 — i 1 1 1 1 — 4 2 i i i - 1 1— ' 0 »' \ - 0 / \ 2 / " / \ <* - 4 i 1 1 1 L_ 0 . 8 2 0 . 8 6 0 . 9 0 0 . 9 4 0 . 9 8 B J D 2 4 4 5 3 5 9 + 254 X8689 l i n e s , r e s p e c t i v e l y . F i g u r e s 6.17, 6.18, and 6.19 show the mean v e l o c i t y curves from the l i n e s of Fe I, S i I, and S I, r e s p e c t i v e l y . F i g u r e s 6.20 and 6.21 show the Ca II X8662 v e l o c i t y curves from the n i g h t s of the 22nd and 25th, r e s p e c t i v e l y . A l l the v e l o c i t i e s from the Ca II X8662, H I X8750, Fe I X8689, and Mg I X8719 l i n e s are l i s t e d i n Table 6.3 while a l l the mean v e l o c i t i e s of the Fe I, S i I, and S I l i n e s are l i s t e d i n Table 6.4. Each mean Fe I v e l o c i t y was d e r i v e d from the weighted mean of the X8689, X8710, X8713, X8757, and X8764 v e l o c i t i e s . The weights have been chosen to be p r o p o r t i o n a l to the square of the mean l i n e depth and i n v e r s e l y p r o p o r t i o n a l to the square of the mean formal e r r o r i n the l i n e - p o s i t i o n measurements. The same weight has a l s o been used f o r each l i n e over a l l three n i g h t s of data. S i m i l a r l y , each mean S i I v e l o c i t y was d e r i v e d from the weighted mean of the X8686, X8728, X8742, and X8752 v e l o c i t i e s . Meanwhile, each mean S I v e l o c i t y was d e r i v e d from the weighted mean of the X8680 and X8695 v e l o c i t i e s . The mean e r r o r i n t r o d u c e d by the u n c e r t a i n t i e s of the HF l i n e p o s i t i o n s was about 30ms~1 f o r the data taken on the ni g h t of the 24th. F i g u r e 6.22 shows the i n d i v i d u a l e r r o r estimate i n the HF d i s p e r s i o n f i t . The r e l a t i v e dependence on the s/n has been removed from the p l o t . The low valu e s shown i n F i g u r e 6.22 are the r e s u l t of the high s/n of the s p e c t r a . In s p i t e of t h i s , one can observe that the e r r o r s are not random. The e r r o r s appear to be p e r i o d i c with the maximum c o i n c i d i n g with the l i g h t minimum of p Pup. T h i s e f f e c t i s most probably caused by the, l i n e - i n t e n s i t y 255 T a b l e 6.3 R e l a t i v e r a d i a l v e l o c i t i e s of p Pup (I) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # Ca II \8662 H I X8750 Fe I X8689 Mg I X8719 1 -1 .118 kms' 1 + 0. 1 06 kms" 1 -1 . 268 kms" 1 - 0 . 885 kms" 1 2 + 0 .245 kms" 1 + 0. 014 kms" 1 + 0. 168 kms" 1 + 0. 186 kms" 1 3 + 1 .484 kms _ 1 + 1 . 710 kms" 1 + 1 . 483 kms" 1 + 1 . 516 kms" 1 4 + 2 .590 kms" 1 + 1 . 973 kms" 1 + 2. 651 kms" 1 + 2. 582 kms" 1 5 + 3 .702 kms" 1 + 3. 425 kms" 1 + 3. 798 kms" 1 + 3. 598 kms" 1 6 + 4 .525 kms" 1 +4. 806 kms" 1 + 4. 560 kms" 1 + 4. 349 kms" 1 7 + 4 .771 kms" 1 + 4. 100 kms" 1 + 4. 856 kms" 1 + 4. 630 kms" 1 8 + 4 .758 kms" 1 + 4. 151 kms" 1 + 4. 629 kms" 1 + 4. 529 kms" 1 9 + 4 .079 kms" 1 + 3. 323 kms" 1 + 3. 846 kms" 1 + 3. 947 kms" 1 10 + 2 .898 kms" 1 + 2. 256 kms" 1 + 2. 756 kms" 1 + 2. 705 kms" 1 1 1 + 1 .521 kms" 1 + 0. 1 36 kms" 1 + 1 . 1 38 kms" 1 + 0. 928 kms" 1 1 2 -0 .213 kms" 1 -1 . 060 kms" 1 - 0 . 602 kms" 1 - o . 828 kms" 1 1 3 -1 .948 kms" 1 - 2 . 650 kms" 1 - 2 . 429 kms" 1 - 2 . 271 kms" 1 14 -3 .220 kms" 1 - 3 . 179 kms" 1 - 3 . 702 kms" 1 - 3 . 873 kms" 1 1 5 -4 .105 kms" 1 - 4 . 1 93 kms" 1 - 4 . 571 kms" 1 - 4 . 802 kms" 1 16 -4 .361 kms" 1 - 5 . 561 kms" 1 - 4 . 837 kms" 1 - 4 . 855 kms" 1 17 -4 . 1 37 kms" 1 - 4 . 270 kms" 1 - 4 . 614 kms" 1 - 4 . 641 kms" 1 18 -4 .644 kms" 1 - 4 . 308 kms" 1 - 4 . 793 kms" 1 - 4 . 600 kms" 1 19 -4 .386 kms" 1 - 4 . 386 kms" 1 - 4 . 1 29 kms" 1 - 4 . 045 kms" 1 20 -2 .791 kms" 1 - 2 . 791 kms" 1 - 2 . 413 kms" 1 - 2 . 402 kms" 1 21 + 0 .448 kms" 1 + 0. 448 kms" 1 - o . 645 kms" 1 - 0 . 738 kms" 1 22 + 1 .915 kms" 1 + 1 . 915 kms" 1 + 1 . 391 kms" 1 + 1 . 484 kms" 1 23 + 2 .845 kms" 1 + 2. 845 kms" 1 + 3. 1 62 kms" 1 + 3. 024 kms" 1 24 + 4 .082 kms" 1 • +4. 082 kms" 1 + 4. 339 kms" 1 + 4. 383 kms" 1 25 + 4 .221 kms" 1 + 4. 221 kms" 1 + 4. 802 kms" 1 + 4. 845 kms" 1 256 26 + 3 .878 kms"1 +3 .878 kms" 1 + 4 .247 kms" 1 +4. 391 kms" 1 27 + 2 .613 kms" 1 + 2 .613 kms" 1 + 2 .497 kms" 1 +2. 715 kms" 1 28 + 0 .781 kms" 1 + 0 .781 kms" 1 + 0 .019 kms" 1 + 0. 235 kms" 1 29 -1 .424 kms" 1 -1 .424 kms" 1 -2 .458 kms" 1 -2. 1 97 kms" 1 30 -3 .007 kms" 1 -3 .007 kms" 1 -4 .099 kms" 1 -3. 879 kms"1 31 -3 .898 kms" 1 -3 .898 kms" 1 -4 .776 kms" 1 -4. 869 kms" 1 32 -3 .893 kms"1 -3 .893 kms"1 -4 .230 kms" 1 -4. 495 kms" 1 33 -2 .970 . kms"1 -2 .970 kms" 1 -3 .045 kms" 1 -3. 1 53 kms" 1 34 -1 .677 kms" 1 -1 .677 kms" 1 -1 .105 kms" 1 -1 . 1 16 kms" 1 35 + 0 .821 kms" 1 + 0 .821 kms" 1 +0 .937 kms" 1 + 0. 908 kms" 1 36 -2 .114 kms" 1 -2 .616 kms" 1 -2 .253 kms" 1 -1 . 895 kms" 1 37 -0 .336 kms" 1 + 1 .087 kms" 1 -0 .362 kms" 1 -o. 556 kms" 1 38 + 1 .383 kms" 1 + 0 .057 kms" 1 + 1 .590 kms" 1 + 1 . 209 kms" 1 39 + 2 .907 kms"1 + 3 .627 kms" 1 + 3 .173 kms" 1 + 3. 1 54 kms"1 40 + 4 .204 kms"1 + 5 .117 kms" 1 + 4 .328 kms" 1 + 4. 487 kms"1 41 + 4 .489 kms" 1 + 4 .478 kms" 1 + 4 .512 kms" 1 + 4. 752 kms" 1 42 + 3 .633 kms" 1 + 2 .262 kms" 1 + 3 .676 kms" 1 + 3. 876 kms" 1 43 + 1 .862 kms" 1 + 1 .544 kms"1 + 1 .870 kms" 1 + 2. 076 kms" 1 44 -0 .553 kms"1 + 0 .930 kms" 1 -0 .637 kms" 1 -0. 214 kms" 1 45 -2 .856 kms"1 -2 .930 kms"1 -3 .124 kms" 1 -2. 837 kms" 1 ********************************************** 257 T a b l e 6.4 R e l a t i v e r a d i a l v e l o c i t i e s of p Pup ( I I ) * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * # mean of Fe I mean of S i I mean of S 1 -1 .119 kms" 1 -0 .855 kms" 1 -0 .950 kms" 2 +0.211 kms" 1 +0.341 kms" 1 +0.462 kms" 3 +1.508 kms" 1 +1.715 kms" 1 + 1 .846 kms" 4 +2.694 kms" 1 +2.956 kms" 1 +3.056 kms" 5 +3.824 kms" 1 +3.968 kms" 1 +4.007 kms" 6 +4.600 kms" 1 +4.695 kms" 1 +4.793 kms" 7 +4.860 kms" 1 +4.976 kms" 1 +5.002 kms" 8 +4.655 kms" 1 +4.871 kms" 1 +4.767 kms" 9 +3.866 kms" 1 +4.107 kms" 1 +4.017 kms" 1 0 +2.793 kms" 1 +2.713 kms" 1 +2.803 kms" 1 1 +1.162 kms" 1 +1.309 kms" 1 +1.244 kms" 1 2 -0.571 kms" 1 -0.651 kms" 1 -0 .603 kms" 1 3 -2 .384 kms" 1 -2 .234 kms" 1 -2 .332 kms" 1 4 -3 .674 kms' 1 -3 .447 kms" 1 -3 .663 kms" 1 5 -4 .526 kms" 1 -4 .310 kms" 1 -4 .518 kms" 1 6 -4.761 kms" 1 -4 .629 kms" 1 -4 .799 kms" 1 7 -4 .544 kms" 1 -4 .256 kms" 1 -4 .629 kms" 18 -4 .772 kms" 1 -4 .768 kms" 1 -4 .686 kms" '19 -4 .097 kms" 1 -3 .964 kms" ' -4 .046 kms" 20 . - 2 .379 kms" 1 -2 .267 kms" 1 -2.211 kms" 21 -0 .592 kms" 1 -0 .378 kms" 1 -0 .440 kms" 22 +1.456 kms" 1 +1.652 kms" 1 +1.585 kms" 23 +3.191 kms" 1 +3.322 kms" 1 +3.283 kms" 24 +4.362 kms" 1 +4.450' kms" 1 +4.460 kms" 25 +4.798 kms ' 1 +4.746 kms" 1 +4.791 kms" 258 26 +4.228 kms" 1 +4.116 kms" 1 •+4.319 kms" 1 27 +2.519 kms" 1 +2.548 kms" 1 +2.551 kms" 1 28 +0.058 kms" 1 +0.176 kms" 1 +0.141 kms' 1 29 -2 .457 kms' 1 -2 .367 kms" 1 -2 .277 kms" 1 30 -4 .094 kms - 1 -3 .948 kms" 1 -3 .972 kms" 1 31 -4 .720 kms" 1 -4 .464 kms" 1 -4 .646 kms" 1 32 -4 .196 kms - 1 -4 .100 kms" 1 -4 .138 kms" 1 33 -2 .964 kms" 1 -2 .767 kms" 1 -2 .862 kms" 1 34 -1.041 kms" 1 -0 .875 kms" 1 -0 .917 kms" 1 35 +0.987 kms" 1 +1.086 kms" 1 +1.146 kms" 1 36 -2 .220 kms" 1 -1 .990 kms" 1 -1 .955, kms" 1 37 -0.361 kms" 1 -0 .178 kms" 1 - 0 . 3 1 9 kms" 1 38 +1.534 kms" 1 +1.807 kms" 1 +1.639 kms" 1 39 +3.155 kms" 1 +3.398 kms" 1 +3.077 kms" 1 40 +4.262 kms" 1 +4.439 kms" 1 +4.151 kms" 1 41 +4.500 kms" 1 +4.512 kms" 1 +4.393 kms" 1 42 +3.689 kms" 1 +3.562 kms" 1 +3.647 kms" 1 43 +1.803 kms" 1 +1.887 kms" 1 +1.718 kms" 1 44 -0.701 kms" 1 -0.901 kms" 1 - 0 . 8 9 5 kms" 1 45 -3 .136 kms" 1 -3 .096 kms" 1 -3 .230 kms" 1 **************************************** 259 v a r i a t i o n s of the s t e l l a r l i n e s . L i n e c a n c e l l a t i o n s by the standard spectrum are imperfect and the r e s u l t i n g r e s i d u a l s would then a f f e c t the HF l i n e - p o s i t i o n d e t e r m i n a t i o n s . A p e r i o d - s e a r c h i n g procedure was a p p l i e d to the Ca II v e l o c i t i e s of a l l three n i g h t s . The Maximum Entropy technique r e q u i r e s evenly spaced data and hence would not be very a p p r o p r i a t e f o r t h i s set of data which spreads over s e v e r a l n i g h t s . The p e r i o d - f i n d i n g computer program used i n t h i s present a n a l y s i s was i n i t i a l l y used by Ninkov [1985] to determine the o r b i t a l p e r i o d of Cygnus X-1. I t i s based on the a l g o r i t h m and computer program given by Morbey [1978]. B a s i c a l l y , a number of t r i a l p e r i o d s are attempted, and i n each case, a phase i s c a l c u l a t e d f o r each v e l o c i t y p o i n t . The optimum p e r i o d would the one which g i v e s the best p l o t of v e l o c i t y a g a i n s t phase a c c o r d i n g to some s i g n i f i c a n c e c r i t e r i a . A p e r i o d of 0.14095 was measured f o r the Ca II v e l o c i t i e s . In view of the short time coverage, t h i s agrees q u i t e w e l l with the p e r i o d of 0.14088141 given by Ponsen [1963] and the value of 0.14088067 d determined by T h u l a s i Doss [1969]. The Ca II X8662 v e l o c i t i e s have a 2K value of 9.08kms"1 while the value f o r the H I X8750 v e l o c i t i e s i s 8.35kms _ 1. The cor r e s p o n d i n g values f o r the Fe I, S i I, and S I v e l o c i t i e s are 9.46, 9.23, and 9.41kms~ 1, r e s p e c t i v e l y . Meanwhile, the Mg I X8719 l i n e has a 2K value of 9.64kms~ 1. These are a l l lower than the v a l u e s of 11 and 11.5kms~1 given by Danzinger and Kuhi [1966] and Campos and Smith [1980], r e s p e c t i v e l y . However, Struve et a l [1956] and 260 F i g u r e 6.22 E r r o r s in the HF d i s p e r s i o n f i t s cr OT cn i_ QJ CL cn — H -CD (TT) n: C D a t  OT cD oo -O CD 1_ CD " O cr cn +-» W L O . . C O CTL C D ZD C L a Q 1 60 X P - _ © / - o - / © 0 / 0 v, 1 1 1 1 CD CO CD + CO LO CO LO CM a ~0 CD C D C O L O 00 (JO O C D C \ J O C D C D C D (^su)]) A i u j e u a o u n AITDOJSA 261 F i g u r e 6.23 V e l o c i t y d i f f e r e n c e ( Ca II curve - Fe I curve ) o CO QJ en QJ O CO CJ) rr ro o QJ I CM CD CO CD OO 00 << CD I—I ro <-> LD 00 CL C D ZD Q_ 0 Q h / Q N Q 1 6 y N . N © ei © / y 0 / 1 i i i i CD in LQ CM CD CD LO CM CD L O CD CD CD CD 1 CD 1 (,.SMI>|) 3 3 U 3 J 3 J J J P A 1 T 3 0 J 3 A r- CD C D C D CD C D C O CD L D C O + CO LD CD LD CM a CO 262 B e s s e l l [1969] have measured the 2K valu e s of 9.7, and 9.5kms~1, r e s p e c t i v e l y . The d i s c r e p a n c y c o u l d not have been the e f f e c t of phase smearing. The exposure time of each spectrum r e p o r t e d here was on l y about 6% of the p e r i o d . F i g u r e 6.23 shows the v e l o c i t y d i f f e r e n c e p l o t between the Ca II and the mean Fe I v e l o c i t i e s . The e f f e c t of the two d i f f e r e n t 2K values i s very evident i n the p l o t . In s p i t e of the amplitude d i f f e r e n c e s , the Ca II X8662 v e l o c i t i e s are i n phase with the v e l o c i t y curves of a l l the other m e t a l l i c l i n e s . The H I X8750 v e l o c i t y curve, however, l a g s a l l the other curves by about 0.003 or 2% of the p e r i o d . T h i s value i s o b t a i n e d by comparing the corresponding v e l o c i t y minima. However, the H I X8750 v e l o c i t y maximum appears to be more i n phase with those of the other v e l o c i t y c u r v e s . T h i s may c a s t doubt on the r e a l i t y of the observed phase-lag phenomenon. But the observed e f f e c t i s not the r e s u l t of a few i s o l a t e d v e l o c i t y p o i n t s . I t i s a r a t h e r l a r g e systematic t r e n d of f i v e c o n s e c u t i v e v e l o c i t y p o i n t s . The H I v e l o c i t i e s from the other n i g h t s are of poorer q u a l i t y because of the lower s/n. And they cannot be used to check the r e a l i t y or the r e p e a t a b i l t y of the observed phenomenon. In any case, the H I curve appears to have a d i f f e r e n t shape than those of the other l i n e s . The phenomenon of the H I v e l o c i t y curve l a g g i n g behind the other curves i s s i m i l a r to the c l a s s i c a l van Hoof e f f e c t found i n other v a r i a b l e s t a r s . T h i s i s d i s c u s s e d f u r t h e r i n Chapter e i g h t . 263 6.7 DISCUSSION Using the 2K value of the Fe I l i n e s , a value of 93kms" 1mag" 1 i s obt a i n e d f o r the 2K/Amv parameter. T h i s would agree with the value of 103±12kms~'mag" 1 obtained by Smith [1982b]. T h i s r a t h e r l a r g e value f o r the 2K/Amv parameter would imply r a d i a l p u l s a t i o n f o r p Pup. The same c o n c l u s i o n was obtained by Campos and Smith [1980] i n t h e i r a n a l y s i s of p Pup's l i n e p r o f i l e s . There i s a di s c r e p a n c y i n the p u b l i s h e d v a l u e s of the lo g g parameter. T h i s has been d i s c u s s e d i n Kurtz [1976]. Using a mean value of 3.53 f o r l o g g and the other parameters from Table 6.1, one can c a l c u l a t e a Q value of 0.03 day f o r p Pup. T h i s value would imply an i d e n t i f i c a t i o n of fundamental r a d i a l p u l s a t i o n f o r p Pup. The value s of 0.044 and 0.0395 day quoted by Breger and Bregman [1975] and Tsvetkov [1982b], r e s p e c t i v e l y , were c a l c u l a t e d u s i n g h i g h e r v a l u e s f o r l o g g. Tsvetkov [1982b] who compared the ^ of p Pup a g a i n s t t h e o r e t i c a l mode-dependent v a l u e s , has a l s o a r r i v e d a t the same c o n c l u s i o n of fundamental r a d i a l p u l s a t i o n . Because of i t s b r i g h t n e s s , sharp l i n e s , and simple v a r i a b l i t i e s , p Pup o f f e r s a r a t h e r unique o p p o r t u n i t y to study the 6 S c u t i p u l s a t i o n . The hig h s/n data has enable the i d e n t i f i c a t i o n of a few a d d i t i o n a l problems a s s o c i a t e d with the s p e c t r o s c o p i c study of 5 S c u t i s t a r s . P r o f i l e v a r i a t i o n s would impose l i m i t a t i o n s on the p r e c i s i o n a t t a i n a b l e i n the v e l o c i t y curve of 5 S c u t i v a r i a b l e s . The p r o f i l e v a r i a t i o n s a f f e c t both the s t e l l a r and HF 264 l i n e - p o s i t i o n d e t e r m i n a t i o n s . These v a r i a t i o n s may not only be the r e s u l t of p u l s a t i o n - g e n e r a t e d s u r f a c e v e l o c i t y f i e l d s . The temperature v a r i a t i o n s over the p u l s a t i o n c y c l e would a l s o cause l i n e - p r o f i l e v a r i a t i o n s and these are i n phase with the l i g h t curve and not the v e l o c i t i e s . These v a r i a t i o n s can a l s o be very complicated as the r e v e r s a l at the core of the Fe I X8689 l i n e has i n d i c a t e d . Chapter 7 THE DELTA SCUTI VARIABLE o 1 ERI 7.1 INTRODUCTION The s t a r o 1 E r i d a n i ( 38 E r i , HR1298 ) i s a b r o a d - l i n e 5 S c u t i v a r i a b l e . H o f f l e i t and Jaschek [1982] quote a v s i n i v alue of 96kms"1 while a value of 98kms~ 1 i s given by B a g l i n et a l . [1973]. Parameters f o r o 1 E r i are summarised i n Table 7.1. Other parameters can be found i n Tsvetkov [1982b], H a l p r i n and Moon [1983], Breger [1979], B a g l i n et a l . [1972], Gupta [1978], A n t o n e l l o et a l . [1981], Breger and Bregman [1975], and H o f f l e i t and Jaschek [1982]. Very few s t u d i e s have been done on o 1 E r i . T h i s s t a r was f i r s t r e p o r t e d to be a photometric v a r i a b l e by Jorgensen et a l . [1971]. A l i g h t amplitude of about 0.02 m was found i n the intermediate-band y and b photometry. Jorgensen and Norgaard-Nielsen [1975] reobserved the s t a r with uvby photometry. They found that the l i g h t curves cannot be represented by a simple p e r i o d i c o s c i l l a t i o n . The l i g h t amplitude i s about 0.05 m and the dominant p e r i o d i s determined to be 0.0815 (1.96 h o u r s ) . There may be a second p e r i o d of 0.1291 . No study of the s p e c t r o s c o p i c v a r i a t i o n s of the s t a r has been r e p o r t e d . 7.2 THE OBSERVATIONS The s t a r o 1 E r i was observed s p e c t r o s c o p i c a l l y with the HF a b s o r p t i o n c e l l at the Canada-France-Hawaii 3.6m t e l e s c o p e on the 25th of January 1983 UT. A time s e r i e s of 265 266 Table 7.1 Parameters f o r o 1 E r i ****************************************** Reference 4 h 9 m 25.321 s HD number : 26574 SAO number : 131019 DM number ': -7 764 R.A. (1950) Dec. (1950) : -6° 57' 59.675" Annual p a r a l l a x : +0.033" Proper motion i n R.A. : +0.004"/yr Proper motion i n Dec. : +0.083"/yr : 199.32° bJI : -38.39° S p e c t r a l type : F2 I I - I I I R a d i a l v e l o c i t y = 11 kms - 1 V s i n i = 98 kms"1 M v = +1.85 T e f f = 7300K lo g g = 3.9 (cgs) Broad band photometry Breger [1979] Eggen [1979] B a g l i n et a l . [1973] Eggen [1979] Breger and Bregman [1975] Breger and Bergman [1975] I r i a t e et a l . [1965] V = 4. 0 5 m B - V = +0 .33 m u - V = +0 .47 m v - R = +0 .32 m v - I = +0 .48 m Intermediate band photometry /3 = 2.730 m Eggen [1979] 267 b - y = 0. 197'" m, = 0 . l 9 2 m c, = 0.789 m 6m, = -0.013 m *************************************** s p e c t r a were obtained under the same c o n d i t i o n s and with the same i n s t r u m e n t a l setups as f o r the data on 20 CVn and p Pup d i s c u s s e d e a r l i e r . S i m i l a r o b serving procedures were used. S t e l l a r and lamp s p e c t r a with and without the imposed HF l i n e s were obtained. These are shown i n F i g u r e 7.1. The mean s t e l l a r + H F spectrum with e i t h e r the s t e l l a r or HF l i n e s n u m e r i c a l l y removed are a l s o shown. The exposure time f o r each spectrum was 650 seconds. T h i s corresponds to about one-tenth the " c y c l e - c o u n t " p e r i o d of 0.0815 day (1.96 h o u r s ) . Each s t e l l a r + H F spectrum has a mean s/n of about 262 in each p o i n t at the continuum. The mid-exposure times of the s p e c t r a are summarised i n Table 7.2. Spectrum #1 and #2 in Table 7.2 are the o 1 E r i s p e c t r a without the imposed HF l i n e s . Approximate r a d i a l v e l o c i t i e s can be measured from these s p e c t r a using the d i s p e r s i o n r e l a t i o n determined f o r spectrum #3. For t h i s set of o b s e r v a t i o n s , the h e l i o c e n t i c times l a g the b a r y c e n t r i c times by about 4.4 seconds. 7.3 THE DATA REDUCTION The data were processed and reduced using the i d e n t i c a l procedure which were a p p l i e d t o the 20 CVn d a t a . In f a c t , as wit h the 20 CVn r e d u c t i o n , the mean s t e l l a r spectrum of the , 268 F i g u r e 7.1 The o 1 E r i spectrum 269 Table 7.2 Mid-exposure times f o r the o 1 E r i s p e c t r a ******************************************* # 25 Jan 83 UT+ B a r y c e n t r i c JD hour angle 1 (04 :53: 26 .22) 2445359. 7059216 1h 23m 45s East 2 (05 :05: 05 .73) 2445359. 7140171 1h 1 2m 03s East 3 (05 :26: 1 1 .87) 2445359. 7286704 Oh 50m 54s East 4 (05 :37: 27 .27) 2445359. 7364868 Oh 39m 36s East 5 (05 :51 • 37 .86) 2445359. 7463308 Oh 25m 23s East 6 (06 :02 55 .48) 2445359. 7541730 Oh 1 4m 04s East 7 (06 : 1 4 1 1 .26) 2445359. 7619939 Oh 02m 46s East 8 (06 :25 26 .90) 2445359. 7698132 Oh 08m 31s West 9 (06 :36 42 .96) 2445359. 7776373 Oh . 1 9m 49s West 10 (06 :47 59 .41 ) 2445359. 7854660 Oh 31m 08s West 1 1 (06 :59 • 1 7 .50) 2445359. 7933136 Oh 42m 27s West 12 (07 : 10 :33 .25) 2445359. 8011341 Oh 53m 45s West *********************************************************** time s e r i e s has been chosen t o be the r a d i a l - v e l o c i t y standard spectrum. T h i s spectrum has the HF l i n e s n u m e r i c a l l y removed and i t p r o v i d e s both the l i n e - s h a p e and p o s i t i o n r e f e r e n c e s i n the a p p l i c a t i o n of the Fahlman-Glaspey d i f f e r e n c e technique. In a d d i t i o n to having a higher s/n, the mean spectrum averaged over one c y c l e w i l l be a more a p p r o p r i a t e standard spectrum i f there are s t e l l a r p u l s a t i o n r e l a t e d l i n e - p r o f i l e v a r i a t i o n s . As i n the case of 20 CVn, any l i n e broadening caused by the v e l o c i t y phase-smearing e f f e c t i n forming the mean spectrum w i l l be s m a l l . I t can be c o r r e c t e d i n a second i t e r a t i o n of the r e d u c t i o n procedure. 2 7 0 F i g u r e 7 . 2 T h e C a I I X 8 6 6 2 v e l o c i t y c u r v e o f o 1 E r i C\J C D L D C O CD r-- C D C M CD 1 1 1 o / p > ei P 1 ? 1 — \ - - \ \ — \ _ \] l l l CD LT) CD + c n L D C O L D C M Q ~ D Q Q CM- CD CM , .SUI>I Ad 3 A J j e j 3 ^ 271 F i g u r e 7.3 The H I X8750 v e l o c i t y curve of o' E r i C M C M C D I LO 0 0 U J 0 0 0 0 + 01 LO CO LO C M Q ~D CD C M C M ,.sui>( Ad 3 A j + e T 3 ^ 272 F i g u r e 7.4 The v e l o c i t y c u r v e of o 1 E r i from weak l i n e s CO (AJ r - CO L O CO CO CO r< cn co CO CO r< CD CO CO CO r< CD CO QJ LU CO 0 0 CD o LO CD CO CD —< O 1 1 1 1 G | i ? / 6 / \ b \ \ V h l l l CD CD LO + co LO CO LO •̂ r csi CD ~D CO CO CD CD CD ,.siu>| AcJ 3AueT3cj 273 7.4 THE RADIAL VELOCITIES The same c r i t e r i a used f o r the 20 CVn r e d u c t i o n were used here to choose the l i n e l i m i t s for both the s t e l l a r and HF l i n e s . A t y p i c a l set of s t e l l a r l i n e l i m i t s i s about 80 p i x e l s i n width. The unmodified d i f f e r e n c e f u n c t i o n from Equation 4.4 was used f o r the s t e l l a r l i n e s i n the a p p l i c a t i o n of the Fahlman-Glaspey technique to measure l i n e s h i f t s . I n d i v i d u a l r e l a t i v e r a d i a l - v e l o c i t y curves were obtained f o r the s t e l l a r l i n e s Ca II X8662 and H I X8750. These are presented i n F i g u r e s 7.2 and 7.3, r e s p e c t i v e l y . The s/n of the s p e c t r a are too low to l e a d to reasonable v e l o c i t y curves f o r the other weaker s t e l l a r l i n e s . In f a c t , even f o r the mean v e l o c i t y curve of the l i n e s S I X8680, X8695, Fe I X8689, and S i I X8728 given i n F i g u r e 7.4, one can b a r e l y d i s c e r n the c y c l i c v a r i a t i o n s . T h i s i s p a r t i a l l y caused by the low 2K v e l o c i t y amplitudes f o r these weaker l i n e s . Table 7.3 l i s t s a l l the v e l o c i t i e s shown i n F i g u r e s 7.2 through 7.4. The mean systemic v e l o c i t y i s 11.947kms" 1. T h i s has been c a l c u l a t e d as the mean v e l o c i t y of a l l the l i n e s averaged over the observed c y c l e . I t i s approximately equal to the value of 11kms - 1 quoted by Eggen [1979]. The mean standard e r r o r i n the HF d i s p e r s i o n f i t s of the s p e c t r a i s ±63ms~ 1. T h i s agrees very w e l l with the va l u e s from the 20 CVn and p Pup data s e t s which have s i m i l a r s/n. The Ca II curve i n F i g u r e 7.2 has an average o n e - s t a n d a r d - d e v i a t i o n u n c e r t a i n t y of ±1.34kms" 1 i n each s t e l l a r l i n e - p o s i t i o n measurement. T h i s i s the formal e r r o r estimate from the Fahlman-Glaspey technique. The l a r g e value 274 Table 7.3 R e l a t i v e r a d i a l v e l o c i t i e s of o 1 E r i ***************************************** # Ca II X8662 H I X8750 mean weak l i n e s 1 -2.068 kms'1 -2.136 kms"1 -0.369 kms"1 2 -0.855 kms"1 -1.060 kms"1 -0.369 kms"1 3 +0.475 kms"1 -0.434 kms"1 +0.028 kms"1 4 +1.168 kms"1 +1.060 kms"1 +0.123 kms"1 5 +2.162 kms - 1 +1.942 kms - 1 +0.757 kms'1 6 +1.968 kms"1 +1.015 kms - 1 +0.370 kms - 1 7 +1.797 kms"1 +2.163 kms"1 +0.298 kms"1 8 +0.517 kms"1 +0.620 kms"1 +0.164 kms"1 9 -0.129 kms~1 -0.504 kms"1 +0.058 kms"1 10 -1.212 kms"1 -1.172 kms"1 -0.638 kms"1 11 -1.849 kms"1 -0.923 kms"1 -0.225 kms'1 12 -1.974 kms"1 -0.571 kms"1 -0.196 kms"1 *********************************************************** here may s i g n i f y the e x i s t e n c e of l i n e - p r o f i l e v a r i a t i o n s . The c o r r e s p o n d i n g v a l u e s f o r the curves i n F i g u r e s 7.3 and 7.4 are ±1.56kms" 1 and ±1.79kms" 1, r e s p e c t i v e l y . Periodogram and Maximum L i k e l i h o o d power-spectrum a n a l y s e s give a mean p e r i o d of 0.088 from the Ca II curve i n F i g u r e 7.2, or a mean frequency of 11.39 day" 1 . T h i s i s d i f f e r e n t from the photometric value of 0.0815 given by Norgaard-Nielsen [1975]. In view of the short d u r a t i o n of the time s e r i e s here, the d i s c r e p a n c y between the two values may not be s i g n i f i c a n t . The e f f e c t of b e a t i n g by the secondary p e r i o d may cause an a p p a r e n t l y d i f f e r e n t primary p e r i o d to be observed over the s h o r t time window. 275 Both the Ca II and H I curves have a 2K amplitude of 4.3kms _ 1. The weak-line curve i n F i g u r e 7.4 has a 2K value of 1.2kms~1 i f one ignores the e f f e c t of the v e l o c i t y p o i n t s at BJD2445359.746 and BJD2445359.785. The r e a l i t y of these two v e l o c i t y p o i n t s may be somewhat u n c e r t a i n i n view of the ra t h e r l a r g e e r r o r i n the v e l o c i t i e s . N e v e r t h e l e s s , they may be the genuine e f f e c t of l i n e - p r o f i l e v a r i a t i o n s . The d i f f e r e n c e s i n the amplitudes between the v a r i o u s v e l o c i t y curves c o u l d not have been caused by the u n c e r t a i n t i e s i n the c h o i c e of the v a r i o u s l i n e l i m i t s . A d i f f e r e n c e of even s e v e r a l p i x e l s i n the l i n e l i m i t s i s too small to a l t e r the r e s u l t of such broad l i n e s . Moreover, the e r r o r i n the c h o i c e of the l i n e l i m i t s should be smal l f o r the mean v e l o c i t y c u rve. The r e a l i t y of the v e l o c i t y d i p at BJD2445359.754 i n the H I curve i s more probable. T h i s d i p of about 1.1kms"1 i s a r e l a t i v e l y l a r g e e f f e c t . A much sm a l l e r d i p may be present i n the Ca II curve. A s i m i l a r type of phenomenon has been r e p o r t e d i n the r a d i a l - v e l o c i t y curves of other 6 S c u t i s t a r s . The "bump" i n the r a d i a l - v e l o c i t y curve i s g e n e r a l l y i n the form of a v e l o c i t y d i p near the v e l o c i t y maximum or a l o c a l v e l o c i t y peak near the v e l o c i t y minimum. T h i s was f i r s t r e p o r t e d i n the v e l o c i t y curve of 14 Aur by C h e v a l i e r et a l . [1968]. The "bumps" have a l s o been observed i n the r a d i a l - v e l o c i t y curves of HR432 and HR515 by V a l t i e r et a l . [1979] and i n 7 Boo by Auvergne et a l . [1979], I t has been p o i n t e d out by Auvergne et a l . [1979] t h a t the v e l o c i t y bumps are a s s o c i a t e d with the s p l i t t i n g of the l i n e 2 7 6 p r o f i l e s . The s p l i t t i n g of the l i n e p r o f i l e i n Cepheids i s commonly known as the "Cheshire-Cat" l i n e s phenomenon. The l i n e s p l i t t i n g and the v e l o c i t y "bumps" have been p r e d i c t e d f o r Cepheid v a r i a b l e s by d e t a i l e d hydrodynamic models (Karp [I975ab]). The l i n e s p l i t t i n g i s caused by the propagation of shock waves i n the s t e l l a r atmosphere. As the shock wave moves outward i n the atmosphere, i t w i l l a c c e l e r a t e because of the d e c r e a s i n g d e n s i t y . A temperature r i s e w i l l r e s u l t . In the region where the weak l i n e s are formed, the gas r e q u i r e s only a very short time to r e t u r n to e q u i l i b r i u m . However, i n the r e g i o n higher i n the atmosphere where the c o r e s of the strong l i n e s are formed, i t would take a much longer time f o r the gas to r e t u r n to e q u i l i b r i u m . Within t h i s time, the shock may t r a v e l a c o n s i d e r a b l e d i s t a n c e and produce a t h i c k l a y e r of heated gas. Hence, l i n e s p l i t t i n g can be observed when the temperature r i s e i s s t r o n g enough at small continuum o p t i c a l depths. T h e o r e t i c a l l y p r e d i c t e d v e l o c i t y "bumps" may be seen i n F i g u r e 13 of Karp [1975b]. The t h e o r e t i c a l c a l c u l a t i o n s by Karp [l975ab], however, are f o r a Cepheid with a p e r i o d of 12 days. No s i m i l a r c a l c u l a t i o n has yet been done f o r the 5 S c u t i s t a r s . The observed bump i s not as pronounced as those observed by Auvergne et a l . [1979] i n 7 Boo. T h i s c o u l d be caused by phase-smearing e f f e c t i n the observed curve. gure 7.5 The Ca II X8662 l i n e p r o f i l e s 0 . 7 0 5 9 2 0.71402 0 . 7 2 8 6 7 0 . 7 3 6 4 9 0 . 7 4 6 3 3 0.75417 0.76199 0.76981 0 . 7 7 7 6 4 0 . 7 8 5 4 7 0.79331 0.80113 M e a n 8 6 6 0 8 6 6 5 8 6 7 0 278 F i g u r e 7.6 The r e s i d u a l s of the Ca II X8662 l i n e p r o f i l e s 0 . 7 0 5 9 2 0.71402 0 . 7 2 8 6 7 0 . 7 3 6 4 9 0 . 7 4 6 3 3 0.75417 0.76199 0.76981 0 . 7 7 7 6 4 0 . 7 8 5 4 7 0.79331 0.80113 8 6 6 0 8 6 6 5 8 6 7 0 279 7.5 THE LINE-PROFILE VARIATIONS Fi g u r e 7.5 shows the region of the Ca II l i n e f o r each spectrum a f t e r c o r r e c t i n g f o r the measured v e l o c i t y s h i f t s . The s p e c t r a have been smoothed by a Gaussian t r a n s f e r f u n c t i o n which has a a of ±0.1A. The mid-exposure time i n f r a c t i o n of days from BJD2445359 i s i n d i c a t e d f o r each spectrum. L i n e - p r o f i l e v a r i a t i o n s are evident from t h i s "stacked" p l o t of the s p e c t r a . Some of the p r o f i l e s have f l a t t e r l i n e c ores than o t h e r s . Some have symmetric l i n e p r o f i l e s while others have the p r o f i l e s skewed towards e i t h e r the short or the long wavelengths. F i g u r e 7.6 shows the r e s i d u a l s a f t e r s u b t r a c t i n g a mean l i n e p r o f i l e from each spectrum. Since the eleven observed s p e c t r a cover about one c y c l i c v a r i a t i o n i n the r a d i a l v e l o c i t i e s , the mean spectrum chosen f o r the s u b t r a c t i o n i s the average of a l l the observed s p e c t r a . The nature of the l i n e - p r o f i l e v a r i a t i o n s i s e v i d e n t i n F i g u r e 7.6. The systematic v a r i a t i o n s are at the l e v e l of about 1% to 2% of the continuum. The dominant v a r i a t i o n s can be c h a r a c t e r i s e d as the r e s u l t of temporal movement of an a b s o r p t i o n f e a t u r e a c r o s s the l i n e p r o f i l e . T h i s f e a t u r e i s r a t h e r weak i n the f i r s t two s p e c t r a as i t has yet to enter the r o t a t i o n a l l y broadened l i n e c o r e . The f e a t u r e can best be seen as i t moves from the short wavelength s i d e of the l i n e core i n spectrum #3 to the long wavelength s i d e of the l i n e core i n spectrum #9. The s t r e n g t h of t h i s f e a t u r e grows as i t moves towards the l i n e c e n t r e and fades as i t moves away from the l i n e c e n t r e . I t i s s t r o n g e s t i n spectrum #5. A f t e r spectrum 280 #9, the f e a t u r e fades as i t moves o f f the l i n e c o r e . One can a l s o see another f e a t u r e s t a r t i n g to enter the l i n e core as the f i r s t f e a t u r e was moving o f f the l i n e c o r e . The f e a t u r e shown i n the "stacked" r e s i d u a l p l o t i s not the tr u e r e p r e s e n t a t i o n of the p r o f i l e v a r i a t i o n s . Because of the short time s e r i e s , the f e a t u r e would a l s o c o n t r i b u t e s i g n i f i c a n t l y towards the mean spectrum that was used f o r the s u b t r a c t i o n . Hence the shapes and p o s i t i o n s of the f e a t u r e shown i n F i g u r e 7.6 can not be i n t e r p r e t e d as i d e n t i c a l to those of the true v a r i a t i o n s . A b e t t e r r e p r e s e n t a t i v e of the l i n e p r o f i l e without the imposed v a r i a t i o n s would be a mean spectrum averaged over s e v e r a l c y c l e s . The type of l i n e - p r o f i l e v a r i a t i o n s e x h i b i t e d by o 1 E r i i s very s i m i l a r to those found i n the Oe s t a r $ Oph (Walker et a l . [1979,1981b], Ninkov et a l . [1983], Vogt and Penrod [1983]) and the 0 Cephei s t a r a V i r (Walker et a l . [1981a,1982], F r a s e r et a l . [1983], Smith [1985]). The p r o f i l e v a r i a t i o n s i n these s t a r s are g e n e r a l l y c o n s i d e r e d to be caused by n o n r a d i a l p u l s a t i o n s . Hence the p r o f i l e v a r i a t i o n s observed i n o 1 E r i s t r o n g l y suggests the presence of n o n r a d i a l p u l s a t i o n modes i n the s t a r . One of the usual ways to i d e n t i f y the p u l s a t i o n mode f o r a set of data such as t h i s i s to f i t the observed p r o f i l e s and t h e i r r e s i d u a l s to t h e o r e t i c a l l y generated c o u n t e r p a r t s . The r a t h e r short time s e r i e s and the small number of s p e c t r a a v a i l a b l e here, however, imply that a unique mode i d e n t i f i c a t i o n may be d i f f i c u l t i f not i m p o s s i b l e . The l a r g e number of parameters 281 necessary i n the f i t would enable a v a r i e t y of modes or combination of modes to f i t the small data s e t , a l l e q u a l l y as w e l l . In any case, mode i d e n t i f i c a t i o n by l i n e p r o f i l e s alone i s treacherous (Smith [1981]). T h i s c o n c l u s i o n does not even take i n t o account any doubt about the a p p r o p r i a t e n e s s and c o r r e c t n e s s of the model. 7.6 DISCUSSION With some simple assumptions, one can make a guess at the p u l s a t i o n mode suggested by the observed v a r i a t i o n s . T h i s i s accomplished by i n t e r p r e t i n g the observed f e a t u r e i n F i g u r e 7.6 to be caused by a p o l e - t o - p o l e azimuthal s t r i p of a n o n r a d i a l l y t r a v e l l i n g p r e s s u r e wave on the s t e l l a r s u r f a c e . I f t h i s i s the case, then one can reason t h a t s i n c e at most two such f e a t u r e s are v i s i b l e i n the p r o f i l e at any one time, there are probably at most three or four such t r a v e l l i n g waves on the s t e l l a r s u r f a c e . T h i s would imply /=3 or 4. Of course, / c o u l d a l s o be 2 or 1. T h i s depends on the r e a l i t y of the weaker second f e a t u r e or any unseen t h i r d or f o u r t h f e a t u r e . A v a r i e t y of assumptions have t o be made in order to allow t h i s i n t e r