UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

The rapidly oscillating Ap star HR 1217 : the effect of a magnetic field on pulsation Cameron, Christopher J. 2004

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2004-0183.pdf [ 5.52MB ]
Metadata
JSON: 831-1.0085153.json
JSON-LD: 831-1.0085153-ld.json
RDF/XML (Pretty): 831-1.0085153-rdf.xml
RDF/JSON: 831-1.0085153-rdf.json
Turtle: 831-1.0085153-turtle.txt
N-Triples: 831-1.0085153-rdf-ntriples.txt
Original Record: 831-1.0085153-source.json
Full Text
831-1.0085153-fulltext.txt
Citation
831-1.0085153.ris

Full Text

The Rapidly Oscillating Ap Star HR 1217 The Effect of a Magnetic Field on Pulsation by Christopher J. Cameron B . S c , Saint Mary's University, 2001 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF MASTER OF SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES (Department of Physics and Astronomy)  We accept this thesis as conforming to the required standard  T H E U N I V E R S I T Y OF BRITISH C O L U M B I A April 1, 2004 © Christopher J. Cameron, 2004  Library Authorization  In presenting this thesis in partial fulfillment 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.  Name of Author (please print)  Date (dd/mm/yyyy)  Title of Thesis:  Degree:  M c < U ,  Department of  o  £ 5,  PlwcV.,  The University of British Columbia Vancouver, B C  /  \ AA ^  Canada  Year:  „,J  £<\.  4  •  I U ,  Abstract  ii  Abstract T h e r a p i d l y oscillating A p ( r o A p ) stars provide a unique o p p o r t u n i t y t o observe a n d test processes t h a t u n t i l r e c e n t l y c o u l d o n l y b e s t u d i e d i n d e t a i l for t h e S u n . A s a class, these s t a r s possess e x t r e m e e x a m p l e s o f c h e m i c a l i n h o m o g e n e i t i e s t h r o u g h t h e effects o f g r a v i t a t i o n a l s e t t l i n g , m i x i n g a n d r a d i a t i v e a c c e l e r a t i o n s ; m a g n e t i c fields t h a t affect b o t h t h e m i c r o - a n d m a c r o s c o p i c p h y s i c s t h a t influence s t e l l a r s t r u c t u r e ; a n d h i g h o v e r t o n e pulsations.  F o r t h e l a t t e r , we m a y use t h e t o o l s d e v e l o p e d for asteroseismic  analysis:  t h e inference o f t h e i n t e r n a l s t r u c t u r a l p r o p e r t i e s o f a s t a r b a s e d o n i n f o r m a t i o n f r o m i t s pulsational instabilities. O n e o f t h e m o s t s t u d i e d o f t h e r o A p s t a r s is H R 1217. T h e r a p i d v a r i a b i l i t y o f t h i s s t a r w a s first d i s c o v e r e d b y K u r t z (1982).  L a t e r , K u r t z et a l . (1989)  p h o t o m e t r i c observations of H R 1217 d u r i n g a g l o b a l c a m p a i g n .  extended  T h e y discovered s i x  p r i m a r y frequencies w i t h p e r i o d s n e a r 6 m i n u t e s a n d a s p a c i n g p a t t e r n t h a t i s r e m i n i s c e n t of p u l s a t i o n s o b s e r v e d i n t h e S u n . T h i s thesis presents a revised frequency analysis of p h o t o m e t r i c d a t a o n H R 1217 o b t a i n e d d u r i n g a W h o l e E a r t h T e l e s c o p e ( W E T ) c a m p a i g n i n l a t e 2000. I n p a r t i c u l a r , we i d e n t i f y a n e w f r e q u e n c y at 2788.94 / / H z w i t h a c h a r a c t e r i s t i c s e c o n d - o r d e r  p-mode  s p a c i n g o f 2.63 / / H z f r o m a n o t h e r f r e q u e n c y p r e v i o u s l y r e p o r t e d b y K u r t z et a l . (2002). E v o l u t i o n a r y a n d p u l s a t i o n a l m o d e l s o f A s t a r s are also p r e s e n t e d w i t h a d i s c u s s i o n o f h o w a m a g n e t i c field c a n p e r t u r b t h e a c o u s t i c frequencies. T o a c c o m p l i s h t h i s , t h e v a r i a t i o n a l m e t h o d u s e d b y C u n h a & G o u g h (2000) i s a d a p t e d t o c a l c u l a t e m a g n e t i c p e r t u r b a t i o n s to the acoustic modes calculated from o u r evolutionary models. These p e r t u r b a t i o n s are u s e d t o e x p l a i n t h e u n e x p e c t e d f r e q u e n c y s p a c i n g o b s e r v e d i n H R 1217.  Contents  iii  Contents Abstract  .  ii  Contents  iii  List of Tables  v  List of Figures Acknowledgements 1  2  vi viii  Introduction  1  1.1  A p Stars and Chemical Abundances  1  1.2  Magnetic Fields and the A p Stars  2  1.2.1  A p Spectra and the Zeeman Effect  2  1.2.2  Some Observable Magnetic Quantities  3  1.3  Diffusion i n A p Stars  6  1.4  Stellar Seismology and the A p Stars  7  1.4.1  Nonradial Oscillations  8  1.4.2  High-Overtone Pulsation and Frequency Spacing  11  1.4.3  The Oblique Pulsator M o d e l  13  1.5  T h e r o A p Star H R 1217  15  1.6  A n Overview of this Thesis  18  Data & Analysis  20  2.1  20  The W h o l e E a r t h Telescope  Contents 2.2  O b s e r v i n g H R 1217 w i t h W E T  21  2.3  P r e p a r i n g the L i g h t Curves  23  2.3.1  26  Frequency Analysis  28  2.5  T h e Unweighted D a t a Results  31  2.5.1  35  E s t i m a t i n g Uncertainties and Significance  T h e Weighted D a t a Results  39  Stellar Evolution and Pulsation Models  48  3.1  Stellar E v o l u t i o n M o d e l s  48  3.1.1  T h e P a r a m e t e r Space  49  3.1.2  M o d e l Properties  52  3.2  3.3  4  R u n Selection  2.4  2.6  3  iv  Pulsation Models  55  3.2.1  58  M o d e l Frequencies  M a g n e t i c Effects  62  3.3.1  65  T h e Frequency Perturbations  Discussion and Conclusions  69  Bibliography  75  A XCOV20 Observing Log  83  B A Listing of the Evolutionary Models  87  C Tables of the Weighted Data Results  102  List of Tables  v  List of Tables 2.1  Results of the frequency analysis of the unweighted data  32  2.2  Uncertainty estimate for the unweighted data  38  2.3  A summary of the weighting parameters for the thirty-four different frequency solutions explored  3.1  T e n models selected from the 569 that fall w i t h i n the Hipparcos  42 luminosity-  error bars  61  A. l  A data log obtained during X C O V 2 0  83  B. l  A listing of the models that fall w i t h i n the Hipparcos luminosity errorbars.  87  C. l  Weighted model number 14  102  C.2  Weighted model number 18  103  C.3  Weighted model number 27  104  C.4  Weighted model number 32  C.5  Average values for weighted reduction models 1 through 11  106  C.6  Average values for weighted reduction models 12 through 15  107  C.7  Average values for weighted reduction models 16 through 19  108  C.8  Average values for weighted reduction models 21 through 14  109  C.9  Average values for weighted reduction models 25 through 28  109  C.10 Average values for weighted reduction models 30 through 32  110  .  105  List of Figures  vi  List of Figures 1.1  A schematic d i a g r a m showing the oblique rotator m o d e l geometry. . . . .  1.2  A s c h e m a t i c r e p r e s e n t a t i o n o f different s p h e r i c a l h a r m o n i c m o d e s  1.3  T h e v a r i a t i o n i n t h e m e a n l i g h t o f H R 1217 t h r o u g h t h e B f i l t e r as a  10  function of the r o t a t i o n phase of the star 1.4  A b o v e is a s c h e m a t i c d i a g r a m o f t h e frequencies f o u n d f r o m t h e  14 1986  o b s e r v a t i o n s o f H R 1217 ( K u r t z et a l . , 1989) 2.1  17  A m a p of the observatories that participated i n the observations of H R 1217 d u r i n g X C O V 2 0  23  2.2  T h e e n d r e s u l t o f t h e QED  2.3  A F o u r i e r s p e c t r u m of d a t a from the runs n o 2 9 0 0 q l (top) a n d joy-002  r e d u c t i o n for r u n m d r l 3 6 at C T I O  25  (bottom) 2.4  5  27  T h e f i n a l l i g h t c u r v e o b t a i n e d after t h e Q E D r e d u c t i o n a n d t h e b a r y c e n t r i c corrections have been t a k e n into account  29  2.5  T h e F o u r i e r s p e c t r u m o f t h e e n t i r e r u n s h o w n i n F i g u r e 2.4  30  2.6  A s c h e m a t i c o f t h e frequencies f o u n d f r o m t h e u n w e i g h t e d d a t a  33  2.7  T h e r a t i o b e t w e e n t h e c o m m o n a m p l i t u d e s ( t o p ) a n d differences i n freq u e n c i e s ( b o t t o m ) f r o m t h i s d a t a set a n d t h e K u r t z et a l . (1989)  data  set 2.8  2.9  34  I m p r o v e m e n t o f t h e s t a n d a r d d e v i a t i o n s of t h e r e s i d u a l s as e a c h f r e q u e n c y is r e m o v e d  36  T h e F W H M of t h e m a i n f r e q u e n c y i n t h e u n w e i g h t e d d a t a  37  List of Figures  vii  2.10 A c o m p a r i s o n b e t w e e n t h e w e i g h t e d a n d u n w e i g h t e d m o d e l s u s i n g E q u a t i o n 2.8  44  2.11 S c h e m a t i c a m p l i t u d e d i a g r a m s for m o d e l s 14 a n d 32  46  2.12 S c h e m a t i c a m p l i t u d e d i a g r a m s for m o d e l s 18 a n d 27  47  3.1  A theoretical H R d i a g r a m showing the extremes of the p a r a m e t e r space .  53  3.2  T h e c o n v e c t i v e core m a s s ( t o p ) a n d t h e c o n v e c t i v e e n v e l o p e m a s s ( b e l o w ) as a f u n c t i o n of age for t h e m o d e l s s h o w n i n F i g u r e 3.1  3.3  . . . .  T h e v a r i a t i o n o f A z / i n / / H z (top) a n d t h e m o d e l age i n G y r  (bottom)  as a f u n c t i o n o f t h e r a t i o X / Z for m o d e l s t h a t f a l l w i t h i n t h e  Hipparcos  errorbars 3.4  56  A n echelle d i a g r a m for t h e 1 . 5 M © m o d e l s w i t h X = 0.700, Z = 0.012 a n d Z = 0.014  3.5  54  59  O n t h e left, a n echelle d i a g r a m for 1 . 6 M the second order spacing  0  models. T h e plot below shows  as a f u n c t i o n o f f r e q u e n c y  60  3.6  E c h e l l e d i a g r a m s for t h e m o d e l s l i s t e d i n T a b l e 3.1  62  3.7  S e c o n d o r d e r s p a c i n g d i a g r a m s for t h e m o d e l s l i s t e d i n T a b l e 3.1  63  3.8  T h e m a g n e t i c p e r t u r b a t i o n s c a l c u l a t e d for m o d e l s 1 t o 5 u s i n g a v a r i a t i o n a l principle  3.9  66  T h e m a g n e t i c p e r t u r b a t i o n s c a l c u l a t e d for m o d e l s 6 t o 10 u s i n g a v a r i a tional principle  67  VUl  Acknowledgements I ' d l i k e t o b e g i n b y t h a n k i n g m y s u p e r v i s o r J a y m i e M a t t h e w s . O v e r t h e p a s t few y e a r s he h a s e x p o s e d m e t o m a n y e x c i t i n g scientific a n d e d u c a t i o n a l o p p o r t u n i t i e s . O f c o u r s e t h e r e were o t h e r n o n - s c i e n t i f i c ; b u t e q u a l l y e d u c a t i o n a l , e x p e r i e n c e s as w e l l . H i s g u i d a n c e a n d patience has m a d e this w o r k possible. T h a n k s J a y m i e . M a n y T h a n k s to M a r g a r i d a C u n h a for h e l p i n g m e u n d e r s t a n d s o m e o f t h e finer p o i n t s o f s t e l l a r p u l s a t i o n t h e o r y a n d for b e i n g t h e s e c o n d r e a d e r o f t h i s thesis. I d o n ' t t h i n k I've ever seen so m a n y e x c l a m a t i o n m a r k s u s e d i n c o m m e n t s before.  T h a n k s M a r g a r i d a ! I w o u l d also l i k e t o t h a n k D a v i d  G u e n t h e r for t h e use o f h i s s t e l l a r e v o l u t i o n a n d p u l s a t i o n software a n d for c o n t i n u i n g t o e n c o u r a g e a n d s h o w interest i n m y w o r k . T o D o n K u r t z , t h a n k y o u for y o u r c o m m e n t s throughout this work. I w o u l d l i k e t o t h a n k S t e v e K a w a l e r a n d R e e d R i d d l e for h e l p i n g m e get a c q u a i n t e d w i t h Q E D a n d t o T h e W h o l e E a r t h T e l e s c o p e c o l l a b o r a t i o n for c o l l e c t i n g , a n d g i v i n g m e access t o , s u c h g r e a t d a t a . A l s o , t h a n k s t o G e r a l d H a n d l e r for h i s c o m m e n t s o n m y earlier d a t a reduction. Special t h a n k s to M o m , D a d , C h e r y l , a n d Jennifer. W i t h o u t y o u none of this w o u l d h a v e b e e n p o s s i b l e . Y o u r s u p p o r t over t h e y e a r s has h e l p e d m e i n b o t h m y p e r s o n a l a n d a c a d e m i c life. T h a n k y o u . I ' d l i k e t o t h a n k m y friends at h o m e a n d at U B C for h e l p i n g m a k e t h i s p r o c e s s a f u n one.  F i n a l l y , I c a n ' t o v e r l o o k D r . M a r k P . F . H u b e r for l i s t e n i n g t o m y r a n t s over t h e  l a s t y e a r a n d for h a n d i n g i n t h i s thesis w h i l e I w a s away.  1  Chapter 1  Introduction 1.1  Ap Stars and Chemical Abundances  In the A p  1  n o m e n c l a t u r e , t h e A represents t h e s p e c t r a l class a n d t h e p s t a n d s for  spectroscopically peculiar. T h e A classification c a n be somewhat m i s l e a d i n g since the group also includes spectral types r a n g i n g from late B to early F w i t h a r a n g e o f 14000 > T ff e  > 7000 K.  temperature  P e c u l i a r i t i e s i n these s t a r s are o b s e r v e d as s p e c t r a l  l i n e s t r e n g t h a n o m a l i e s t h a t are s i g n i f i c a n t l y different f r o m t h e m a j o r i t y o f o t h e r s t a r s a n d are i n t e r p r e t e d as a b u n d a n c e e n h a n c e m e n t s or d e p l e t i o n s o n t h e surface o f t h e s t a r . N e a r t h e m a i n sequence t h e r e e x i s t s a v a r i e t y of c h e m i c a l l y p e c u l i a r s t a r s .  Each  g r o u p is d e f i n e d b y a specific t e m p e r a t u r e range, m a g n e t i c field s t r e n g t h , s p e c t r a l l i n e anomalies, a n d their pulsational properties. temperatures  A s an example, the hotter A p stars w i t h  i n t h e r a n g e o f 10000 - 14000 K , c a n e x h i b i t e i t h e r S i o r H g a n d M n  p e c u l i a r i t i e s i n t h e i r s p e c t r a . T h e difference here is t h e A p S i s t a r s e x h i b i t a m a g n e t i c field  while the A p H g M n stars generally do not. W e c a n d i v i d e t h e A p s t a r s i n t w o g e n e r a l categories b a s e d o n t h e i r t e m p e r a t u r e  the observed abundances.  T h e c o o l e r s t a r s w i t h T jj e  < 10000 K  and  are classified as A p  S r C r E u s t a r s a n d s h o w v e r y s t r o n g l i n e s t r e n g t h s for t h e r a r e e a r t h e l e m e n t s  2  Sr, C r ,  a n d E u . I n a d d i t i o n , these s t a r s g e n e r a l l y have w e a k O l i n e s . T h e h o t t e r A p s t a r s w i t h 1  A n alternate classification scheme for these stars is provided by Preston (1974). Under this scheme,  the Ap stars discussed above are classified as CP2: magnetic Ap and Bp stars; including the He weak and He strong stars. In this thesis I will use the Ap nomenclature because it is more widely used in North America and because the nomenclature for the variable stars follow that of the Ap classification. elements L a to Er 2  Chapter  T ff e  1.  2  Introduction  > 10000 K show strong S i lines and are known as A p S i stars. These stars sometimes  show strong C a lines as well as some enhancement i n the lines of the iron peak elements . 3  B o t h hot and cool A p stars seem to have weak He lines, if they are present at a l l . A s a comparison, these peculiar line strengths correspond to photospheric overabundances up to 10 times the solar value and under abundances down to 1 0 5  - 2  times solar value.  A n overview of the A p class of stars is provided by Wolff (1983) and K u r t z & Martinez (2000).  1.2 Magnetic Fields and the Ap Stars O f the stars located near the upper m a i n sequence, magnetic fields are measured only i n the chemically peculiar ones.  In the case of the A p stars, the fields appear to be  global and predominately dipolar w i t h strengths that range from approximately 0.3 k G to 30 k G (Landstreet, 1992a). These magnetic fields seem to be intricately connected to the chemical peculiarities and the variability of these stars; b o t h of which w i l l be discussed later i n this chapter. A general discussion of magnetic fields i n stars is given by M a t h y s (1989) and Landstreet (1992a; 1992b; and 1993).  1.2.1  Ap Spectra and the Zeeman Effect  T h e perturbations to stellar spectra produced by line transitions of an atom i n the presence of a magnetic field are described by the Zeeman effect.  If an a t o m is not i n the  presence of a magnetic field there exists a set of discrete energy transitions that can occur as electrons change their energy levels. There is also coupling between an electron's spin and orbital angular momentum; known as L S coupling, that add to give rise to a larger number of possible energy states available to the electron. If the t o t a l angular momentum from the L S coupling is represented by J , then i n the presence of a magnetic field there becomes 2 J + 1 available states for the electron. E a c h of these magnetic substates are 3  elements Sc to N i  Chapter 1. Introduction  3  r e p r e s e n t e d b y t h e i n t e g e r q u a n t u m n u m b e r m. If t h e difference i n m a g n e t i c s u b s t a t e s i s A m = 0, t h e n t h e y a r e r e f e r r e d t o as TT components.  If A m =  ± 1 , t h e n t h e s u b s t a t e s a r e k n o w n as a c o m p o n e n t s .  The n  c o m p o n e n t is s y m m e t r i c a l l y spread a r o u n d the u n p e r t u r b e d wavelength, w h i l e the a c o m p o n e n t s h a v e w a v e l e n g t h s t h a t are o n e i t h e r side o f t h e u n p e r t u r b e d w a v e l e n g t h . T h e s p a c i n g o f t h e o c o m p o n e n t s is p r o p o r t i o n a l t o t h e m a g n e t i c field s t r e n g t h for fields w i t h s t r e n g t h s u p t o a few t e n s o f  kiloGauss.  T h e relation between the spacing of the  7r  a n d a c o m p o n e n t s is g i v e n b y (cf. L a n d s t r e e t 1992b)  AA = A e5/47rmc 2  (1.1)  2  w h e r e A i s t h e u n s p l i t w a v e l e n g t h , B i s t h e m a g n e t i c field, a n d e / m a n d c a r e t h e c h a r g e to-mass r a t i o of the electron a n d the speed of light, respectively. T h i s r e l a t i o n assumes t h a t t h e L a n d e factor; c a l c u l a t e d f r o m t h e s p i n q u a n t u m n u m b e r s , i s e q u a l t o o n e . E v e n i f t h e Z e e m a n c o m p o n e n t s o f a l i n e a r e n o t f u l l y r e s o l v e d , as i s a l m o s t a l w a y s t h e case, t h e TT a n d a c o m p o n e n t s t e n d t o b r o a d e n t h e s p e c t r a l l i n e . I f t h i s b r o a d e n i n g c a n b e m e a s u r e d t h e effective field s t r e n g t h m a y b e e s t i m a t e d . H o w e v e r , t h e r e a r e o t h e r effects t h a t c o m p e t e t o b r o a d e n a s p e c t r a l l i n e s u c h as t h e r m a l a n d D o p p l e r b r o a d e n i n g a n d m i c r o - a n d m a c r o s c o p i c t u r b u l e n c e . T h e Z e e m a n b r o a d e n i n g d o m i n a t e s for a s m a l l subset o f s t a r s w i t h B > 10 k G a n d p r o j e c t e d v e l o c i t i e s v s i n ( i ) < 10 k m / s ( L a n d s t r e e t , 1992b).  1.2.2  Some Observable Magnetic Quantities  T h e s e n s i t i v i t y t o t h e m a g n e t i c field v a r i e s f r o m l i n e t o l i n e .  B y observing lines from  different e l e m e n t s , o n e c a n d e d u c e t h e m a g n e t i c field s t r e n g t h b y c o m p a r i n g t h e w i d t h s of these l i n e s t o e a c h o t h e r . I n t h i s way, t h e Z e e m a n c o m p o n e n t s c a n p r o v i d e a m e a s u r e of t h e m a g n e t i c field s t r e n g t h i n t e g r a t e d over t h e s t e l l a r d i s c .  T h i s q u a n t i t y is k n o w n  as t h e m e a n field m o d u l u s a n d m a y b e d e n o t e d as < B > . F o r e x a m p l e , M a t h y s et a l . (1997) present m e a s u r e s o f t h e m a g n e t i c field m o d u l u s for 4 0 A p s t a r s .  Chapter 1.  Introduction  4  T h e s p e c t r a l l i n e s i n f l u e n c e d b y a m a g n e t i c field also possess i n t e r e s t i n g p o l a r i z a t i o n properties.  I n f o r m a t i o n gained f o r m the p o l a r i z a t i o n of spectral lines c a n further  u s e d t o set l i m i t s o n m a g n e t i c field g e o m e t r y .  be  I f a s p e c t r a l l i n e is o b s e r v e d i n b o t h left  a n d r i g h t - c i r c u l a r l y p o l a r i z e d ( L C P a n d R C P respectively) light, the c o n t r i b u t i o n to the s p e c t r a l l i n e f r o m e a c h o f t h e o c o m p o n e n t s is different.  T h e difference i n t h e p o s i t i o n  of t h e m e a n w a v e l e n g t h of t h e l i n e i n R C P a n d L C P l i g h t p r o v i d e s a m e a s u r e m e n t o f the line-of-sight component  o f t h e m a g n e t i c field a v e r a g e d over t h e s t e l l a r d i s c .  This  m e a s u r e m e n t is k n o w n as t h e m e a n l o n g i t u d i n a l field s t r e n g t h a n d is r e p r e s e n t e d b y Bi. F o r m o s t A p s t a r s , Bi a n d i t s v a r i a t i o n w i t h r o t a t i o n o f t h e s t a r are t h e o n l y m a g n e t i c o b s e r v a t i o n s a v a i l a b l e b e c a u s e t h e y are r e l a t i v e l y easy t o o b t a i n ( L a n d s t r e e t , 1993). T h i s is b e c a u s e t h e m e a s u r e m e n t is m o s t sensitive t o m o d e s t fields w i t h s i m p l e s t r u c t u r e s a n d n o a priori i n f o r m a t i o n a b o u t t h e l i n e p r o f i l e is n e e d e d . I n fact, t h e v a r i a t i o n i n B\ w i t h r o t a t i o n p h a s e (j) is p e r i o d i c a n d is g i v e n b y t h e s i m p l e a n a l y t i c e x p r e s s i o n  Bi(4>) where B  p  « OABp (cos/3cosi + s i n s i n  i  cos  0)  (1.2)  is t h e p o l a r field s t r e n g t h ; a s s u m e d t o b e a c e n t r e d d i p o l e , i is t h e a n g l e b e t w e e n  t h e r o t a t i o n a x i s a n d t h e l i n e o f sight, a n d t h e m a g n e t i c p o l e is i n c l i n e d t o t h e r o t a t i o n a x i s o f t h e s t a r b y t h e a n g l e (3. T h i s e x p r e s s i o n m a y b e d e r i v e d f r o m o b l i q u e r o t a t o r m o d e l o f S t i b b s (1950) a s s u m e s a l i m b d a r k e n i n g coefficient of u n i t y .  and  T h e g e o m e t r y o f t h e o b l i q u e r o t a t o r is  s h o w n i n F i g u r e 1.1. A s t h e s t a r r o t a t e s , t h e angle f r o m t h e m a g n e t i c field p o l e t o t h e l i n e o f sight changes.  T h u s , the change i n the aspect of the magnetic  field  leads to a  m o d u l a t i o n o f Bi w i t h t h e r o t a t i o n o f t h e s t a r . I f t h e r e is a s t r o n g field ( > 1 k G ) we m a y also o b t a i n i n f o r m a t i o n a b o u t i t s g e o m e t r y f r o m t h e t r a n s v e r s e field c o m p o n e n t .  The  7r c o m p o n e n t o f a Z e e m a n t r i p l e t w i l l u s u a l l y s a t u r a t e before t h e o c o m p o n e n t s .  The  i n t e g r a t e d l i n e p r o f i l e w i l l t h e n leave a net l i n e a r p o l a r i z a t i o n (e.g., L e r o y et a l . , 1993). F r o m t h i s net l i n e a r p o l a r i z a t i o n one c a n , i n s o m e cases, o b t a i n u n i q u e v a l u e s for i, (3 a n d Bp d e f i n e d i n E q u a t i o n 1.2. T h i s c h a r a c t e r i s t i c o f t h e Z e e m a n c o m p o n e n t s is k n o w n as b r o a d b a n d l i n e a r p o l a r i z a t i o n .  Chapter 1.  5  Introduction  Figure 1.1:  A schematic d i a g r a m showing the oblique rotator m o d e l geometry. T h e r o t a t i o n a x i s a n d d i p o l e m a g n e t i c a x i s are l a b e l e d R a n d  M  r e s p e c t i v e l y . T h e l i n e o f sight t o E a r t h is t o w a r d t h e r i g h t . A n g l e s 4>, j3 a n d i are defined i n S e c t i o n 1.2.2.  T h e r e are o t h e r t e c h n i q u e s for e x t r a c t i n g m a g n e t i c field i n f o r m a t i o n f r o m  spectra.  T h e m o m e n t t e c h n i q u e , for e x a m p l e , fits s p e c t r a l l i n e s m e a s u r e d w i t h different p o l a r i z a t i o n s (i.e., different S t o k e s p a r a m e t e r s ) w i t h l i n e m o m e n t s d e f i n e d b y (A — A o )  n + 1  . Here,  A a n d Ao are t h e i n t e g r a l over t h e l i n e profile a n d t h e c e n t r o i d o f t h e w a v e l e n g t h respect i v e l y (see M a t h y s 1989). T h e m o m e n t s o f different S t o k e s p a r a m e t e r s y i e l d i n f o r m a t i o n a b o u t t h e field g e o m e t r y .  F o r i n s t a n c e , t h e first m o m e n t o f t h e difference b e t w e e n  the  i n t e n s i t y i n L C P a n d R C P (Stokes V ) l i g h t recovers t h e m e a n l o n g i t u d i n a l field ( M a t h y s , 1988).  Chapter 1. Introduction  6  1.3 Diffusion in Ap Stars I n t h e c o n t e x t o f A p stars, t h e s e p a r a t i o n o f e l e m e n t s as a m e a n s t o e x p l a i n t h e o b s e r v e d a b u n d a n c e p a t c h e s w a s first s t u d i e d as e a r l y as t h e l a t e 1960s a n d e a r l y 1970s (e.g., M i c h a u d , 1 9 7 0 ) . I n p r i n c i p l e , a n y element t h a t i s h e a v i e r t h a n t h e s u r r o u n d i n g ( m a i n l y H y d r o g e n ) m i x t u r e w i l l s i n k u n d e r t h e influence o f g r a v i t y . E x c e p t i o n s o c c u r for e l e m e n t s t h a t h a v e a b s o r p t i o n l i n e s at t h e w a v e l e n g t h s n e a r t h e l o c a l flux m a x i m u m . I n these cases, t h e e l e m e n t s m a y b e l e v i t a t e d u p w a r d t o w a r d t h e s t e l l a r surface i f t h e r a d i a t i v e forces are g r e a t e r t h a n t h e g r a v i t a t i o n a l force. O n c e t h e e l e m e n t s r e a c h a n e q u i l i b r i u m p o s i t i o n b e t w e e n g r a v i t a t i o n a l a n d r a d i a t i v e forces t h e y m a y a c c u m u l a t e i n sufficient  amounts  a n d cause a b u n d a n c e a n o m a l i e s at these l o c a t i o n s . R a r e E a r t h e l e m e n t s , s u c h as S r , m a y a c c u m u l a t e a t t h e surface o f A p s t a r s i n t h i s w a y . G e n e r a l r e v i e w s o f d i f f u s i o n t h e o r y may  b e f o u n d i n V a u c l a i r & V a u c l a i r (1982) a n d M i c h a u d & The  Proffitt  (1993).  process of element separation described above is, however, a fragile one. If there  are t u r b u l e n t o r c o n v e c t i v e v e l o c i t i e s i n t h e u p p e r a t m o s p h e r e s o f these s t a r s t h a t e x c e e d t h e d i f f u s i o n v e l o c i t i e s o f a few c m / s , t h e a b u n d a n c e a n o m a l i e s w i l l s i m p l y b e m i x e d a w a y ( M i c h a u d , 1976). W h i l e t h e m a i n sequence A s t a r s d o n o t h a v e l a r g e c o n v e c t i v e zones, t h e A g i a n t s d o . T h i s e x p l a i n s w h y t h e A - t y p e g i a n t s loose t h e i r a b u n d a n c e p e c u l i a r i t i e s . It s h o u l d also b e n o t e d t h a t diffusion t i m e scales a r e o n t h e o r d e r o f « s m a l l fraction of the evolutionary timescale of a near solar-mass star. s e p a r a t i o n m a y o c c u r e a r l y i n t h e e v o l u t i o n o f these s t a r s .  10 yrs; a 6  Thus, chemical  Diffusion theory predicts a  t h e t h i n l a y e r o f a b u n d a n c e a n o m a l i e s a t t h e surface o f these s t a r s . A s n o t e d b y K u r t z & M a r t i n e z (2000), i f the observed abundances of the rare E a r t h elements i n A p stars (over a b u n d a n c e s o f ~ 1 0 s o l a r values) were r e p r e s e n t a t i v e o f t h e i n t e r i o r m e t a l c o n t e n t 4  of these s t a r s , t h e y w o u l d c o n t a i n n e a r l y a l l o f these e l e m e n t s i n t h e U n i v e r s e . S i n c e A p s t a r s represent a p p r o x i m a t e l y o n e i n a h u n d r e d t h o u s a n d stars, t h i s c a n ' t p o s s i b l y b e t h e case. I n o r d e r for t h e diffusion m e c h a n i s m i n these s t a r s t o b e efficient, t h e r e m u s t b e a s t a b i l i z i n g m e c h a n i s m a g a i n s t t u r b u l e n c e as d i s c u s s e d a b o v e .  It i s b e l i e v e d t h a t t h e  Chapter 1.  Introduction  7  m a g n e t i c field p r o v i d e s t h i s s t a b i l i z i n g m e c h a n i s m for these s t a r s . It a l s o h a s t h e a d d e d effect o f c a u s i n g h o r i z o n t a l l y d i s t r i b u t e d a b u n d a n c e s o n t h e s t e l l a r s u r f a c e n e a r r e g i o n s of h o r i z o n t a l m a g n e t i c field ( M i c h a u d et a l . , 1981). T h i s o c c u r s w h e n i o n s o f a n e l e m e n t are l e v i t a t e d t o d e p t h s w h e r e t h e m a g n e t i c field b e c o m e s i m p o r t a n t c o m p a r e d t o t h e r a d i a t i v e a n d g r a v i t a t i o n a l forces.  These ions are t h e n d i s t r i b u t e d b o t h v e r t i c a l l y a n d  h o r i z o n t a l l y a t t h e s t e l l a r surface. M a g n e t i c fields c a n d e - s a t u r a t e s o m e l i n e s o f a n e l e m e n t t h r o u g h t h e Z e e m a n effect. T h i s r e s u l t s f r o m t h e Z e e m a n s p l i t l i n e s b e i n g e x c i t e d at s l i g h t l y different  frequencies  t h a n t h e s a m e l i n e s p r o d u c e d w h e n n o m a g n e t i c field i s p r e s e n t . W h e n a m a g n e t i c field is i n d u c e d , these l i n e s w i l l n o l o n g e r have t h e o p t i m a l w i d t h s a n d energies for r a d i a t i v e forces t o o v e r c o m e g r a v i t y . T h e r e f o r e , s o m e elements w i l l b e c o m e less s u s c e p t i b l e t o r a d i a t i v e a c c e l e r a t i o n s w h e n t h e r e are m a g n e t i c influences. T h e p o l a r i z a t i o n o f t h e Z e e m a n c o m p o n e n t s also affects t h e h o r i z o n t a l a c c e l e r a t i o n o f elements; a d d i n g t o t h e d i s t r i b u t i o n of e l e m e n t s o b s e r v e d for t h e A p s t a r s ( B a b e l & M i c h a u d , 1991, a n d M i c h a u d , 1 9 9 6 ) .  1.4 Stellar Seismology and the Ap Stars R a p i d variability  4  o f s t a r l i g h t w a s first o b s e r v e d i n t h e S u n . L e i g h t o n et a l . (1962)  f o u n d t h a t s p a t i a l l y i n c o h e r e n t w a v e p a t t e r n s e x i s t o n t h e surface o f t h e S u n w i t h p e r i o d s n e a r 5 m i n u t e s . U l r i c h (1970), L e i b a c h e r k, S t e i n (1971) were t h e first t o i n t e r p r e t these o s c i l l a t i o n s as s o u n d waves p r o d u c e d i n t h e s o l a r i n t e r i o r t h a t r e s o n a t e i n a c o u s t i c c a v i t i e s g e n e r a t e d b y changes i n t h e l o c a l s o u n d s p e e d . T h e s e s o u n d waves p r o d u c e s t a n d i n g w a v e p a t t e r n s as t h e y p r o p a g a t e a r o u n d t h e i n t e r i o r o f t h e s u n w i t h i n these a c o u s t i c c a v i t i e s . S i n c e these e a r l y b e g i n n i n g s , m i l l i o n s o f p - m o d e s h a v e b e e n o b s e r v e d i n t h e S u n . T h e n a m e p — mode c o m e s f r o m t h e fact t h a t p r e s s u r e i s t h e r e s t o r i n g force for t h e s o u n d waves t h a t a r e r e s p o n s i b l e for t h e o b s e r v e d o s c i l l a t i o n s . T h e s e p - m o d e s h a v e p r o v e d t o b e a p o w e r f u l t o o l for s t u d y i n g t h e i n t e r n a l p r o p e r t i e s o f t h e S u n t h r o u g h 4  helioseismology.  The term rapid is in comparison to other classic pulsators, such as Cepheid Variables, that can have  oscillation periods of months.  Chapter 1.  Introduction  8  T h e field o f h e l i o s e i s m o l o g y is a k i n t o s t u d y i n g i n t e r n a l p r o p e r t i e s o f t h e E a r t h u s i n g seismology.  A few e x a m p l e s o f successful inferences o n t h e s o l a r i n t e r i o r m a d e  from  helioseismology include a n estimate of the internal r o t a t i o n rate, the d e p t h of the solar c o n v e c t i v e z o n e , c o n s t r a i n t s o n e l e m e n t diffusion, a n d t h e r u n o f s o u n d s p e e d . A recent r e v i e w o f h e l i o s e i s m o l o g y is p r o v i d e d b y C h r i s t e n s e n - D a l s g a a r d (2002). U n t i l t h e l a t e 1970s, o b s e r v a t i o n s o f s t a r s h a d n o t s h o w n p - m o d e s w i t h p e r i o d s a n d f r e q u e n c y s p a c i n g s i m i l a r t o t h o s e o b s e r v e d i n t h e S u n . T h i s c h a n g e d w h e n K u r t z (1978, 1982) d i s c o v e r e d t h e r a p i d l y o s c i l l a t i n g A p s t a r s .  These stars exhibit r a p i d variability  i n t h e i r l i g h t c u r v e s w i t h p e r i o d s b e t w e e n 5 a n d 15 m i n u t e s a n d s e m i - a m p l i t u d e s u n d e r approximately 8 mi/Zzmagnitudes through a Johnson B  filter.  T o d a t e , t h e r e are 32  k n o w n r o A p s t a r s : K u r t z & M a r t i n e z (2000) r e p o r t 31 a n d t h e m o s t recent, H D 12098, is r e p o r t e d b y G i r i s h et a l . (2001).  S o m e of these s t a r s are m u l t i - p e r i o d i c a n d e x h i b i t  frequency p a t t e r n s similar to those observed i n the S u n . T h e best e x a m p l e of this is the star o f t h i s thesis, H R 1217 (see S e c t i o n 1.5).  S o m e d e t a i l e d r e v i e w s o f r o A p s t a r s are  p r o v i d e d b y K u r t z (1990), M a t t h e w s (1991) a n d K u r t z & M a r t i n e z ( 2 0 0 0 ) . I n g e n e r a l , t h e A p s t a r s s h o w t h r e e t y p e s of v a r i a b i l i t y . T h e s e i n c l u d e s p e c t r a l l i n e strength,  m a g n e t i c field s t r e n g t h ,  and photometric (integrated light) variations.  The  p h o t o m e t r i c variations c a n further be s u b - d i v i d e d into l o n g - t e r m ( L T ) variations a n d r a p i d o s c i l l a t i o n s (the r o A p s t a r s ) .  E a c h o f these f o r m s o f v a r i a b i l i t y s e e m t o b e i n -  t e r c o n n e c t e d t h r o u g h t h e c o m p l e x g e o m e t r y o f t h e m a g n e t i c field, a b u n d a n c e p a t t e r n s , s t e l l a r r o t a t i o n a n d i n c l i n a t i o n o f t h e s t a r t o t h e observer. I w i l l b e g i n w i t h a d i s c u s s i o n of n o n r a d i a l o s c i l l a t i o n s a n d t h e i r relevance t o t h e r o A p s t a r s . T h e c o n n e c t i o n b e t w e e n the magnetic, spectral, L T a n d r a p i d variations a n d the geometry that links t h e m w i l l then be discussed.  1.4.1  Nonradial Oscillations  N o n r a d i a l o s c i l l a t i o n s are d i s c u s s e d i n d e t a i l b y U n n o et a l . ( 1 9 8 9 ) . A b r i e f i n t r o d u c t i o n t o areas o f p a r t i c u l a r r e l e v a n c e t o t h e r o A p s t a r s is p r e s e n t e d b e l o w .  Chapter 1.  Introduction  9  T o first o r d e r , a s l o w l y r o t a t i n g , n o n - m a g n e t i c s t a r i s s p h e r i c a l l y s y m m e t r i c . P e r t u r bations t o this spherical symmetry c a n be described b y spherical harmonics,  Y {9,(j)). m  l  Here 9 a n d 0 are t h e u s u a l angular coordinates o f t h e spherical c o o r d i n a t e system. T h e s t u d y o f o s c i l l a t i o n s i n s t a r s c a n b e d e s c r i b e d as s m a l l p e r t u r b a t i o n s t o p h y s i c a l v a r i a b l e s w i t h i n t h e s t a r s u c h as p r e s s u r e o r d e n s i t y o r i n r a d i a l d i s p l a c e m e n t s £ . I n g e n e r a l , w e r  m a y e x p a n d r a d i a l d i s p l a c e m e n t s i n t e r m s o f s p h e r i c a l h a r m o n i c s as  (1.3) n,£,m  where the n is the r a d i a l order of the oscillations a n d corresponds t o the n u m b e r of r a d i a l n o d e s (zeroes) i n t h e e i g e n f u n c t i o n £ e, r>n  o e is the eigenfrequency o f t h e oscillations, a n d n  t is time. T h e spherical harmonic provides the angular description of the modes t h r o u g h the i n d i c e s £ a n d m.  T h e i n d e x £ i s c o m m o n l y c a l l e d t h e degree o f t h e m o d e a n d d i v i d e s  t h e surface i n t o r e g i o n s o s c i l l a t i n g i n o p p o s i t e p h a s e .  I f £ = 0, t h e o s c i l l a t i o n m o d e i s  r a d i a l ; o r more precisely, spherically s y m m e t r i c . T h e i n d e x m is t h e a z i m u t h a l order of t h e m o d e a n d r e p r e s e n t s t h e n u m b e r o f l o n g i t u d i n a l n o d e s o n a sphere.  Physically, the  order m c a n be related t o t h e phase velocity o f a wave t h r o u g h  (1.4)  T h e sign of m indicates the direction i n w h i c h the wave travels.  A standing wave i n  the l o n g i t u d i n a l d i r e c t i o n c a n be formed b y t h e s u p e r p o s i t i o n o f two waves t r a v e l i n g i n o p p o s i t e d i r e c t i o n s s i n c e t h e b a c k g r o u n d s t a t e i s , t o first o r d e r , s p h e r i c a l l y s y m m e t r i c . In terms of q u a n t i z a t i o n of t h e indices, m m a y take integer values f r o m —£ t o + £ a n d £ i s a n o n n e g a t i v e integer. I f m = 0 a l l n o d a l lines o n a s p h e r e a r e l i n e s o f l a t i t u d e , w h i l e i f m = £, t h e y a r e a l l l i n e s o f l o n g i t u d e . T h e s e a r e k n o w n as z o n a l a n d s e c t o r a l m o d e s r e s p e c t i v e l y . I f m takes o n v a l u e s b e t w e e n these e x t r e m e s , t h e m o d e s a r e k n o w n as t e s s e r a l m o d e s a n d o f t h e £ n o d a l lines, £— \m\ o f w h i c h a r e l i n e s o f l a t i t u d e . E x a m p l e s are d r a w n s c h e m a t i c a l l y i n F i g u r e 1.2 for t h e cases £ = 3 a n d m = 0 t o 3. N o t e t h a t 5  5  Image taken from the Delta Scuti Network homepage: http://www.deltascuti.net/DeltaScutiWeb/indexl .html  Chapter 1.  Introduction  H * f »3  |m|»2  Figure 1.2: A schematic representation of different spherical harmonic modes. At the top left £ = 3 and ra = 0 and at the lower right £ = 3 and ra = 3. The top right and lower left modes represent £ = 3 and ra = 1 and 2, respectively. The blue (-) and yellow (+) represent regions that oscillate opposite in phase. Note there are £ — |ra| lines of latitude. This image was taken from the Delta Scuti Network homepage. both  £ ,n£ r  and o e in Equation 1 . 3 are independent of ra. This results in a degeneracy n  that may be lifted due to perturbations from slow rotation or magnetic field. For example slow, uniform, rotation causes frequencies to split according to the equation <im  = -rn(l-C ,t)n n  (1.5)  where il is the angular rotation frequency of the star measured in the rotating frame  Chapter 1. Introduction and  C e Ut  11  is k n o w n as t h e L e d o u x c o n s t a n t  ( L e d o u x , 1951).  T h e L e d o u x constant  is  m o d e l d e p e n d e n t a n d is o f t h e o r d e r l / £ . N o w e a c h f r e q u e n c y m a y b e d e s c r i b e d i n a n r  i n e r t i a l reference f r a m e b y t h e u n p e r t u r b e d f r e q u e n c y  o  f r o m E q u a t i o n 1.3 p l u s  o °^ r  im  f r o m E q u a t i o n 1.5. I n t h e case o f t h e r o A p stars, t h e s t r o n g m a g n e t i c field p r o d u c e s p r e s s u r e s t h a t are g r e a t e r t h a n t h e l o c a l gas p r e s s u r e near t h e s t e l l a r surface.  A traditional perturbation  a p p r o a c h i n d e s c r i b i n g t h e effect of a m a g n e t i c field o n a p u l s a t i o n m o d e is n o l o n g e r valid.  I n d i v i d u a l o s c i l l a t i o n m o d e s c a n n o t b e d e s c r i b e d b y a s i n g l e set o f n , £, a n d  m  values. Instead, each m o d e must be expanded i n a n infinite s u m of spherical h a r m o n i c s t o a c c o u n t for t h e c o u p l i n g b e t w e e n t h e m a g n e t i c field a n d p u l s a t i o n g e o m e t r y  (e.g.,  D z i e m b o w s k i k G o o d e , 1996, B i g o t et a l , 2 0 0 0 , a n d C u n h a & G o u g h , 2 0 0 0 ) . T h e effects of t h e s t r o n g m a g n e t i c field o n t h e  normal m o d e s (i.e., t h o s e c a l c u l a t e d i n t h e case o f  n o m a g n e t i c field o r r o t a t i o n ) w i l l b e d i s c u s s e d i n S e c t i o n 3.3 i n t e r m s o f a v a r i a t i o n a l a p p r o a c h u s e d b y C u n h a &; G o u g h (2000).  1.4.2  High-Overtone Pulsation and Frequency Spacing  O s c i l l a t i o n frequencies o f a few m i Z Z z H e r t z ( m H z ) are c o n s i s t e n t w i t h p - m o d e frequencies 6  i n non-degenerate stars described b y large n values. C u r r e n t l y , p h o t o m e t r i c observations of p u l s a t i n g s t a r s y i e l d o n l y t h e c o n t r i b u t i o n s o f I m o d e s w i t h s m a l l v a l u e s . because the c o n t r i b u t i o n from modes w i t h £ >  T h i s is  3 gets s m o o t h e d o u t w h e n t h e l i g h t  f r o m a s t e l l a r d i s c is i n t e g r a t e d . T h e h i g h o v e r t o n e s o f s u c h l o w - d e g r e e p u l s a t i o n s h a v e a s y m p t o t i c b e h a v i o r t h a t m a y b e e x p l o i t e d for s t a r s l i k e t h e r o A p s t a r s . If a s p h e r i c a l l y s y m m e t r i c s t a r p u l s a t e s a d i a b a t i c a l l y w i t h p - m o d e s o f n  £, we m a y  use t h e a s y m p t o t i c t h e o r y of T a s s o u l (1980, 1990) t o d e s c r i b e t h e frequencies b y  i V « Au(n + £/2 + e) + T  (1.6)  w h e r e e is a c o n s t a n t , a n d T is a s t r u c t u r a l l y d e p e n d e n t  q u a n t i t y t h a t is a n o r d e r o f  m a g n i t u d e s m a l l e r t h a n t h e first t e r m . 6  It is c o n v e n i e n t t o i n t r o d u c e t h e f r e q u e n c y  A period of 5 minutes corresponds to a frequency of 3.3 mHz  v  Chapter 1. Introduction  12  since i t i s u s u a l l y f o u n d i n t h e o b s e r v a t i o n a l l i t e r a t u r e o n p u l s a t i n g s t a r s .  T h e relation  b e t w e e n t h e a n g u l a r f r e q u e n c y d e f i n e d i n e q u a t i o n 1.3 a n d t h a t o f E q u a t i o n 1.6 i s s i m p l y  u = o/(2n). The  f a c t o r Au i n E q u a t i o n 1.6 w i l l b e u s e d e x t e n s i v e l y i n t h i s t h e s i s . It i s r e l a t e d t o  t h e t i m e i t t a k e s s o u n d t o cross t h e d i a m e t e r o f t h e s t a r a n d i s u s u a l l y r e f e r r e d t o as t h e large spacing. T h i s frequency is w r i t t e n i n terms o f the s o u n d speed c t h r o u g h , - i  Au = w h e r e R is t h e r a d i u s o f t h e s t a r .  2 [ . Jo  dr/c  (1.7)  Since this q u a n t i t y is a p p r o x i m a t e l y related t o t h e  m e a n density of the star, it m a y be w r i t t e n i n terms of the stellar mass M , a n d radius via the relation Au = (0.205 ± 0.011) f where G is the g r a v i t a t i o n a l constant.  j  Hz  (1.8)  T h e n u m e r i c a l f a c t o r arises f r o m m o d e l c a l c u l a -  t i o n s p e r f o r m e d b y G a b r i e l et a l . (1985). M a t t h e w s et a l . (1999) r e w r o t e E q u a t i o n 1.8 i n t e r m s o f t h e s t a r s effective t e m p e r a t u r e T ff, a n d l u m i n o s i t y L , t o g i v e e  Au = (6.64 ± 0.36) x 10- M T^ L~ ie  1/2  Hz  3/4  f  (1.9)  w i t h L a n d M i n s o l a r u n i t s a n d T / / i n degrees K e l v i n . e  It s h o u l d b e n o t e d t h a t t h e r e is a n a m b i g u i t y i n t h e i n t e r p r e t a t i o n o f t h e o b s e r v e d f r e q u e n c y s p a c i n g d e p e n d i n g o n t h e £ v a l u e s . I f t h e m o d e s h a v e o r d e r s t h a t differ f r o m e a c h o t h e r b y o n e a n d £ v a l u e s t h a t a l t e r n a t e b e t w e e n e v e n and o d d v a l u e s , E q u a t i o n 1.6 y i e l d s a f r e q u e n c y s p a c i n g o f a p p r o x i m a t e l y Au/2.  If however, consecutive £ values are  e i t h e r a l l e v e n or o d d , t h e frequencies w i l l b e s p a c e d b y a b o u t Au; t h e l a r g e s p a c i n g . T h e r e i s also a s e c o n d o r d e r s p a c i n g defined b y 8^  = u(n,£)  — u(n — 1,£ + 2) . B y  w r i t i n g T f r o m E q u a t i o n 1.6 i n t e r m s o f t h e s o u n d s p e e d , t h i s s p a c i n g y i e l d s  S^ocAu The  f -^-dr R  Jo  r  dr  details l e a d i n g t o this e q u a t i o n m a y be found i n the w o r k of T a s s o u l (1990).  (1.10)  Chapter 1.  Introduction  13  T h i s s p a c i n g i s referred t o as t h e s m a l l s p a c i n g .  Close t o t h e center o f a star, t h e  l e a d i n g c o n t r i b u t i o n t o E q u a t i o n 1.10 c o m e s f r o m t h e 1 / r f a c t o r i n t h e i n t e g r a n d .  Thus,  t h i s s p a c i n g m a y b e u s e d t o infer p r o p e r t i e s o f t h e i n t e r i o r s o f s t a r s (e.g., P r o v o s t , 1984, a n d G u e n t h e r & D e m a r q u e 1997). F o r t h e r o A p s t a r s , however, t h e d i a g n o s t i c p o w e r o f m a y b e l i m i t e d b e c a u s e t h e m a g n e t i c p e r t u r b a t i o n s t o t h e frequencies m a y b e o f t h e s a m e o r d e r , o r l a r g e r t h a n , t h e s m a l l s p a c i n g ( D z i e m b o w s k i & G o o d e , 1996).  1.4.3  The Oblique Pulsator Model  T h e L T v a r i a b i l i t y m e n t i o n e d i n S e c t i o n 1.4 is s h o w n for H R 1217 a l o n g w i t h t h e v a r i a t i o n of i t s m a g n e t i c field i n F i g u r e 1.3. It is c l e a r l y s h o w n t h a t t h e m e a n l i g h t v a r i a t i o n o f the star is i n phase w i t h t h e magnetic field.  T h e m o d u l a t i o n o f t h e m a g n e t i c field i s  d e s c r i b e d w e l l b y t h e o b l i q u e rotator m o d e l o f S t i b b s (see S e c t i o n 1.2.2). K u r t z  (1982)  suggested t h a t t h e L T v a r i a b i l i t y a n d r a p i d variations m a y b e m o d u l a t e d i n a s i m i l a r w a y t h r o u g h w h a t h e c a l l e d t h e o b l i q u e pulsator m o d e l .  I n this model, the pulsation  axis of the star is aligned w i t h t h e magnetic axis o f the star; b o t h of w h i c h are i n c l i n e d to t h e r o t a t i o n axis. A s t h e star rotates t h e aspect o f t h e p u l s a t i o n a n d magnetic axis v a r y ; l e a d i n g t o m o d u l a t i o n w i t h t h e r o t a t i o n o f t h e s t a r . T h e g e o m e t r y i s t h e s a m e as i n F i g u r e 1.1, b u t n o w t h e m a g n e t i c a x i s a n d p u l s a t i o n a x i s a r e o n e i n t h e s a m e . C o n s i d e r a n a x i s y m m e t r i c (m = 0) p u l s a t i o n m o d e w i t h f r e q u e n c y o. K u r t z  (1982)  showed that i n t h e geometry of the oblique pulsator, t h e l u m i n o s i t y v a r i a t i o n w i t h rotat i o n m a y b e e x p r e s s e d as  oc pp(cosa)  cos [at + <p ]  (1-H)  p  Ii  w h e r e P™ i s t h e a s s o c i a t e d L e g e n d r e p o l y n o m i a l , <p i s a n a r b i t r a r y p h a s e , a n d t h e a n g l e p  a is t h e v a r y i n g a n g l e b e t w e e n t h e m a g n e t i c field a x i s a n d t h e l i n e o f s i g h t . T h i s a n g l e h a s t h e s a m e f u n c t i o n a l d e p e n d e n c e as i n t h e o b l i q u e r o t a t o r m o d e l o f E q u a t i o n 1.2. I n t h e case o f a d i p o l e m o d e (£ = 1), t h e L e g e n d r e p o l y n o m i a l is e q u a l t o cos a a n d E q u a t i o n 1.11 m a y b e e x p a n d e d as ^  « A cos (at + <p ) + Ai [cos ({a + Q.}t + </? ) + cos ({a - £l}t + <p )] 0  p  p  p  (1-12)  Chapter 1. Introduction  14  F i g u r e 1 . 3 : T h e v a r i a t i o n i n t h e m e a n l i g h t o f H R 1217 t h r o u g h t h e B f i l t e r as a function of the r o t a t i o n phase of the star (upper panel). T h e m a g n e t i c field v a r i a t i o n defined b y B\ i n S e c t i o n 1.2.2 as a f u n c t i o n o f r o t a t i o n phase is s h o w n i n the lower panel. P u l s a t i o n d a t a a n d m a g n e t i c field d a t a were t a k e n f r o m K u r t z (1982) a n d P r e s t o n  (1972)  r e s p e c t i v e l y . T h i s figure w a s o b t a i n e d f r o m M a t t h e w s ( 1 9 9 1 ) .  T h e a m p l i t u d e s a r e g i v e n b y AQ = c o s i c o s / ? a n d A\ — ( s i n i s i n / ? ) / 2 . F r o m t h e a b o v e r e l a t i o n s , w e see t h a t a s i n g l e d i p o l e m o d e i s s p l i t i n t o a t r i p l e t exactly spaced b y the r o t a t i o n p e r i o d of the star. I n general, this m o d e l predicts t h a t a m o d e w i t h degree £ i s s p l i t i n t o 2 ^ + 1 frequencies. s t r u c t u r e i n t h e i r frequencies.  T h e r o A p s t a r s d o e x h i b i t s u c h fine  Chapter 1.  Introduction  15  F r e q u e n c i e s m a y also b e s p l i t b y r o t a t i o n t h r o u g h E q u a t i o n 1.5. T h e s e frequencies are not e x a c t l y spaced b y the r o t a t i o n frequency of the star because of the L e d o u x constant. C a l c u l a t i o n s o f t h e L e d o u x c o n s t a n t for A s t a r m o d e l s b y S h i b a h a s h i & T a k a t a (1993) suggest a v a l u e o f C (  «  n>  10~ . 3  A s t r i n g e n t o b s e r v a t i o n a l c o n s t r a i n t is p r o v i d e d b y  K u r t z et a l . (1997) f r o m t h e f r e q u e n c y s p a c i n g o f t h e r o A p s t a r H R 3 8 3 1 . T h e y p l a c e a n upper limit of 1 0 ~ o n C f 6  n  at t h e 3 a confidence l e v e l . T h e c o i n c i d e n c e o f m a g n e t i c a n d  p u l s a t i o n p h a s e a l o n g w i t h t h e a b o v e c o n s t r a i n t are s t r o n g i n d i c a t i o n s t h a t t h e o b s e r v e d fine s t r u c t u r e i n t h e f r e q u e n c y s p e c t r u m of t h e r o A p s t a r s is n o t t h e r e s u l t o f r o t a t i o n a l l y perturbed m-modes. A s h o r t c o m i n g of t h i s s i m p l e m o d e l p u t f o r t h b y K u r t z (1982) is i n t h e a m p l i t u d e a s y m m e t r y o b s e r v e d b e t w e e n s p l i t frequencies. F r o m E q u a t i o n 1.12, w e see t h a t t h e s p l i t frequencies s h o u l d h a v e t h e s a m e a m p l i t u d e . T h i s is n o t t h e case for t h e A p s t a r s . T h i s p r o b l e m is a v o i d e d i f one t a k e s i n t o a c c o u n t t h e effects o f b o t h r o t a t i o n a n d m a g n e t i c field o n t h e frequencies t h r o u g h t h e C o r i o l i s a n d L o r e n t z forces.  T h i s correction by  D z i e m b o w s k i & G o o d e (1985) c o r r e c t l y p r e d i c t s b o t h t h e r o t a t i o n a l l y s p l i t frequencies and the amplitude asymmetries. T h e m o s t recent c o n t r i b u t i o n t o t h e o b l i q u e r o t a t o r m o d e l c o m e s f r o m B i g o t & D z i e m b o w s k i ( 2 0 0 2 ) . T h e y u s e d a n o n - p e r t u r b a t i v e a p p r o a c h t o s h o w t h a t t h e c e n t r i f u g a l force is i m p o r t a n t i n d e t e r m i n i n g t h e f r e q u e n c y shifts w h i l e t h e C o r i o l i s force i s d o m i n a n t i n determining the amplitude asymmetries.  These authors also show t h a t the p u l s a t i o n ,  r o t a t i o n , a n d m a g n e t i c a x i s are a l l i n c l i n e d t o e a c h o t h e r .  1.5  The roAp Star H R 1217  T h e focus o f t h e s i s is t h e A p s t a r H R 1217; a . k . a H D 24712 or D O E r i . T h i s w a s one of t h e first A p s t a r s t o b e i d e n t i f i e d as a r o A p s t a r ( K u r t z , 1982) a n d h a s s i n c e b e c o m e one o f t h e m o s t s t u d i e d .  Recently, h i g h - q u a l i t y d a t a has p r o v i d e d i n f o r m a t i o n about  t h e a b u n d a n c e , m a g n e t i c a n d p h o t o m e t r i c c h a r a c t e r i s t i c s o f H R 1217. T h e m o s t recent p h o t o m e t r i c d a t a is p r e s e n t e d i n C h a p t e r 2 o f t h i s w o r k . A r e v i e w o f s o m e o f t h e o t h e r  Chapter 1.  Introduction  16  recent observations of HR 1217 is presented in this section. As introduced in Section 1.1, the spectral anomalies in Ap stars are believed to be the result of abundance enhancements on the surface of the star. The first thorough abundance analysis HR 1217 was performed by Ryabchikova et al. (1997). Their results are consistent with the idea that the abundance enhancements on the surfaces of these stars are patchy. In particular they find the mean chemical abundances vary with the magnetic and rotational phase of the star. Since the abundances are believed to be concentrated near the magnetic poles, this spectral variation may be described by the oblique rotator model discussed in Section 1.2.2. When compared to the Sun, the rare earth elements are the most overabundant. With the exception of Mg, they also show the largest change over the ~ 12 day rotation period of this star. The iron peak elements, on the other hand, are under abundant. The exception in this case being in Co, which is overabundant. Using the techniques outlined in Section 1.2, Bagnulo et al. (1995) were able to model the magnetic field geometry for HR 1217. Their results yield values of 137°, 150°, and 3.9 kG for i, (3 and B respectively. Estimated uncertainties on the above angular p  measurements are « 2 — 3° and the uncertainly of the polar field strength is « 5%. The mean longitudinalfield,Bi, for HR 1217 varies between « 0.5 and 1.5 kG (Preston, 1972). This is illustrated in the lower panel of Figure 1.3. Agreement between theory and observations of the magnetic field for Ap stars is currently among the best in the case of HR 1217 (Bagnulo et al, 1995, and Bagnulo 1998). The frequency spacing of HR 1217 is a perfect illustration of the difficulties that arise in identifying oscillation modes. In 1986, a global observation campaign headed by Don Kurtz and Jaymie Matthews collected photometric data for HR 1217 (Kurtz et al., 1989). Their goal was to achieve as much continuous coverage of the star as possible so that gaps in their data would not affect their frequency analysis. They achieved a 29% duty cycle with 325 hrs of data spanning a 46-day period. A schematic diagram of the principal frequencies that they identified is found in Figure 1.4. As discussed in Section 1.4.2, there exists an ambiguity in identifying A f from the  Chapter 1.  Introduction  17 l  i  r-  [/LtHz] 0.8  68.04  67.91  68.14  84.88  •§ 0 . 6  hv/2  [/xHz]  33.38  34.66  0.4  33.25  34.89  49.99  0.2 _i  2.65  i  ii  J  2.7  i  i  _J  2.75  i i i  2.8  v(mHz)  F i g u r e 1 . 4 : A b o v e is a s c h e m a t i c d i a g r a m o f t h e frequencies f o u n d f r o m t h e 1986 o b s e r v a t i o n s o f H R 1 2 1 7 ( K u r t z et a l , 1989).  T h e axis are  a m p l i t u d e i n rmZZzmagnitudes a n d frequencies i n m H z .  T h etwo  p o s s i b l e v a l u e s o f e i t h e r Au o f « 68 / x H z (blue) o r Au/2 o f « 34 / x H z ( b l a c k ) a r e s h o w n . N o t e t h e strange s p a c i n g o f 4 9 . 9 9 a n d 8 4 . 8 8 / x H z i n d i c a t i n g a p o s s i b l e missing  observed frequency spacing.  f r e q u e n c y a t t h e r e d l a b e l ?.  B o t h Au a n d Au/2  a r e p r e s e n t e d i n F i g u r e 1.4.  If the  m o d e s "are a l t e r n a t i n g b e t w e e n e v e n a n d o d d £ v a l u e s w e w o u l d e x p e c t t o see a Au o f « 68 ^xHz. I n fact, t h e a l t e r n a t i n g s p a c i n g o f 33 a n d 34 ^tHz a d d t o t h i s c o n c l u s i o n . I f t h e  Chapter 1.  Introduction  18  m o d e s were a l l e v e n o r o d d , we w o u l d e x p e c t t h a t t h e s p a c i n g b e t w e e n a d j a c e n t  modes  i n F i g u r e 1.4 w o u l d r e m a i n t h e s a m e . T h e p o s s i b i l i t y t h a t t h e m o d e s are a l l o f t h e s a m e degree c a n n o t b e r u l e d o u t . W h a t is n o t s h o w n i n F i g u r e 1.4 is t h e fine s t r u c t u r e s p a c i n g a r o u n d a l l b u t t h e first f r e q u e n c y ( r e a d f r o m t h e left). E a c h o f t h e frequencies, 2 t h r o u g h 6, are a c t u a l l y t r i p l e t s w i t h f r e q u e n c y s p a c i n g o f a p p r o x i m a t e l y 0.9 / / H z ; t h e r o t a t i o n a l f r e q u e n c y o f t h e s t a r . T h e o r i g i n o f these s p l i t t i n g s was d i s c u s s e d i n S e c t i o n 1.4.3.  Suffice  i t t o say, t h e u n e x p e c t e d f r e q u e n c y s p a c i n g b e t w e e n t h e l a s t t w o frequencies i n F i g u r e 1.4 does n o t h e l p d e t e r m i n e i f we are o b s e r v i n g t h e l a r g e s p a c i n g o r h a l f o f t h e l a r g e s p a c i n g . It w a s o n l y i n t h e p a s t few years; a l m o s t a decade after K u r t z et a l . (1989) r e l e a s e d t h e i r r e s u l t s , t h a t t h e s p a c i n g c o n t r o v e r s y s e e m e d t o b e r e s o l v e d . U s i n g E q u a t i o n 1.9, M a t t h e w s et a l . (1999) were a b l e t o c a l c u l a t e a p a r a l l a x for H R 1217 b a s e d o n a n i n f e r r e d l a r g e s p a c i n g of 68 / i H z . T h e i r p r e d i c t e d p a r a l l a x o f IT = 19.23 ± 0.54 mas t o b e c o n s i s t e n t w i t h t h e recent Hipparcos Av/2  o f 34 uHz  was shown  p a r a l l a x o f IT = 20.41 ± 0.84 mas.  seems t o h a v e b e e n r u l e d o u t b y a n o b s e r v a t i o n m a d e  Thus, a  independently  from asteroseismic analysis. A n o t h e r recent success i n t h e i n t e r p r e t a t i o n o f t h e f r e q u e n c y s p a c i n g o f H R 1217 w a s p r o v i d e d b y C u n h a (2001). S h e p r e d i c t e d t h a t m a g n e t i c d a m p i n g c o u l d b e t h e cause o f t h e m i s s i n g f r e q u e n c y i n t h e 1986 d a t a set. S h e also s h o w e d t h a t s o m e frequencies m a y b e s h i f t e d b y a p p r o x i m a t e l y 10-20 / / H z because o f m a g n e t i c field effects ( C u n h a & G o u g h 2000, a n d C u n h a 2 0 0 1 ) . I n 2000, H R 1217 w a s s e l e c t e d t o b e o b s e r v e d i n a n o t h e r g l o b a l c a m p a i g n (see C h a p t e r 2 ) . A p r e l i m i n a r y d a t a r e d u c t i o n for t h i s d a t a set d i d find a n e w f r e q u e n c y at a p p r o x i m a t e l y 2.79 m H z ( K u r t z et a l . , 2002); a d d i n g f u r t h e r e v i d e n c e t h a t t h e m a g n e t i c field a n d t h e o s c i l l a t i o n s i n r o A p s t a r s are i n t e r c o n n e c t e d .  1.6  A n Overview of this Thesis  P r o g r e s s i n t h e s t u d y o f t h e A p a n d r o A p s t a r s h a s b e e n s t e a d i l y i n c r e a s i n g over t h e p a s t few y e a r s w i t h t h e d e v e l o p m e n t o f n e w o b s e r v a t i o n a l a n d t h e o r e t i c a l t o o l s . I n t h i s t h e s i s I a t t e m p t t o t i e t o g e t h e r t h e m o s t recent p h o t o m e t r i c o b s e r v a t i o n s o f t h e r o A p s t a r H R  Chapter 1.  Introduction  19  1217 w i t h t h e l a t e s t g r i d o f A s t a r e v o l u t i o n a r y a n d p u l s a t i o n m o d e l s . I n C h a p t e r 2 I outline the r e d u c t i o n a n d frequency analysis of the p h o t o m e t r i c d a t a o n H R 1217 o b t a i n e d b y t h e W h o l e E a r t h T e l e s c o p e c o l l a b o r a t i o n i n l a t e 2 0 0 0 . A n i n t r o d u c t i o n t o t h e g l o b a l o b s e r v a t i o n c o n c e p t a n d d a t a a c q u i s i t i o n is p r e s e n t e d i n S e c t i o n s 2.1 a n d 2.2, r e s p e c t i v e l y . T h e r e d u c t i o n p r o c e d u r e s a p p l i e d t o t h e r a w t i m e - s e r i e s d a t a a n d t h e f r e q u e n c y a n a l y s i s f o l l o w s i n S e c t i o n s 2.3 a n d 2.4. T h e r e s u l t s o f t h e a n a l y s i s i s t h e n p r e s e n t e d i n S e c t i o n 2.5 a n d c o m p a r e d t o r e s u l t s o b t a i n e d u s i n g a v a r i e t y o f d a t a w e i g h t i n g schemes i n S e c t i o n 2.6. T h e o r e t i c a l e v o l u t i o n a r y a n d p u l s a t i o n m o d e l s are p r e s e n t e d i n C h a p t e r 3. A l t h o u g h t h e m a i n f o c u s o f t h i s t h e s i s is H R 1217, t h e e v o l u t i o n a r y m o d e l s c a l c u l a t e d for t h i s w o r k cover a l a r g e e n o u g h p a r a m e t e r space t o b e r e l e v a n t t o o t h e r r o A p s t a r s .  Specific  i m p r o v e m e n t s t o t h e p u l s a t i o n m o d e l s (see S e c t i o n s 3.2 t h r o u g h 3.3.1) i n c l u d e b o t h n o n a d i a b a t i c ( e n e r g y g a i n s a n d loses t h r o u g h r a d i a t i v e processes) a n d m a g n e t i c effects. A f i n a l d i s c u s s i o n c o m p a r i n g t h e r e s u l t s o f t h e d a t a a n a l y s i s a n d t h e s t e l l a r m o d e l s is presented i n the final chapter of this work.  20  Chapter 2  Data & Analysis 2.1  The Whole Earth Telescope  T h e W h o l e E a r t h T e l e s c o p e ( W E T ) is a c o l l a b o r a t i o n o f a s t r o n o m e r s a n d f a c i l i t i e s f r o m a r o u n d t h e w o r l d w h o s e c o l l e c t i v e g o a l is t o o b t a i n h i g h - q u a l i t y , c o n t i n u o u s , t i m e - s e r i e s p h o t o m e t r y o f v a r i a b l e stars.  O r i g i n a l l y , W E T was organized to s t u d y variable, degen-  e r a t e s t a r s . Its goals were e x p a n d e d t o i n c l u d e t y p e s o f v a r i a b l e s s u c h as 5 S c u t i , r o A p , c a t a c l y s m i c s , a n d s u b - d w a r f B stars. T h e m a i n a d v a n t a g e o f a g l o b a l c a m p a i g n over s i n g l e - s i t e o b s e r v a t i o n s i s t h e r e d u c t i o n of d a t a g a p s i n a g i v e n set o f t i m e series m e a s u r e m e n t s .  S u c h g a p s are c o m m o n w i t h  s i n g l e - s i t e o b s e r v a t i o n s b e c a u s e a n y p a r t i c u l a r s t a r is o n l y v i s i b l e for a c e r t a i n f r a c t i o n of the night.  I n a g l o b a l c a m p a i g n , o b s e r v a t o r i e s are d i s t r i b u t e d i n l o n g i t u d e , so o n e  s i t e c a n s t a r t o b s e r v a t i o n s w h e n a n o t h e r site h a s f i n i s h e d (see F i g u r e 2.1). I n p r i n c i p l e , c o n t i n u o u s coverage of a c h o s e n t a r g e t c a n b e o b t a i n e d .  O f course, w e a t h e r  and/or  e q u i p m e n t p r o b l e m s m a y a l s o l e a d t o a loss o f d a t a . T h e effect o f t h e s e m i s s i n g d a t a i s seen as a l i a s i n g i n f r e q u e n c y a n a l y s i s of v a r i a b l e stars. T h e g a p s p r o d u c e a d d i t i o n a l p e a k s i n the F o u r i e r s p e c t r u m of the data, m a k i n g identification of a real oscillation frequency difficult. T h e m o r e f a c i l i t i e s t h a t p a r t i c i p a t e i n a W E T r u n , t h e g r e a t e r t h e p o s s i b i l i t y for l o n g e r o b s e r v i n g t i m e o n a t a r g e t . T h i s increases t h e f r e q u e n c y r e s o l u t i o n o f t h e  data  a n d h e l p s i d e n t i f y fine s t r u c t u r e i n t h e F o u r i e r s p e c t r u m . T h i s fine s t r u c t u r e is c r u c i a l for m o d e i d e n t i f i c a t i o n , r o t a t i o n a l i n f o r m a t i o n , a n d i n t h e case o f t h e r o A p stars, i n f o r m a t i o n a b o u t t h e m a g n e t i c field. F o r r e v i e w s o f W E T see W i n g e t (1993), K a w a l e r et a l . (1995)  Chapter 2. Data & Analysis  21  and references therein.  2.2  Observing H R 1217 with W E T  High-speed photometry is the preferred technique for observing r o A p stars. T h i s technique involves a continual monitoring of the star, w i t h no observations of comparison stars. B y doing this, the observations are non-differential. A constant change between observing a target star and a comparison star results i n a loss of d a t a on the rapid light variations exhibited by r o A p stars. T h e use of C C D s for differentially observing r o A p stars is also problematic because it is usually difficult to find a bright comparison star i n a small C C D field. Continuous viewing of a target star relies heavily on precise auto-guiding and properly balanced instruments. Otherwise, a star that is centred i n the diaphragm can drift; causing either a loss of light or light contamination from nearby stars. If such drifts are periodic, spurious frequencies may appear i n the data. Since H R 1217 is a bright star; w i t h a magnitude of B « 6.3, background light is not a major problem. Thus, isolating the star i n a diaphragm w i t h a diameter less than 20 arcseconds w i l l not significantly reduce the ratio of sky-light to star-light. T h e principal source of noise i n the frequency range of interest is scintillation (short-period sky transparency variations). T h e rapid oscillations of r o A p stars; w i t h periods of approximately 5 minutes, are not affected by sky variations that occur w i t h periods of a half-hour or more.  A good review of the general technique for observing r o A p stars w i t h W E T is  provided by K u r t z k M a r t i n e z (2000). In late 2000, the W h o l e E a r t h Telescope launched its twentieth extended  coverage  campaign, X C O V 2 0 . H R 1217 was included as one of the primary targets of this cam1  paign, which ran from Nov. 6 to Dec. 11. D u r i n g X C O V 2 0 , a variety of telescopes and instruments were used. T h e telescope apertures ranged from 0.6 m to 2.1 m and instruments included one-, two- and three-channel photo-electric photometers.  In particular,  information on XCOV20 may be found on the World Wide Web at: http://wet.iitap.iastate.edu/xcov20/  Chapter 2. Data & Analysis observations at C T I O  2  22  used a two-channel photometer, while observations at B A O , 3  N a i n i T a l , Teide , M a u n a K e a , and M c D o n a l d 4  5  6  7  used three-channel  photometers.  T h e single-channel photometers were used at S A A O , P e r t h , and S S O . E a c h of 8  9  1 0  these sites are shown o n the map i n Figure 2.1. T h e three-channel photometers and sky monitoring procedures are described by K l e i n m a n et a l . (1996). Observations w i t h oneand two-channel photometers sampled the sky a number of times for each run, while the three-channel photometers were able to continuously monitor the sky. E a c h site observed the target using a Johnson B filter w i t h 10 sec integration times. Since H R 1217 is a bright star, larger telescopes required neutral density filters to keep count rates below 10 s 6  - 1  i n order to avoid saturating, or damaging, the phototubes.  Photoelectric pho-  tometers currently provide a higher level of precision than C C D photometers when count rates are this high. Comparison stars were not specified for the multi-channel photometric observations. T h e selection of these stars was at the discretion of each individual observer. A complete log for a l l H R 1217 observations can be found i n Table A . l  1 1  of A p p e n d i x  A . T h i s log includes the observatories that participated, the dates of participation and the telescope that was used.  K u r t z et a l . (2003) review the X C O V 2 0 observing r u n  w i t h emphasis o n H R 1217. T h e need for high-precision d a t a results i n some runs being selected for the final data analysis while others had to be discarded. R u n s were not used if they exhibited noise levels that were too high for the sub-millimagnitude precision required to identify low-amplitude frequencies i n the data. D a t a selection a n d further detail on reduction procedures are outlined i n the following sections of this chapter. Cerro Tololo Inter-American Observatory, Chile Beijing Astronomical Observatory, China Uttar Pradesh State Observatory, Naini Tal Manora Peak, India Observatorio del Teide (Tenerife), Spain Mauna Kea Observatory, Hawaii, U.S.A McDonald Observatory, Texas, U.S.A. South African Astronomical Observatory, South Africa Perth Observatory, Australia Sliding Spring Observatory, Australia This data log was adapted from the one on the XCOV20 website  2  3  4  5  6  7  8  9  1 0  1 1  Chapter 2. Data & Analysis  23  .60.  Figure 2.1:  A map of the observatories that participated i n the observations of H R 1217 during X C O V 2 0 . Further information about the locations may be found i n Section 2.2, a n d i n Table A . l  2.3  Preparing the Light Curves  E a c h individual r u n was reduced following the procedures outlined b y Nather et a l . (1990). T h e standard reduction software for W E T is called QED. T h i s software performs various photometric reduction tasks such as: deadtime corrections, sky interpolation and subtraction for multiple channels, sky extinction corrections, a n d p o l y n o m i a l fitting of light curves. It is important to use a standard reduction package t o reduce each r u n so that numerical artifacts from the use of multiple programs do not appear.  This  uniform reduction also allows for easy comparison between individually reduced runs. T h e reference manual for QED was written by R . E . Nather i n 2000 and can be found on the W o r l d W i d e W e b . A n overview of the reduction procedure for a n i n d i v i d u a l r u n is 1 2  described below. 12  http://bullwinkle.as.utexas.edu/~wet/contents.htm  Chapter 2. Data &  Analysis  24  Q E D begins by reading a file containing the star name, the observatory, the instrument used, the integration time used, default extinction coefficients and the raw counts as a function of time from the beginning of the run. T h e data is displayed on the screen so the user can identify bad data by eye. T h e selection of bad data is subjective, and the person performing this initial step uses individual observing logs to help identify such points. Typically, b a d d a t a points include those collected while clouds pass i n front of the star, or points that have counts inconsistent w i t h the rest of the r u n due to the star drifting i n the diaphragm. T h e sky measurements are obtained by observing the sky; i.e., a measurement while there is no star is i n the diaphragm. These sky measurements are marked where indicated i n the observing logs. T h e deadtime corrections and the sky subtractions are then performed on the data. T h e deadtime is the recovery time after a photon hit when the photomulitplier tube and the preamp are unable to register another pulse. .Thus, this correction takes into account the time lost between measurements.  E a c h of the channels has a deadtime correction  applied and the default deadtime constants for each instrument are used. T h e deadtime correction to the counts is given by C = C o / ( l — G V d ) , where Co is the original count, t  C  t  is the corrected count, and tj is the deadtime constant.  changed i n QED  These constants can be  if measurements are available. Typically, constants are on the order  of 50 nanoseconds. Once the deadtime correction is applied, a linear trend interpolated from the sky measurements is subtracted from the stellar data. A s a star is observed throughout the night, its measured brightness changes due to varying airmass. T h e airmass is simply the thickness of the atmosphere as a function of the zenith angle z. T h i s change i n brightness is known as the atmospheric extinction X a n d it is related to the zenith angle by (Hiltner, 1962) X = sec{z) - 0 . 0 0 1 8 1 6 7 ( s e c ( ^ ) - 1 ) - 0 . 0 0 2 8 7 5 ( s e c ( , z ) - 1 ) - 0 . 0 0 0 8 0 8 3 ( s e c ( ^ ) - 1 ) 2  3  (2.1)  Once the atmospheric extinction is calculated, the counts are corrected using the equation logio(C /Ci) s  = k X/2.5. ex  In this relation, C j and C correspond to the instrus  mental counts and the corrected counts respectively. T h e extinction coefficient is given  Chapter 2. Data & Analysis by k  ex  ble.  25  a n d i s a d j u s t e d i n o r d e r t o r e m o v e as m u c h o f t h e e x t i n c t i o n t r e n d as i s p o s s i -  T h u s , i f t h e extinction correction perfectly subtracts brightness variations caused  by airmass, t h e light curve w o u l d have n o l o n g - t e r m t r e n d . F o r light curves where t h e extinction correction doesn't remove such long t e r m trends completely, p o l y n o m i a l s of v a r y i n g degree were s u b t r a c t e d f r o m t h e c u r v e .  T h e degree o f t h e p o l y n o m i a l n e v e r  exceeds 3, a n d i n m o s t cases, a p a r a b o l i c o r l i n e a r fit w a s sufficient. /home/wet/People/chris/redb/MDR136.1 C  *-Qod_hiioderJor_run_mdrl36 QuHL_0_copy SJ«o«.00 2:23:30<RUN.mdrl38 DrtwuC hrl217 Ic2c3c_l 0.000.0.000 F2.O.OO0LO.OOO F3.0.000-0.000 F4_D.000_0.000 KoLPIJlJ-l  rci.i?so  B]ed_24B IBM.6061070  '.Qed.0p6rati0M.on_niii.rndr 136 1_1.17$0.80_D 2-l-1760_60_D 3.1.1760.00J) 1.3.1.17B0.T 2.4-1.1T8Q.T 1.3.1.1780.0.0846508^ Z.4.1.1700.0.00088386 J * 3_l.1760.9-~ 1.3.1.1760.1.8 4.1.1780.9.2,4.1.1760.1_S Uom_-70.81M0jBt.-30.18500_ uraJ.82178jiee.-12.09672_(»tUS4.184 l-l-1760-0.ee.a 1,2.733.17_30000_+ 1.2.1557.1 g.40DO0_+ 1.2.1577.37.26000.+ 1_3__38-_2J 1.2_439.51_20000_-  Fa*: . : Jun 9 04:21 2002  b.0015  I  • '  '  I '  I  • I '  I  '  I '  I  I  I  Baryc_412.2414.1.0602 #_Who;_mdr #_Wh trn :_ctio |_TeLtcp:.l.Bm #J>rogrmi.{JuiltO.U #_PhoU.muP3B  0  2000  4000 6000 PrequencyOiHz)  6000  10000  F i g u r e 2.2: T h e e n d r e s u l t o f t h e QED r e d u c t i o n for r u n m d r l 3 6 a t C T I O . T h e t o p panels show the channel 1 (top) a n d 2 d a t a a n d t h e lower panels show t h e Fourier transforms o f each. A g a i n , channel 1 is t h e t o p F o u r i e r p l o t . T h e r e d u c t i o n d e t a i l s f r o m t h e QED o u t p u t a r e printed o n the Figure.  T h e r e s u l t s o f t h e a b o v e r e d u c t i o n p r o c e d u r e s u s i n g QED a r e s h o w n i n F i g u r e 2.2 for t h e r u n m d r l 3 6 (see T a b l e A . l ) . A t w o - c h a n n e l p h o t o m e t e r w a s u s e d a n d e a c h o f t h e r e d u c e d c h a n n e l s c a n b e seen i n t h e t o p t w o p a n e l s o f t h e F i g u r e . T h e F o u r i e r t r a n s f o r m s  Chapter 2. Data & Analysis  26  of e a c h o f t h e c h a n n e l s is s h o w n i n t h e l o w e r t w o p a n e l s . T h e s e F o u r i e r p l o t s s h o w p o w e r at l o w frequencies f r o m r e m a i n i n g l o n g - t e r m t r e n d s i n t h e d a t a . S m a l l h e l i o c e n t r i c c o r r e c t i o n s t o t h e t i m e s c a u s e d b y changes i n t h e E a r t h ' s p o s i t i o n w i t h respect to the star due to its o r b i t a l m o t i o n h a d to be t a k e n i n t o account. f i n a l s t e p after t h e QED  As a  r e d u c t i o n , t h e t i m e d a t a for e a c h r u n are p l a c e d i n t h e s a m e  i n e r t i a l f r a m e a n d t h e c o u n t s are n o r m a l i z e d a b o u t t h e m e a n c o u n t . A s u i t a b l e reference f r a m e for a l l o f t h e t i m e s is t h e b a r y c e n t e r o f t h e S o l a r S y s t e m . T h u s , e a c h r u n h a s i t s t i m e s c o r r e c t e d w i t h reference t o t h e b a r y c e n t e r o f t h e S o l a r S y s t e m a n d l i s t e d s i n c e a time T . For X C O V 2 0 , T 0  0  = 2451854.5 b a r y c e n t r i c J u l i a n D a t e ( B J E D ) . T h e f i n a l r e s u l t  is t h e n b i n n e d i n 4 0 sec b i n s t o increase s i g n a l t o noise a n d t o decrease c o m p u t a t i o n a l t i m e d u r i n g t h e f r e q u e n c y a n a l y s i s . T h i s d a t a t h e n goes t h r o u g h a s e l e c t i o n p r o c e d u r e described i n the following section.  2.3.1  Run Selection  B e f o r e f r e q u e n c y a n a l y s i s c o u l d b e g i n , r u n s w i t h t h e b e s t s i g n a l - t o - n o i s e h a d t o b e sel e c t e d . F i r s t , l o w - f r e q u e n c y noise w a s r e m o v e d b y f i t t i n g a n d s u b t r a c t i n g s i n u s o i d s w i t h frequencies b e l o w a p p r o x i m a t e l y 1 m H z . T h e frequencies a n d a m p l i t u d e s were s e l e c t e d u s i n g t h e software Period98 c a n b e f o u n d i n S e c t i o n 2.4.  1 3  ( S p e r l , M . 1998). M o r e d e t a i l s o n t h i s software p a c k a g e L o w - f r e q u e n c y noise i n t h e d a t a is c a u s e d b y l o n g - t e r m  s k y v a r i a t i o n s a n d o t h e r i n s t r u m e n t a l d r i f t s . S u c h frequencies s h o u l d h a v e i d e a l l y b e e n r e m o v e d b y t h e e x t i n c t i o n c o r r e c t i o n a n d a l o w - o r d e r p o l y n o m i a l fit; h o w e v e r , a m p l i tudes w i t h periods greater t h a n a half-hour do r e m a i n i n some runs. T h e r e m o v a l of such frequencies does n o t affect t h e frequencies a b o v e 2 m H z t h a t are i m p o r t a n t i n r o A p s t a r studies. T h e F o u r i e r s p e c t r u m o f e a c h r u n was e x a m i n e d a n d t h o s e t h a t h a d a s i g n a l - t o noise g r e a t e r t h a n a p p r o x i m a t e l y t h r e e at t h e f r e q u e n c y w i t h t h e l a r g e s t a m p l i t u d e were s e l e c t e d . F i g u r e 2.3 s h o w s e x a m p l e s o f a selected r u n a n d a r e j e c t e d o n e . T h e n o i s e l e v e l 13  http://www. astro. univie.ac.at/~dsn/dsn/Period98/current  Chapter 2. Data & Analysis  F i g u r e 2 . 3 : A Fourier spectrum of data from the runs no2900ql (top) a n d joy002 ( b o t t o m ) . It is clear t h a t t h e a m p l i t u d e a n d n o i s e levels for t h e n o 2 9 0 0 q l are m u c h b e t t e r t h a n t h o s e i n t h e j o y - 0 0 2 r u n . T h e r e i s also a l a r g e f r e q u e n c y at a p p r o x i m a t e l y 7 2 0 c y c l e s / d a y (8.3 m H z ) i n t h e j o y - 0 0 2 r u n c a u s e d b y a p e r i o d i c d r i v e e r r o r i n t h e telescope.  27  Chapter 2. Data &  Analysis  28  a n d a m p l i t u d e o f t h e m a i n f r e q u e n c y at a p p r o x i m a t e l y 2.7 m H z is c l e a r l y different for b o t h r u n s . A l t h o u g h t h e a m p l i t u d e s of r o A p o s c i l l a t i o n s c a n v a r y over t h e r o t a t i o n c y c l e of t h e s t a r , t h e l o w s i g n a l - t o - n o i s e i n t h e j o y - 0 0 2 ( b o t t o m p l o t i n F i g u r e 2.3) r u n m a k e s i t u n a c c e p t a b l e for f u r t h e r f r e q u e n c y a n a l y s i s . A f t e r b a d r u n s were e l i m i n a t e d , t h e r e m a i n i n g d a t a c o n s i s t s o f 330.5 h r s o f p h o t o m e t r y over a p e r i o d o f 35 d a y s , as p l o t t e d at a c o m p r e s s e d scale i n F i g u r e 2.4.  This  covers s l i g h t l y less t h a n t h r e e r o t a t i o n cycles of H R 1217, w h i c h h a s a r o t a t i o n p e r i o d o f a p p r o x i m a t e l y 12.5 d a y s . T h e r e s u l t i n g d u t y c y c l e for t h e e n t i r e r u n is 3 3 % .  2.4  Frequency Analysis  A l l f r e q u e n c y a n a l y s i s w a s p e r f o r m e d u s i n g t h e software p a c k a g e Period98 T h i s p a c k a g e uses a n o n l i n e a r least s q u a r e s  fitting  ( S p e r l , 1998).  m e t h o d to calculate the amplitudes,  frequencies, a n d p h a s e s t h a t m i n i m i z e t h e r e s i d u a l s o f a g i v e n fit t o t h e d a t a . T h e f i t t i n g f u n c t i o n is a s i n u s o i d o f t h e f o r m  (2.2)  w h e r e Z is a z e r o - p o i n t offset t h a t takes i n t o a c c o u n t a n y l i n e a r t r e n d s i n t h e d a t a . a n d fa are t h e a m p l i t u d e a n d p h a s e o f t h e i  th  Period98  Ai  f r e q u e n c y t/j, r e s p e c t i v e l y .  does n o t r e l y o n p r e w h i t e n i n g t o c a l c u l a t e a f r e q u e n c y s o l u t i o n for  the  l i g h t c u r v e . P r e w h i t e n i n g is t h e process w h e r e b y a p e r i o d i c f u n c t i o n is r e m o v e d f r o m t h e d a t a a n d t h e r e s i d u a l s are u s e d for t h e n e x t s t e p i n t h e r e d u c t i o n p r o c e d u r e .  By  d o i n g t h i s , e a c h f r e q u e n c y d e p e n d s o n t h e p r e v i o u s l y d e t e r m i n e d frequencies t h r o u g h t h e residuals.  T h i s p r o c e s s is u s e d t o d e t e r m i n e w h i c h frequencies are i n c l u d e d i n t h e  fit.  T h e f r e q u e n c y w i t h t h e largest a m p l i t u d e i n t h e F o u r i e r s p e c t r u m is i d e n t i f i e d a n d u s e d as a guess f r e q u e n c y for t h e least s q u a r e s c a l c u l a t i o n . T h e r e s i d u a l s f r o m t h i s fit are t h e n u s e d t o i d e n t i f y t h e n e x t h i g h e s t frequency. A t e a c h step, a l l i d e n t i f i e d frequencies, a m p l i t u d e s a n d p h a s e s are i m p r o v e d v i a t h e least s q u a r e s c a l c u l a t i o n . T h i s p r o c e s s o f parameter identification a n d improvement continues u n t i l the amplitudes i n the Fourier  Chapter 2. Data & Analysis  29 HR 1217 330.5hrs of coverage  20  1 £  ~i  1  -i  T  r-  1  i  1  10  r  (tin r piiHi i mil  0  a.  6 -10 20 10  -J  L_  1  10  J—i—i—i—i  J—i  i  i  t  i  i  i  i  i • • 15  i  i  16  .*  .  20  E_i—1—i—i  24  i  l  i  i  26  17  i  21  i  i  28  i  i  i  22  i  30  32  34  BJED (2451854.5 +) [days]  F i g u r e 2 . 4 : T h e final l i g h t c u r v e o b t a i n e d after t h e Q E D r e d u c t i o n a n d t h e barycentric corrections have been t a k e n into account.  T h e light  c u r v e s h o w s 330.5 h r s o f d a t a w h i c h c o r r e s p o n d s t o a d u t y c y c l e o f a p p r o x i m a t e l y 3 3 % . T h e a m p l i t u d e s are m e a s u r e d i n u n i t s o f millimagnitude.  s p e c t r u m a r e a t t h e l e v e l o f t h e noise. A m o v i n g average t e c h n i q u e is u s e d t o e s t i m a t e t h e noise l e v e l i n t h e F o u r i e r s p e c t r u m . Period98  c a l c u l a t e s t h e average o f t h e a m p l i t u d e s i n t h e F o u r i e r s p e c t r u m i n a g i v e n  frequency b i n a n d t h e n moves along to the next frequency b i n . If the noise was w h i t e (frequency i n d e p e n d e n t ) ,  one c o u l d s i m p l y m o d e l it w i t h a G a u s s i a n centered o n the  frequency w i t h the highest a m p l i t u d e .  However, W E T d a t a obtained from  different  i n s t r u m e n t s , u n d e r v a r y i n g seeing c o n d i t i o n s , c o n t r i b u t e d i f f e r e n t l y t o t h e noise o f t h e  Chapter 2. Data & Analysis 2550  30  2800  2650  2700  2750  2800 i '  '.  "1—'  i/[ Hz]  i  i  2850 2900 i i i i i i i  i  M  :  Window  1  220  1  1  1  1  1  1  ..J  I  230  i  I  240  i  .  i  V  1 250  i^cycles/day]  F i g u r e 2 . 5 : T h e F o u r i e r s p e c t r u m o f t h e e n t i r e r u n s h o w n i n F i g u r e 2.4. T h e lower p a n e l s h o w s t h e w i n d o w f u n c t i o n f o r t h i s d a t a . T h e w i n d o w was calculated from the Fourier transform o f a sinusoid w i t h t h e largest a m p l i t u d e f r e q u e n c y i d e n t i f i e d i n t h e d a t a a n d s a m p l e d a t t h e s a m e t i m e s as t h e d a t a set.  d a t a set. T h e r e f o r e , i f t h e b o x size i s c h o s e n a p p r o p r i a t e l y , t h e average o f t h e frequencies i n e a c h b o x r e p r e s e n t s t h e noise s p e c t r u m w e l l . T h e first s e a r c h o f F o u r i e r space w a s p e r f o r m e d i n t h e r a n g e f r o m 0 m H z t o t h e N y q u i s t f r e q u e n c y o f 12.5 m H z . A l l p o w e r i n t h e F o u r i e r s p e c t r u m w a s i d e n t i f i e d b e t w e e n 2.5 m H z a n d 2.9 m H z as w a s f o u n d i n e a r l i e r o b s e r v a t i o n s (e.g., K u r t z & S e e m a n , 1983, a n d K u r t z e t a l . , 1989). I n o r d e r t o s p e e d t h e F o u r i e r c a l c u l a t i o n s t h e l a t t e r f r e q u e n c y range was explored a n d sampled at 1 x 1 0 ~ m H z intervals. T h e results o f t h e analysis 5  Chapter 2. Data & Analysis  31  w i l l b e p r e s e n t e d i n S e c t i o n s 2.5 a n d 2.6. T h e noise o f t h e s p e c t r u m w a s c a l c u l a t e d a t i n t e r v a l s o f 3 x 1 0 ~ m H z u s i n g a frequency b i n size o f 23 / J H Z (2 c y c l e s / d a y ) . T h i s b i n 3  size w a s c h o s e n so t h a t o n l y t h e noise n e a r t h e f r e q u e n c y b e i n g c o n s i d e r e d w a s u s e d i n the calculation. .  2.5  The Unweighted Data Results  T h e r e s u l t s o f t h e a b o v e r e d u c t i o n p r o c e d u r e s for t h e u n w e i g h t e d d a t a set a r e s h o w n i n T a b l e 2 . 1 . P r e s e n t e d a r e 21 frequencies c o n s i s t i n g o f 8 p r i m a r y frequencies a n d t h e i r r o t a t i o n a l s p l i t t i n g s . F r e q u e n c i e s l a b e l e d V\ t h r o u g h v  5  a n d u are consistent w i t h those 7  i d e n t i f i e d b y K u r t z et a l . (1989) i n t h e 1986 c a m p a i g n . T h i s a n a l y s i s a l s o recovers t h e new f r e q u e n c y o f 2791.48 / / H z t h a t w a s p r e d i c t e d b y C u n h a (2001) a n d r e p o r t e d b y K u r t z et a l . (2002) (see S e c t i o n 1.5). T h i s f r e q u e n c y i s l a b e l e d v (old) e  for t h i s d a t a set.  I n a d d i t i o n , a p r e v i o u s l y u n k n o w n f r e q u e n c y is r e p o r t e d i n t h i s w o r k a t 2 7 8 8 . 9 4 / / H z . T h i s f r e q u e n c y fits t h e a l t e r n a t i n g s p a c i n g p a t t e r n o f 34.5 a n d 3 3 . 5 / / H z a n d i s l a b e l e d as u (new) &  i n T a b l e 2 . 1 . T h e s p a c i n g b e t w e e n u (new) 6  a n d u^{old) i s 2.63 / / H z a n d t h e  s p a c i n g b e t w e e n u (old) a n d u i s a p p r o x i m a t e l y 15 / / H z . e  7  T h i s n e w f r e q u e n c y w a s n o t p r e v i o u s l y i d e n t i f i e d b y K u r t z et a l . (2002) t h e y e x a m i n e d a smaller subset o f t h e data.  because  I n c l u d i n g m o r e d a t a (i.e., e x t e n d i n g t h e  t i m e coverage) increases t h e f r e q u e n c y r e s o l u t i o n b e c a u s e t h e d i s c r e t e F o u r i e r t r a n s f o r m is w e i g h t e d e x p o n e n t i a l l y b y t h e n u m b e r o f d a t a p o i n t s . o f a d a t a set p r o d u c e s s h a p e r s p e c t r a l features.  Thus, increasing the length  A numerical analysis b y L o u m o s &  D e e m i n g (1978) also s h o w e d t h e s p e c t r a l r e s o l u t i o n A / o f a d a t a set o f l e n g t h A T i s g i v e n a p p r o x i m a t e l y b y A / = 1 . 5 / A T . A n a p p l i c a t i o n o f t h i s r e l a t i o n for u n c e r t a i n t y estimates is given i n Section 2.5.1. A s c h e m a t i c a m p l i t u d e s p e c t r a for these r e s u l t s i s g i v e n i n F i g u r e 2 . 6 .  Chapter 2. Data & Analysis  Table  2.i:  weighted  32  Results data.  of the The  frequency  frequencies,  analysis of the  un-  amplitudes  the  and  p h a s e s a r e s h o w n . T h e r o t a t i o n a l s p a c i n g s 5v a n d t h e i n fered l a r g e s p a c i n g s , A ^ , a r e a l s o s h o w n . v§(old)  was the  s i x t h f r e q u e n c y f o u n d p r e v i o u s l y b y K u r t z et a l . (2002) and  ue(new) i s a n e w f r e q u e n c y i d e n t i f i e d i n t h i s s t u d y .  Amp.(mma)  *  2.61953721  0.247873599  0.061262411  2.62052628  0.101942968  0.19455008  v  +  (mHz)  <Si/(mHz)  Ai/(mHz)  9.8907E-04  V2-V\  0.03340692  -  2.6519788  0.239812675  0.006703652  ^2  2.65294413  0.793304662  0.145744578  +  2.65389542  0.181838918  0.300326255  9.6533E-04  9.5129E-04 0.03454836  -  2.68644247  0.332793541  0.200445295  "3  2.68749249  0.554168633  0.11110511  +  2.68841904  0.215781733  0.336983675  9.2655E-04  -  2.72006539  0.408958037  0.505309948  8.6172E-04  2.72092711  1.14018239  0.782933145  +  2.72182605  0.436426134  0.015917406  8.9894E-04  -  2.75432171  0.218422957  0.114667745  1.0080E-03  2.75532974  0.278092265  0.224817247  2.75623738  0.119315631  0.489636487  2.78893673  0.126677281  0.606594168  +  2.78996948  0.090747676  0.507306041  1.0327E-03  -  2.7906266  0.130919596  0.197095448  9.4253E-04  1.0500E-03  V4-V3  0.03343462  0.03440263  +  9.0764E-04  i/6 (new)-i/5 0.03360699  ug  (new)  VQ  (old)  2.79156913  0.193687427  0.37786162  +  2.79213576  0.070257486  0.092062776  -  2.80562143  0.190737069  0.450445087  7  2.80656663  0.120611222  0.485536548  VQ (old)-i/6 (new) 0.0026324  5.6663E-04 V7-v&  u  (old)  0.0149975 9.4520E-04  T h e a v e r a g e v a l u e o f t h e fine f r e q u e n c y s t r u c t u r e i s 9 . 2 7 x 1 0  - 4  m H z with a standard  Chapter 2. Data &  Analysis  33 T  1  1  1  1  1  1  1  1 r  r  0.8  V, 0.6  0.4  0.2  llll  .  2.65  2.7  2650  2.75  2.8"  •/(mHz) 2700  2800 '  "[^Hz]  0.8 E Jj, ¥ 3  0.6  E 0.4  0.2  J  230  L_Ld_  235 i/[cycles/day]  240  F i g u r e 2.6: A schematic of the frequencies found from the unweighted data (top). T h e frequencies are listed i n Table 2.1. O n the b o t t o m , the amplitudes of the frequencies and the noise level after a l l frequencies are prewhitened. T h e (  ) line represent four times the noise level  and the (. . .) line represents three times the noise level.  Chapter 2. Data &  Analysis  34  e r r o r o n t h e m e a n o f « 1%. A s s u m i n g H R 1217 is a n o b l i q u e r o t a t o r , t h e fine s t r u c t u r e s h o u l d be e x a c t l y spaced b y the r o t a t i o n frequency of the star.  This would imply a  r o t a t i o n p e r i o d o f 12.5 d a y s ± 1 % ; c o n s i s t e n t w i t h v a l u e s o f t h e r o t a t i o n p e r i o d d e r i v e d i n t h e l i t e r a t u r e . F o r e x a m p l e , K u r t z & M a r a n g (1987) use p h o t o m e t r y t o d e d u c e a p e r i o d of 12.4572 ± 0.0003 d a y s f r o m t h e l o n g t e r m v a r i a b i l i t y of H R 1217. B a g n u l o et a l . (1995) use l i n e a r l y a n d c i r c u l a r l y p o l a r i s e d l i g h t d a t a a n d o b t a i n a p e r i o d o f 12.4610 ±  0.0011  d a y s f r o m t h e m a g n e t i c field v a r i a t i o n s .  (mHz)  , (mHz)  F i g u r e 2 . 7 : T h e r a t i o b e t w e e n t h e c o m m o n a m p l i t u d e s ( t o p ) a n d differences i n frequencies ( b o t t o m ) f r o m t h i s d a t a set a n d t h e K u r t z et a l . (1989) d a t a set. T h e frequencies are c o n s i s t e n t w i t h e a c h o t h e r w h i l e t h e a m p l i t u d e s o f t h e t h i r d a n d f o u r t h f r e q u e n c y ( r e a d f r o m t h e left) are c l e a r l y different.  T h e net a m p l i t u d e a n d f r e q u e n c y  differences  are s h o w n i n t h e u p p e r r i g h t o f e a c h p l o t .  W h i l e t h e d e r i v e d frequencies are c o n s i s t e n t w i t h t h e 1986 d a t a , t h e a m p l i t u d e s differ  Chapter 2. Data & significantly.  Analysis  35  F i g u r e 2.7 c o m p a r e s t h e difference b e t w e e n t h e p r i n c i p l e a m p l i t u d e s a n d  frequencies t h a t were c o m m o n t o b o t h t h i s d a t a set a n d t h e 1986 d a t a . T h e a m p l i t u d e s o f frequencies u  3  by  a n d v± s h o w t h e largest difference b e t w e e n t h e t w o d a t a sets. T h e y differ  a p p r o x i m a t e l y 6 0 % a n d 40%, respectively.  H o w e v e r , t h e net a m p l i t u d e  difference  b e t w e e n t h e 6 frequencies is s m a l l , at —0.070 ± 0.192 m m a g . A s e a c h o f t h e frequencies w a s i d e n t i f i e d a n d p r e w h i t e n e d , t h e s t a n d a r d d e v i a t i o n o f t h e r e s i d u a l s w a s c a l c u l a t e d . F r e q u e n c i e s were r e m o v e d u n t i l t h e s t a n d a r d d e v i a t i o n o f the residuals approached a constant value.  T h e s e r e s u l t s m a y b e f o u n d i n F i g u r e 2.8.  A f t e r the twenty-first frequency was removed, the s t a n d a r d d e v i a t i o n of the is i m p r o v e d b y less t h a n a 0 . 1 % .  residuals  E s t i m a t e s of the uncertainty a n d significance of the  d e r i v e d frequencies w i l l b e d i s c u s s e d i n t h e n e x t s e c t i o n .  2.5.1  Estimating Uncertainties and Significance  U n c e r t a i n t y e s t i m a t e s for t h e f r e q u e n c y a n a l y s i s were c a r r i e d o u t i n t h r e e different w a y s . T h e first is t h a t u s e d b y K u r t z & W e g n e r (1979) t o e s t i m a t e t h e f r e q u e n c y r e s o l u t i o n o f t w o c l o s e l y s p a c e d frequencies.  T h e y s t a t e t h a t t h e G a u s s i a n s t a n d a r d d e v i a t i o n asso-  c i a t e d w i t h e a c h f r e q u e n c y u n c e r t a i n t y is a p p r o x i m a t e l y o n e - s i x t h t h a t o f t h e f r e q u e n c y r e s o l u t i o n e s t i m a t e d b y L o u m o s h D e e m i n g (1978). T h i s m a y b e c a l c u l a t e d u s i n g  A  ' =s i r  ( 2  '  3 )  w h e r e A T is t h e l e n g t h o f t h e o b s e r v i n g r u n . U s i n g t h i s m e t h o d , t h e f r e q u e n c y u n c e r t a i n t y is e s t i m a t e d as A / « 8 x 1 0  - 5  m H z for A T = 35.12 d a y s .  T h e s e c o n d m e t h o d is c l o s e l y r e l a t e d t o t h e first a n d e s t i m a t e s t h e n u m b e r o f i n d e p e n d e n t frequencies f r o m t h e f u l l - w i d t h - h a l f - m a x i m u m ( F W H M ) o f t h e m a i n p e a k i n t h e F o u r i e r S p e c t r u m ( A l v a r e z et a l , 1998). F r o m F i g u r e 2.9, t h e F W H M be 6 x 1 0  - 4  is e s t i m a t e d  m H z . T h i s is a p p r o x i m a t e l y a n o r d e r o f m a g n i t u d e a b o v e t h e  to  uncertainty  e s t i m a t e f r o m E q u a t i o n 2.3. T h e t h i r d technique estimates the u n c e r t a i n t y i n phase, a m p l i t u d e a n d  frequency.  T h i s m e t h o d a s s u m e s t h e t i m e s are c e r t a i n a n d t h e c o u n t s are s u b j e c t t o r a n d o m noise.  Chapter 2. Data & Analysis  36  Figure 2.8: Improvement of the standard deviations of the residuals as each of the k frequencies are removed. E a c h of the frequencies are listed i n Table 2.1. T h e frequencies are prewhitened i n order of decreasing amplitude. T h e inset is a blow-up of the standard deviation for the low amplitude frequency improvements.  Assuming cross-terms between frequencies are small (i.e., a l l frequencies may be considered independent from each other), Montgomery & O'Donoghue (1999) and Breger et al. (1999) derive analytic equations for the uncertainty of the amplitude, phase and frequency for a least squares fit. These equations are, respectively,  a = (2/iV) /V(m) 1  a  (2.4)  Chapter 2. Data &  Analysis  37 2720  I 1__J I I  234.95  I  235  I I  2721  1  III  2722  I I I I . . .  235.05  u[c/d]  235.1  I I I I  235.15  F i g u r e 2 . 9 : T h e F W H M o f t h e m a i n f r e q u e n c y i n t h e u n w e i g h t e d d a t a . T h i s is u s e d as a n e s t i m a t e for t h e u n c e r t a i n t y o f t h e frequencies d e r i v e d from the r e d u c t i o n of the unweighted data.  a, = ( 2 / i V ) / 1  2  ^  (2.5) QJ  a  » =  ( 6 / A  °  1 / 2  ( 2  -  6 )  T h e a m p l i t u d e b e i n g c o n s i d e r e d is r e p r e s e n t e d b y a, t h e l e n g t h o f t h e r u n is T, t h e n u m b e r of d a t a p o i n t s N, a n d t h e r o o t - m e a n - s q u a r e o f t h e r e s i d u a l s is a(m).  T h e s e e q u a t i o n s are  a p p l i e d t o t h e r e s i d u a l s o f t h e s i n u s o i d fit as e a c h f r e q u e n c y is p r e w h i t e n e d . T h e e r r o r e s t i m a t e s for t h i s p r o c e d u r e are l i s t e d i n T a b l e 2.2. A c c o r d i n g t o M o n t g o m e r y h O ' D o n o g h u e (1999), i f t h e r a n d o m noise is c o r r e l a t e d w i t h time, the uncertainties c o u l d be underestimated. T h e y urge t h a t these values o n l y b e t a k e n as a l o w e r l i m i t o f t h e u n c e r t a i n t y a n d suggest t h a t t h e a c t u a l r e s u l t m a y b e a n  Chapter 2. Data & Analysis  38  order of magnitude higher. If this is the case, and all of the uncertainties i n Table 2.2 are increased by an order of magnitude, they would be consistent w i t h the previous uncertainty estimates for this data. K u r t z et a l . (2002) derive uncertainties formally from their least squares analysis during a preliminary data reduction. T h e y obtain a frequency uncertainty that is «  10  - 5  m H z . G i v e n the above information, a conservative estimate o n the frequency  uncertainties would be 1 x 10~ m H z . 4  Table 2.2:  Uncertainty estimate for the unweighted data us-  ing Equations 2.4, 2.5, and 2.6. These values are taken as lower limits only.  av  (mHz)  CT^ (mHz)  <7a(mma)  2.49E-06  0.0141  0.0054  +  2.42E-06  0.0137  0.0052  4.91E-06  -  2.44E-06  0.0140  0.0053  5.11E-06  (TAi/(mHz)  V2-V\  5.16E-06 2.67E-06  0.0153  0.0058  +  2.40E-06  0.0138  0.0052  5.07E-06  -  2.44B-06  0.0142  0.0053  5.00E-06  2.56E-06  0.0148  0.0055  2.39E-06  0.0139  0.0052  VZ-V2  5.23E-06  +  4.95E-06  f4-f3  5.29E-06  -  +  2.46E-06  0.0144  0.0053  2.73E-06  0.0160  0.0059  2.48E-06  0.0146  0.0054  5.18E-06  5.21E-06  1/5-1/4  5.13E-06  -  2.34E-06  0.0139  0.0051  "5  2.40E-06  0.0143  0.0052  +  2.30E-06  0.0137  0.0050  4.70E-06  (new)  2.28E-06  0.0137  0.0049  4.55E-06  +  2.27E-06  0.0137  0.0049  -  2.28E-06  0.0137  0.0049  (old)  2.33E-06  0.0140  0.0050  +  2.27E-06  0.0137  0.0049  4.74E-06  1/6 (new)-f5  4.68E-06 UQ  continued on next page  4.61E-06  1/6 (old)-i/6  (new)  4.61E-06 4.60E-06  Chapter 2. Data & Analysis  39 Table 2.2: continued  a  (mHz)  <7 (mma)  04,  iTfo(mHz)  2.28E-06  0.0138  0.0049  4.54E-06  2.26E-06  0.0137  0.0049  v  a  (TA„(mHz) 1/7-1/6  1/7  (old)  4.59E-06  E s t i m a t e s for t h e s i g n a l t o noise at w h i c h a f r e q u e n c y m a y c o n f i d e n t l y b e i d e n t i f i e d are c a l c u l a t e d b y K u s c h n i g et a l . (1997). K u s c h n i g u s e d a m p l i t u d e s p e c t r a w i t h s i m u l a t e d noise i n c o n j u n c t i o n w i t h the H u b b l e Space Telescope's F i n e G u i d a n c e Sensor guide star d a t a t o s h o w t h a t a s i g n a l t o noise o f 3.6 w o u l d p r o d u c e a 9 9 % c o n f i d e n c e l e v e l for f r e q u e n c y i d e n t i f i c a t i o n , w h i l e a s i g n a l t o noise o f 4.0 w o u l d p r o d u c e a 9 9 . 9 % c o n f i d e n c e l e v e l . T h e s e levels are c o n s i s t e n t w i t h those s u g g e s t e d b y B r e g e r et a l . (1993) a n d t h o s e d i s c u s s e d b y A l v a r e z et a l . (1998), a n d B r e g e r et a l . (1999). T h o s e a u t h o r s also s t a t e t h a t t h i s m e t h o d is e q u i v a l e n t t o t h e S c a r g l e false a l a r m p r o b a b i l i t y test ( S c a r g l e , 1982) w h i c h assigns a c o n f i d e n c e l e v e l for a n i d e n t i f i e d f r e q u e n c y a s s u m i n g w h i t e n o i s e .  Following  these a u t h o r s , a s i g n i f i c a n t confidence level for a f r e q u e n c y i d e n t i f i c a t i o n i s o b t a i n e d w h e r e t h e a m p l i t u d e o f t h e f r e q u e n c y is at least 3.5 t i m e s t h e noise (see F i g u r e 2.6).  2.6 The  The Weighted Data Results  d a t a c o l l e c t e d d u r i n g a W E T c a m p a i g n h a s noise c h a r a c t e r i s t i c s t h a t are a c o m b i -  n a t i o n o f noise f r o m e a c h i n d i v i d u a l o b s e r v i n g r u n .  I n order to o b t a i n the best signal  t o noise, o n e m u s t c o n s i d e r e i t h e r r e m o v i n g s o m e o f t h e d a t a , or w e i g h t i n g t h e  data.  R e m o v i n g d a t a m a y not be the most desirable alternative since gaps i n the light curve p r o d u c e aliases i n t h e F o u r i e r s p e c t r u m . T h i s , i n t u r n , confuses f r e q u e n c y i d e n t i f i c a t i o n . It is a l s o i m p o r t a n t t o n o t e t h a t t h e F o u r i e r t r a n s f o r m is w e i g h t e d b y t h e n u m b e r o f points considered. T h u s , lowering the number of points b y r e m o v i n g d a t a w i l l also i n crease t h e noise i n t h e c a l c u l a t e d s p e c t r u m .  I n t h i s thesis, different w e i g h t i n g schemes  w i l l b e c o n s i d e r e d i n a n a t t e m p t t o p r o d u c e t h e b e s t f r e q u e n c y s o l u t i o n for t h e H R 1217  Chapter 2. Data & Analysis data.  40  T w o w e i g h t i n g schemes d i s c u s s e d i n H a n d l e r (2003) w i l l b e d e s c r i b e d , as w e l l as  t h e m o d i f i c a t i o n s t o these m e t h o d s u s e d i n t h i s w o r k t o a n a l y z e t h e H R 1 2 1 7 d a t a . T h e first w e i g h t i n g scheme i s s i g m a - c u t o f f w e i g h t i n g . I n t h i s m e t h o d , e a c h d a t a p o i n t is w e i g h t e d b a s e d o n t h e r e s i d u a l s f r o m a g i v e n fit. E a c h w e i g h t i s a s s i g n e d b y t h e relations  Wi = 1 Wi = (Ka /ai)  x  res  where  K a n d x a r e free  if  G < Ka  if  a > Ka  {  res  (2.7)  res  p a r a m e t e r s , Oi is t h e r m s r e s i d u a l o f t h e  i  th  point and  a  res  is the  average s t a n d a r d d e v i a t i o n o f t h e r e s i d u a l s .  oi K —  F o r e x a m p l e , R o d r i g u e z et a l . (2003) choose v a l u e s  1.0 a n d  x=  2.0.  Period98  uses t h i s m e t h o d t o a s s i g n w e i g h t s t o d a t a ; h o w e v e r , K a n d x a r e fixed a t 1 a n d 2, r e s p e c t i v e l y , a n d i t i s u p t o t h e user t o choose t h e c u t - o f f a m p l i t u d e  a . res  F r a n d s e n et a l .  (2001) use t h i s f u n c t i o n o f Period98 as one o f t h e w e i g h t i n g schemes i n t h e i r p a p e r . t h e y a n d H a n d l e r (2003) c a u t i o n t h a t t h e v a l u e o f cr  re3  Both  s h o u l d b e chosen wisely. If i t is  n o t , b a d d a t a t h a t falls w i t h i n t h e c u t - o f f c o u l d b e g i v e n f u l l w e i g h t , o r g o o d d a t a c o u l d b e g i v e n l o w w e i g h t . T h i s m e t h o d i s also l i m i t e d b y i t s d e p e n d e n c e o n a p r e d e t e r m i n e d solution t o calculate residuals. A n o t h e r m e t h o d o f d a t a w e i g h t i n g d e s c r i b e d i n H a n d l e r (2003) i s l i g h t c u r v e v a r i a n c e w e i g h t i n g . I n t h i s m e t h o d , t h e d a t a are b o x c a r s m o o t h e d t o r e m o v e t h e p o i n t - t o - p o i n t differences i n i n t e n s i t y . weights to the data.  T h e inverse o f t h e s m o o t h i n g f u n c t i o n i s t h e n u s e d t o a s s i g n  B y doing this, night-to-night variations i n the d a t a c a n be taken  i n t o a c c o u n t . T h e free p a r a m e t e r s i n t h i s m e t h o d w o u l d b e t h e size o f t h e t i m e b i n b e i n g considered a n d t h e exponent b y w h i c h y o u weight the data; i.e., t h e x i n E q u a t i o n 2.7. T h e m a i n d i s a d v a n t a g e o f t h i s m e t h o d is t h a t t h e s m o o t h i n g f u n c t i o n c a n b e g r e a t l y affected b y o u t l y i n g p o i n t s . T h i s m a y , i n t u r n , r e s u l t i n a n i m p r o p e r w e i g h t a s s i g n m e n t . I n t h i s s t u d y , four w e i g h t i n g m e t h o d s b a s e d o n t h e a b o v e schemes were t e s t e d . E a c h w e i g h t i n g p r o c e d u r e is s u m m a r i z e d as:  method 1 : The  Period98  w e i g h t i n g scheme i s u s e d w i t h a v a r i e t y o f c u t o f f  Chapter 2. Data & Analysis amplitudes a  r e s  41  .  m e t h o d 2 : T h e d a t a w e i g h t s are a s s i g n e d u s i n g E q u a t i o n 2.7 w i t h t h e average s t a n d a r d e r r o r of t h e r e s i d u a l s f r o m a fit. T h i s fit is o b t a i n e d f r o m t h e r e s u l t s o f t h e u n w e i g h t e d d a t a a n a l y s i s (see S e c t i o n 2.5). W h e n E q u a t i o n 2.7 is a p p l i e d t o d a t a , t h e free p a r a m e t e r s are v a r i e d t o e x p l o r e t h e i r effect o n t h e f i n a l s o l u t i o n . m e t h o d 3 : T h e average s t a n d a r d e r r o r o f t h e r e s i d u a l s t o a fit are c a l c u l a t e d a t v a r i o u s t i m e b i n s before E q u a t i o n 2.7 is a p p l i e d . T h i s m e t h o d is s i m i l a r t o a p p l y i n g w e i g h t s f r o m a s m o o t h i n g filter i n t h a t t h e c h a n g e i n t h e noise f r o m t i m e b i n t o t i m e b i n is t a k e n i n t o a c c o u n t . m e t h o d 4 : T h e w e i g h t s are c a l c u l a t e d a n d a p p l i e d u s i n g E q u a t i o n 2.7 as e a c h n e w f r e q u e n c y is a d d e d . B y d o i n g t h i s , we d o n ' t r e l y o n r e s i d u a l s f r o m a f i n a l fit t o the data. T h i s m e t h o d a t t e m p t s to overcome the dependence o n a predetermined s o l u t i o n b y considering the weight to be a parameter t h a t changes w i t h each step i n the frequency analysis.  E a c h of the results o b t a i n e d f r o m the above w e i g h t i n g schemes are c o m p a r e d t o t h e r e s u l t s f r o m t h e u n w e i g h t e d d a t a t h r o u g h a s t a t i s t i c xlomp d e f i n e d b y  Xcomp  (2.8) ££=_(/. -  //ito) /( jo) 2  c 7  w h e r e v a l u e s w i t h t h e s u b s c r i p t 0 refers t o t h o s e o b t a i n e d f r o m t h e u n w e i g h t e d d a t a , r e p r e s e n t s a n o b s e r v e d t i m e - s e r i e s d a t a p o i n t , ff  it  /  is a p o i n t c a l c u l a t e d u s i n g E q u a t i o n  2.2, a n d t h e s t a n d a r d e r r o r o f t h e r e s i d u a l s f r o m t h e fit for a g i v e n d a t a b i n are r e p r e s e n t e d b y a. M o d e l s w i t h v a r i o u s t i m e b i n s a n d c u t - o f f a m p l i t u d e s were c o n s i d e r e d . S o m e o f t h e m o d e l s c o n s i d e r e d m o d e s t o b e t r i p l e t s , q u i n t e t s , or a n a l t e r n a t i n g c o m b i n a t i o n o f b o t h . O t h e r m o d e l s were c o n s t r u c t e d a s s u m i n g t h a t t h e frequencies are s p l i t b y e x a c t l y t h e r o t a t i o n p e r i o d o f t h e s t a r . I n t h e l a t t e r case, frequencies were a d j u s t e d after e a c h least s q u a r e s i t e r a t i o n t o b e s p l i t w i t h a f r e q u e n c y o f 9.2897 x 1 0  - 4  m H z ; t h e average o f t h e  Chapter 2. Data & Analysis  42  m e a s u r e m e n t s b y B a g n u l o et a l . (1995) a n d K u r t z k, M a r a n g (1987). T h e frequencies, a m p l i t u d e s a n d p h a s e s were t h e n i m p r o v e d u s i n g Period^.  A l l models are s u m m a r i z e d  i n Table 2.3.  Table 2.3:  A s u m m a r y o f t h e w e i g h t i n g p a r a m e t e r s for t h e  34 different f r e q u e n c y s o l u t i o n s e x p l o r e d .  T h e param-  eters a r e d e f i n e d i n E q u a t i o n 2 . 7 . T h e c o l u m n l a b e l e d method refers t o t h e w e i g h t i n g m e t h o d s d e s c r i b e d i n Sect i o n 2.6. T h e points /bin  c o l u m n is t h e n u m b e r o f d a t a  p o i n t s p e r t i m e b i n . If method 1 i s u s e d , t h e c u t o f f a m plitude is presented instead of the number of d a t a points p e r b i n . See t h e comments c o l u m n for specific r e d u c t i o n details.  model  #  K  X  points/bin  method  or cr ea{mrnag]  method = 1  500  2  r  1  1.0  1.0  comment  The number of points in each time bin are varied; as are the weighting factor K and exponent x.  2  1.0  1.0  200  2  3  1.0  1.0  800  2  4  1.0  1.0  1100  2  5  2.0  1.0  500  2  6  2.0  1.0  800  2  7  2.0  1.0  1100  2  8  1.0  2.0  200  2  9  2.0  2.0  200 .  2  9  1.0  1.0  100  4  New weights are calculated as each new frequency is identified.  10  1.0  1.0  100  4  11  2.0  1.0  100  4  no weight  1  12  Calculate a solution where frequencies alternate between triplets and quintuplets. The we and 1/7 frequencies are described by triplets in this solution.  13  2.0  1.0  5  1  See model 12 comment (New Weight).  14  2.0  1.0  1  1  See model 12 comment (New Weight).  continued on next page  Chapter 2. Data &  Analysis  43  Table 2.3: continued model  #  K  X  points/bin or  o [mmag] res  method  comment  method = 1  15  2.0  1.0  0.8  1  See model 12 comment (New Weight).  16  -  -  no weight  1  Calculate a solution where all frequencies  17  2.0  1.0  5  1  See model 16 comment (New Weight).  18  2.0  1.0  2  1  See model 16 comment (New Weight).  19  2.0  1.0  1  1  See model 16 comment (New Weight).  20  -  -  no weight  1  Calculate a solution with all frequencies  are triplets.  as quintuplets except for the triplet ve frequencies. 21  -  -  no weight  1  Calculate a solution where all frequencies are triplets and attemp to force ve(new) to be a quintuplet.  22  -  -  no weight  1  Adjust i/e(new) to be a quintuplet.  23  2.0  1.0  1  1  See model 21 comment (New Weight).  24  2.0  1.0  5  1  See model 21 comment (New Weight).  25  -  -  no weight  1  Calculate a solution where the unweighted frequenies are adjusted to be exactly rotationally split.  26  2.0  1.0  27  2.0  28  2.0  29  -  5  1  See model 25 comment (New Weight).  1.0  1  1  See model 25 comment (New Weight).  1.0  0.8  1  See model 25 comment (New Weight).  -  no weight  1  Take the solution from model 20 and continually remove frequencies that have amplitudes below 0.06 mmag. Then recalculate the solution with the omitted frequencies.  30  -  -  no weight  1  31  2.0  1.0  5  1  Weight model 30 solution.  32  2.0  1.0  1  1  Weight model 30 solution.  33  -  -  no weight  1  The unweighted data is forced to be  Adjust frequencies to be exactly rotationally split.  rotationally split; however, the amplitudes, frequencies and phases are not improved using Period98. 34  -  -  no weight  1  See comment 33. Now the amplitudes and phases are improved using Period98.  A l l m o d e l s are c o m p a r e d t o t h e u n w e i g h t e d d a t a t h r o u g h E q u a t i o n 2 . 8 . T h e r e s u l t s  Chapter 2. Data & Analysis  44  are s h o w n i n F i g u r e 2.10. A l o w e r xlomp s t a t i s t i c i m p l i e s l o w e r r e s i d u a l s for t h e f i t . I n a l l cases e x p l o r e d , t h e i m p r o v e m e n t i n t h e r e s i d u a l s w a s less t h a n 1%. A n a l t e r e d v e r s i o n o f t h e u n w e i g h t e d d a t a w a s u s e d t o test t h e s e n s i t i v i t y o f t h e xlomp s t a t i s t i c t o a p o o r f i t . I n t h i s case; m o d e l n u m b e r 3 3 , t h e u n w e i g h t e d d a t a h a d i t s fine s t r u c t u r e s p a c e d e x a c t l y b y t h e r o t a t i o n p e r i o d o f t h e s t a r w h i l e t h e a m p l i t u d e s a n d p h a s e s were n o t a l t e r e d . T h e r e s u l t s i n d i c a t e a xlomp t h a t is a p p r o x i m a t e l y 5 % worse t h a n i n t h e o t h e r m o d e l s . M o d e l n u m b e r s 25 a n d 34 s h o u l d b e c o m p a r e d t o m o d e l 3 3 . B o t h o f these m o d e l s force t h e frequencies o f t h e fine s p l i t t i n g for t h e u n w e i g h t e d d a t a t o b e t h e r o t a t i o n a l f r e q u e n c y o f t h e s t a r . T h e e x c e p t i o n i n m o d e l 25 is t h a t t h e frequencies, a m p l i t u d e s a n d p h a s e s were t h e n i m p r o v e d u s i n g Period98. improved using  F o r m o d e l 34, o n l y t h e a m p l i t u d e s a n d p h a s e s were t h e n  Period98.  T h e lowest p e a k s i n F i g u r e 2.10 c o r r e s p o n d t o m o d e l s 14, 18, 2 7 a n d 3 2 . S c h e m a t i c a m p l i t u d e d i a g r a m s s h o w i n g t h e w e i g h t e d noise are p r e s e n t e d i n F i g u r e s 2.11 a n d 2 . 1 2 . I n a l l cases c o n s i d e r e d , v  7  shows o n l y a significant doublet structure.  E a c h of the other  frequencies s h o w , at least, s i g n i f i c a n t t r i p l e t s t r u c t u r e . C o m p l e t e d a t a t a b l e s for these 4 models are given i n A p p e n d i x C . In the other  s t a n d a r d e r r o r w a s c o m p u t e d for m o d e l s w i t h t h e s a m e frequen-  cies, b u t w i t h different w e i g h t p a r a m e t e r s .  T h e r e s u l t s s h o w t h a t t h e frequencies o n l y  v a r y b y a p p r o x i m a t e l y 1 0 ~ w h i l e t h e a m p l i t u d e s differ b y less t h a n a h a l f o f a p e r c e n t . 6  C h a n g i n g t h e p a r a m e t e r s p a c e f o r t h e d a t a w e i g h t s h a s l i t t l e effect o n t h e o u t c o m e o f t h e d a t a . T h e c o m p u t e d s t a n d a r d e r r o r s for a l l cases a r e also p r e s e n t e d i n A p p e n d i x C .  Chapter 2.. Data &  -I 0  1  1  1  1  1 10  Analysis  1 1 1 1 1 20 Model Number n  1  45  1  1  1  1 30  1  1—I  0.898 - - J 0  i  i  i  i  I 10  i i i , I 20 Model Number n  ,  ,  ,  i  1_ 30  F i g u r e 2 . 1 0 : A c o m p a r i s o n between the weighted a n d unweighted models u s i n g E q u a t i o n 2.8. T h e p l o t at t h e t o p s h o w s t h e s t a t i s t i c for t h e 34 m o d e l s d e s c r i b e d i n T a b l e 2.3. T h e r i g h t p l o t is a n e x p a n d e d v i e w of t h e first 32 m o d e l s .  I n a l l cases, t h e m o d e l s v a r y f r o m  u n w e i g h t e d m o d e l b y n o m o r e t h a n 5%.  the  Chapter  2. Data  &  Analysis  2700  1.2  1 h  0.8  0.6  0.4  0.2  -i  L_J  1  230 '0  230  Figure 2.11: Schematic  i  i  i  I  i  i  235 i/[cycles/day] 2700  L  V  \4  240 27S0  235 i/[oycles/day]  240  amplitude diagrams for models 14 (top) and 32 (bot-  tom). T h e (  ) line represent four times the noise level and the  (. . .) line represents three times the noise level. T h e noise is calculated from the weighted residuals.  Chapter  2. Data  &  Analysis  47  230  235 y[cycles/day]  230  Figure 2.12: Schematic  235 u[cycles/day]  240  amplitude diagrams for models 18 (top) and 27 (bot-  tom). T h e (  ) line represent four times the noise level and the  (. . .) line represents three times the noise level. T h e noise is calculated from the weighted residuals.  48  Chapter 3  Stellar Evolution and P u l s a t i o n Models A g r i d o f e v o l u t i o n a r y a n d p u l s a t i o n m o d e l s o f A t y p e s t a r s is c a l c u l a t e d e x p l o r i n g a variety of parameters.  T h e effect o f a m a g n e t i c field o n t h e r e s u l t i n g p u l s a t i o n frequencies  is also e s t i m a t e d for these m o d e l s .  I n t h i s c h a p t e r , t h e p r o c e d u r e s for c a l c u l a t i n g t h e  m o d e l s are d i s c u s s e d a n d t h e r e s u l t s o f t h e c a l c u l a t i o n s are i n t r o d u c e d .  3.1  Stellar Evolution Models  T h e s t e l l a r e v o l u t i o n m o d e l s were c a l c u l a t e d u s i n g t h e Y a l e S t e l l a r E v o l u t i o n C o d e w i t h Rotation (YREC7)  i n i t s n o n - r o t a t i n g c o n f i g u r a t i o n ( G u e n t h e r et a l . , 1 9 9 2 ) .  YREC7  solves t h e m e c h a n i c a l , c o n s e r v a t i o n , a n d e n e r g y t r a n s p o r t e q u a t i o n s o f s t e l l a r s t r u c t u r e u s i n g t h e H e n y e y r e l a x a t i o n scheme ( H e n y e y et a l , 1 9 6 4 ) . A d e t a i l e d d i s c u s s i o n o f t h e equations of stellar structure c a n be found i n the text of K i p p e n h a h n & Weigert (1994). T h e d e n s i t y o f a s t e l l a r m o d e l is r e l a t e d t o t h e o t h e r m a t e r i a l f u n c t i o n s ; i.e., t h e t e m p e r a t u r e a n d pressure o b t a i n e d from the stellar structure  equations,  through  the  e q u a t i o n o f s t a t e ( E O S ) . Y R E C 7 i n t e r p o l a t e s b e t w e e n O P A L E O S t a b l e s ( R o g e r s 1986, a n d R o g e r s , S w e n s o n , &; Iglesias, 1996) w i t h different c o m p o s i t i o n s i n o r d e r t o o b t a i n t h e a p p r o p r i a t e d e n s i t i e s for a p a r t i c u l a r m o d e l . T h e o p a c i t y r o u t i n e s n e e d e d t o solve t h e energy transport equations of stellar structure are interpolated f r o m two separate tables. If t h e t e m p e r a t u r e o f a m a s s s h e l l i n a g i v e n m o d e l is g r e a t e r t h a n 6 0 0 0 K , t h e  OPAL  o p a c i t y t a b l e s are u t i l i z e d (Iglesias & R o g e r s , 1996). If t h e t e m p e r a t u r e o f a s h e l l is less t h a n 15000 K , t h e l o w - t e m p e r a t u r e ( m o l e c u l a r ) o p a c i t y t a b l e s o f A l e x a n d e r & F e r g u s o n  Chapter 3. Stellar Evolution  and Pulsation  Models  49  (1994) are u s e d t o o b t a i n t h e o p a c i t y . I n r e g i o n s o f t h e m o d e l w h e r e t h e s e o v e r l a p , a l i n e a r r a m p f u n c t i o n is u s e d t o l i n k t h e t w o o p a c i t y t a b l e s .  temperatures A l l tables  are  c a l c u l a t e d for a s o l a r e l e m e n t a l a b u n d a n c e s (Grevesse et a l . , 1996). F r o m t h i s s t r u c t u r a l i n f o r m a t i o n , t h e n u c l e a r r e a c t i o n n e t w o r k ; e.g., t h e p r o t o n - p r o t o n o r c a r b o n - n i t r o g e n - o x y g e n c h a i n s , is c a l c u l a t e d for e a c h m o d e l s h e l l t o d e t e r m i n e  the  e n e r g y g e n e r a t e d v i a n u c l e a r b u r n i n g u s i n g t h e n u c l e a r cross s e c t i o n s o f B a h c a l l et a l . (2001).  T h e new interior elemental abundances  are c a l c u l a t e d f r o m t h i s p r o c e s s  and  another m o d e l is evolved f r o m this complete m o d e l t o a later t i m e step. I n t h i s s t u d y , m o d e l s are e v o l v e d f r o m t h e zero age m a i n sequence ( Z A M S ) t o t h e a p p r o a c h o f t h e base o f t h e r e d g i a n t b r a n c h (see F i g u r e 3.1).  E a c h m o d e l generated  h a s a p p r o x i m a t e l y 3000 shells e v e n l y d i s t r i b u t e d a m o n g t h e i n t e r i o r , e n v e l o p e a n d t h e a t m o s p h e r e . T h e m o d e l i n t e r i o r represents t h e i n n e r ~ 9 9 % o f t h e m o d e l b y m a s s w h i l e t h e e n v e l o p e m a k e s u p t h e o t h e r 1%.  T h e o v e r l y i n g a t m o s p h e r e is c a l c u l a t e d a s s u m -  i n g a f r e q u e n c y i n d e p e n d e n t t e m p e r a t u r e - o p t i c a l d e p t h ( T — r ) r e l a t i o n ; k n o w n as t h e E d d i n g t o n gray approximation.  3.1.1  The Parameter Space  T o b e g i n t o s p e c i f y a s t e l l a r m o d e l , t h e mass, l u m i n o s i t y , effective t e m p e r a t u r e a n d c o m p o s i t i o n o f t h e s t a r are r e q u i r e d .  I n m o s t cases, t h e r e are 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  t h a t y i e l d i n f o r m a t i o n a b o u t t h e effective t e m p e r a t u r e w i t h a t o m i c n u m b e r s g r e a t e r t h a n 2) c o n t e n t .  a n d the heavy m e t a l  (elements  T h e l u m i n o s i t y is also e s t i m a t e d  from  p h o t o m e t r i c o b s e r v a t i o n s , or a p p a r e n t m a g n i t u d e a n d p a r a l l a x m e a s u r e m e n t s ; w h i l e t h e m a s s is o n l y t i g h t l y c o n s t r a i n e d i f t h e s t a r b e l o n g s t o a b i n a r y f r o m w h i c h a c o n f i d e n t o r b i t a l g e o m e t r y m a y b e d e r i v e d . T h e r e is also t h e p o t e n t i a l t o use a s t e r o s e i s m i c o b servations to further constrain the evolutionary status of a star t h r o u g h parameters like t h e l a r g e s p a c i n g A i / (see E q u a t i o n 1.7).  E v e n i f a stellar modeler h a d i n f o r m a t i o n o n  a l l these o b s e r v a b l e q u a n t i t i e s , t h e s t a n d a r d t r e a t m e n t o f c o n v e c t i v e e n e r g y  transport  a d d s a n o t h e r free p a r a m e t e r k n o w n as t h e m i x i n g l e n g t h p a r a m e t e r a ( B o h m - V i t e n s e ,  Chapter 3. Stellar Evolution 1958).  and Pulsation  Models  50  T h i s p a r a m e t e r sets t h e n u m b e r o f p r e s s u r e s c a l e - h e i g h t s a c o n v e c t i v e  element  rises before r e l e a s i n g i t s h e a t t o t h e s u r r o u n d i n g p l a s m a . R e c e n t l y , C u n h a et a l . (2003) e x a m i n e d a set o f s t e l l a r m o d e l s f o r H R 1217 i n h o p e s of d e t e r m i n i n g i t s e v o l u t i o n a r y s t a t u s . T h e i r r e s u l t s a r e c o m p l i m e n t a r y t o t h o s e o f t h i s thesis a n d t h e r e i s a n o v e r l a p i n t h e choice o f p a r a m e t e r s t o e x p l o r e . T h e effective t e m p e r a t u r e o f H R 1217 is e s t i m a t e d f r o m t w o sources.  Ryabchikova  et a l . (1997) use t h e i r s p e c t r a l s y n t h e s i s c o d e t o e s t i m a t e a n effective t e m p e r a t u r e o f T ff  = 7250; w i t h o u t a q u o t e d u n c e r t a i n t y . T h e o t h e r e s t i m a t e o f t h e T ff  e  e  comes from  t h e S t r o m g r e n p h o t o m e t r y o f M o o n k D w o r e t s k y (1985). M a t t h e w s et a l . (1999) u s e d t h i s p h o t o m e t r y t o e s t i m a t e a n effective t e m p e r a t u r e o f 7400 ± 100 K . I n t h i s w o r k , w e c o m b i n e these t o r e s u l t s t o o b t a i n T f ef  =  K . T h i s is identical t o the constraints  7400^200  o n effective t e m p e r a t u r e u s e d b y C u n h a et a l . (2003). T h e m a s s o f H R 1217 i s e s t i m a t e d f r o m p r e v i o u s A s t a r m o d e l s t o b e a p p r o x i m a t e l y 2.0 ± 0 . 5 M © (e.g., S h i b a h a s h i k  S a i o , 1985, a n d M a t t h e w s et a l , 1999).  c o n s t r a i n t f o r t h e m a s s o f H R 1217 o f 1.8 ± O . 3 M it is a r i g i d rotator.  0  A tighter  i s d e r i v e d b y W a d e (1997) a s s u m i n g  I n t h i s s t u d y , t h e masses were v a r i e d f r o m 1.3 t o 1 . 8 M  0  i n 0.5M©  steps. T h e l u m i n o s i t y o f H R 1217 i s also e s t i m a t e d u s i n g t w o different a p p r o a c h e s . first  is f r o m t h e o b s e r v e d l a r g e s p a c i n g o f w 6 8 fxEz.  The  U s i n g E q u a t i o n 1.9 w i t h t h e  a b o v e e s t i m a t e s o n T ff, t h e m a s s , a n d t h e large s p a c i n g , a a s t e r o s e i s m i c l u m i n o s i t y e  is c a l c u l a t e d t o b e 8.2tl'^L . Q  I f w e observe h a l f t h e l a r g e s p a c i n g o f « 3 4 / . H z , t h e  l u m i n o s i t y is estimated t o be 2 O . 7 l ; g . L . 4  0  T h e second m e t h o d is a direct c a l c u l a t i o n  of t h e s t a r s l u m i n o s i t y f r o m t h e Hipparcos l u m i n o s i t y o f 7.8 ± O . 7 L  0  f r o m t h e Hipparcos  parallax.  M a t t h e w s et a l . (1999) d e r i v e a  p a r a l l a x ; c o n s i s t e n t w i t h a Au « 6 8 / i H z .  T h e m e t a l l i c i t y o f H R 1217 i s e s t i m a t e d f r o m t h e s p e c t r o s c o p y o f R y a b c h i k o v a et a l . (1997), w h i c h i n d i c a t e s [Fe/H]  « 0.32 ± 1 6 % . A s s u m i n g F e i s a t r a c e r o f t h e i n t e r i o r  m e t a l c o n t e n t o f a s t a r , a h e a v y m e t a l m a s s f r a c t i o n o f Z — 0.008 c a n b e e s t i m a t e d f r o m the relations  X + Y + Z=l,  (3.1)  Chapter 3. Stellar Evolution  and Pulsation  Models  51  (3.2) and  AY _ (n - Y ) AZ (Z* - Z )  (3.3)  p  e  w h e r e X a n d Y are t h e m a s s f r a c t i o n s o f h y d r o g e n a n d h e l i u m a n d AY I AZ is t h e G a l a c t i c enrichment parameter.  T h e subscripts p, * a n d © denote p r i m o r d i a l , stellar a n d solar  v a l u e s , r e s p e c t i v e l y . T o c o m p l e t e t h e c a l c u l a t i o n , v a l u e s o f (Z/X)® et a l , 1996), Y  p  = 0.0244 ( G r e v e s s e  = 2.232 ± 0.003 ( O l i v e & S t e i g m a n , 1995), a n d AY j AZ  = 2.5 ( B r e s s a n  et a l . , 1994) were a d o p t e d . A d i f f i c u l t y arises for t h e A p s t a r s i n t h a t t h e i r i n f e r r e d m e t a l c o n t e n t f r o m s p e c t r a d r a m a t i c a l l y changes over t h e m a g n e t i c p h a s e o f t h e s t a r .  R y a b c h i k o v a et a l . (1997)  g i v e v a l u e s o f m e t a l l i c i t y for b o t h m a x i m u m a n d m i n i m u m m a g n e t i c phases.  I f i t is  ( n a i v e l y ) a s s u m e d t h a t t h e average o f a l l o f t h e m e t a l s for e a c h o f t h e m a g n e t i c m a x i m u m a n d m i n i m u m phases represent t h e i n t e r i o r h e a v y m e t a l c o n t e n t o f H R 1217, one c a n c a l c u l a t e f r o m E q u a t i o n s 3.1 - 3.3, a Z t h a t v a r i e s b e t w e e n 0.019 a n d 0.024. T h u s , t h e i n f e r r e d i n t e r i o r Z for H R 1217 c a n v a r y over a large r a n g e d e p e n d i n g o n h o w t h e surface m e t a l l i c i t y is u s e d . It s h o u l d b e n o t e d t h a t t h e r e is n o r e a s o n t o b e l i e v e t h a t t h e surface m e t a l c o n t e n t o f t h e A p s t a r s is r e p r e s e n t a t i v e o f t h e i n t e r i o r m e t a l c o n t e n t . H o w e v e r , for completeness, Z  =  0.008 t o 0.022 i n steps o f 0.002 is u s e d for t h e e v o l u t i o n a r y m o d e l s ;  along w i t h a n estimated hydrogen mass fraction of X  =  0.700 t o 0.740 i n s t e p s o f 0.020.  T h e l a r g e e x t e n t o f t h e p a r a m e t e r space i n c o m p o s i t i o n e n c o m p a s s e s t h e u n c e r t a i n t i e s f r o m t h e o b s e r v e d (Z/X) , Q  Y , AY/AZ, p  a n d [Fe/H] above.  F i n a l l y , m i x i n g l e n g t h p a r a m e t e r s o f a = 1.4, 1.6, a n d 1.8 are u s e d i n t h e m o d e l g r i d . If a s t a r possesses a c o n v e c t i v e envelope, a m a y b e u s e d t o set t h e a d i a b a t i c t e m p e r a t u r e g r a d i e n t at t h e base o f t h e c o n v e c t i v e z o n e . S i n c e t h e r a d i u s is a f u n c t i o n o f l u m i n o s i t y a n d T ff, v a r y i n g a w i l l s l i g h t l y change t h e r a d i u s o f t h e m o d e l ( G u e n t h e r et a l . , 1 9 9 2 ) . e  S i n c e t h e e n v e l o p e s o f A s t a r s are e s s e n t i a l l y r a d i a t i v e , a n y s t r u c t u r a l c h a n g e s  from  t h e different v a l u e s o f a are s m a l l w h e n c o m p a r e d t o t h e t o t a l p a r a m e t e r s p a c e b e i n g considered.  Chapter 3. Stellar Evolution  3.1.2  and Pulsation  Models  52  Model Properties  T o get a n i n d i c a t i o n o f t h e p a r a m e t e r s p a c e e x p l o r e d , t h e e x t r e m e s o f m a s s a n d c o m p o s i t i o n are p l o t t e d o n t h e t h e o r e t i c a l H e r t z s p r u n g - R u s s e l ( H R ) d i a g r a m i n F i g u r e 3.1. S h o w n are t h e e r r o r b o x e s c a l c u l a t e d f r o m t h e Hipparcos  l u m i n o s i t y , a n d b o t h t h e as-  t e r o s e i s m i c l u m i n o s i t i e s . It is clear t h a t t h e e r r o r b a r s for t h e a s t e r o s e i s m i c l u m i n o s i t y c a l c u l a t e d f r o m a Av/2  ~ 3 4 fiRz are o n l y c r o s s e d b y h i g h m a s s m o d e l s w i t h Z v a l u e s  t h a t are g r e a t e r t h a n t h e s o l a r v a l u e of Z «  0.02.  It is also e v i d e n t t h a t t h e  change  i n t h e h e a v y m e t a l c o n t e n t o f t h e m o d e l s p r o d u c e s t h e largest c h a n g e i n t h e p o s i t i o n o f t h e m o d e l o n t h e H R d i a g r a m . T h e r e were a p p r o x i m a t e l y 1 0 s t e l l a r m o d e l s 5  generated  for t h e m o d e l g r i d . D e t a i l e d m o d e l s ( c o n t a i n i n g i n f o r m a t i o n n e c e s s a r y for t h e p u l s a t i o n a n a l y s i s ) were o u t p u t i n age steps of 0.05 G y r f r o m t h e Z A M S , d e c r e a s i n g t h e of m o d e l s b y a n o r d e r o f m a g n i t u d e .  O f t h o s e m o d e l s , 569 f a l l w i t h i n t h e  number Hipparcos  l u m i n o s i t y e r r o r b o x ( H L E B ) . T h e m a j o r i t y o f these m o d e l s h a v e a m a s s o f «  1.6M  0  a n d a m e t a l l i c i t y r a n g e f r o m Z « 0.014 t o Z « 0.020. A f u l l l i s t i n g o f a l l o f these m o d e l s can be found i n T a b l e B . l of A p p e n d i x B . It is also i n t e r e s t i n g t o l o o k at t h e c o n v e c t i v e p r o p e r t i e s o f t h e c a l c u l a t e d m o d e l s since convection has a n influence o n b o t h the e v o l u t i o n a r y status a n d p u l s a t i o n a l properties of a s t a r , as w e l l as i m p l i c a t i o n s for t h e surface c h e m i c a l i n h o m o g e n e i t i e s o f A p s t a r s . F i g u r e 3.2 s h o w s t h e e v o l u t i o n o f b o t h t h e c o n v e c t i v e core m a s s a n d t h e  convective  e n v e l o p e m a s s for t h e m o d e l s s h o w n i n t h e H R d i a g r a m ( F i g u r e 3.1). M o d e l s w i t h h i g h e r Z a n d X h a v e c o n v e c t i v e cores t h a t last l o n g e r t h a n t h o s e w i t h t h e l o w e r Z a n d X m a s s fractions. I n general, a m o d e l w i t h a given mass t h a t has a higher m e t a l content w i l l take l o n g e r t o r e a c h t h e s a m e e v o l u t i o n a r y s t a t e as a m o d e l w i t h a l o w e r m e t a l c o n t e n t .  The  s a m e is t r u e for m o d e l s w i t h a l a r g e r h y d r o g e n m a s s f r a c t i o n ; a l t h o u g h t o a lesser e x t e n t . I n essence, t h e m o d e l s w i t h t h e lower Z c o n t e n t c o n t a i n m o r e n u c l e a r fuel; i n c r e a s i n g e n e r g y p r o d u c t i o n i n t h e n u c l e a r b u r n i n g core. F o r a g i v e n Z , t h e m o d e l s w i t h a h i g h e r X h a v e less efficient e n e r g y t r a n s p o r t i n t h e i n t e r i o r b e c a u s e o f t h e i r s m a l l e r c o n v e c t i v e cores.  Chapter 3. Stellar Evolution and Pulsation Models T  '  •  1  '  1  '  53 '  '  '  1  '  '  '  r  Log(T ) eff  F i g u r e 3 . 1 : A theoretical H R d i a g r a m showing the extremes of the parameter space. T h e i n n e r r e d e r r o r b o x a r o u n d <g) r e p r e s e n t s t h e  Hipparcos  l u m i n o s i t y w h i l e the error boxes a r o u n d the s y m b o l s o a n d * represent t h e l u m i n o s i t i e s c a l c u l a t e d f r o m a Au o f 68 / / H z a n d Au/2 / / H z . A l s o s h o w n are t w o l i n e s o f c o n s t a n t Au.  34  T h e m e t a l l i c i t y has  t h e greatest effect o f t h e e v o l u t i o n a r y s t a t u s o f t h e s t e l l a r m o d e l : s h i f t i n g t h e t r a c k s t o w a r d t h e lower r i g h t for l o w e r Z v a l u e s .  T h e m a s s o f t h e c o n v e c t i v e envelope is i n s i g n i f i c a n t i n these m o d e l s . I n fact, i t is o n l y a few o f t h e m o d e l s s h o w n i n F i g u r e 3.2 t h a t e n v e l o p e c o n v e c t i o n is a l a r g e f r a c t i o n o f t h e e n v e l o p e m a s s ( « 1% o f t h e m o d e l m a s s ) .  T h e s e m o d e l s m a y b e m o r e efficient at  m i x i n g away patches of chemical inhomogeneities; r u l i n g out low mass a n d h i g h Z models as c a n d i d a t e s A p s t a r s . F o r e v o l u t i o n a r y m o d e l s t h a t f a l l w i t h i n t h e H L E B , t h e s p r e a d i n Au  is s m a l l . T h i s  Chapter 3. Stellar Evolution  and Pulsation  54  Models  -=1.311.U-l.SM...  33 \  U-l.BM.-  S  Z-0.008  o 33  Z-0.022  o o o > a o  0.2  h  CJ  2  3  age [Gyr.] 1 • ' ' 1i i i i I 1  20.008 \  _  "^0.006 V  H-1.3M.  '  •  1 '  '  T  _  r  _  U-l.SU.  s  <o  i  Z = 0.022 X = 0.700  -  X = 0.740  a.  > c "0.004  a> _> *J o  -  0) C  u00.002  1 'X ; L/! 1  i  i  i  i 1 • • 2  ''i  i! i i 3  age [Gyr.]  Figure 3.2:  T h e c o n v e c t i v e core m a s s (top) a n d t h e c o n v e c t i v e e n v e l o p e m a s s ( b e l o w ) as a f u n c t i o n o f age for t h e m o d e l s s h o w n i n F i g u r e  3.1.  T h e masses are n o r m a l i z e d t o t h e m a s s o f t h e s t a r . T h e e n v e l o p e r e p r e s e n t s a p p r o x i m a t e l y t h e o u t e r 5% o f t h e m o d e l r a d i u s .  is i l l u s t r a t e d i n F i g u r e 3.3.  T h e e r r o r b a r s i n F i g u r e 3.3 r e p r e s e n t t h e m a x i m u m a n d  m i n i m u m v a l u e s o f Au a n d X / Z for a g i v e n m a s s . T h e o n l y m o d e l s t h a t d o n o t r e p r o d u c e  Chapter 3. Stellar Evolution  and Pulsation  Models  t h e o b s e r v e d A z / o f 68 / x H z are t h e l o w Z , 1 . 3 M  0  55  models. Once again, the models that  s e e m t o b e s t r e p r o d u c e t h e o b s e r v e d large s p a c i n g are t h o s e w i t h a m a s s o f 1.5 t o 1 . 6 M © a n d a c o m p o s i t i o n o f ( X , Z ) « (0.700, 0.017). It s h o u l d b e n o t e d t h a t t h e l a r g e s p a c i n g s p r e s e n t e d i n t h i s p l o t are c a l c u l a t e d d i r e c t l y f r o m t h e i n t e g r a l d e f i n i t i o n i n E q u a t i o n 1.7, a n d n o t f r o m t h e s p a c i n g o f t h e c a l c u l a t e d o s c i l l a t i o n frequencies.  Spacings calculated  u s i n g these t w o m e t h o d s differ f r o m e a c h o t h e r b y a few / x H z ; d u e t o t h e i n t e g r a l a v e r a g i n g of t h e s o u n d s p e e d a n d n o n a d i a b a t i c effects. A l s o s h o w n o n t h e r i g h t i n F i g u r e 3.3 is t h e s p r e a d i n age for e a c h o f t h e m o d e l s t h a t f a l l i n t h e H L E B . A s is e x p e c t e d , t h e m o d e l s w i t h t h e lowest m a s s a n d t h e h i g h e s t X / Z r a t i o are t h e o l d e s t .  3.2  Pulsation Models  T h e p u l s a t i o n c a l c u l a t i o n s are c a r r i e d o u t u s i n g t h e n o n a d i a b a t i c p u l s a t i o n p a c k a g e o f D a v i d G u e n t h e r ; J I G 8 ( G u e n t h e r , 1994).  J I G 8 solves t h e s i x l i n e a r i z e d , n o n a d i a b a t i c  equations of n o n r a d i a l stellar oscillations using the Henyey r e l a x a t i o n m e t h o d (Henyey et a l . , 1 9 6 4 ) . T h e s e s i x c o m p l e x p a r t i a l d i f f e r e n t i a l e q u a t i o n s d e s c r i b e t h e r a d i a l d e p e n dence of the v e r t i c a l a n d h o r i z o n t a l displacement vectors, the L a g r a n g i a n p e r t u r b a t i o n s t o t h e e n t r o p y a n d t h e r a d i a t i v e l u m i n o s i t y , as w e l l as t h e E u l e r i a n p e r t u r b a t i o n s t o t h e g r a v i t a t i o n a l p o t e n t i a l a n d its r a d i a l derivative. T h e t i m e dependence of the  perturbed  q u a n t i t i e s is p e r i o d i c t h r o u g h t h e f u n c t i o n exp(iut); w h e r e t h e u is t h e c o m p l e x e i g e n frequency.  A complete i n t r o d u c t i o n to b o t h the adiabatic a n d n o n a d i a b a t i c , n o n r a d i a l  o s c i l l a t i o n e q u a t i o n s c a n b e f o u n d i n U n n o et a l . (1989). T h e n o n a d i a b a t i c e q u a t i o n s are s o l v e d u s i n g t h e E d d i n g t o n a p p r o x i m a t i o n ( S a i o &; C o x , 1 9 8 0 ) . R a d i a t i v e losses a n d g a i n s are t a k e n i n t o a c c o u n t t h r o u g h t h e r a d i a t i v e f l u x i n the entropy equation  w h e r e T , s, t, a n d p are t h e t e m p e r a t u r e , e n t r o p y , t i m e , a n d t h e d e n s i t y , r e s p e c t i v e l y . T h e v e c t o r F is t h e s u m o f t h e r a d i a t i v e a n d c o n v e c t i v e fluxes. I f t h e p e r t u r b a t i o n s  to  Chapter  3. Stellar  Evolution  and Pulsation  Models  100  100  Figure  3.3: T h e variation of Az/ i n / / H z (top) and the model age i n G y r (bottom) as a function of the ratio X / Z for models that fall w i t h i n the Hipparcos  errorbars. T h e errorbars on the plot represent the  m a x i m u m and m i n i m u m values for each mass.  Chapter 3. Stellar Evolution  and Pulsation  57  Models  t h e d i v e r g e n c e o f t h e c o n v e c t i v e f l u x are i g n o r e d , F b e c o m e s ; i n t h e E d d i n g t o n a p p r o x i mation, P r -  _  < L [47r fe«  3«p  + _!.T£l  V  A-KK  (3.5)  at]  where c is the speed of light, a is the r a d i a t i o n density constant, a n d K is the opacity. T h e p r o c e s s b y w h i c h J I G 8 o b t a i n s t h e e i g e n f u n c t i o n s a n d frequencies i n v o l v e s five steps o u t l i n e d b y G u e n t h e r ( 1 9 9 4 ) .  A f t e r a n i n i t i a l guess o f t h e o s c i l l a t i o n f r e q u e n c y  a n d a degree £ o f t h e m o d e , t h e i n n e r t u r n i n g p o i n t r i s a p p r o x i m a t e d f r o m t h e r u n o f t  — u/y/£(£  s o u n d s p e e d c i n t h e m o d e l u s i n g t h e r e l a t i o n c (r )/r s  s  s h e l l is set a t r a d i u s w h e r e t h e wave n u m b e r i s 1 0  t  - 8  t  + 1). T h e innermost  times smaller then the value of inner  t u r n i n g p o i n t . T h e a d i a b a t i c o s c i l l a t i o n e q u a t i o n s are s o l v e d w i t h t h e o u t e r m e c h a n i c a l b o u n d a r y c o n d i t i o n r e m o v e d ; p r o v i d i n g a d i s c r i m i n a n t t o o b t a i n a n i n i t i a l guess for t h e a d i a b a t i c e i g e n f u n c t i o n s (see U n n o et a l . , 1989). O n c e t h i s i n i t i a l guess i s o b t a i n e d , t h e complete linearized adiabatic equations are solved t h r o u g h t h e r e l a x a t i o n m e t h o d . N e x t , t h e n o n a d i a b a t i c o s c i l l a t i o n e q u a t i o n s are s o l v e d w i t h t h e o u t e r m e c h a n i c a l b o u n d a r y c o n d i t i o n r e m o v e d u s i n g t h e r e a l eigenfrequency o b t a i n e d f r o m t h e a d i a b a t i c c a l c u l a t i o n s . A n i n i t i a l guess for t h e i m a g i n a r y p a r t o f t h e n o n a d i a b a t i c f r e q u e n c y i s t h e n c a l c u l a t e d f r o m t h e w o r k a n d k i n e t i c e n e r g y i n t e g r a l s b y use o f t h e r e l a t i o n W(R )/wk  =  e  —4nr). H e r e , 77 i s k n o w n as t h e g r o w t h r a t e o f t h e m o d e a n d i t i s t h e r a t i o b e t w e e n t h e i m a g i n a r y a n d r e a l p a r t s o f t h e o s c i l l a t i o n frequency. T h e w o r k a n d k i n e t i c e n e r g y o f t h e mode are given, respectively b y W(r) = - 4 7 r r I m ( £ p * £ ) 2  2  r  (3.6)  and  f  M  1  k = 2 ld J  w  U  o  I  5  v  I  2 d  m  (-) 3  7  w h e r e 8p* is t h e L a g r a n g i a n pressure v a r i a t i o n , a n d £ i s t h e v e r t i c a l p a r t o f t h e r a d i a l r  d i s p l a c e m e n t , Sr. F i n a l l y , t h e e i g e n f u n c t i o n s a n d eigenfrequencies o b t a i n e d i n t h e l a t t e r s t e p a r e u s e d t o calculate the s o l u t i o n t o the full linearized nonadiabatic n o n r a d i a l oscillation equations using the Henyey method.  Chapter 3. Stellar Evolution  3.2.1  and Pulsation  Models  58  Model Frequencies  Since p h o t o m e t r i c stellar p u l s a t i o n observations do not give disc-resolved i n f o r m a t i o n , the most i m p o r t a n t diagnostic of stellar structure comes from the frequency spacing described i n S e c t i o n s 1.4.2 a n d 1.5. I n p a r t i c u l a r , t h e l a r g e s p a c i n g Au gives i n f o r m a t i o n a b o u t t h e m e a n d e n s i t y o f t h e s t a r a n d t h e s e c o n d o r d e r s p a c i n g 8^  provides information about  t h e m e a n m o l e c u l a r w e i g h t n e a r t h e s t e l l a r core; i.e., age i n f o r m a t i o n . I n t h i s s e c t i o n , we present t h e p u l s a t i o n r e s u l t s for a l l s t e l l a r m o d e l s t h a t lie w i t h i n t h e  HLEB.  T h e m a t c h i n g o f m o d e l frequencies t o o b s e r v e d frequencies is a s u b j e c t i v e  process  since t h e r e are u s u a l l y m o r e m o d e s e x c i t e d i n m o d e l s t h a n are o b s e r v e d . T h e m o d e l t h a t r e p r o d u c e s t h e o b s e r v e d o s c i l l a t i o n s p e c t r u m o f a s t a r is n o t u n i q u e . R e c e n t l y , h o w e v e r , G u e n t h e r & B r o w n (2004) h a v e d e v e l o p e d a m e t h o d t o i m p r o v e t h e m a t c h i n g o f m o d e l s to observations.  T h e i r m e t h o d involves m i n i m i z i n g a x  2  of m o d e l frequencies a n d a n o b s e r v e d s p e c t r u m .  statistic between a large grid  O n c e their m o d e l g r i d is e x p a n d e d to  a m o r e d i v e r s e p a r a m e t e r space, t h i s m e t h o d p r o m i s e s t o b e a m u c h m o r e q u a n t i t a t i v e m e t h o d of m o d e identification. T h e c o m p l e x i t y r e s u l t i n g i n t h e l a r g e n u m b e r o f o s c i l l a t i o n m o d e s e x c i t e d i n a set of s t e l l a r m o d e l s is i l l u s t r a t e d i n F i g u r e 3.4.  T h e p l o t o f Au  vs.  u is k n o w n as a n  echelle d i a g r a m . A n echelle d i a g r a m s h o w s r i d g e s o f p o w e r for a g i v e n set o f o s c i l l a t i o n frequencies.  F o r example, lines of c o m m o n £ r u n vertically o n the d i a g r a m , while the  h i g h e r o r d e r p - m o d e frequencies (larger n values) are s o r t e d h o r i z o n t a l l y . I n F i g u r e 3.4, t h e n o n a d i a b a t i c frequencies a n d l a r g e s p a c i n g are p l o t t e d for a l l o f t h e 1 . 5 M © m o d e l s w i t h h y d r o g e n m a s s f r a c t i o n o f 0.700 a n d m e t a l m a s s f r a c t i o n s o f 0.012 a n d 0.014. T h e c a l c u l a t e d frequencies a l t e r n a t e b e t w e e n e v e n a n d o d d £ v a l u e s as t h e o r d e r o f t h e m o d e s is i n c r e a s e d .  T h e v e r t i c a l d o t t e d l i n e s are frequencies i d e n t i f i e d i n  t h i s s t u d y for H R 1217 (see S e c t i o n 2.5). S e l e c t i n g t h e p r o p e r s t e l l a r m o d e l , a n d e v e n t h e c o r r e c t degree o f t h e m o d e , is v e r y difficult. I n fact, e v e n t h o u g h t h e r e are m o r e m o d e s e x c i t e d for t h e Z = 0.014 m o d e l s , t h e r e are a s i g n i f i c a n t n u m b e r o f m o d e s c a l c u l a t e d f r o m t h e Z = 0.012 m o d e l s n e a r Au «  68 / i H z .  Chapter 3. Stellar Evolution  and Pulsation  Models  59  90  60 2400  1 = 0  M = 1.5 M , X = 0.700  1= 1 1=2  Z = 0.013  0  Z =  o  0.014  2600  2800  3000  i/[AtHz]  Figure 3.4: A n echelle d i a g r a m for t h e 1 . 5 M © m o d e l s w i t h X = 0.700, Z 0.012 a n d Z = 0.014.  T h e v e r t i c a l b l a c k l i n e s are t h e  i d e n t i f i e d for H R 1217 i n S e c t i o n 2.5.  N o t e the modes  =  frequencies alternate  b e t w e e n b o t h e v e n a n d o d d degrees.  T h e d e p e n d e n c e o f t h e f r e q u e n c y s p a c i n g o n age a n d c o m p o s i t i o n is s h o w n i n F i g u r e 3.5. T h e echelle left d i a g r a m i n t h i s figure s h o w 1 . 6 M m o d e l s w i t h different c o m p o s i t i o n s 0  a n d ages. T h e m o d e l w i t h Z = 0.016, X = 0.720 a n d a n age o f 1.05 G y r c o m e s closest  Chapter  3. Stellar  Evolution  1 •  •  •  1 • •  1  1  1  and Pulsation 1  1  1  .  i  i  |  Models  60  ,  J5.BO Gyr  • • "» " *  •  •  0.90  •a  7. = 0.014 Z = 0.016  •* * * * • " "  .•  0.80  Z = 0.020 X =700  A  -  X =720  °" ° ° *.»  8  -  X =740  ° ° ° ° ° ° »!  .  _  # = Model Age M = 160 M  0 5  e  a = l.B  -  -  1.10 -o  0  °  °  °  °  - ° °°° 0  o „  0  0  {  1.20  1  -  °  a  ° °„ ° ° = > o o o o  8  1  2600  2B00  ,  .  .  1  .  3000 v( Hz)  2600  Figure  2800 "(MHZ)  M  3 . 5 : O n t h e left, a n echelle d i a g r a m for 1 . 6 M  Q  models.  These models  were s e l e c t e d b a s e d o n t h e d i v e r s i t y o f t h e c o m p o s i t i o n a n d of t h e m o d e l s for t h i s m a s s .  age  T h e plot below shows the second or-  der s p a c i n g 5(2) as a f u n c t i o n o f frequency.  T h e number labels o n  t h e p l o t i n d i c a t e t h e m o d e l s age; w h e r e t h e o p e n s y m b o l s are t h e y o u n g e r m o d e l s . T h e l e g e n d for t h e p l o t o n t h e left a p p l i e s t o b o t h plots. t o t h e 68 / i H z ; h o w e v e r , t h e y o u n g e r m o d e l s w i t h a s o l a r Z o f 0.020 a n d a l o w e r X  =  0.700 are a l s o close. T h e o l d e r m o d e l s s h o w a decrease i n t h e l a r g e s p a c i n g ; i n d i c a t i n g a n i n c r e a s i n g s t e l l a r r a d i u s (decrease i n m e a n d e n s i t y ) as t h e s t a r e v o l v e s t h r o u g h  the  error box. T h e p l o t o n t h e r i g h t i n F i g u r e 3.5 s h o w s t h e s e c o n d o r d e r s p a c i n g as a f u n c t i o n o f frequency. It s h o u l d b e n o t e d t h a t t h i s is less t h a n a b o u t 10 / i H z for a l l o f t h e p u l s a t i o n m o d e l s c a l c u l a t e d for t h i s t h e s i s . T h i s d i a g n o s t i c d i a g r a m s h o w s t h e s a m e g e n e r a l f e a t u r e s  Chapter 3. Stellar Evolution  and Pulsation  Models  61  as t h o s e i n a n echelle d i a g r a m ; h o w e v e r , t h e m o d e l s w i t h a s m a l l s p a c i n g o f « 3 / / H z m a y b e i m p o r t a n t for t h e i n t e r p r e t a t i o n o f t h e f r e q u e n c y a n a l y s i s o f S e c t i o n s 2.5 a n d 2.6.  Table 3.1-. T e n m o d e l s s e l e c t e d f r o m t h e 569 t h a t f a l l w i t h i n t h e Hipparcos  luminosity error bars. These models sam-  p l e a r a n g e o f age, c o m p o s i t i o n , m a s s a n d m i x i n g l e n g t h . T h e full m o d e l listing m a y be found i n A p p e n d i x A .  del #  MQ  Log(T  e / /  )  Log(L/Z,©)  Age(Gyrs)  Av(fj.Hz)  X  Z  a  1  1.3  3.86815  0.897607  2.05  64.6873  0.7  0.008  1.6  2  1.3  3.85765  0.928366  2.15  57.6018  0.7  0.008  1.8  3  1.4  3.85757  0.851844  1.4  67.8035  0.7  0.01  1.8  4  1.4  3.86471  0.867916  1.5  68.8062  0.72  0.008  1.8  5  1.5  3.8739  0.865959  0.85  75.6791  0.7  0.014  1.4  6  1.5  3.86287  0.882933  1.2  68.4562  0.72  0.012  1.4  7  1.6  3.86042  0.880244  0.8  70.1876  0.7  0.02  1.8  8  1.6  3.87367  0.900453  0.9  73.6316  0.74  0.014  1.8  9  1.7  3.8715  0.90147  0.5  74.71  0.72  0.022  1.8  10  1.7  3.86716  0.888114  0.6  74.3194  0.74  0.02  1.8  T a b l e 3.1 selects t e n m o d e l s w i t h a r a n g e i n m a s s , age, c o m p o s i t i o n a n d m i x i n g l e n g t h p a r a m e t e r f r o m t h e 569 m o d e l s t h a t f a l l w i t h i n t h e H L E B . B o t h echelle d i a g r a m s a n d  5^  vs. v d i a g r a m s are s h o w n for e a c h of these m o d e l s i n F i g u r e s 3.6 a n d 3.7, r e s p e c t i v e l y . T h e echelle d i a g r a m s o f F i g u r e 3.6 s h o w t h e s a m e g e n e r a l t r e n d s as d i s c u s s e d e a r l i e r . T h e i n t e r e s t i n g t h i n g t o n o t e is t h a t m o d e l s 6 - 1 0 s l o w l y i n c r e a s e i n Au.  T h i s is c a u s e d b y  b o t h t h e y o u n g e r age o f t h e m o r e m a s s i v e m o d e l s , as w e l l as, t h e i r l o w e r m e a n d e n s i t i e s . T h e m o d e l s w i t h t h e s e c o n d o r d e r s p a c i n g o f 3 / / H z are a l s o e i t h e r o l d e r , or h a v e a m a s s less t h a n s~  1.5M . 0  Chapter 3. Stellar Evolution  2600  and Pulsation  Models  2800  i/OiHz)  62  2600  2800 K(MHZ)  F i g u r e 3.6: Echelle diagrams for the models listed i n Table 3.1.  T h e model  number is listed at the top of the diagram.  3.3  Magnetic Effects  T h e effects of the magnetic field on b o t h the oscillation frequencies and the eigenfunctions have been described by a number of authors (e.g., Shibahashi & Takata, 1993, Dziembowski & Goode, 1996, Bigot et al., 2000, and C u n h a h G o u g h , 2000). In the case of r o A p stars, pressures induced by the magnetic field dominate over gas pressure near the surface of the star. T h e usual perturbation techniques for calculating the magnetic effects on stellar oscillations are no longer valid. However, there has been recent success i n the interpretation of r o A p oscillations i n terms of a variational principle ( C u n h a , 2001). B y only calculating the perturbed eigenmodes and not the eigenfunctions, the method of C u n h a k. G o u g h (2000) is an attractive alternative to a perturbation analysis of magneto-acoustic modes. T h e work of C u n h a & G o u g h (2000) follows from that of C a m p b e l l & Papaloizou  Chapter 3. Stellar Evolution  2600  and Pulsation  Models  2800  63  2600  2800 I/(MHZ)  F i g u r e 3.7: S e c o n d o r d e r s p a c i n g d i a g r a m s for t h e m o d e l s l i s t e d i n T a b l e 3.1. T h e m o d e l n u m b e r i s l i s t e d at t h e t o p o f t h e d i a g r a m .  (1986). T h e s e a u t h o r s d i v i d e t h e s t a r i n t o a t h i n o u t e r b o u n d a r y l a y e r a n d t h e i n t e r i o r . I n t h e b o u n d a r y layer, t h e L o r e n t z forces are c o m p a r a b l e t o ; o r l a r g e r t h a n , t h e g a s p r e s s u r e , w h i l e i n t h e i n t e r i o r t h e field i s e s s e n t i a l l y force free. S i n c e t h e m a g n e t i c b o u n d a r y l a y e r is t h i n , a p l a n e - p a r a l l e l a p p r o x i m a t i o n m a y b e u s e d . It i s a l s o a s s u m e d t h a t t h e effects o f t h e m a g n e t i c field c a n b e c a l c u l a t e d l o c a l l y at e a c h l a t i t u d e . I n t h i s case, t h e field o n l y v a r i e s i n t h e v e r t i c a l d i r e c t i o n a n d h a s c o m ponents B =  (B ,0,B ). x  T h e m a g n e t o - a c o u s t i c waves a r e d e s c r i b e d u s i n g a h o r i z o n t a l  z  (k ,k ,0).  wavenumber k =  x  y  W i t h this information, the adiabatic, magnetically n o n -  diffusive, p u l s a t i o n e q u a t i o n s m a y t h e n b e w r i t t e n as ( C a m p b e l l & P a p a l o i z o u , 1986, a n d C u n h a & G o u g h , 2000)  -u pu 2  = i | k | W + (B • V ) — - A ? _ L (B • V ) ( V - 0 Mo Mo k | 2  (3.8)  Chapter 3. Stellar Evolution  -  W  and Pulsation  > = (B • V )  -w tf, = ^  Models  f +  2  Mo  64  Mo I K- I  (B • V) (V-0  (3-9)  - <?V- ( t f ) - — [(B • V) (V-0] + (B • V ) ^  2  (3.10)  2  MO  PO  where W  .Vp U+*)  (V.O-( - )< '^ B  +  i  =  \  M  u  V  /  B  (3.11)  Mo  and 9  In  = i ^  (3.12)  t h e a b o v e r e l a t i o n s , t h e d i s p l a c e m e n t v e c t o r £ is d e c o m p o s e d i n t o a v e r t i c a l c o m p o n e n t  £ = £ • e , a c o m p o n e n t t h a t i s p e r p e n d i c u l a r t o t h e w a v e n u m b e r v = £-(e x k ) / | k |, z  z  z  a n d a c o m p o n e n t t h a t is p a r a l l e l t o t h e w a v e n u m b e r u — (£ • k ) / | k |. T h e l o c a l g r a v i t a t i o n a l a c c e l e r a t i o n i s d e n o t e d b y g a n d t h e first a d i a b a t i c e x p o n e n t i s 7. A l l o t h e r s y m b o l s have their u s u a l meanings. In  t h e deep i n t e r i o r , a m a g n e t o - a c o u s t i c  mode completely decouples into a pure  A l f v e n i c m o d e a n d a pure acoustic mode. T h i s was verified a n a l y t i c a l l y b y R o b e r t s & S o w a r d (1983) a n d n u m e r i c a l l y b y C a m p b e l l & P a p a l o i z o u (1986). T h u s , i n t h e  JWKB  approximation, the magnetic modes i n the interior m a y be described b y t h e functions (cf.  C u n h a & G o u g h 2000)  K»,  u ) mz  ^ P  1/4  (C,  D)exp  n  p pu \  sr) ~  . f  z  p- '\C ,D+)exp l  +  where C , D , C + , a n d D  +  2 2  0  iz  (popu> \ 2  i  .k B z x  2  dz — i-  x  +  .k B z x  x  (3.13)  are c o m p l e x constants. T h e i n w a r d p r o p a g a t i n g A l f v e n waves  are e x p e c t e d t o d i s s i p a t e before t h e y a r e reflected b a c k t o w a r d t h e surface o f t h e s t a r ( R o b e r t s & S o w a r d , 1983). W i t h t h i s i n m i n d , t h e c o n s t a n t s C  +  and D  +  i n Equation  3.13 m a y t h e n b e set t o zero t o assure t h a t n o o u t w a r d l y p r o p a g a t i n g m a g n e t i c w a v e s occur i n the interior. In The  t h e i n t e r i o r , t h e v e r t i c a l c o m p o n e n t o f t h e u n c o u p l e d m o d e s is e s s e n t i a l l y a c o u s t i c . a m p l i t u d e o f t h i s v e r t i c a l m o d e m a y b e r e p r e s e n t e d a s y m p t o t i c a l l y b y (cf. C u n h a  Chapter 3. Stellar Evolution  and Pulsation  Models  65  k G o u g h 2000) Ax '  1 2  6*  ^  " ^7/2  c  o  P  s  (^J  x  d  z  z  +  P  S  )  (  3  -  1  4  )  1  were 5  P  is a p h a s e a n d K is t h e v e r t i c a l a c o u s t i c w a v e n u m b e r . T h e c o o r d i n a t e z a n d z*  r e p r e s e n t t h e d e p t h i n t h e b o u n d a r y layer a n d t h e p o s i t i o n o f t h e b a s e o f t h e b o u n d a r y layer, r e s p e c t i v e l y . T h e n u m e r i c a l s o l u t i o n s o f t h e s y s t e m of E q u a t i o n s 3.8 t o 3.10 are m a t c h e d t o t h e a s y m p t o t i c r e l a t i o n s 3.13 a n d 3.14 a t e a c h l a t i t u d e t o o b t a i n v a l u e s for t h e phases 5 . P  magnetic  T h e p u r e l y a c o u s t i c case, i.e, B = 0 i n E q u a t i o n s 3.8 t o 3.10, is also m a t c h e d  o n t o t h e a s y m p t o t i c r e l a t i o n s t o o b t a i n t h e u n p e r t u r b e d p h a s e s 5 Q. P h a s e shifts A5 p  are  P  t h e n o b t a i n e d f r o m t h e difference b e t w e e n these 2 p h a s e s at e a c h l a t i t u d e . W i t h t h e a b o v e i n f o r m a t i o n , t h e v a r i a t i o n a l m e t h o d o f C u n h a k G o u g h (2000) c a n b e used to estimate the  first-order  f r e q u e n c y shifts o f t h e e i g e n m o d e s c a u s e d b y a m a g n e t i c  f i e l d . T h e f r e q u e n c y shifts m a y b e c a l c u l a t e d f r o m  A5  Au  P  (3.15)  p  w h e r e t h e average o f t h e p h a s e shifts A5  P  f*  over a s p h e r e is  A6 {Y ) sm6d9 m  p  2  e  (  3  -  1  6  )  T h e q u a n t i t i e s w i t h a s u b s c r i p t o f zero are t h o s e e x p e c t e d i n t h e case o f n o m a g n e t i c field.  A complete description of the n u m e r i c a l procedure used to calculate the frequency  shifts is o u t l i n e d i n t h e a p p e n d i c e s o f C u n h a k G o u g h (2000).  3.3.1  The Frequency Perturbations  T h e p e r t u r b a t i o n t o t h e a c o u s t i c o s c i l l a t i o n s are c a l c u l a t e d u s i n g a n a d a p t e d v e r s i o n o f t h e c o d e u s e d b y C u n h a k G o u g h (2000). T h i s n e w v e r s i o n o f t h e c o d e w a s u p d a t e d b y C u n h a ( p r i v a t e c o m m u n i c a t i o n , 2003) t o r e a d t h e o u t p u t f r o m t h e e v o l u t i o n a r y m o d e l s p r e s e n t e d i n S e c t i o n 3 . 1 . F r e q u e n c y s h i f t s w e r e c a l c u l a t e d for m o d e l s 1 t o 10 i n T a b l e 3.1 u s i n g E q u a t i o n 3.15.  T h e m a g n e t i c p e r t u r b a t i o n s t o frequencies f r o m 9 0 0 t o 3 1 0 0  Chapter 3. Stellar Evolution  and Pulsation  Models Models  40  20  I  ' ' ' I I ' model: 1 2 3 4 5 1  '  1  1  1  66  1 - 5 40  1  1=1  h  4  <  V  -20  05  -40 \I  .  .  1000  . I  2000 f(yuHz)  20  _  .  1  1  1  3000  1000  2000  3000  1000  2000 i/(/itHz)  3000  1  -  15  N  X  ^  10 _  <  ^  5  ^\\\ /  oh  V  \ N \ / /  " 1 , , , , 1 , , , 1000 2000 3000 y(/iHz)  Figure 3.8: T h e  m a g n e t i c p e r t u r b a t i o n s c a l c u l a t e d for m o d e l s 1 t o 5 u s i n g a  v a r i a t i o n a l p r i n c i p l e . T h e u p p e r p l o t s s h o w t h e r e a l f r e q u e n c y shift for m o d e s w i t h degrees £ =  1 (left) a n d £ =  2 (right). T h e lower  p l o t s s h o w t h e i m a g i n a r y p a r t o f t h e f r e q u e n c y shifts for t h e s a m e degrees.  uRz are s h o w n i n F i g u r e s 3.8 a n d 3.9 a s s u m i n g a d i p o l a r m a g n e t i c field w i t h a p o l a r s t r e n g t h o f 4.0 k G . T h i s f r e q u e n c y r a n g e a n d m a g n e t i c field s t r e n g t h i s c o n s i s t e n t m e a s u r e m e n t s o f H R 1217 (see S e c t i o n 1.5).  with  Chapter 3. Stellar Evolution  and Pulsation  Models Models  1000  2000  3000  67  6-10  1000  2000  Figure 3 . 9 : T h e magnetic perturbations calculated for models 6 to 10 using a variational principle. T h e upper plots show the real frequency shift for modes w i t h degrees £ — 1 (left) and £ = 2 (right). T h e lower plots show the imaginary part of the frequency shifts for the same degrees.  E a c h of Figures 3.8 and 3.9 show the real and the imaginary frequency shifts for modes w i t h a degree of £ — 1 and 2. Note that there is very little difference i n the real frequency shifts for modes of different degree. T h e imaginary part of the £ = 2 modes  Chapter 3. Stellar Evolution  and Pulsation  Models  68  are; h o w e v e r , c l e a r l y l o w e r . T h i s w a s also s h o w n for t h e case o f a p o l y t r o p e s t e l l a r m o d e l b y C u n h a & G o u g h (2000). I n general, i f the i m a g i n a r y part of the frequency is positive, the m o d e loses energy.  magneto-acoustic  T h e s m a l l e r a m p l i t u d e of t h e i m a g i n a r y p a r t for t h e I = 2 m o d e s  suggest t h a t t h e y are m u c h less s u s c e p t i b l e t o d a m p i n g f r o m t h e m a g n e t i c effect d e p e n d s o n t h e g e o m e t r y o f t h e m o d e a n d t h e m a g n e t i c f i e l d .  field.  This  C u n h a Sz G o u g h  (2000) s h o w t h a t t h e m a x i m u m e n e r g y loss o f a m a g n e t o - a c o u s t i c m o d e o c c u r s a p p r o x i m a t e l y at t h e l a t i t u d e w h e r e B B Z  magnetic  field.  X  is a m a x i m u m .  T h i s o c c u r s at 4 5 ° for a d i p o l a r  M o d e s o f h i g h e r degree I decrease i n a m p l i t u d e as t h e y a p p r o a c h t h e  e q u a t o r m o r e q u i c k l y t h a n m o d e s o f lower degree.  W h e n t h e p h a s e shifts are a v e r a g e d  over a s p h e r e , t h e n e t effect i s a s m a l l e r i m a g i n a r y f r e q u e n c y s h i f t for m o d e s o f l a r g e r t. S i m p l y p u t , these m o d e s h a v e l o w e r a m p l i t u d e s n e a r t h e l a t i t u d e w h e r e t h e m a x i m u m o f e n e r g y loss o c c u r s . It is b e l i e v e d t h a t t h e r e s u l t i n g f r e q u e n c y shifts are c a u s e d b y e n e r g y loss t h r o u g h t h e c o u p l i n g o f i n w a r d l y p r o p a g a t i n g m a g n e t i c s l o w w a v e s a n d t h e a c o u s t i c modes ( C u n h a & G o u g h , 2000). T h e frequency j u m p s change very little between the models. T h i s makes it difficult to use t h e m a g n e t i c p e r t u r b a t i o n s as a d i s c r i m i n a n t b e t w e e n different e v o l u t i o n a r y m o d e l s . It is i m p o r t a n t t o n o t i c e t h e f r e q u e n c y shifts b e t w e e n m o d e s o f different degrees are o f t h e s a m e o r d e r as t h e s e c o n d - o r d e r s p a c i n g d e s c r i b e d i n S e c t i o n 3 . 2 . 1 . T h i s m a y h a m p e r efforts t o use a n o b s e r v e d 8^ 1996).  as a d i a g n o s t i c o f s t e l l a r s t r u c t u r e ( D z i e m b o w s k i & G o o d e ,  69  Chapter 4  Discussion and Conclusions E a r l y p h o t o m e t r i c o b s e r v a t i o n s o n t h e r o A p s t a r H R 1217 (e.g., K u r t z et a l . , 1989) s h o w e d a p u l s a t i o n p a t t e r n t h a t w a s c h a r a c t e r i s t i c o f h i g h - o r d e r p - m o d e s o b s e r v e d i n t h e S u n (see F i g u r e 1.4). T h e m a i n differences b e t w e e n t h e s o l a r o s c i l l a t i o n s p e c t r u m a n d t h a t o f H R 1217 were t h e o b s e r v e d a m p l i t u d e s a n d f r e q u e n c y s p a c i n g s . I n g e n e r a l , t h e a m p l i t u d e s of t h e r o A p s t a r s are g r e a t e r t h a n t h e s o l a r o s c i l l a t i o n a m p l i t u d e s . F r e q u e n c y s p a c i n g s suggest a A i / o f a p p r o x i m a t e l y 135 / / H z for t h e s u n (e.g., A i n d o w et a l . , 1 9 8 8 ) , w h i l e t h e i n f e r r e d l a r g e s p a c i n g for H R 1217 i s h a l f t h i s v a l u e a t 68 / / H z , c o n s i s t e n t w i t h a l o w e r m e a n d e n s i t y o f t h e A p s t a r . T h e d a t a r e d u c t i o n p r e s e n t e d i n C h a p t e r 2 for t h e X C O V 2 0 c a m p a i g n o n H R 1217 yields results t h a t are consistent w i t h those o b t a i n e d b y ( K u r t z et a l . , 1989) i n t h e 1986 c a m p a i g n . T h e t w o n o t a b l e e x c e p t i o n s a r e t h e a m p l i t u d e s o f t h e o s c i l l a t i o n m o d e s (see F i g u r e 2.7) a n d t h e n e w frequencies a t 2788.94 a n d 2 7 9 1 . 5 7 / / H z (see T a b l e 2 . 1 ) . T h e l a t t e r o f these n e w frequencies w a s also i d e n t i f i e d i n a p r e l i m i n a r y r e d u c t i o n o f t h e X C O V 2 0 d a t a b y K u r t z et a l . ( 2 0 0 2 ) . The  a m p l i t u d e s o f t h e t h i r d a n d f o u r t h frequencies i d e n t i f i e d i n t h e 1986 d a t a differ  f r o m t h o s e i d e n t i f i e d i n t h i s s t u d y (see S e c t i o n 2.5) b y & 6 0 % a n d 4 0 % r e s p e c t i v e l y . A l t h o u g h t h e a m p l i t u d e s b e t w e e n t h e X C O V 2 0 d a t a a n d t h e 1986 d a t a a r e c l e a r l y different, t h e n e t a m p l i t u d e difference (defined b y V J ^ o o o mmag.  —  ^1986])  i s o n l y —0.07 ± 0.192  T h i s suggests t h e r e i s a n e x c h a n g e o f p o w e r ( a m p l i t u d e s q u a r e d ) b e t w e e n t h e  observed modes while the net power is conserved. T h e exact m e c h a n i s m governing the e x c h a n g e o f p o w e r b e t w e e n i n d i v i d u a l m o d e s i s n o t k n o w n . It i s a l s o i n t e r e s t i n g t h a t t h e n e t f r e q u e n c y difference b e t w e e n t h e c o m m o n frequencies i d e n t i f i e d i n b o t h d a t a sets is o n l y  [^2000 — ^1986] = 0.56 ± 0.96 / / H z . T h i s c o r r e s p o n d s t o a n a v e r a g e  frequency  Chapter 4- Discussion  and  Conclusions  70  difference o f 0.09 ± 0 . 9 6 / / H z . T h e p e r s i s t e n c e o f these o b s e r v e d frequencies suggests t h a t t h e m o d e l i f e t i m e s i n H R 1217 are s t a b l e over t h e 14 y e a r t i m e g a p b e t w e e n o b s e r v a t i o n s . M o d e s e l e c t i o n a n d l i f e t i m e are p a r t i c u l a r l y i n t e r e s t i n g i n r o A p s t a r s . F o r e x a m p l e , t h e r o A p s t a r H R 3831 e x h i b i t s o s c i l l a t i o n m o d e s t h a t are s t a b l e over 20 y e a r s o f observ a t i o n s ( K u r t z et a l . , 1997) w h i l e t h e r o A p s t a r H D 60435 s h o w s s o m e m o d e l i f e t i m e s of less t h a n 7 d a y s ( M a t t h e w s et a l . , 1987). A l o n g b a s e l i n e for o b s e r v a t i o n s o n o t h e r r o A p s t a r s w o u l d b e i n v a l u a b l e t o o u r u n d e r s t a n d i n g o f m o d e e x c i t a t i o n ; w h i c h is s t i l l u n c e r t a i n (e.g., B a l m f o r t h et a l . , 2 0 0 1 ) . T h e f r e q u e n c y a n a l y s i s p r e s e n t e d i n S e c t i o n s 2.5 a n d 2.6 suggest t h a t t h e a m p l i t u d e l i m i t at w h i c h frequencies c a n c o n f i d e n t l y b e i d e n t i f i e d f r o m t h e c u r r e n t t i m e - s e r i e s red u c t i o n has been reached.  O n c e e a c h of t h e 21 frequencies h a d b e e n r e m o v e d b y t h e  u n w e i g h t e d d a t a a n a l y s i s , t h e r e s i d u a l s i m p r o v e d b y less t h a n « 0 . 1 % (see F i g u r e 2.8). T h e i m p r o v e m e n t between the unweighted frequency analysis a n d each of the weighted f r e q u e n c y r e d u c t i o n s p r e s e n t e d i n S e c t i o n 2.6 is also less t h a n 5 % (see F i g u r e 2 . 1 0 ) . W h i l e t h e w e i g h t e d n o i s e levels i n t h e F o u r i e r s p e c t r a o f F i g u r e s 2.11 a n d 2.12 are c l e a r l y l o w e r t h a n t h e noise of F i g u r e 2.6, t h e g e n e r a l r e s u l t s are t h e s a m e .  It i s n o t c l e a r h o w t h e  w e i g h t i n g o f t h e frequencies affects t h e d e t e r m i n a t i o n o f t h e c o n f i d e n c e levels d i s c u s s e d i n S e c t i o n 2 . 5 . 1 . R e g a r d l e s s , a l m o s t a l l o f t h e frequencies c a n b e d e s c r i b e d b y a t r i p l e t fine s t r u c t u r e . T h e e x c e p t i o n is t h e f r e q u e n c y u . 7  Even when v  7  is f o r c e d t o b e  fitted  as  a r o t a t i o n a l l y s p l i t t r i p l e t , t h e a m p l i t u d e o f one o f t h e c o m p o n e n t s is a l w a y s less t h a n 0.035 m m a g (see T a b l e s O l a n d C . 2 ) . T h i s is a l w a y s b e l o w t h e noise; e v e n w h e n i t is w e i g h t e d . A d o u b l e t s t r u c t u r e is n o t p r e d i c t e d b y t h e o b l i q u e p u l s a t o r m o d e l . It s h o u l d b e n o t e d t h a t l e a s t - s q u a r e s fits c a l c u l a t e d u s i n g Period98  always converged  close t o t h e s a m e a m p l i t u d e s , frequencies a n d p h a s e s . T h i s is i l l u s t r a t e d i n T a b l e s C . 5 t h r o u g h C . 1 0 o f A p p e n d i x C . T h e frequencies, a m p l i t u d e s , a n d p h a s e s were a v e r a g e d a n d a s t a n d a r d e r r o r o f t h e m e a n w a s c a l c u l a t e d for f r e q u e n c y s o l u t i o n s f r o m T a b l e 2.3 w i t h t h e s a m e i d e n t i f i e d frequencies. T h e s t a n d a r d e r r o r o n t h e frequencies, a n d p h a s e s are o f t h e o r d e r 1 0 to the robustness of  - 6  Period98.  mHz, 10  - 2  mmag, and 1 0  - 2  amplitudes  r a d i a n s . T h i s is a t e s t a m e n t  Chapter 4- Discussion  and  Conclusions  71  T h e absence o f t h e f r e q u e n c y u(old) (see T a b l e 2.1) i n t h e 1986 d a t a i s e x p l a i n e d b y C u n h a (2001) as s t r o n g m a g n e t i c d a m p i n g o f t h e o s c i l l a t i o n m o d e . T h e m o d e l s p r e s e n t e d i n S e c t i o n 3.3.1 (see F i g u r e s 3.8 a n d 3.9) are i n a g r e e m e n t w i t h h e r r e s u l t s for a p o l y t r o p e s t e l l a r m o d e l . T h e s e m o d e l s s h o w a n increase o f d a m p i n g t o w a r d t h e s e v e n t h f r e q u e n c y i n T a b l e 2.1 (2806.57 / / H z ) . T h i s m a y e x p l a i n t h e o b s e r v e d s p a c i n g b e t w e e n  frequencies  6 a n d 7 o f ~ 15 / / H z ; - 2 0 / / H z a w a y f r o m t h e e x p e c t e d v a l u e . A s d i s c u s s e d i n S e c t i o n 3.3.1, t h e m a g n e t i c d a m p i n g o f m o d e s decreases w i t h i n c r e a s i n g degree (larger £ v a l u e s ) . T h i s is a consequence o f t h e m o d e a n d m a g n e t i c field g e o m e t r y as w e l l as t h e w e i g h t i n g o f t h e m o d e s b y t h e v a r i a t i o n a l m e t h o d . T h e m o d e s w i t h t h e l a r g e r £ v a l u e s have s m a l l e r a m p l i t u d e s n e a r l a t i t u d e s o f m a x i m u m m a g n e t i c d a m p i n g . T h e n e t d a m p i n g o f t h e h i g h e r degree m o d e s i s less w h e n t h e average o f t h e m a g n e t i c effects i s c a l c u l a t e d for a l l l a t i t u d e s . D a m p i n g increases w i t h i n c r e a s i n g i m a g i n a r y f r e q u e n c y shift. T h e i m a g i n a r y shifts s h o w n i n F i g u r e s 3.8 a n d 3.9 a r e less for t h e £ = 2 modes t h a n the corresponding £ — 1 modes. T h e s p a c i n g b e t w e e n b o t h o f t h e n e w m o d e s (u(old) a n d u(new))  is approximately  2.63 / / H z . T h i s i s c o n s i s t e n t w i t h t h e s e c o n d - o r d e r f r e q u e n c y s p a c i n g s c a l c u l a t e d for t h e s t e l l a r m o d e l s i n S e c t i o n 3.2.1 (see F i g u r e s 3.5 a n d 3 . 7 ) . I f t h i s i s a s m a l l s p a c i n g , E q u a t i o n 1.10 shows; b y d e f i n i t i o n , t h a t t h e m o d e s differ i n degree b y 2. C o n s i d e r a s i n g l e m o d e o f l o w degree w i t h a n a m p l i t u d e t h a t i s s t a b i l i z e d b e l o w o b s e r v a b l e levels through magnetic damping.  If this m o d e exchanges power between itself a n d a m o d e  of l a r g e r degree, t h e a m p l i t u d e o f t h e n e w m o d e m a y g r o w t o o b s e r v a b l e levels b e c a u s e i t i s less d a m p e d b y m a g n e t o - a c o u s t i c i n t e r a c t i o n s . T h i s c o n j e c t u r e i s s u p p o r t e d b y t h e observation of two closely spaced modes i n the X C O V 2 0 d a t a while there is no evidence for s i m i l a r m o d e s i n t h e 1986 d a t a set. H o w e v e r , t h i s i s difficult t o p r e d i c t e m p i r i c a l l y since t h e m e c h a n i s m s for m o d e e x c i t a t i o n a n d d a m p i n g a r e s t i l l u n c e r t a i n . W h i l e t h e o b l i q u e p u l s a t o r m o d e l s d i s c u s s e d i n S e c t i o n 1.4.3 a r e n o t u s e d t o p r e d i c t the amplitude asymmetries observed i n this work, a prediction o f the models is tested u s i n g t h e fine f r e q u e n c y s p a c i n g o f t h e X C O V 2 0 d a t a .  T h e oblique pulsator model, i n  a l l i t s f o r m s , p r e d i c t s t h a t t h e frequencies a r e e x a c t l y s p l i t b y t h e r o t a t i o n f r e q u e n c y o f  Chapter 4- Discussion the star.  and  Conclusions  72  I f t h e m o d e s are d e s c r i b e d i n s t e a d b y t r a v e l i n g waves ( m ^ 0) d i s c u s s e d i n  S e c t i o n 1.4.1, o n e w o u l d e x p e c t d e p a r t u r e s f r o m t h e e x a c t s p a c i n g b e c a u s e o f t h e L e d o u x c o n s t a n t o f E q u a t i o n 1.5. B y a v e r a g i n g t h e difference b e t w e e n t h e fine s p a c i n g s 5u o f T a b l e 2.1 a n d t h e r o t a t i o n f r e q u e n c y o f 9.2897 x 1 0  - 4  m H z ( t h e average r o t a t i o n p e r i o d  d e r i v e d f r o m B a g n u l o et a l . (1995) a n d K u r t z & M a r a n g ( 1 9 8 7 ) ; see S e c t i o n 2 . 6 ) , t h e L e d o u x c o n s t a n t i s e s t i m a t e d t o b e 7.2 x 1 0 ~ ± 1 1 % . T h i s i s s t i l l 2 o r d e r s o f m a g n i t u d e 5  lower t h a n t h e value o f C j n  «  1 0 ~ calculated b y Shibahashi k 3  T a k a t a (1993).  It i s  also i n a g r e e m e n t w i t h t h e v a l u e o f C t < 0.0006 o b t a i n e d b y K u r t z et a l . (1989) f r o m n%  t h e fine s t r u c t u r e o b s e r v e d i n t h e 1986 d a t a set. T h u s , i t i s u n l i k e l y t h a t t h e o b s e r v e d r o t a t i o n a l s p l i t t i n g i s d e s c r i b e d b y E q u a t i o n 1.5. T h e d i a g n o s t i c p o w e r o f a s e c o n d - o r d e r s p a c i n g for t h e r o A p s t a r s h a s b e e n q u e s t i o n e d b y s o m e a u t h o r s (e.g., D z i e m b o w s k i k G o o d e , 1996, a n d B a l m f o r t h et a l . , 2 0 0 1 ) . T h e r e a s o n i s t h a t t h e m a g n e t i c p e r t u r b a t i o n s b e t w e e n m o d e s o f d i f f e r i n g degree i s o f t h e s a m e o r d e r as t h e s m a l l s p a c i n g s .  However, it is still interesting t o d r a w some conclusions  f r o m t h e c a l c u l a t e d s m a l l s p a c i n g s o f S e c t i o n 3.2.1 i n l i g h t o f t h e p o s s i b l e o b s e r v e d s m a l l s p a c i n g o f 2 . 6 3 pRz b e t w e e n frequencies v{old) a n d v{new).  In particular, the  e v o l u t i o n a r y m o d e l s t h a t r e p r o d u c e t h e o b s e r v e d s e c o n d - o r d e r s p a c i n g o f 2.63 / / H z h a v e a mass of « 1 . 6 M  0  a n d a c o m p o s i t i o n o f ( X , Z ) ~ (0.720, 0.017) (see F i g u r e s 3.5 a n d  3.7). I f t h e c a l c u l a t e d l a r g e s p a c i n g i s also c o m p a r e d t o t h e o b s e r v e d v a l u e o f 68 / / H z , the e v o l u t i o n a r y m o d e l properties w o u l d be consistent w i t h those f r o m t h e s m a l l s p a c i n g e s t i m a t e s ; i.e, a s u b - s o l a r Z , s o l a r X , a n d a m a s s o f a p p r o x i m a t e l y 1 . 6 M  0  (see F i g u r e s 3.5  a n d 3.6). T h e s e r e s u l t s a r e also c o n f i r m e d b y t h e w o r k o f C u n h a et a l . (2003). H o w e v e r , n o o n e o s c i l l a t i o n m o d e l agrees u n a m b i g u o u s l y w i t h t h e o b s e r v e d o s c i l l a t i o n s p e c t r u m o f H R 1217. M o d e l s w i t h a l o w e r g l o b a l m e t a l l i c i t y c o n f i r m t h e r e s u l t s o f L e b r e t o n et a l . (1999). It i s u s u a l l y a s s u m e d t h a t s t a r s i n t h e s o l a r n e i g h b o r h o o d , l i k e H R 1217, h a v e a s o l a r metallicity.  L e b r e t o n et a l . (1999) c a l c u l a t e a m e t a l l i c i t y r a n g e o f —1 < [Fe/H]  < 0.3  for a s a m p l e o f s t a r s w i t h d i s t a n c e s less t h a n 3 0 p c . W h i l e t h e r e i s n o w a y t o a b s o l u t e l y d e t e r m i n e t h e i n t e r i o r m e t a l c o n t e n t o f a n y s t a r f r o m t h e o b s e r v e d surface  abundances,  Chapter 4- Discussion  and  Conclusions  73  it is e s p e c i a l l y difficult for t h e A p s t a r s b e c a u s e o f t h e r e p e c u l i a r s p e c t r a l features.  The  o l d a s s u m p t i o n o f s o l a r m e t a l l i c i t y m o d e l s for n e a r b y s t a r s m a y h a v e t o b e r e v i s e d . T h e m o d e l s t h a t f a l l w i t h i n t h e l u m i n o s i t y e r r o r b o x d e f i n e d b y t h e Hipparcos w i t h a m a s s less t h a n 1 . 5 M convection.  Q  parallax  a n d a m e t a l l i c i t y greater t h a n solar e x h i b i t large envelope  I n t h e case o f a 1 . 3 M  0  model with composition ( X , Z) =  (0.740, 0 . 0 2 2 ) ,  t h e e n v e l o p e is a l m o s t c o m p l e t e l y c o n v e c t i v e b y t h e t i m e i t enters t h e d e f i n e d e r r o r b o x . T h e s e m o d e l s h a v e left t h e m a i n sequence a n d are a p p r o a c h i n g t h e b a s e o f t h e r e d g i a n t b r a n c h . W h i l e s m a l l c o n v e c t i v e zones (<C 0 . 1 % M * ) e x i s t i n o t h e r m o d e l s , t h e c o n v e c t i v e e n v e l o p e m o d e l s p r e s e n t e d i n F i g u r e 3.2 c a n c l e a r l y b e r u l e d o u t as A p m o d e l c a n d i d a t e s . The  r e a s o n b e i n g t h a t a n y surface i n h o m o g e n e i t i e s w o u l d b e m i x e d a w a y b y t h e on-set  of s u c h d e e p surface c o n v e c t i v e zones. M o d e l s p r e s e n t e d i n t h i s s t u d y are a v a l u a b l e asset for c o n s t r a i n i n g t h e p h y s i c s o f Ap  a n d r o A p stars.  m o d e l atmospheres.  Future improvements will include better approximations to  the  S p e c i f i c a l l y , t h e use o f a n E d d i n g t o n g r e y a t m o s p h e r e i n t h e m o d e l  calculations predicts a n i s o t h e r m a l acoustic cut-off frequency t h a t is b e l o w the observed frequencies for H R 1217. M o d i f i c a t i o n s t o t h e T - r r e l a t i o n s for r o A p s t a r s h a v e b e e n u s e d b y G a u t s c h y et a l . (1998) a n d A u d a r d et a l . (1998). I n t h e first case, a T - r w a s a r t i f i c i a l l y m o d i f i e d t o cause a steeper t e m p e r a t u r e g r a d i e n t n e a r t h e s t e l l a r surface; m i m i c k i n g t h e effect o f a c h r o m o s p h e r e .  H o w e v e r , A u d a r d et a l . (1998) use m o d e l a t m o s p h e r e s  a surface c o m p o s i t i o n specific t o H R 1217 b u t w i t h a g l o b a l Z = 0.02.  Both  with  groups  were successful i n r a i s i n g t h e a c o u s t i c cut-off so t h a t t h e c a l c u l a t e d frequencies a n d t h e o b s e r v e d frequencies were i n a g r e e m e n t . M o d e l atmospheres have been calculated b y W e r n e r Weiss of the U n i v e r s i t y of V i e n n a ( p r i v a t e c o m m u n i c a t i o n ) for a n H R 1217 surface c o m p o s i t i o n ( R y a b c h i k o v a et a l . , 1997) a n d w e are c u r r e n t l y a d a p t i n g t h e m so t h e y c a n b e u s e d w i t h Y R E C 7 . i m p r o v e m e n t s over t h e e a r l i e r w o r k o f A u d a r d et a l . (1998). m o d e l s w i l l n o t b e r e s t r i c t e d t o a c o n s t a n t surface g r a v i t y .  T h e r e are t w o  First, the evolutionary T h e effect t h i s w i l l h a v e  is e x p e c t e d t o b e s m a l l ; h o w e v e r , i t is c o n s i s t e n t w i t h a n e v o l v i n g sequence o f s t e l l a r m o d e l s . T h e s e c o n d , a n d m o r e i m p o r t a n t , is t h e a d d i t i o n o f m u l t i p l e c o m p o s i t i o n s t h a t  Chapter 4- Discussion  and  Conclusions  74  reflect t h e o b s e r v e d c h a n g e i n a b u n d a n c e p a t t e r n s as H R 1217 r o t a t e s . T h u s , a n u m b e r o f i n d e p e n d e n t e v o l u t i o n a r y t r a c k s w i l l b e c a l c u l a t e d a l l o w i n g u s t o m o d e l t h e frequencies w i t h a n a z i m u t h a l dependence.  These new models w i l l t h e n also be subject to  the  m a g n e t i c f r e q u e n c y p e r t u r b a t i o n a n a l y s i s p r e s e n t e d i n S e c t i o n 3.3. T h e q u a l i t y o f p h o t o m e t r i c d a t a o f H R 1217 is also e x p e c t e d t o i m p r o v e i n t h e n e a r future since the M O S T  1  micro-satellite (aperture  ( W a l k e r et a l . , 2003, a n d M a t t h e w s et a l . , 2000) s p a c e s e i s m o l o g y =  15 c m ) w a s s u c c e s s f u l l y l a u n c h e d o n J u n e 30, 2 0 0 3 .  This  p r o j e c t is C a n a d a ' s first space telescope a n d i t s p r i m a r y f u n c t i o n s are a s t e r o s e i s m o l o g y a n d t h e d e t e c t i o n o f reflected l i g h t f r o m e x o p l a n e t s .  M O S T is a l r e a d y s u r p a s s i n g i t s  e x p e c t e d p e r f o r m a n c e a n d v i e w i n g t a r g e t s w i t h a p h o t o m e t r i c p r e c i s i o n o f ~ a few p a r t s p e r - m i l l i o n i n t h e f r e q u e n c y r a n g e r e l e v a n t for r o A p s t a r s a n d w i t h d u t y c y c l e s o f 9 9 % (Matthews, private communication). A s a comparison, the X C O V 2 0 d a t a covered a d u t y c y c l e o f « 3 3 % ; m a k i n g i t a h i g h l y successful g r o u n d b a s e d s t u d y , a n d r e a c h e d a noise l e v e l o f « 0.1 m m a g ; a l m o s t 2 o r d e r s of m a g n i t u d e g r e a t e r t h a n  MOST.  I n l a t e 2004, M O S T w i l l observe H R 1217 for j u s t over 3 weeks.  T h i s should be  sufficient t o c o v e r 3 r o t a t i o n p e r i o d s o f t h e s t a r p r o v i d i n g a n a d e q u a t e t i m e - b a s e for h i g h f r e q u e n c y r e s o l u t i o n . S i n c e H R 1217 w i l l b e o b s e r v e d o n l y 4 y e a r s after t h e  XCOV20  observations, there could be interesting results o n the m o d e lifetimes of the star. n e w m o d e l s presented i n this thesis c o m b i n e d w i t h i m p r o v e d observations from  The space  should shed new light o n the physics governing the structure a n d e v o l u t i o n of A p a n d r o A p stars.  1  Microvariability & Oscillations of STars or Microvariabilite et Oscillations STellaire  75  Bibliography A i n d o w , A . , E l s w o r t h , Y . P . , Isaak, G . R . , M c L e o d , C . P . , N e w , R . , & v a n d e r  Raay,  H . B . 1988, i n S e i s m o l o g y o f t h e S u n a n d S u n - l i k e S t a r s , E S A S P - 2 8 6 , 157 A l e x a n d e r , D . R . & F e r g u s o n , J . W . 1994, A p J , 437, 879 A l v a r e z , M . , H e r n a n d e z , M . M . , M i c h e l , E . , J i a n g , S. Y . , B e l m o n t e , J . A . , C h e v r e t o n , M . , Massacrier, G . , L i u , Y . Y . , L i , Z . P., G o u p i l , M . J., Cortes, T . R . , Mangeney, A . , D o l e z , N . , V a l t i e r , J . C , V i d a l , I., S p e r l , M . , k T a l o n , S. 1998, A & A , 340, 149 A u d a r d , N . , K u p k a , F . , M o r e l , P . , P r o v o s t , J . , k W e i s s , W . W . 1998, A & A , 3 3 5 , 9 5 4 B o h m - V i t e n s e , E . 1958, Z e i t s c h r i f t fur A s t r o p h y s i c s , 46, 108 Babel, J. k  M i c h a u d , G . 1991, A & A , 241, 493  B a g n u l o , S. 1998, C o n t r i b u t i o n s o f t h e A s t r o n o m i c a l O b s e r v a t o r y  S k a l n a t e P l e s o , 27,  431  B a g n u l o , S., L a n d i d e g l ' I n n o c e n t i ,  E . , L a n d o l f i , M . , & L e r o y , J . L . 1995, A k  A , 295,  459 Bahcall, J . N . , Pinsonneault, M . H . , k  B a s u , S. 2 0 0 1 , A p J , 555, 990  B a l m f o r t h , N . J . , C u n h a , M . S., D o l e z , N . , G o u g h , D . O . , k V a u c l a i r , S. 2 0 0 1 , M N R A S , 3 2 3 , 362 B i g o t , L . k D z i e m b o w s k i , W . 2002, A & A , 3 9 1 , 235  Chapter 4- Discussion  and  Conclusions  76  B i g o t , L . , P r o v o s t , J . , B e r t h o m i e u , G . , D z i e m b o w s k i , W . A . , & G o o d e , P . R . 2000, A  k A , 356, 218 Breger, M . , Handler, G . , G a r r i d o , R . , A u d a r d , N . , Z i m a , W . , Paparo, M . , Beichbuchner, F., Zhi-Ping, L . , Shi-Yang, J., Zong-Li, L . , A i - Y i n g , Z., Pikall, H . , Stankov, A . , G u z i k , J . A . , S p e r l , M . , K r z e s i n s k i , J . , O g l o z a , W . , P a j d o s z , G . , Z o l a , S., T h o m a s s e n , T . , S o l heim, J . - E . , Serkowitsch, E . , Reegen, P., R u m p f , T . , Schmalwieser, A . , k  Montgomery,  M . H . 1999, A & A , 349, 225 B r e g e r , M . , S t i c h , J . , G a r r i d o , R . , M a r t i n , B . , J i a n g , S. Y . , L i , Z . P . , H u b e , D . P . , Ostermann, W . , Paparo, M . , k Bressan, A . , Chiosi, C . , k Campbell, C. G . k  Scheck, M . 1993, A & A , 2 7 1 , 4 8 2  F a g o t t o , F . 1994, A p J S , 94, 63  P a p a l o i z o u , J . C . B . 1986, M N R A S , 220, 5 7 7  Christensen-Dalsgaard,  J . 2002, R e v i e w s o f M o d e r n P h y s i c s , 74, 1073  C u n h a , M . S. 2 0 0 1 , M N R A S , 325, 373 C u n h a , M . S., F e r n a n d e s , J . M . M . B . , k M o n t e i r o , M . J . P . F . G . 2 0 0 3 , M N R A S , 3 4 3 , 831 C u n h a , M . S. k  G o u g h , D . 2000, M N R A S , 319, 1020  D z i e m b o w s k i , W . k G o o d e , P . R . 1985, A p J , 296, L 2 7 Dziembowski, W . A . k Frandsen, entoft, M.  G o o d e , P . R . 1996, A p J , 458, 338  S., P i g u l s k i , A . , N u s p l , J . , B r e g e r , M . , B e l m o n t e , J . A . , D a l l , T . H . , A r -  T . , Sterken,  C., Medupe,  J. P. F . G . , Barban,  T., Gupta,  C , Chevreton,  S. K . , P i n h e i r o ,  F . J. G . , Monteiro,  M . , Michel, E . , Benko, J . M . , Barcza,  S.,  S z a b o , R . , K o l a c z k o w s k i , Z . , K o p a c k i , G . , k U d o v i c h e n k o , S. N . 2 0 0 1 , A & A , 3 7 6 , 175 G a b r i e l , K . , Noels, A . , Scuflaire, R . , k  M a t h y s , G . 1985, A & A , 143, 206  Chapter 4- Discussion  and  Conclusions  Gautschy, A . , Saio, H . , k Harzenmoser,  77  H . 1998, M N R A S , 3 0 1 , 31  G i r i s h , V . , S e e t h a , S., M a r t i n e z , R , J o s h i , S., A s h o k a , B . N . , K u r t z , D . W . , C h a u b e y , U . S., G u p t a , S. K . , k  S a g a r , R . 2 0 0 1 , A & A , 380, 142  G r e v e s s e , N . , N o e l s , A . , k S a u v a l , A . J . 1996, i n A S P C o n f . Ser. 99: C o s m i c A b u n d a n c e s , 117 G u e n t h e r , D . B . 1994, A p J , 422, 400 Guenther, D . B . k  B r o w n , K . I. T . 2004, A p J , 600, 419  G u e n t h e r , D . B . k D e m a r q u e , P . 1997, A p J , 484, 9 3 7 Guenther, D . B . , Demarque,  P., K i m , Y . - C , k  Pinsonneault,  M . H . 1992, A p J , 387,  372 H a n d l e r , G . 2 0 0 3 , B a l t i c A s t r o n o m y , 12, 253 Henyey, L. G . , Forbes, J . E . , k  G o u l d , N . L . 1964, A p J , 139, 306  H i l t n e r , W . A . 1962, A s t r o n o m i c a l t e c h n i q u e s . ( C h i c a g o , U n i v e r s i t y P r e s s [1962]) Iglesias, C . A . k  R o g e r s , F . J . 1996, A p J , 464, 943  K a w a l e r , S. D . , O ' B r i e n , M . S., C l e m e n s , J . O , N a t h e r , R . E . , W i n g e t , D . E . , W a t s o n , T . K . , Y a n a g i d a , K . , D i x s o n , J . S., B r a d l e y , P . A . , W o o d , M . A . , S u l l i v a n , D . J . , K l e i n m a n , S. J . , M e i s t a s , E . , L e i b o w i t z , E . M . , M o s k a l i k , P . , Z o l a , S., P a j d o s z ,  G.,  Krzesinski, J., Solheim, J.-E., Bruvold, A . , O'Donoghue, D . , K a t z , M . , Vauclair, G . , D o l e z , N . , C h e v r e t o n , M . , B a r s t o w , M . A . , K a n a a n , A . , K e p l e r , S. O . , G i o v a n n i n i , O . , Provencal, J . L . , k  H a n s e n , C . J . 1995, A p J , 450, 350  K i p p e n h a h n , R . k W e i g e r t , A . 1994, S t e l l a r S t r u c t u r e a n d E v o l u t i o n ( S t e l l a r S t r u c t u r e a n d E v o l u t i o n , X V I , 468 p p . 192 figs.. S p r i n g e r - V e r l a g B e r l i n H e i d e l b e r g N e w Y o r k . A l s o Astronomy and Astrophysics Library)  Chapter 4- Discussion  and  Conclusions  K l e i n m a n , S. J . , N a t h e r , R . E . , k  78  P h i l l i p s , T . 1996, P A S P , 108, 356  K u r t z , D . k W e g n e r , G . 1979, A p J , 232, 5 1 0 K u r t z , D . W . 1978, I n f o r m a t i o n a l B u l l e t i n o n V a r i a b l e S t a r s , 1436, 1 — . 1982, M N R A S , 200, 807 — . 1990, A R A k A , 28, 6 0 7 K u r t z , D . W . , K a w a l e r , S. D . , R i d d l e , R . L . , R e e d , M . D . , C u n h a , M . S., W o o d , M . , S i l v e s t r i , N . , W a t s o n , T . K . , D o l e z , N . , M o s k a l i k , P . , Z o l a , S., P a l l i e r , E . , G u z i k , J . A . , M e t c a l f e , T . S., M u k a d a m , A . , N a t h e r , R . E . , W i n g e t , D . E . , S u l l i v a n , D . J . , S u l l i v a n , T., Sekiguchi, K . , Jiang, X . J . , Shobbrook, R . R . , B i r c h , P . V . , A s h o k a , B . N . , Seetha, S., J o s h i , S., G i r i s h , V . , O ' D o n o g h u e , D . , H a n d l e r , G . , M u e l l e r , M . , G o n z a l e z P e r e z , J . M . , S o l h e i m , J . E . , J o h a n n e s s e n , F . , U l l a , A . , K e p l e r , S. O . , K a n a a n , A . , d a C o s t a , A . , Fraga, L . , Giovannini, O., Matthews, J . M . , Cameron, C , Vauclair, G . , Nitta, A . , k  K l e i n m a n , S. J . 2003, B a l t i c A s t r o n o m y , 12, 105  K u r t z , D . W . , K a w a l e r , S. D . , R i d d l e , R . L . , R e e d , M . D . , C u n h a , M . S., W o o d , M . , S i l v e s t r i , N . , W a t s o n , T . K . , D o l e z , N . , M o s k a l i k , P . , Z o l a , S., P a l l i e r , E . , G u z i k , J . A . , M e t c a l f e , T . S., M u k a d a m , A . S., N a t h e r , R . E . , W i n g e t , D . E . , S u l l i v a n , D . J . , S u l l i v a n , T . , S e k i g u c h i , K . , J i a n g , X . , S h o b b r o o k , R . , A s h o k a , B . N . , S e e t h a , S., J o s h i , S., O ' D o n o g h u e ,  D . , Handler, G . , Mueller, M . , Gonzalez Perez, J . M . , Solheim, J . -  E . , J o h a n n e s s e n , F . , U l l a , A . , K e p l e r , S. O . , K a n a a n , A . , d a C o s t a , A . , F r a g a , L . , Giovannini, O., k  Kurtz, D. W . k  M a t t h e w s , J . M . 2002, M N R A S , 330, L 5 7  M a r a n g , F . 1987, M N R A S , 229, 285  K u r t z , D . W . k M a r t i n e z , P . 2000, B a l t i c A s t r o n o m y , 9, 253 K u r t z , D . W . , M a t t h e w s , J . M . , Martinez, P., Seeman, J . , Cropper, M . , Clemens, J . C , K r e i d l , T . J . , S t e r k e n , C , S c h n e i d e r , H . , W e i s s , W . W . , K a w a l e r , S. D . , k K e p l e r , S. O . 1989, M N R A S , 240, 881  Chapter 4- Discussion  and Conclusions  79  K u r t z , D . W . k Seeman, J . 1983, M N R A S , 205, 11 K u r t z , D . W . , van W y k , F . , Roberts, G . , Marang, F . , Handler, G . , Medupe, R . , k Kilkenny, D . 1997, M N R A S , 287, 69 Kuschnig, R . , Weiss, W . W . , Gruber, R . , Bely, P. Y . , k Jenkner, H . 1997, A & A , 328, 544 Landstreet, J . D . 1992a, A k A R , 4, 35 — . 1992b, Reviews of M o d e r n Astronomy, 5, 105 Landstreet, J . D . 1993, i n A S P Conf. Ser. 44: I A U C o l l o q . 138: Peculiar versus N o r m a l Phenomena i n A - t y p e and Related Stars, 218 Lebreton, Y . , Perrin, M . N . , Cayrel, R., B a g l i n , A . , k Fernandes, J . 1999, A & A , 350, 587 Ledoux, P. 1951, A p J , 114, 373 Leibacher, J . W . k Stein, R . F . 1971, Astrophys. Lett., 7, 191 Leighton, R . B . , Noyes, R . W . , k Simon, G . W . 1962, A p J , 135, 474 Leroy, J . L . , Landstreet, J . D . , Deglinnocenti, E . L . , k Landolfi, M . 1993, i n A S P Conf. Ser. 44: I A U C o l l o q . 138: Peculiar versus N o r m a l Phenomena i n A - t y p e and Related Stars, 274 Loumos, G . L . k Deeming, T . J . 1978, Astrophys. and Space Science, 56, 285 M a t h y s , G . 1988, A & A , 189, 179 — . 1989, Fundamentals of Cosmic Physics, 13, 143 Mathys, G . , Hubrig, S., Landstreet, J . D . , Lanz, T . , k Manfroid, J . 1997, A k A S , 123, 353  Chapter 4- Discussion  and Conclusions  80  Matthews, J . M . 1991, P A S P , 103, 5 Matthews, J . M . , K u r t z , D . W . , k Martinez, P. 1999, A p J , 511, 422 Matthews, J . M . , Kuschnig, R . , Walker, G . A . H . , Pazder, J . , Johnson, R . , Skaret, K . , Shkolnik, E . , Lanting, T . , Morgan, J . P., k Sidhu, S. 2000, i n A S P Conf. Ser. 203: I A U C o l l o q . 176: T h e Impact of Large-Scale Surveys on Pulsating Star Research, 74-75 Matthews, J . M . , Wehlau, W . H . , k K u r t z , D . W . 1987, A p J , 313, 782 M i c h a u d , G . 1970, A p J , 160, 641 M i c h a u d , G . 1976, i n I A U Colloq. 32: Physics of A p Stars, 81 M i c h a u d , G . 1996, i n I A U Symp. 176: Stellar Surface Structure, 321 M i c h a u d , G . , C h a r l a n d , Y . , k Megessier, C . 1981, A & A , 103, 244 M i c h a u d , G . J . k Proffitt, C . R . 1993, i n A S P Conf. Ser. 44: I A U C o l l o q . 138: Peculiar versus N o r m a l Phenomena i n A - t y p e and Related Stars, 439 Montgomery, M . - H . k O'Donoghue, D . 1999, D S S N (Vienna), 13 M o o n , T . T . k Dworetsky, M . M . 1985, M N R A S , 217, 305 Nather, R . E . , Winget, D . E . , Clemens, J . C , Hansen, C . J . , k Hine, B . P. 1990, A p J , 361, 309 Olive, K . A . k Steigman, G . 1995, A p J S , 97, 49 Preston, G . W . 1972, A p J , 175, 465 — . 1974, A R A k A , 12, 257 Provost, J . 1984, i n I A U Symp. 105: Observational Tests of the Stellar E v o l u t i o n Theory, 47 Roberts, P . H . k Soward, A . M . 1983, M N R A S , 205, 1171  Chapter 4- Discussion  and Conclusions  Rodriguez, E . , Costa, V . , Handler, G . , k  81 G a r c i a , J . M . 2 0 0 3 , A & A , 399, 253  R o g e r s , F . J . 1986, A p J , 3 1 0 , 723 R o g e r s , F . J . , S w e n s o n , F . J . , k Iglesias, C . A . 1996, A p J , 456, 902 Ryabchikova, T . A . , Landstreet,  J . D . , Gelbmann, M . J., Bolgova, G . T., Tsymbal,  V . V . , k W e i s s , W . W . 1997, A & A , 327, 1137 Saio, H . k  C o x , J . P . 1980, A p J , 236, 549  S c a r g l e , J . D . 1982, A p J , 263, 835 Shibahashi, H . k  S a i o , H . 1985, P A S J , 37, 245  Shibahashi, H . k  T a k a t a , M . 1993, P A S J , 4 5 , 6 1 7  S t i b b s , D . W . N . 1950, M N R A S , 110, 395 T a s s o u l , M . 1980, A p J S , 4 3 , 469 — . 1990, A p J , 358, 313 U l r i c h , R . K . 1970, A p J , 162, 993 U n n o , W . , O s a k i , Y . , A n d o , H . , S a i o , H . , & S h i b a h a s h i , H . 1989, N o n r a d i a l o s c i l l a t i o n s of s t a r s ( N o n r a d i a l o s c i l l a t i o n s of stars, T o k y o : U n i v e r s i t y o f T o k y o P r e s s , 1989, 2 n d ed.)  V a u c l a i r , S. k V a u c l a i r , G . 1982, A R A k A , 20, 37 W a d e , G . A . 1997, A & A , 325, 1063 W a l k e r , G . , M a t t h e w s , J . , K u s c h n i g , R . , J o h n s o n , R . , R u c i n s k i , S., P a z d e r , J . , B u r l e y , G . , W a l k e r , A . , S k a r e t , K . , Zee, R . , G r o c o t t , S., C a r r o l l , K . , S i n c l a i r , P . , S t u r g e o n , D . , k  H a r r o n , J . 2003, P A S P , 115, 1023  Chapter 4- Discussion  and  Conclusions  82  Winget, D . E . 1993, i n A S P Conf. Ser. 42: G O N G 1992. Seismic Investigation of the Sun and Stars, 331 Wolff, S. C . 1983, T h e A-stars:  Problems and perspectives. M o n o g r a p h series on.  nonthermal phenomena i n stellar atmospheres (The A-stars: Problems and perspectives. M o n o g r a p h series on nonthermal phenomena i n stellar atmospheres)  Appendix A. XCOV20  Observing Log  83  Appendix A  X C O V 2 0 Observing L o g T h i s a p p e n d i x c o n t a i n s t h e o b s e r v i n g l o g for H R 1 2 1 7 d u r i n g X C O V 2 0 .  See t h e t a b l e  c a p t i o n for d e t a i l s .  Table  A.i:  A data log obtained during X C O V 2 0 .  The  l o g c o n t a i n s r u n n a m e s , telescopes u s e d , t h e n u m b e r o f p o i n t s collected, a n d c o m m e n t s from observers. T h e r u n s m a r k e d w i t h a * are r u n s t h a t were u s e d i n t h e f r e q u e n c y a n a l y s i s . See S e c t i o n 2.3.1 for s e l e c t i o n c r i t e r i a .  Run Name  Telescope  Date  Start T i m e  (UT)  (UT)  # of  Observer Comments  Points  mdrl36*  C T I O 1.5m  6-Nov-00  2:23:30  1760  2001 year in original data!  mdrl37*  C T I O 1.5m  9-Nov-OO  1:55:20  2524  vignetting present, re-reduced to  mdrl38*  C T I O 1.5m  10-Nov-OO  1:59:50  1756  sky in channel 2  mdrl39*  C T I O 1.5m  12-Nov-OO  2:16:00  2422  sky in CH2, dome problems in curve  mdrl40*  C T I O 1.5m  13-Nov-OO  1:34:30  2580  moon!  mdrl41*  C T I O 1.5m  14-Nov-OO  1:27:40  333  a few cycles between clouds  sa-od044*  S A A O 1.9m  14-Nov-00  21:03:00  1886  One channel - great run!  correct (somewhat)  mdrl42*  C T I O 1.5m  15-Nov-OO  1:28:10  2645  very good run - look at those beats!  sa-od045*  S A A O 1.9m  15-Nov-OO  19:20:00  2551  typical S A A O data - outstanding  teideOl*  Teide 0.8m  16-Nov-00  0:42:10  1342  nice c h l  mdrl43*  C T I O 1.5m  16-Nov-00  1:23:00  2732  another very good run  teideN03  Teide 0.8m  17-Nov-OO  0:43:40  2164  entire run through clouds  mdrl44*  C T I O 1.5m  17-Nov-OO  1:18:20  2730  good run  nol700q2*  Hawaii 0.6m  17-Nov-OO  7:28:00  1375  good night for first half  nol700q3*  Hawaii 0.6m  17-Nov-OO  12:34:20  196  good second half of night!  teideN04*  Teide 0.8m  17-Nov-OO  22:09:10  4366  5-sec integrations no sky at beginning before moon  continued on next page  Appendix  A. XCOV20  Observing  84  Log  Table A . l : continued  Run Name  nol800ql*  Telescope  Date  Start T i m e  (UT)  (UT)  # of  Observer C o m m e n t s  Points  H a w a i i 0.6m  18-Nov-00  7:22:30  2360  nice r u n - some clouds  sa-od047*  S A A O 1.9m  18-Nov-00  23:29:00  798  short b u t sweet  teiden05*  Teide 0.8m  18-Nov-00  22:53:20  3892/2  reduction problems  nol900q2*  H a w a i i 0.6m  19-Nov-OO  10:14:20  1423  N i c e run! subtracted 8066 cts from ch3  sa-od048*  S A A O 1.9m  19-Nov-00  18:55:00  2611  some dome glitches a n d 1% d r o p i n counts reason u n k n o w n  teideN06*  Teide 0.8m  19-Nov-OO  22:05:30  4419  5 sec integrations...nice set  asm-0079*  M c D o n a l d 2.1m  20-Nov-OO  3:58:40  2115  poor data  no2000ql*  H a w a i i 0.6m  20-Nov-00  7:37:00  2249  low a m p part of r o t a t i o n cycle  pvblll8  B i c k l e y 0.6m  18-Nov-00  12:28:31  2220  Single channel d a t a from P e r t h  pvblll9  B i c k l e y 0.6m  19-Nov-OO  12:15:13  1495  sa-od049*  S A A O 1.9m  20-Nov-OO  18:51:00  2651  D a r r a g h ' s L a s t S t a n d - G o o d night - no fit a p p l i e d a n d 17pt. s k y chunk so wrote i n two chunks  pvbll20 asm-0081* pvbll21  B i c k l e y 0.6m  20-Nov-OO  12:19:12  2454  M c D o n a l d 2.1m  21-Nov-OO  9:06:10  349  not used  B i c k l e y 0.6m  21-Nov-OO  12:49:17  2320  curious peaks at 200,250, 400, 1200 s...  sa-m0002*  S A A O 0.75m  21-Nov-OO  19:08:00  106  counts  9 m i l l i o n at 10 sec  int. times sa-m0003*  S A A O 0.75m  21-Nov-OO  19:26:50  4857  5sec integrations - counts  jxj-0121*  B A O 0.85m  22-Nov-OO  14:47:50  1538  transparency variations - not used  sa-m0004*  S A A O 0.75m  22-Nov-OO  18:28:20  5524  5 sec integrations - some  4 m i l l i o n per point  intermittent clouds joy-002  M c D o n a l d 2.1m  23-Nov-OO  3:59:50  1958  Transparency variations (?)  teideN08*  Teide 0.8m  22-Nov-OO  22:07:20  1364  5 sec integrations, h u m i d i t y at end  no2300ql*  H a w a i i 0.6m  23-Nov-OO  7:15:50  1751  Intervening clouds  sa-m0005  S A A O 0.75m  23-Nov-OO  18:11:50  892  ssoll23a*  sso sso  23-Nov-OO  12:42:28  467  not used: clouds, t i m i n g i n question  23-Nov-OO  13:49:19  191  not used: clouds, t i m i n g i n question  joy-005  M c D o n a l d 82"  24-Nov-00  4:08:30  2161  not used  teidenlO*  tenerife 80cm  23-Nov-OO  22:05:40  3943  pvbll23*  B i c k l e y 0.6m  23-Nov-OO  12:15:41  1298  S A A O 0.75m  24-Nov-00  18:18:00  5646  P e r i o d i c i t y i n the s k y at  Tenerife  25-Nov-OO  2:03:30  1229  Huge seeing  ssoll23b*  saOm0006* teidenll*  but good d a t a nonetheless continued on next page  Appendix A. XCOV20 Observing Log  85  Table A . l : continued  Run  Name  pvbll24*  Telescope  Date  Start T i m e  (UT)  (UT)  Perth  24-Nov-00  12:18:40  joy-009  M c D o n a l d 2.1m  25-Nov-00  jxj-0122  B A O 0.85m  25-Nov-OO  no2500ql*  H a w a i i 0.6m  25-Nov-00  teidenl2*  tenerife 80cm  25-Nov-OO  joy-012*  M c D o n a l d 2.1m  26-Nov-00  jxj-0123  B A O 0.85m  26-Nov-00  no2600q2*  H a w a i i 0.6m  jxj-0124*  B A O 0.85m  sa-m0007*  S A A O 0.75m M c D o n a l d 82"  no2700ql*  #of  Observer  Comments  Points 2494  good first part  4:22:50  1567  clouds - not used  15:41:20  563  clouds - not used  7:03:10  2457  very good r u n  22:09:20  4040  long, good r u n  3:55:50  1497  good d a t a  13:00:00  329  26-Nov-00  6:59:30  2370  another nice r u n  26-Nov-00  14:14:50  1536  strong transparency variations  26-Nov-00  18:28:40  5408  5 sec integrations, a nice r u n !  27-Nov-OO  4:03:00  1956  clouds, c h l / c h 2 d i v i s i o n  H a w a i i 0.6m  27-Nov-00  6:38:00  2460  another nice r u n  sa-m0008*  S A A O 0.75m  27-Nov-OO  18:27:50  5464  scattered o n ends, b u t good signal  teidenl3*  Teide 0.8m  26-Nov-OO  21:50:30  316  a very short r u n , 5 sec integrations  joy-016*  jxj-0127*  B A O 0.85m  27-Nov-OO  13:44:10  1733  beautiful!  teidenl4*  Teide 0.8m  27-Nov-OO  22:28:20  2910  Steve d i d this one...isn't it nice? :)  joy-020*  M c D o n a l d 82"  28-Nov-00  4:04:20  2315  clouds, non-standard reduction t o b r i n g o u t signal  no2800ql  H a w a i i 0.6m  28-Nov-00  6:45:00  886  m o s t l y clouds  sa-h-046*  S A A O 74"  28-Nov-00  18:54:30  2674  good r u n , about a n hour  ssoll27  S S O 0.6m  27-Nov-OO  10:03:13  859  there's signal here!  jxj-0130*  B A O 0.85m  28-Nov-00  16:23:50  768  lots of clouds, b u t d i v i d e d  teidenl5*  Tenerife 0.8 m  28-Nov-00  22:01:50  3976  large seeing b u t signal O K  M c D o n a l d 2.1 m  29-Nov-00  4:00:40  1895  m i d d l e part o n clouds  H a w a i i 0.6m  29-Nov-00  6:41:00  2479  single channel b u t beautifull !  sa-gh465*  S A A O 74"  29-Nov-OO  20:30:30  1942  teideN16*  Tenerife 0.8m  29-Nov-00  21:18:50  4092  M c D o n a l d 2.1 m  30-Nov-00  3:54:20  1899  t r i m m e d from e n d  by channel 2 to get a signal  joy-025* no2900ql*  joy-028*  first half clouds, second half gorgeous! a dataset so good even a theorist can reduce it! :)  no3000ql*  M a u n a K e a 24"  30-Nov-00  6:40:50  2461  single-channel  jxj-0131  B A O 0.85m  29-Nov-OO  12:58:40  801  run t h r o u g h clouds - unusable  jxj-0134  B A O 0.85m  30-Nov-OO  13:22:20  833  Too m a n y clouds. Rejected.  sa-gh466*  S A A O 74"  30-Nov-00  19:30:20  618  first of four pieces of t h e night  sa-gh467*  S A A O 74"  30-Nov-OO  21:19:50  311  second piece  sa-gh468*  S A A O 74"  30-Nov-OO  22:15:20  848  t h i r d piece  sa-gh469*  S A A O 74"  l-Dec-00  0:39:00  560  last piece  teidenl7*  Teide 0.8m  30-Nov-OO  21:17:50  4276  m i d d l e part missing because  continued on next page  Appendix A. XCOV20  Observing Log  86  Table A . l : continued  Run  Name  Telescope  Date  Start T i m e  (UT)  (UT)  #of  Observer  Comments  Points m i r r o r loose  ssc-1201  S S O 0.6m  l-Dec-00  9:47:57  1938  needed lots of low-f filtering  sa-gh470*  S A A O 74"  l-Dec-00  18:57:20  1259  first part of night  sa-gh471*  S A A O 74"  l-Dec-00  22:30:00  1348  second part of night  M c D o n a l d 82"  l-Dec-00  3:48:50  1591  too many clouds dangit! target not centered  joy-031 teidenl8*  Tenerife  l-Dec-00  22:04:00  2780  sa-gh472*  S A A O 74"  2-Dec-00  18:46:50  1020  sa-gh473*  S A A O 74"  2-Dec  21:38:30  1600  teidenlQ*  Teide 0.8m  2-Dec  21:46:50  3631  not useable  sa-gh474*  S A A O 74"  3-Dec  18:35:20  2713  good r u n , a m p l i t u d e is decreaseing  sa-gh475*  S A A O 74"  4-Dec-00  21:51:00  1530  clouds, transparency variations  tsm-0087*  M c D o n a l d 82"  5-Dec-00  5:50:40  1100  electronics problems, no signal :(  tsm-0089*  M c D o n a l d 2.1m  6-Dec-00  3:47:00  1448  observer never d i d s k y i n c h l at a l l  N a i n i t a l 40in.  6-Dec-00  18:55:00  859  rOO-022* rOO-025*  UPSO lm  7-Dec-00  16:53:20  842  short r u n of reasonable q u a l i t y  teiden22*  Teide 0.8m  8-Dec-00  21:37:40  3788  p o o r conditions to start, then cloud  teiden24*  Teide 0.8m  9-Dec-00  21:01:50  1386  5 sec. good short r u n  teiden27*  Teide 0.8m  1 l-Dec-00  0:20:40  1741  5 sec. good short r u n  87  Appendix B  A Listing of the Evolutionary Models T h i s a p p e n d i x p r o v i d e s a l i s t i n g o f p a r a m e t e r s for t h e 569 m o d e l s t h a t fell w i t h i n t h e e r r o r b o x d e f i n e d b y t h e H i p p a r c o s l u m i n o s i t y o f 7.8 ± 0.7 LQ a n d t h e effective t e m p e r a t u r e of T ff e  = 7400^200 K . A d i s c u s s i o n of these p a r a m e t e r s m a y b e f o u n d i n S e c t i o n 3 . 1 . 1 .  Table B . i : A l i s t i n g o f t h e m o d e l s t h a t f a l l w i t h i n t h e  Hip-  parcos luminosity errorbars. T h e table lists the M a s s i n S o l a r u n i t s , t h e effective t e m p e r a t u r e i n degrees K e l v i n , t h e l o g a r i t h m o f the l u m i n o s i t y i n S o l a r u n i t s , t h e age i n u n i t s of 1 0 years, the large s p a c i n g i n / z H z , the h y d r o g e n 9  a n d heavy m e t a l mass fractions, a n d the m i x i n g l e n g t h p a r a m e t e r for e a c h m o d e l .  MQ  Log(T  e / /  )  Log(L/L )  Age(Gyrs)  Q  Av(fj.Hz)  X  Z  a  1.3  3.85765  0.928366  2.15  57.6018  0.7  0.008  1.8  1.3  3.86255  0.916046  2.1  60.6058  0.7  0.008  1.8  1.3  3.86822  0.897114  2.05  64.8293  0.7  0.008  1.8  1.3  3.85737  0.928035  2.15  57.4146  0.7  0.008  1.6  1.3  3.86241  0.91588  2.1  60.4919  0.7  0.008  1.6  1.3  3.86815  0.897607  2.05  64.6873  0.7  0.008  1.6  1.3  3.85733  0.927571  2.15  57.3839  0.7  0.008  1.4  1.3  3.86238  0.915361  2.1  60.4818  0.7  0.008  1.4  1.3  3.868  0.897284  2.05  64.6136  0.7  0.008  1.4  1.4  3.8575  0.875949  1.6  64.9489  0.72  0.008  1.8  1.4  3.86122  0.871991  1.55  66.8804  0.72  0.008  1.8  1.4  3.86471  0.867916  1.5  68.8062  0.72  0.008  1.8  1.4  3.86798  0.863722  1.45  70.7379  0.72  0.008  1.8  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/L )  Age(Gyrs)  0  1.4  3.87098  0.859405  1.4  X  Z  a  72.6281  0.72  0.008  1.8  Av(nHz)  1.4  3.87374  0.855005  1.35  74.4692  0.72  0.008  1.8  1.4  3.85702  0.875949  1.6  64.5369  0.72  0.008  1.6  1.4  3.86092  0.871991  1.55  66.5959  0.72  0.008  1.6  1.4  3.86451  0.867916  1.5  68.6111  0.72  0.008  1.6  1.4  3.86783  0.863722  1.45  70.5892  0.72  0.008  1.6  1.4  3.87087  0.859405  1.4  72.5052  0.72  0.008  1.6  1.4  3.87364  0.855005  1.35  74.3607  0.72  0.008  1.6  1.4  3.86077  0.871985  1.55  66.4611  0.72  0.008  1.4  1.4  3.8644  0.867917  1.5  68.5018  0.72  0.008  1.4  1.4  3.86774  0.863723  1.45  70.4938  0.72  0.008  1.4  1.4  3.87079  0.859405  1.4  72.4196  0.72  0.008  1.4  1.4  3.87357  0.855005  1.35  74.2826  0.72  0.008  1.4  1.4  3.85757  0.851844  1.4  67.8035  0.7  0.01  1.8  1.4  3.85985  0.922619  1.5  60.8632  0.7  0.008  1.8  1.4  3.86449  0.917968  1.45  63.1424  0.7  0.008  1.8  1.4  3.86899  0.913406  1.4  65.4752  0.7  0.008  1.8  1.4  3.87314  0.908921  1.35  67.7485  0.7  0.008  1.8  1.4  3.85958  0.922623  1.5  60.6338  0.7  0.008  1.6  1.4  3.86432  0.917972  1.45  62.9949  0.7  0.008  1.6  1.4  3.86887  0.913409  1.4  65.3644  0.7  0.008  1.6  1.4  3.87304  0.908925  1.35  67.6572  0.7  0.008  1.6  1.4  3.85949  0.922435  1.5  60.5577  0.7  0.008  1.4  1.4  3.86426  0.91779  1.45  62.9375  0.7  0.008  1.4  1.4  3.86877  0.913329  1.4  65.2847  0.7  0.008  1.4  1.4  3.87298  0.908855  1.35  67.5987  0.7  0.008  1.4  1.45  3.85783  0.896504  1.6  63.8919  0.74  0.008  1.8  1.45  3.86165  0.892743  1.55  65.8174  0.74  0.008  1.8  1.45  3.86524  0.888781  1.5  67.7605  0.74  0.008  1.8  1.45  3.86857  0.884659  1.45  69.6902  0.74  0.008  1.8  1.45  3.87162  0.880411  1.4  71.5685  0.74  0.008  1.8  1.45  3.87441  0.876023  1.35  73.4015  0.74  0.008  1.8  1.45  3.85743  0.896512  1.6  63.543  0.74  0.008  1.6  1.45  3.8614  0.892746  1.55  65.5866  0.74  0.008  1.6  1.45  3.86507  0.888784  1.5  67.5971  0.74  0.008  1.6  1.45  3.86844  0.884662  1.45  69.56  0.74  0.008  1.6  1.45  3.87151  0.880414  1.4  71.4574  0.74  0.008  1.6  1.45  3.87432  0.876023  1.35  73.3052  0.74  0.008  1.6  1.45  3.85727  0.896504  1.6  63.3988  0.74  0.008  1.4  1.45  3.86128  0.892743  1.55  65.47  0.74  0.008  1.4  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/L ) 0  Age(Gyrs)  Au(fiHz)  X  Z  a  1.45  3.86497  0.888781  1.5  67.4989  0.74  0.008  1.4  1.45  3.86836  0.884659  1.45  69.4738  0.74  0.008  1.4  1.45  3.87144  0.880411  1.4  71.3801  0.74  0.008  1.4  1.45  3.87426  0.876023  1.35  73.233  0.74  0.008  1.4  1.45  3.85793  0.8708  1.4  66.8921  0.72  0.01  1.8  1.45  3.86091  0.86701  1.35  68.55  0.72  0.01  1.8  1.45  3.86366  0.863251  1.3  70.1623  0.72  0.01  1.8  1.45  3.86612  0.858976  1.25  71.7555  0.72  0.01  1.8  1.45  3.86856  0.854837  1.2  73.3787  0.72  0.01  1.8  1.45  3.85746  0.8708  1.4  66.4672  0.72  0.01  1.6  1.45  3.86058  0.86701  1.35  68.228  0.72  0.01  1.6  1.45  3.86343  0.863251  1.3  69.9228  0.72  0.01  1.6  1.45  3.86594  0.858976  1.25  71.5686  0.72  0.01  1.6  1.45  3.86841  0.854837  1.2  73.2248  0.72  0.01  1.6  1.45  3.85728  0.8708  1.4  66.2956  0.72  0.01  1.4  1.45  3.86043  0.86701  1.35  68.0886  0.72  0.01  1.4  1.45  3.86331  0.863251  1.3  69.8024  0.72  0.01  1.4  1.45  3.86583  0.858976  1.25  71.4594  0.72  0.01  1.4  1.45  3.86833  0.854837  1.2  73.1265  0.72  0.01  1.4  1.45  3.87396  0.928899  1.35  66.9595  0.72  0.008  1.8  1.45  3.87388  0.928899  1.35  66.8791  0.72  0.008  1.6  1.45  3.87382  0.928899  1.35  66.8173  0.72  0.008  1.4  1.45  3.85888  0.869731  1.25  67.4304  0.7  0.012  1.8  1.45  3.86176  0.865865  1.2  69.0743  0.7  0.012  1.8  1.45  3.86449  0.86184  1.15  70.7197  0.7  0.012  1.8  1.45  3.86705  0.85774  1.1  72.36  0.7  0.012  1.8  1.45  3.86945  0.853583  1.05  73.9823  0.7  0.012  1.8  1.45  3.85844  0.869731  1.25  67.0264  0.7  0.012  1.6  1.45  3.86145  0.865865  1.2  68.7697  0.7  0.012  1.6  1.45  3.86426  0.86184  1.15  70.4924  0.7  0.012  1.6  1.45  3.86688  0.85774  1.1  72.1824  0.7  0.012  1.6  1.45  3.86931  0.853583  1.05  73.8338  0.7  0.012  1.6  1.45  3.85827  0.869731  1.25  66.8639  0.7  0.012  1.4  1.45  3.86132  0.865865  1.2  68.6354  0.7  0.012  1.4  1.45  3.86415  0.86184  1.15  70.3754  0.7  0.012  1.4  1.45  3.86678  0.85774  1.1  72.0775  0.7  0.012  1.4  1.45  3.86922  0.853583  1.05  73.738  0.7  0.012  1.4  1.45  3.85832  0.919151  1.35  61.7557  0.7  0.01  1.8  1.45  3.86254  0.915222  1.3  63.8007  0.7  0.01  1.8  1.45  3.86648  0.911116  1.25  65.8569  0.7  0.01  1.8  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued Log(T  e / /  )  Log(L/L )  Age(Gyrs)  0  X  Z  a  1.45  3.87003  0.906915  1.2  67.8436  0.7  0.01  1.8  1.45  3.87337  0.902535  1.15  69.8247  0.7  0.01  1.8  1.45  3.85797  0.919152  1.35  61.4599  0.7  0.01  1.6  1.45  3.86233  0.915223  1.3  63.6147  0.7  0.01  1.6  1.45  3.86633  0.911118  1.25  65.7233  0.7  0.01  1.6  1.45  3.86991  0.906916  1.2  67.7348  0.7  0.01  1.6  1.45  3.87327  0.902536  1.15  69.7304  0.7  0.01  1.6  1.45  3.85782  0.919151  1.35  61.3306  0.7  0.01  1.4  1.45  3.86222  0.915222  1.3  63.5119  0.7  0.01  1.4  1.45  3.86624  0.911116  1.25  65.636  0.7  0.01  1.4  1.45  3.86984  0.906915  1.2  67.657  0.7  0.01  1.4  1.45  3.87321  0.902535  1.15  69.6618  0.7  0.01  1.4  1.5  3.8587  0.888643  1.4  66.2434  0.74  0.01  1.8  1.5  3.86171  0.884874  1.35  67.9063  0.74  0.01  1.8  1.5  3.86452  0.880972  1.3  69.5574  0.74  0.01  1.8  1.5  3.86715  0.876985  1.25  71.1936  0.74  0.01  1.8  1.5  3.86951  0.872786  1.2  72.7824  0.74  0.01  1.8  1.5  3.87177  0.868646  1.15  74.3589  0.74  0.01  1.8  1.5  3.87383  0.864429  1.1  75.884  0.74  0.01  1.8  1.5  3.85831  0.888656  1.4  65.8896  0.74  0.01  1.6  1.5  3.86145  0.884881  1.35  67.6495  0.74  0.01  1.6  1.5  3.86433  0.880979  1.3  69.3654  0.74  0.01  1.6  1.5  3.867  0.876993  1.25  71.0392  0.74  0.01  1.6  1.5  3.86938  0.872792  1.2  72.6488  0.74  0.01  1.6  1.5  3.87166  0.868652  1.15  74.2391  0.74  0.01  1.6  1.5  3.87374  0.864435  1.1  75.7747  0.74  0.01  1.6  1.5  3.85817  0.888663  1.4  65.7509  0.74  0.01  1.4  1.5  3.86132  0.884879  1.35  67.5261  0.74  0.01  1.4  1.5  3.86423  0.880977  1.3  69.2558  0.74  0.01  1.4  1.5  3.86691  0.876992  1.25  70.941  0.74  0.01  1.4  1.5  3.8693  0.872788  1.2  72.5581  0.74  0.01  1.4  1.5  3.87159  0.868649  1.15  74.1561  0.74  0.01  1.4  1.5  3.87367  0.864432  1.1  75.6969  0.74  0.01  1.4  1.5  3.85727  0.890614  1.3  65.4885  0.72  0.012  1.8  1.5  3.86035  0.8869  1.25  67.1448  0.72  0.012  1.8  1.5  3.86318  0.883  1.2  68.7772  0.72  0.012  1.8  1.5  3.86586  0.879004  1.15  70.4083  0.72  0.012  1.8  1.5  3.86832  0.875007  1.1  71.9974  0.72  0.012  1.8  1.5  3:87064  0.870876  1.05  73.5765  0.72  0.012  1.8  1.5  3.87279  0.86665  1  75.13  0.72  0.012  1.8  continued on next page  Appendix B. A Listing  of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/L ) 0  Age(Gyrs)  Au(fiHz)  X  Z  a  1.5  3.8748  0.862408  0.95  76.6515  0.72  0.012  1.8  1.5  3.86003  0.886835  1.25  66.8467  0.72  0.012  1.6  1.5  3.86299  0.882919  1.2  68.5757  0.72  0.012  1.6  1.5  3.86573  0.878904  1.15  70.2628  0.72  0.012  1.6  1.5  3.86817  0.875035  1.1  71.8421  0.72  0.012  1.6  1.5  3.87051  0.870867  1.05  73.4485  0.72  0.012  1.6  1.5  3.87269  0.866648  75.0159  0.72  0.012  1.6  1.5  3.87471  0.862395  0.95  76.5461  0.72  0.012  1.6  1.5  3.85988  0.886847  1.25  66.7075  0.72  0.012  1.4  1.5  3.86287  0.882933  1.2  68.4562  0.72  0.012  1.4  1.5  3.86563  0.878914  1.15  70.1577  0.72  0.012  1.4  1.5  3.86809  0.875035  1.1  71.7493  0.72  0.012  1.4  1.5  3.87044  0.87087  1.05  73.3627  0.72  0.012  1.4  1.5  3.87262  0.866654  74.9362  0.72  0.012  1.4  1.5  3.87465  0.862396  0.95  76.4702  0.72  0.012  1.4  1.5  3.86851  0.928683  1.25  65.8202  0.72  0.01  1.8  1.5  3.87204  0.92441  1.2  67.8106  0.72  0.01  1.8  1.5  3.86839  0.928683  1.25  65.7117  0.72  0.01  1.6  1.5  3.87194  0.92441  1.2  67.7188  0.72  0.01  1.6  1.5  3.86832  0.928683  1.25  65.6342  0.72  0.01  1.4  1.5  3.87188  0.92441  1.2  67.6512  0.72  0.01  1.4  1.5  3.85908  0.890462  1.15  66.2585  0.7  0.014  1.8  1  1  1.5  3.86202  0.886593  1.1  67.9038  0.7  0.014  1.8  1.5  3.86479  0.882576  1.05  69.5472  0.7  0.014  1.8  1.5  3.86736  0.878627  71.1523  0.7  0.014  1.8  1.5  3.86972  0.874471  0.95  72.7368  0.7  0.014  1.8  1.5  3.87197  0.870235  0.9  74.3154  0.7  0.014  1.8  1.5  3.87406  0.865959  0.85  75.865  0.7  0.014  1.8  1.5  3.8587  0.890515  1.15  65.9006  0.7  0.014  1.6  1.5  3.86176  0.886648  1.1  67.6398  0.7  0.014  1.6  1.5  3.86459  0.882608  1.05  69.3468  0.7  0.014  1.6  1.5  3.8672  0.878653  70.9899  0.7  0.014  1.6  1.5  3.8696  0.874471  0.95  72.6039  0.7  0.014  1.6  1.5  3.87186  0.870235  0.9  74.1976  0.7  0.014  1.6  1.5  3.87397  0.865959  0.85  75.7569  0.7  0.014  1.6  1.5  3.85854  0.890462  1.15  65.7599  0.7  0.014  1.4  1.5  3.86164  0.886593  1.1  67.5215  0.7  0.014  1.4  1.5  3.86449  0.882576  1.05  69.2466  0.7  0.014  1.4  1.5  3.86712  0.878627  70.899  0.7  0.014  1.4  1.5  3.86952  0.874471  72.5136  0.7  0.014  1.4  continued on next page  1  1  1 0.95  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued M  Log(T  )  Log(L/L )  Age(Gyrs)  3.87179  0.870235  0.9  1.5  3.8739  0.865959  1.5  3.87091  0.926943  1.5  3.87416  0.922573  1.5  3.87081  0.926943  1.5  3.87408  1.5  3.87074  Q  1.5  e / /  X  Z  a  74.1147  0.7  0.014  1.4  0.85  75.6791  0.7  0.014  1.4  1.1  67.065  0.7  0.012  1.8  1.05  68.986  0.7  0.012  1.8  1.1  66.9671  0.7  0.012  1.6  0.922573  1.05  68.9015  0.7  0.012  1.6  0.926943  1.1  66.8963  0.7  0.012  1.4  0  1.5  3.87402  0.922573  1.05  68.838  0.7  0.012  1.4  1.55  3.85746  0.855907  1.15  70.6776  0.74  0.014  1.8  1.55  3.85949  0.852278  1.1  71.9939  0.74  0.014  1.8  1.55  3.85906  0.852278  1.1  71.5591  0.74  0.014  1.6  1.55  3.85891  0.852254  1.1  71.3906  0.74  0.014  1.4  1.55  3.85814  0.906712  1.3  65.0552  0.74  0.012  1.8  1.55  3.86115  0.90298  1.25  66.6864  0.74  0.012  1.8  1.55  3.86399  0.899172  1.2  68.3096  0.74  0.012  1.8  1.55  3.86665  0.895184  1.15  69.931  0.74  0.012  1.8  1.55  3.8691  0.891157  1.1  71.5133  0.74  0.012  1.8  1.55  3.87137  0.887073  1.05  73.0606  0.74  0.012  1.8  1.55  3.87349  0.882926  1  74.5796  0.74  0.012  1.8  1.55  3.85777  0.906712  1.3  64.7198  0.74  0.012  1.6  1.55  3.8609  0.90298  1.25  66.4415  0.74  0.012  1.6  1.55  3.86381  0.899172  1.2  68.1254  0.74  0.012  1.6  1.55  3.8665  0.895184  1.15  69.7831  0.74  0.012  1.6  1.55  3.86898  0.891157  1.1  71.3852  0.74  0.012  1.6  1.55  3.87127  0.887073  1.05  72.9476  0.74  0.012  1.6  1.55  3.8734  0.882926  1  74.4758  0.74  0.012  1.6  1.55  3.85763  0.906686  1.3  64.5847  0.74  0.012  1.4  1.55  3.86079  0.902952  1.25  66.3279  0.74  0.012  1.4  1.55  3.86371  0.899143  1.2  68.0274  0.74  0.012  1.4  1.55  3.86641  0.895184  1.15  69.6886  0.74  0.012  1.4  1.55  3.8689  0.891157  1.1  71.2988  0.74  0.012  1.4  1.55  3.8712  0.887073  1.05  72.8676  0.74  0.012  1.4  1.55  3.87334  0.882926  1  74.401  0.74  0.012  1.4  1.55  3.85714  0.859941  1.05  70.0967  0.72  0.016  1.8  1.55  3.85922  0.85626  1  71.4261  0.72  0.016  1.8  1.55  3.86118  0.852536  0.95  72.7393  0.72  0.016  1.8  1.55  3.85877  0.85626  1  70.981  0.72  0.016  1.6  1.55  3.86081  0.852535  0.95  72.3597  0.72  0.016  1.6  1.55  3.8586  0.85626  1  70.8024  0.72  0.016  1.4  1.55  3.86067  0.852536  0.95  72.2031  0.72  0.016  1.4  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/X, )  Age(Gyrs)  Q  X  Z  a  1.55  3.85723  0.909133  1.2  64.4636  0.72  0.014  1.8  1.55  3.86026  0.90557  1.15  66.0613  0.72  0.014  1.8  1.55  3.86309  0.901847  1.1  67.6516  0.72  0.014  1.8  1.55  3.86577  0.897882  1.05  69.2581  0.72  0.014  1.8  1.55  3.86828  0.893962  1  70.8379  0.72  0.014  1.8  1.55  3.87059  0.889845  0.95  72.3882  0.72  0.014  1.8  1.55  3.87277  0.885666  0.9  73.9256  0.72  0.014  1.8  1.55  3.87481  0.881453  0.85  75.4312  0.72  0.014  1.8  1.55  3.85997  0.905564  1.15  65.7864  0.72  0.014  1.6  1.55  3.86289  0.901845  1.1  67.4478  0.72  0.014  1.6  1.55  3.86561  0.897882  1.05  69.0986  0.72  0.014  1.6  1.55  3.86816  0.893962  1  70.7027  0.72  0.014  1.6  1.55  3.87048  0.889845  0.95  72.2691  0.72  0.014  1.6  1.55  3.87268  0.885666  0.9  73.8191  0.72  0.014  1.6  1.55  3.87472  0.881453  0.85  75.3334  0.72  0.014  1.6  1.55  3.85983  0.905596  1.15  65.6521  0.72  0.014  1.4  1.55  3.86277  0.901884  1.1  67.3287  0.72  0.014  1.4  1.55  3.86551  0.897918  1.05  68.9925  0.72  0.014  1.4  1.55  3.86807  0.893995  1  70.6058  0.72  0.014  1.4  1.55  3.87041  0.889845  0.95  72.1866  0.72  0.014  1.4  1.55  3.87261  0.885666  0.9  73.7422  0.72  0.014  1.4  1.55  3.87466  0.881453  0.85  75.2612  0.72  0.014  1.4  1.55  3.85718  0.865042  0.95  69.5384  0.7  0.018  1.8  1.55  3.85935  0.861363  0.9  70.8981  0.7  0.018  1.8  1.55  3.86138  0.857611  0.85  72.2392  0.7  0.018  1.8  1.55  3.86327  0.853773  0.8  73.5562  0.7  0.018  1.8  1.55  3.8589  0.861363  0.9  70.4601  0.7  0.018  1.6  1.55  3.86103  0.857611  0.85  71.8715  0.7  0.018  1.6  1.55  3.86299  0.853773  0.8  73.249  0.7  0.018  1.6  1.55  3.85874  0.861363  0.9  70.288  0.7  0.018  1.4  1.55  3.86089  0.857611  0.85  71.7202  0.7  0.018  1.4  1.55  3.86287  0.853773  0.8  73.1135  0.7  0.018  1.4  1.55  3.85994  0.909967  1.05  65.472  0.7  0.016  1.8  1.55  3.86286  0.906075  1  67.1018  0.7  0.016  1.8  1.55  3.86562  0.902089  0.95  68.7333  0.7  0.016  1.8  1.55  3.86821  0.89794  0.9  70.3609  0.7  0.016  1.8  1.55  3.87061  0.893779  0.85  71.9499  0.7  0.016  1.8  1.55  3.87285  0.889562  0.8  73.5117  0.7  0.016  1.8  1.55  3.85963  0.909967  1.05  65.1873  0.7  0.016  1.6  1.55  3.86265  0.906075  1  66.8935  0.7  0.016  1.6  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued Log(T  e / /  )  Log(L/L )  Age(Gyrs)  0.902089  0.95  0  1.55  3.86546  1.55  3.86808  0.89794  1.55  3.8705  0.893779  1.55  3.87276  0.889562  1.55  3.8595  0.909967  1.55  3.86254  0.906075  1.55  3.86537  AV(IJ,HZ)  X  Z  a  68.5729  0.7  0.016  1.6  0.9  70.226  0.7  0.016  1.6  0.85  71.8318  0.7  0.016  1.6  0.8  73.4064  0.7  0.016  1.6  1.05  65.0596  0.7  0.016  1.4  1  66.784  0.7  0.016  1.4  0.902089  0.95  68.4745  0.7  0.016  1.4  1.55  3.868  0.89794  0.9  70.1374  0.7  0.016  1.4  1.55  3.87043  0.893779  0.85  71.7501  0.7  0.016  1.4  1.55  3.87269  0.889562  0.8  73.3303  0.7  0.016  1.4  1.6  3.85784  0.873785  1.05  69.7927  0.74  0.016  1.8  1.6  3.85988  0.870186  1  71.0932  0.74  0.016  1.8  1.6  3.86179  0.866522  0.95  72.3761  0.74  0.016  1.8  1.6  3.8636  0.862854  0.9  73.6413  0.74  0.016  1.8  1.6  3.86531  0.859104  0.85  74.8966  0.74  0.016  1.8  1.6  3.86641  0.854883  0.8  75.9517  0.74  0.016  1.8  1.6  3.85738  0.873785  1.05  69.3439  0.74  0.016  1.6  1.6  3.8595  0.870186  1  70.7117  0.74  0.016  1.6  1.6  3.86149  0.866522  0.95  72.0553  0.74  0.016  1.6  1.6  3.86336  0.862854  0.9  73.3726  0.74  0.016  1.6  1.6  3.86511  0.859104  0.85  74.6695  0.74  0.016  1.6  1.6  3.86623  0.854883  0.8  75.7441  0.74  0.016  1.6  1.6  3.8572  0.873785  1.05  69.1632  0.74  0.016  1.4  1.6  3.85936  0.870186  1  70.5541  0.74  0.016  1.4  1.6  3.86136  0.866522  0.95  71.9153  0.74  0.016  1.4  1.6  3.86324  0.862854  0.9  73.2447  0.74  0.016  1.4  1.6  3.865  0.859104  0.85  74.5509  0.74  0.016  1.4  1.6  3.86614  0.854883  0.8  75.6298  0.74  0.016  1.4  1.6  3.85849  0.923694  1.2  64.3305  0.74  0.014  1.8  1.6  3.86144  0.920105  1.15  65.9082  0.74  0.014  1.8  1.6  3.86421  0.916358  1.1  67.4857  0.74  0.014  1.8  1.6  3.86683  0.912452  1.05  69.065  0.74  0.014  1.8  1.6  3.86928  0.908516  1  70.6182  0.74  0.014  1.8  1.6  3.87155  0.904496  0.95  72.1435  0.74  0.014  1.8  1.6  3.87367  0.900453  0.9  73.6316  0.74  0.014  1.8  1.6  3.85817  0.923694  1.2  64.0363  0.74  0.014  1.6  1.6  3.86122  0.920105  1.15  65.693  0.74  0.014  1.6  1.6  3.86405  0.916358  1.1  67.3201  0.74  0.014  1.6  1.6  3.86669  0.912452  1.05  68.9279  0.74  0.014  1.6  1.6  3.86916  0.908516  1  70.4989  0.74  0.014  1.6  continued on next page  Appendix  B. A Listing  of the Evolutionary  Models  Table B . l : continued Log(T  e / /  )  Log(L/L )  Age(Gyrs)  Q  X  Z  a  72.0366  0.74  0.014  1.6  1.6  3.87146  0.904496  0.95  1.6  3.87359  0.900453  0.9  73.5351  0.74  0.014  1.6  1.6  3.85803  0.923694  1.2  63.9062  0.74  0.014  1.4  1.6  3.8611  0.920105  1.15  65.5815  0.74  0.014  1.4  1.6  3.86395  0.916358  1.1  67.2213  0.74  0.014  1.4  1.6  3.86661  0.912452  1.05  68.8386  0.74  0.014  1.4  1.6  3.86909  0.908516  1  70.417  0.74  0.014  1.4  1.6  3.87139  0.904496  0.95  71.9607  0.74  0.014  1.4  1.6  3.87353  0.900453  0.9  73.4632  0.74  0.014  1.4  1.6  3.85767  0.878412  0.95  69.2064  0.72  0.018  1.8  1.6  3.85975  0.874753  0.9  70.5244  0.72  0.018  1.8  1.6  3.86172  0.871049  0.85  71.8266  0.72  0.018  1.8  1.6  3.86358  0.867309  0.8  73.1143  0.72  0.018  1.8  1.6  3.85719  0.878412  0.95  68.7535  0.72  0.018  1.6  1.6  3.85937  0.874753  0.9  70.139  0.72  0.018  1.6  1.6  3.86142  0.871049  0.85  71.5049  0.72  0.018  1.6  1.6  3.86334  0.867309  0.8  72.8471  0.72  0.018  1.6  1.6  3.85702  0.878412  0.95  68.5725  0.72  0.018  1.4  1.6  3.85922  0.874753  0.9  69.9831  0.72  0.018  1.4  1.6  3.86129  0.871049  0.85  71.3663  0.72  0.018  1.4  1.6  3.86323  0.867309  0.8  72.7208  0.72  0.018  1.4  1.6  3.85802  0.926903  1.1  63.8287  0.72  0.016  1.8  1.6  3.86104  0.923197  1.05  65.4296  0.72  0.016  1.8  1.6  3.86387  0.919343  1  67.0315  0.72  0.016  1.8  1.6  3.86655  0.915407  0.95  68.6281  0.72  0.016  1.8  1.6  3.86904  0.911376  0.9  70.2034  0.72  0.016  1.8  1.6  3.87136  0.907276  0.85  71.7458  0.72  0.016  1.8  1.6  3.87352  0.903129  0.8  73.2589  0.72  0.016  1.8  1.6  3.85768  0.926903  1.1  63.5175  0.72  0.016  1.6  1.6  3.8608  0.923197  1.05  65.2041  0.72  0.016  1.6  1.6  3.8637  0.919343  1  66.8601  0.72  0.016  1.6  1.6  3.86641  0.915407  0.95  68.4883  0.72  0.016  1.6  1.6  3.86893  0.911376  0.9  70.0826  0.72  0.016  1.6  1.6  3.87126  0.907276  0.85  71.638  0.72  0.016  1.6  1.6  3.87344  0.903129  0.8  73.1617  0.72  0.016  1.6  1.6  3.85753  0.926903  1.1  63.3833  0.72  0.016  1.4  1.6  3.86068  0.923197  1.05  65.0899  0.72  0.016  1.4  1.6  3.8636  0.919343  1  66.7603  0.72  0.016  1.4  1.6  3.86633  0.915407  0.95  68.3984  0.72  0.016  1.4  1.6  3.86886  0.911376  0.9  70.0002  0.72  0.016  1.4  continued on next page  Appendix B. A Listing  of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/L ) 0  1.6  3.8712  0.907276  1.6  3.87338  1.6  3.85828  1.6  3.86042  0.880244  1.6  3.85783  0.883993  1.6  3.86007  0.880261  1.6  3.85767  0.883976  1.6  3.85993  0.880244  1.6  3.86179  0.928187  1.6  3.86466  0.924254  1.6  3.86734  0.920222  1.6  3.86986  0.916113  1.6  3.86158  0.928187  1.6  3.8645  0.924254  1.6  3.86721  1.6  3.86975  Age(Gyrs)  Av'pHz)  X  Z  a  0.85  71.5616  0.72  0.016  1.4  0.903129  0.8  73.0904  0.72  0.016  1.4  0.883976  0.85  68.841  0.7  0.02  1.8  0.8  70.1876  0.7  0.02  1.8  0.85  68.4121  0.7  0.02  1.6  0.8  69.8307  0.7  0.02  1.6  0.85  68.2481  0.7  0.02  1.4  0.8  69.6879  0.7  0.02  1.4  0.95  65.2113  0.7  0.018  1.8  0.9  66.8359  0.7  0.018  1.8  0.85  68.4449  0.7  0.018  1.8  0.8  70.0399  0.7  0.018  1.8  0.95  65.0071  0.7  0.018  1.6  0.9  66.6799  0.7  0.018  1.6  0.920222  0.85  68.3156  0.7  0.018  1.6  0.916113  0.8  69.9274  0.7  0.018  1.6  1.6  3.86147  0.928187  0.95  64.9001  0.7  0.018  1.4  1.6  3.86441  0.924254  0.9  66.5841  0.7  0.018  1.4  1.6  3.86713  0.920222  0.85  68.2287  0.7  0.018  1.4  1.6  3.86968  0.916113  0.8  69.8485  0.7  0.018  1.4  1.65  3.85826  0.890965  0.95  68.9859  0.74  0.018  1.8  1.65  3.8602  0.887234  0.9  70.2487  0.74  0.018  1.8  1.65  3.86213  0.883633  0.85  71.5199  0.74  0.018  1.8  1.65  3.8639  0.880199  0.8  72.7274  0.74  0.018  1.8  1.65  3.86678  0.871864  0.7  75.0978  0.74  0.018  1.8  1.65  3.86517  0.875609  0.75  73.8728  0.74  0.018  1.8  1.65  3.86829  0.868118  0.65  76.2971  0.74  0.018  1.8  1.65  3.86954  0.86402  0.6  77.4363  0.74  0.018  1.8  1.65  3.87208  0.856272  0.5  79.7346  0.74  0.018  1.8  1.65  3.87086  0.860187  0.55  78.5947  0.74  0.018  1.8  1.65  3.87302  0.852268  0.45  80.758  0.74  0.018  1.8  1.65  3.85785  0.890965  0.95  68.59  0.74  0.018  1.6  1.65  3.86187  0.883633  0.85  71.241  0.74  0.018  1.6  1.65  3.85987  0.887234  0.9  69.9142  0.74  0.018  1.6  1.65  •3.86369  0.880199  0.8  72.4954  0.74  0.018  1.6  1.65  3.86499  0.875609  0.75  73.6627  0.74  0.018  1.6  1.65  3.86663  0.871864  0.7  74.9147  0.74  0.018  1.6  1.65  3.86815  0.868118  0.65  76.1343  0.74  0.018  1.6  1.65  3.86941  0.86402  0.6  77.2841  0.74  0.018  1.6  1.65  3.87074  0.860187  0.55  78.4532  0.74  0.018  1.6  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/L©)  Age(Gyrs)  Au(^Hz)  X  Z  a  1.65  3.87197  0.856272  0.5  79.6023  0.74  0.018  1.6  1.65  3.87292  0.852268  0.45  80.6305  0.74  0.018  1.6  1.65  3.8577  0.890965  0.95  68.4283  0.74  0.018  1.4  1.65  3.85974  0.887234  0.9  69.7705  0.74  0.018  1.4  1.65  3.86358  0.880199  0.8  72.3759  0.74  0.018  1.4  1.65  3.86175  0.883633  0.85  71.1122  0.74  0.018  1.4  1.65  3.86489  0.875609  0.75  73.5494  0.74  0.018  1.4  1.65  3.86653  0.871864  0.7  74.8082  0.74  0.018  1.4  1.65  3.86934  0.86402  0.6  77.1869  0.74  0.018  1.4  1.65  3.86807  0.868118  0.65  76.0334  0.74  0.018  1.4  1.65  3.87067  0.860187  0.55  78.3599  0.74  0.018  1.4  1.65  3.87191  0.856272  0.5  79.5123  0.74  0.018  1.4  1.65  3.87286  0.852268  0.45  80.5424  0.74  0.018  1.4  1.65  3.86754  0.928044  0.95  68.6027  0.74  0.016  1.8  1.65  3.86992  0.924115  0.9  70.1192  0.74  0.016  1.8  1.65  3.87426  0.916134  0.8  73.0762  0.74  0.016  1.8  1.65  3.87216  0.920137  0.85  71.6134  0.74  0.016  1.8  1.65  3.86742  0.928044  0.95  68.48  0.74  0.016  1.6  1.65  3.86982  0.924115  0.9  70.0109  0.74  0.016  1.6  1.65  3.87208  0.920137  0.85  71.5155  0.74  0.016  1.6  1.65  3.87418  0.916134  0.8  72.9862  0.74  0.016  1.6  1.65  3.86734  0.928044  0.95  68.3969  0.74  0.016  1.4  1.65  3.86975  0.924115  0.9  69.9344  0.74  0.016  1.4  1.65  3.87202  0.920137  0.85  71.4434  0.74  0.016  1.4  1.65  3.87413  0.916134  0.8  72.9198  0.74  0.016  1.4  1.65  3.85753  0.858615  0.7  72.6221  0.72  0.022  1.8  1.65  3.85909  0.855098  0.65  73.7511  0.72  0.022  1.8  1.65  3.86039  0.851173  0.6  74.8307  0.72  0.022  1.8  1.65  3.857  0.858615  0.7  72.0963  0.72  0.022  1.6  1.65  3.85863  0.855098  0.65  73.2726  0.72  0.022  1.6  1.65  3.85998  0.851173  0.6  74.3905  0.72  0.022  1.6  1.65  3.85846  0.855098  0.65  73.0813  0.72  0.022  1.4  1.65  3.85983  0.851173  0.6  74.215  0.72  0.022  1.4  1.65  3.8585  0.895305  0.85  68.6089  0.72  0.02  1.8  1.65  3.86053  0.89167  0.8  69.8964  0.72  0.02  1.8  1.65  3.86248  0.887946  0.75  71.1833  0.72  0.02  1.8  1.65  3.86432  0.884185  0.7  72.4607  0.72  0.02  1.8  1.65  3.86607  0.880392  0.65  73.7256  0.72  0.02  1.8  1.65  3.86771  0.876596  0.6  74.9731  0.72  0.02  1.8  1.65  3.86893  0.872333  0.55  76.0978  0.72  0.02  1.8  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued Log(T 1.65  e / /  )  Log(L/L )  Age(Gyrs)  Q  Av(fiHz)  X  Z  a  3.87169  0.864517  0.45  78.4759  0.72  0.02  1.8  1.65  3.87041  0.868548  0.5  77.3078  0.72  0.02  1.8  1.65  3.87298  0.860641  0.4  79.6426  0.72  0.02  1.8  1.65  3.85809  0.89533  0.85  68.2174  0.72  0.02  1.6  1.65  3.86022  0.89167  0.8  69.5744  0.72  0.02  1.6  1.65  3.86223  0.887946  0.75  70.9184  0.72  0.02  1.6  1.65  3.86412  0.884185  0.7  72.2395  0.72  0.02  1.6  1.65  3.8659  0.880392  0.65  73.537  0.72  0.02  1.6  1.65  3.86757  0.876596  0.6  74.8072  0.72  0.02  1.6  1.65  3.8688  0.872333  0.55  75.9447  0.72  0.02  1.6  1.65  3.87029  0.868548  0.5  77.1676  0.72  0.02  1.6  1.65  3.87159  0.864517  0.45  78.3445  0.72  0.02  1.6  1.65  3.87288  0.860641  0.4  79.5188  0.72  0.02  1.6  1.65  3.85794  0.89533  0.85  68.0606  0.72  0.02  1.4  1.65  3.86009  0.89167  0.8  69.4358  0.72  0.02  1.4  1.65  3.86212  0.887946  0.75  70.7929  0.72  0.02  1.4  1.65  3.86402  0.884185  0.7  72.1248  0.72  0.02  1.4  1.65  3.86581  0.880392  0.65  73.43  0.72  0.02  1.4  1.65  3.86748  0.876596  0.6  74.7065  0.72  0.02  1.4  1.65  3.86872  0.872333  0.55  75.8465  0.72  0.02  1.4  1.65  3.87022  0.868548  0.5  77.075  0.72  0.02  1.4  1.65  3.87281  0.860641  0.4  79.4327  0.72  0.02  1.4  1.65  3.87152  0.864517  0.45  78.2553  0.72  0.02  1.4  1.65  3.87007  0.92741  0.8  69.8258  0.72  0.018  1.8  1.65  3.87234  0.923318  0.75  71.3384  0.72  0.018  1.8  1.65  3.87446  0.919198  0.7  72.8226  0.72  0.018  1.8  1.65  3.86997  0.92741  0.8  69.7189  0.72  0.018  1.6  1.65  3.87225  0.923318  0.75  71.2417  0.72  0.018  1.6  1.65  3.87438  0.919198  0.7  72.7338  0.72  0.018  1.6  1.65  3.86991  0.92741  0.8  69.6431  0.72  0.018  1.4  1.65  3.87219  0.923318  0.75  71.1713  0.72  0.018  1.4  1.65  3.87433  0.919198  0.7  72.6679  0.72  0.018  1.4  1.65  3.85819  0.918735  0.85  65.8182  0.7  0.022  1.8  1.65  3.86064  0.915033  0.8  67.2334  0.7  0.022  1.8  1.65  3.86298  0.9113  0.75  68.6429  0.7  0.022  1.8  1.65  3.86446  0.906274  0.7  69.8677  0.7  0.022  1.8  1.65  3.86856  0.89851  0.6  72.6495  0.7  0.022  1.8  1.65  3.86657  0.902398  0.65  71.2674  0.7  0.022  1.8  1.65  3.87044  0.894579  0.55  74.0153  0.7  0.022  1.8  1.65  3.87203  0.89035  0.5  75.3075  0.7  0.022  1.8  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued MQ  Log(T  e / /  )  Log(L/L ) 0  Age(Gyrs)  Av(fj,Hz)  X  Z  a  1.65  3.87371  0.886314  0.45  76.6406  0.7  0.022  1.8  1.65  3.85783  0.918735  0.85  65.4753  0.7  0.022  1.6  1.65  3.86037  0.915033  0.8  66.9652  0.7  0.022  1.6  1.65  3.86278  0.9113  0.75  68.4332  0.7  0.022  1.6  1.65  3.86429  0.906274  0.7  69.6819  0.7  0.022  1.6  1.65  3.86642  0.902398  0.65  71.1103  0.7  0.022  1.6  1.65  3.87033  0.894579  0.55  73.8921  0.7  0.022  1.6  1.65  3.86844  0.89851  0.6  72.5128  0.7  0.022  1.6  1.65  3.87193  0.89035  0.5  75.1921  0.7  0.022  1.6  1.65  3.87362  0.886314  0.45  76.5342  0.7  0.022  1.6  1.65  3.86025  0.915033  0.8  66.8418  0.7  0.022  1.4  1.65  3.85768  0.918735  0.85  65.3347  0.7  0.022  1.4  1.65  3.86267  0.9113  0.75  68.3235  0.7  0.022  1.4  1.65  3.86419  0.906274  0.7  69.5776  0.7  0.022  1.4  1.65  3.86836  0.89851  0.6  72.4232  0.7  0.022  1.4  1.65  3.86634  0.902398  0.65  71.0142  0.7  0.022  1.4  1.65  3.87026  0.894579  0.55  73.8067  0.7  0.022  1.4  1.65  3.87187  0.89035  0.5  75.1115  0.7  0.022  1.4  1.65  3.87356  0.886314  0.45  76.4574  0.7  0.022  1.4  1.65  3.87358  0.928031  0.65  71.3719  0.7  0.02  1.8  1.65  3.8735  0.928031  0.65  71.2819  0.7  0.02  1.6  1.65  3.87344  0.928031  0.65  71.2143  0.7  0.02  1.4  1.7  3.87422  0.893294  0.4  76.9724  0.72  0.022  1.4  1.7  3.87133  0.90147  0.5  74.5169  0.72  0.022  1.4  1.7  3.87295  0.897583  0.45  75.7939  0.72  0.022  1.4  1.7  3.86962  0.905362  0.55  73.2113  0.72  0.022  1.4  1.7  3.8678  0.909232  0.6  71.8816  0.72  0.022  1.4  1.7  3.86586  0.913075  0.65  70.5273  0.72  0.022  1.4  1.7  3.8638  0.916886  0.7  69.1476  0.72  0.022  1.4  1.7  3.86161  0.920658  0.75  67.7426  0.72  0.022  1.4  1.7  3.85928  0.924385  0.8  66.3148  0.72  0.022  1.4  1.7  3.87427  0.893294  0.4  77.0461  0.72  0.022  1.6  1.7  3.8714  0.90147  0.5  74.5966  0.72  0.022  1.6  1.7  3.873  0.897583  0.45  75.87  0.72  0.022  1.6  1.7  3.86969  0.905362  0.55  73.2953  0.72  0.022  1.6  1.7  3.86787  0.909232  0.6  71.9716  0.72  0.022  1.6  1.7  3.86594  0.913075  0.65  70.6225  0.72  0.022  1.6  1.7  3.86389  0.916886  0.7  69.2505  0.72  0.022  1.6  1.7  3.86171  0.920658  0.75  67.8562  0.72  0.022  1.6  1.7  3.8594  0.924385  0.8  66.4404  0.72  0.022  1.6  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued )  Log(L/L©)  Age(Gyrs)  Log(T  1.7  3.87435  0.893294  0.4  1.7  3.8715  0.90147  1.7  3.87309  0.897583  1.7  3.8698  0.905362  1.7  3.868  0.909232  1.7  3.86609  0.913075  1.7  3.86193  0.920658  1.7  3.86406  1.7  3.85969  1.7 1.7  X  Z  a  77.1483  0.72  0.022  1.8  0.5  74.71  0.72  0.022  1.8  0.45  75.9758  0.72  0.022  1.8  0.55  73.4184  0.72  0.022  1.8  0.6  72.1076  0.72  0.022  1.8  0.65  70.7766  0.72  0.022  1.8  0.75  68.0785  0.72  0.022  1.8  0.916886  0.7  69.4326  0.72  0.022  1.8  0.924385  0.8  66.721  0.72  0.022  1.8  3.85732  0.928007  0.85  65.3619  0.72  0.022  1.8  3.87081  0.876273  0.45  77.4638  0.74  0.02  1.4  1.7  3.87206  0.872455  0.4  78.5961  0.74  0.02  1.4  1.7  3.86957  0.880248  0.5  76.3344  0.74  0.02  1.4  1.7  3.86694  0.888114  0.6  74.0574  0.74  0.02  1.4  1.7  3.86813  0.883961  0.55  75.1525  0.74  0.02  1.4  1.7  3.86532  0.891834  0.65  72.827  0.74  0.02  1.4  1.7  3.86359  0.895551  0.7  71.5678  0.74  0.02  1.4  1.7  3.85978  0.902875  0.8  68.9728  0.74  0.02  1.4  1.7  3.86174  0.899232  0.75  70.2833  0.74  0.02  1.4  1.7  3.85769  0.906474  0.85  67.6378  0.74  0.02  1.4  1.7  3.87212  0.872455  0.4  78.6818  0.74  0.02  1.6  1.7  3.87088  0.876273  0.45  77.5526  0.74  0.02  1.6  1.7  3.86965  0.880248  0.5  76.4263  0.74  0.02  1.6  1.7  3.86702  0.888114  0.6  74.1571  0.74  0.02  1.6  1.7  3.86821  0.883961  0.55  75.2498  0.74  0.02  1.6  1.7  3.86541  0.891834  0.65  72.9326  0.74  0.02  1.6  1.7  3.86185  0.899232  0.75  70.4055  0.74  0.02  1.6  1.7  3.86369  0.895551  0.7  71.6808  0.74  0.02  1.6  1.7  3.8599  0.902875  0.8  69.1072  0.74  0.02  1.6  1.7  3.85783  0.906474  0.85  67.7887  0.74  0.02  1.6  1.7  3.87099  0.876273  0.45  77.6848  0.74  0.02  1.8  MQ  e / /  Ai/(nHz)  1.7  3.87222  0.872455  0.4  78.8056  0.74  0.02  1.8  1.7  3.86976  0.880248  0.5  76.5656  0.74  0.02  1.8  1.7  3.86834  0.883961  0.55  75.4013  0.74  0.02  1.8  1.7  3.86716  0.888114  0.6  74.3194  0.74  0.02  1.8  1.7  3.86557  0.891834  0.65  73.1171  0.74  0.02  1.8  1.7  3.86388  0.895551  0.7  71.8951  0.74  0.02  1.8  1.7  3.86209  0.899232  0.75  70.661  0.74  0.02  1.8  1.7  3.8602  0.902875  0.8  69.4141  0.74  0.02  1.8  1.7  3.85821  0.906474  0.85  68.1559  0.74  0.02  1.8  continued on next page  Appendix B. A Listing of the Evolutionary  Models  Table B . l : continued  Log(r )  Log(L/L )  Age(Gyrs)  3.86192  0.855628  0.5  1.7  3.8606  0.859189  1.7  3.85931  0.862978  1.7  3.85782  1.7  M 1.7  76.7749  0.74  0.022  a 1.  75.7297  0.74  0.022  1.  0.55  74.637  0.74  0.022  1.  0.6  73.5504  0.74  0.022  1.  0.866448  0.65  72.4199  0.74  0.022  1.  3.86324  0.852096  0.45  76.9168  0.74  0.022  1.  1.7  3.86205  0.855628  0.5  75.8786  0.74  0.022  1.  1.7  3.85946  0.862978  0.6  73.7227  0.74  0.022  1.  1.7  3.86074  0.859189  0.55  74.7964  0.74  0.022  1.  1.7  3.85798  0.866448  0.65  72.6091  0.74  0.022  1.  1.7  3.86351  0.852096  0.45  77.2376  0.74  0.022  1.  1.7  3.86236  0.855628  0.5  76.2322  0.74  0.022  1.  1.7  3.86109  0.859189  0.55  75.1875  0.74  0.022  1.  1.7  3.85844  0.866448  0.65  73.08  0.74  0.022  1.  1.7  3.85986  0.862978  0.6  74.1491  0.74  0.022  1.  1.75  3.87365  0.903068  0.4  76.4893  0.74  0.022  1.  1.75  3.87078  0.910972  0.5  74.0836  0.74  0.022  1.  1.75  3.8722  0.906883  0.45  75.2845  0.74  0.022  1.  1.75  3.86737  0.918494  0.6  71.5552  0.74  0.022  1.  1.75  3.86914  0.914748  0.55  72.8352  0.74  0.022  1.  1.75  3.86408  0.926875  0.7  69.0688  0.74  0.022  1.  1.75  3.8655  0.922215  0.65  70.2498  0.74  0.022  1.  1.75  3.87226  0.906883  0.45  75.3618  0.74  0.022  1.  1.75  3.87371  0.903068  0.4  76.5625  0.74  0.022  1.  1.75  3.86921  0.914748  0.55  72.9197  0.74  0.022  1.  1.75  3.87084  0.910972  0.5  74.1629  0.74  0.022  1.  1.75  3.86558  0.922215  0.65  70.3442  0.74  0.022  1.  1.75  3.86745  0.918494  0.6  71.6436  0.74  0.022  1.  1.75  3.86417  0.926875  0.7  69.1673  0.74  0.022  1.  1.75  3.87235  0.906883  0.45  75.4687  0.74  0.022  1.  1.75  3.87379  0.903068  0.4  76.6634  0.74  0.022  1.  1.75  3.86931  0.914748  0.55  73.0422  0.74  0.022  1.  1.75  3.87094  0.910972  0.5  74.2772  0.74  0.022  1.  1.75  3.86433  0.926875  0.7  69.3334  0.74  0.022  1.  1.75  3.86572  0.922215  0.65  70.4959  0.74  0.022  1.  1.75  3.86757  0.918494  0.6  71.7794  0.74  0.022  1.  Q  1.7  e//  3.86313  0  0.852096  0.45  AufaHz)  X  Z  102  Appendix C  Tables of the Weighted D a t a Results T h e results f r o m t h e weighted d a t a analysis are presented i n T a b l e s C . l , C . 2 , C . 3 , a n d C . 4 for t h e four m o d e l s w i t h t h e lowest xLmp s t a t i s t i c . E a c h o f t h e m o d e l a r e i d e n t i f i e d b y n u m b e r i n t h e t a b l e c a p t i o n w h i l e a d e s c r i p t i o n o f h o w e a c h m o d e l w a s r e d u c e d is d e s c r i b e d i n S e c t i o n 2.6 a n d T a b l e 2 . 3 . T a b l e s C . 5 , C . 6 , C . 7 , C . 8 , C . 9 , a n d C . 1 0 present t h e average frequency, a n d p h a s e f o r m o d e l s w i t h t h e s a m e i d e n t i f i e d frequencies, b u t w i t h different parameters.  T h e s t a n d a r d e r r o r o f t h e m e a n for e a c h o f t h e frequencies,  amplitude weighting amplitudes  a n d p h a s e s a r e d e n o t e d as a. M o d e l s are i d e n t i f i e d b y n u m b e r i n t h e t a b l e c a p t i o n a n d r e d u c t i o n d e s c r i p t i o n s m a y b e f o u n d i n S e c t i o n 2.6 a n d T a b l e 2 . 3 .  Tabled:  -  +  W e i g h t e d m o d e l n u m b e r 14. <5i/(mHz)  v (mHz)  Amp.(mma)  2.61858507  0.070913352  0.910846  2.61953486  0.24294598  0.0675517  2.62054495  0.095604089  0.192343  0.00101009  2.65118266  0.049705656  0.499794  0.00078301 0.00098078  4>  Ai/(mHz)  0.00094979  -  2.65196567  0.227132022  0.0337749  1/2  2.65294645  0.785205032  0.136531  + ++  2.65389001  0.198684201  0.305039  0.00094356  2.65491013  0.036362386  0.295139  0.00102012  0.03341159  V3-L>2  +  2.68645653  0.316776759  0.182504  2.68749324  0.554052254  0.100974  2.68840557  0.202862201  0.373698  0.00091233  2.71912802  0.050704868  0.24473  0.0009345  continued on next page  0.00103671  0.03454679  0.03343382  Appendix  C. Tables of the Weighted Data  Results  Table C . l : continued v (mHz)  Amp.(mma)  4>  dV(mHz)  -  2.72006252  0.382050618  0.518275  0.00086454  1/4  2.72092706  1.16528474  0.7818  + ++  2.72182564  0.411449615  0.0173451  0.00089858  2.7227747  0.095959262  0.204671  0.00094906  _  2.75431902  0.206336308  0.111059  0.00101107  "5  2.75533009  0.273609593  0.2087  +  2.75623487  0.122168294  0.499628  0.00090478  2.78801304  0.06013802  0.351141  0.00091362  (new)  2.78892666  0.103978465  0.648353  +  2.79001282  0.096200533  0.417312  0.00108616  2.79059069  0.131995516  0.25712  0.00093511  1/6 (old)  2.7915258  0.210137403  0.42461  +  2.79250869  0.085426427  0.612155  0.00098289  2.80563375  0.18459977  0.430217  0.00094644  2.80658019  0.129121204  0.486159  2.80739686  0.034644142  0.762129  Ai/(mHz)  I/5-f4  0.03440303  i/6 (new)-i/5  1/6  0.03359657  i/6 (old)-i/6 (new)  0.00259914  1/7-1/6 (old)  +  Table c.2:  -  +  0.01505439  0.00081667  Weighted model number 18.  v (mHz)  Amp.(mma)  4>  <Si/(mHz)  2.61856429  0.071942551  0.964752  0.0009714  2.61953569  0.229725434  0.06479  2.62053759  0.093576944  0.185547  0.0010019  Ai/(mHz)  1/2-1/1  0.03341124  -  2.65197721  0.237377855  0.006794  2.65,294693  0.778877511  0.138822  +  2.65388537  0.194361604  0.316978  0.00093844  -  2.68644101  0.326210402  0.208817  0.00105424  2.68749525  0.55849757  0.100642  2.68839972  0.207372074  0.384754  0.00096972  1/3-  V2  0.03454832  +  0.00090447  1/4-^3  0.03343165  -  2.72006234  0.383081687  continued on next page  0.513257  0.00086456  ix C. Tables of the Weighted Data Results Table C.2: continued <5i/(mHz)  v (mHz)  Amp.(mma)  4>  1/4  2.7209269  1.16752688  0.781976  +  2.72182919  0.421655594  0.014391  0.00090229  -  2.75431064  0.208573433  0.117343  0.00101379  Ai/(mHz)  1/5-1/4  0.03439753 2.75532443  0.272340984  0.225663  +  2.75623806  0.113454944  0.478726  0.00091363  -  2.78800841  0.051362814  0.372008  0.00092015  1/6 (new)  2.78892856  0.116984091  0.642873  +  2.79002184  0.098770964  0.396236  0.00109328  -  2.79063517  0.123071617  0.17611  0.00089349  1/6 (old)  2.79152866  0.213303151  0.423763  +  2.79251626  0.081582989  0.606488  -  2.80563794  0.187002768  0.423267  1/7  2.80655839  0.127268129  0.526702  +  2.807416  0.028527146  0.740054  i/6 (new)-i/5  0.03360413  1/6 (old)-i/6 (new)  0.0026001  0.0009876 1/7-1/6 (old)  0.00092045  0.01502973  0.00085761  Table c.3: W e i g h t e d m o d e l n u m b e r 2 7 . v (mHz)  Amp.(mma)  *  <5i/(mHz)  1/1  2.61952756  0.250389276  0.075044  0.001022  +  2.62055003  0.097967802  0.188145  -  2.65197412  0.236102849  0.010989  1/2  2.65294523  0.779581638  0.13877  +  2.65389634  0.198998402  0.293861  -  2.6864492  0.318197961  0.196758  "3  2.68749237  0.554835233  0.105008  +  2.68840277  0.204518648  0.377414  Ai/(mHz)  1/2-1/1  0.03341767 0.000971  0.000951 0.03454714 0.001043  0.00091  1/4-1/3  0.03343572  -  2.72005924  0.387951466  0.52163  1/4  2.72092809  1.16502003  0.780566  +  2.72182907  0.425400013  0.014838  0.000869  0.000901  1/5-1/4  0.03440585 continued on next page  Appendix  C. Tables of the Weighted Data  Results  Table C.3: continued v (mHz)  Amp.(mma)  2.75431424 2.75533394  4>  <5t/(mHz)  0.202921595  0.121033  0.00102  0.27371851  0.207484  2.75623869  0.120983186  0.480159  0.000905  2.78894119  0.110023328  0.610227  0.000952  +  2.78989309  0.089420136  0.629435  -  2.79062019  0.133141748  0.20376  2.79152475  0.206709227  0.420454  +  2.79244265  0.09004229  0.718838  -  2.80563819  0.182137724  0.428567  2.80657825  0.135530035  0.482141  -  +  Ai/(mHz)  i/6 (new)-i/5  0.03360725 (new)  1/6  1/6  (old)  0.000905  i/6 (old)-i/6 (new)  0.00258356 0.000918 1/7-1/6  0.00094  (old)  0.0150535  Table c.4: W e i g h t e d m o d e l n u m b e r 3 2 . 1/  -  + ++  (mHz)  Amp.(mma)  4>  <5i/(mHz) 0.00091106  2.61862362  0.073309533  0.832223  2.61953468  0.239755041  0.069636  2.62048027  0.095932955  0.245718  0.00094559  2.62118363  0.080067584  0.011993  0.00070336  Ai/(mHz)  0.03341106 -  2.6511999  0.051266074  0.460183  0.00079055  -  2.65199045  0.229358057  0.990978  0.00095529  2.65294574  0.781589995  0.138796  + ++  2.65389168  0.197025132  0.305247  0.00094594  2.65493441  0.039233571  0.275253  0.00104273  -  2.68558661  0.082521756  0.099311  0.0008698  -  2.68645641  0.304507647  0.17997  0.0010341  2.68749051  0.556092148  0.10715  2.68841601  0.201709362  0.351061  1/3-1/2  0.03454477  +  0.0009255  1/4-^3  0.03343682 -  2.72005692  0.378040442  0.523409  1/4  2.72092733  1.16704816  0.78134  2.72182519  0.412164347  0.017549  0.00089786  2.72280218  0.09867953  0.143867  0.00097699  + ++  continued on next page  0.00087041  ix C. Tables of the Weighted Data Results Table C.4: continued  1/  (mHz)  Amp.(mma)  <5i/(mHz)  <t>  Ai/(mHz)  0.03441038 2.75431941  0.205991964  2.75533771  0.270062018  0.19682  2.75623916  0.121934717  0.491413  2.78894448  0.110591288  0.599474  +  2.78997949  0.095712189  0.48221  0.00103501  -  2.79058457  0.134939722  0.261878  0.00094464  (old)  2.79152921  0.212994084  0.416727  +  2.79247997  0.083587537  0.67673  0.00095076  -  2.80563634  0.183135677  0.431134  0.00094312  1/7  2.80657946  0.13235342  0.483335  -  +  0.113945  0.0010183  0.00090145  1/6 (new)-i/5  0.03360677 v% (new)  1/6 ( o l d ) - f 6  (new)  0.00258473  1/6  1/7-1/6  (old)  0.01505025  Table c.5: Average values for weighted reduction models 1 through 11. Also Shown are the standard errors on the average, a, for each of the frequencies, amplitudes and phases.  v (mHz)  A m p l i t u d e (mma)  & Amp  <t>  &<t>  2.61953545  7 900E-07  0.24601249  7 373E-04  0.0633009  1 051E-03  2.62052083  1 070E-06  0.10171523  1 705E-03  0.19650982  1 356E-03  2.65198565  7 500E-07  0.24279207  5 547E-04  0.9961617  1 035E-03  2.65294337  2 400E-07  0.79240263  6 705E-04  0.14624444  4 458E-04  2.65389228  3 400E-07  0.18579648  5 661E-04  0.30279192  5 390E-04  2.68644238  1 140E-06  0.330598  1 029E-03  0.20020376  1 827E-03  2.68749022  3 700E-07  0.56121862  7 310E-04  0.11432059  6 970E-04  2.68841076  8 400E-07  0.21633655  5 988E-04  0.35386428  1 600E-03  2.72006512  2 700E-07  0.4069277  1 241E-03  0.50648844  4 798E-04  2.72092748  8 000E-08  1.14889014  1 484E-03  0.78240085  1 675E-04  2.72182369  5 500E-07  0.43446591  3 040E-04  0.01875269  5 798E-04  2.75431871  6 400E-07  0.21439223  6 065E-04  0.11760015  1 002E-03  2.7553339  5 900E-07  0.27781844  5 303E-04  0.21567683  1 273E-03  2.75623393  1 330E-06  0.11794064  6 995E-04  0.4960227  2 322E-03  2.78893447  4 400E-07  0.12777089  3 318E-04  0.61323953  8 992E-04  ix C. Tables of the Weighted Data  Results  Table C.5: continued v (mHz)  0V  Amplitude (mma)  °~Amp  2.78996491  2.890E-06  0.09356487  6.730E-04  0.50497055  5.201E-03  2.79062867  1.100E-06  0.13355158  6.760E-04  0.19025911  1.706E-03  2.79156089  1.230E-06  0.20052089  6.845E-04  0.38543555  1.287E-03  <t>  2.7921505  2.400E-06  0.07815215  1.272E-03  0.09016077  3.053E-03  2.80562377  7.200E-07  0.19279967  8.135E-04  0.44659683  1.029E-03  2.80656481  6.400E-07  0.12456012  9.508E-04  0.48891917  1.161E-03  Table c.6: A v e r a g e v a l u e s for w e i g h t e d r e d u c t i o n m o d e l s 12 t h r o u g h 15. A l s o S h o w n a r e t h e s t a n d a r d e r r o r s o n t h e average, o, for e a c h o f t h e frequencies, a m p l i t u d e s a n d phases.  v (mHz)  Ov  Amplitude (mma)  OAmp  2.6185801  5.46E-06  0.0715267  0.001161  0.91292524  0.0068766  2.619539  3.92E-06  0.2424247  0.002655  0.06225838  0.006325  2.6205368  9.79E-06  0.0976105  0.001203  0.19342275  0.0057129  2.6511872  6.01E-06  0.0647797  0.012309  0.494468  0.007732  2.6519675  1.86E-06  0.226293  0.0012  0.03131885  0.0015268  2.6529458  5.2E-07  0.7874028  0.002408  0.13963825  0.0025338  0~<f,  2.653890  6.7E-07  0.1903922  0.008158  0.30810699  0.0033934  2.6549232  8.68E-06  0.0481614  0.008691  0.28563949  0.0040056  2.6864514  6.72E-06  0.3262788  0.006929  0.187787  0.0091915  2.6874924  1.34E-06  0.5533767  0.001739  0.10520358  0.0040416  2.6884134  6.67E-06  0.2080756  0.003902  0.35549724  0.0135296  2.719132  3.29E-06  0.050237  0.001858  0.20023625  0.0308297  2.7200642  1.49E-06  0.390004  0.006373  0.51269579  0.0043499  2.7209268  4.1E-07  1.156417  0.007015  0.78290647  0.0009107  2.7218266  5.4E-07  0.4154691  0.003349  0.01652785  0.0004565  2.7227788  5.79E-06  0.0916642  0.003555  0.19412225  0.015008  2.7543182  2.23E-06  0.2086525  0.002428  0.1121475  0.0025403  2.7553289  2.6E-06  0.2761036  0.001343  0.21627975  0.0083184 0.0029707  2.7562337  1.61E-06  0.120785  0.001266  0.50185174  2.788000  1.24E-05  0.0534726  0.004968  0.3860465  0.029895  2.7889254  1.37E-06  0.1096075  0.005586  0.64184302  0.0052977  2.7900126  2.77E-06  0.0961226  0.001073  0.42164949  0.0084355  2.7905953  5.41E-06  0.1327671  0.001003  0.25295049  0.0077471  2.7915325  7.71E-06  0.2017605  0.006026  0.41866401  0.0086059  2.7925239  1.52E-05  0.0771367  0.007067  0.58520973  0.0246078  Appendix  C. Tables of the Weighted Data  Results  108  Table C.6: continued v (mHz)  Amplitude (mma)  QAmp  4>  0>  2.8056314  3.1E-06  0.1854348  0.001802  0.43582401  2.8065777  3.34E-06  0.1242794  0.003883  0.48549101  0.0051416  2.807390  1.03E-05  0.0377643  0.002778  0.77594125  0.0130536  0.0061031  Table c.7: A v e r a g e v a l u e s for w e i g h t e d r e d u c t i o n m o d e l s 16 t h r o u g h 19. A l s o S h o w n a r e t h e s t a n d a r d e r r o r s o n t h e average, cf, for e a c h o f t h e frequencies, a m p l i t u d e s a n d phases.  v (mHz)  <7„  Amplitude (mma)  2 61856365  2 610E-06  0.07097502  1 044E-03  0.95333153  4 762E-03  2 61953688  3 200E-06  0.2359844  2 497E-03  0.06417342  4 773E-03  2 62054586  3 710B-06  0.0981698  1 705E-03  0.17652375  3 844E-03  2 65197802  2 260E-06  0.23576348  1 570E-03  0.00600668  2 442E-03  2 65294647  5 800E-07  0.78556252  3 354E-03  0.13977875  2 047E-03  2 65388799  9 100E-07  0.19226289  3 603E-03  0.31249276  1 947E-03  2 68644285  4 650E-06  0.32979065  5 685E-03  0.20419475  6 233E-03  2 68749428  9 300E-07  0.55675828  1 288E-03  0.10356157  3 047E-03  2 68840957  8 020E-06  0.20968853  3 247E-03  0.36267751  1 678E-02  2 72006321  1 410E-06  0.39359266  6 682E-03  0.51109952  3 760E-03  2.720927  3 0OOE-O7  1.15567088  7 099E-03  0.78218025  7 558E-04  2 72182894  3 500E-07  0.42885065  4 093E-03  0.0144197  5 332E-04  4>  2 75431371  1 730E-06  0.21081457  2 863E-03  0.11894425  2 114E-03  2 75532389  2 540E-06  0.27500349  2 543E-03  0.22870174  7 207E-03  2 75623894  5 500E-07  0.11634202  1 571E-03  0.48223901  1 813E-03  2 78798771  1 421E-05  0.05151148  2 720E-03  0.41204751  2 952E-02  2 78892875  3 801E-03  0.63687551  4 427E-03  1 030E-06  0.11496046  2 79001665  3 120E-06  0.09732255  1 377E-03  0.40953773  8 861E-03  2 79062533  4 330E-06  0.12959906  2 632E-03  0.19501725  7 734E-03  2 79153204  6 010E-06  0.20236254  6 278E-03  0.42214176  5 745E-03  2 79252291  1 587E-05 .  0.07779086  6 570E-03  0.58961576  2 539E-02  2 80563474  2 750E-06  0.18785006  1 122E-03  0.42921099  5 395E-03  2 80656528  3 320E-06  0.1226374  3 468E-03  0.50999427  7 254E-03  2 80740094  7 880E-06  0.03561373  3 603E-03  0.765073  1 064E-02  ix C. Tables of the Weighted Data Results Table c.8: A v e r a g e v a l u e s for w e i g h t e d r e d u c t i o n m o d e l s 21 t h r o u g h 24. A l s o S h o w n are t h e s t a n d a r d e r r o r s o n t h e average, o, for e a c h of t h e frequencies, a m p l i t u d e s a n d phases.  1/ ( m H z )  a  v  A m p l i t u d e (mma)  oAmp  2.61856  3.32E-06  0.071109  0 000871  0.952817  0.003281  2 619539  2.51E-06  0.236962  0 001243  0.061373  0.004041  2 620543  3.93E-06  0.100365  0 000319  0.178258  0.000697  2 651981  1.83E-06  0.23488  0 001981  0.502181  0.287136  2 652946  3.1E-07  0.790163  0 003414  0.14174  0.001667  2 653889  3.4E-07  0.189002  0 003843  0.310469  0.002024  2 686444  3.6E-06  0.330803  0 004905  0.200294  0.004694  2 687494  6.9E-07  0.557191  0 001435  0.105834  0.002632  2 688414  5.52E-06  0.211807  0 002841  0.351072  0.011941  2 720066  7.3E-07  0.399857  0 005238  0.505701  0.002535 0.000868  2 720927  4E-07  1.148638  0 005753  0.782802  2 721828  3.6E-07  0.434166  0 003309  0.016203  0.00082  2 754317  1.19E-06  0.213565  0 002504  0.119555  0.001758  2 755326  2.17E-06  0.277703  0 001866  0.225756  0.006226  2 756234  1.59E-06  0.119914  0 001552  0.495347  0.000796  2 787999  1.22E-05  0.05118  0 004149  0.379469  0.026376  2 788923  1.42E-06  0.115093  0 003382  0.649199  0.007537  2 789982  4.66E-06  0.097075  0 001393  0.48186  0.008765  2 790619  2.82E-06  0.131418  0 001352  0.20935  0.004125  2 791539  5.39E-06  0.202949  0 005398  0.421275  0.006661  2 805624  1.49E-06  0.19008  0 002272  0.447171  0.003409  2 806567  2.06E-06  0.118159  0 004019  0.50231  0.005409  2 807406  1.43E-05  0.034288  0 003765  0.740786  0.026875  Table c.9: A v e r a g e v a l u e s for w e i g h t e d r e d u c t i o n m o d e l s 25 t h r o u g h 28. A l s o S h o w n are t h e s t a n d a r d e r r o r s o n t h e average, o, for e a c h o f t h e frequencies, a m p l i t u d e s a n d phases.  v (mHz)  0V  Amplitude (mma)  °~Amp  2.619531  4.11E-06  0.249794  0.002486  0.070423  0.005858  2.620545  5.16E-06  0.099087  0.001556  0.184552  0.001297  2.651977  2.73E-06  0.23728  0.002204  0.007028  0.002996  4>  c<t>  ix C. Tables of the Weighted Data  Results  Table C.9: continued v (mHz)  Ov  Amplitude (mma)  °" Amp  2.652945  5.7E-07  0.78296  0.00294  0.141938  0.002486  2.653896  1.24E-06  0.1935  0.005232  0.295626  0.003961  2.686445  6.26E-06  0.325477  0.006056  0.200329  0.008523  2.687492  9.6E-07  0.554609  0.001527  0.108359  0.003112  2.688411  7.1E-06  0.209002  0.004507  0.357545  0.014579  2.720061  1.39E-06  0.396077  0.005775  0.516621  0.003966  2.720928  3.8E-07  1.156686  0.006538  0.781549  0.000724  2.721829  6E-07  0.428799  0.003657  0.014375  0.000309  2.754314  1.04E-06  0.206781  0.003963  0.120556  0.000893  </>  2.75533  4.96E-06  0.276679  0.001751  0.218244  0.011929  2.756242  1.27E-06  0.118345  0.002162  0.477665  0.001553  2.788937  5.03E-06  0.115846  0.006419  0.615988  0.007687  2.78995  2.91E-05  0.091216  0.001361  0.530571  0.051962  2.790627  4.29E-06  0.13455  0.001285  0.193457  0.00707  2.79153  6.63E-06  0.201207  0.004239  0.417806  0.005454  2.792489  2.34E-05  0.07973  0.008848  0.64849  0.035226  2.805636  3.07E-06  0.183588  0.001291  0.42937  0.003845  2.806574  4.47E-06  0.129968  0.003565  0.487405  0.00602  Table c.iO: A v e r a g e v a l u e s for w e i g h t e d r e d u c t i o n m o d e l s 30 t h r o u g h 3 2 . A l s o S h o w n are t h e s t a n d a r d e r r o r s o n t h e average, o, for e a c h o f t h e frequencies,  amplitudes  a n d phases.  v (mHz)  ov  Amplitude (mma)  ""Amp  2.618615  4.53E-06  0.072739  0 000822  0 842523  0.007433  0(f,  2.619542  3.75E-06  0.237434  0 001716  0 057926  0.005873  2.620466  7.51E-06  0.096504  0 000338  0 259676  0.007361  2.621176  3.76E-06  0.089528  0 004765  0 021728  0.00491  2.651208  3.9E-06  0.074912  0 012117  0 448473  0.006704  2.651994  1.71E-06  0.229328  0 000249  0 988625  0.001177  2.652945  4.1E-07  0.785146  0 003256  0 142981  0.0021  2.653892  2.8E-07  0.185178  0 006099  0 308044  0.001411  2.654934  1.94E-06  0.057818  0 009549  0 296519  0.0109  2.685572  7.32E-06  0.083156  0 002637  0 107762  0.005915  2.686444  6.4E-06  0.321998  0 008771  0 195068  0.00815  2.687489  9.7E-07  0.557244  0 001609  0 113338  0.00335  2.68843  6.9E-06  0.212632  0 005463  0 321584  0.015154  ix C. Tables of the Weighted Data Results Table C.10: continued V (mHz)  o-  Amplitude (mma)  v  G Amp  <t>  <T0  2.72006  1.62E-06  0.390504  0 006558  0.515212  0 004099  2 720927  4.2E-07  1.153283  0 007155  0.783116  0 000936  2 721827  9.2B-07  0.417881  0 003188  0.015499  0 001042  2 722815  6.16E-06  0.091935  0 003758  0.114753  0 014753  2 754318  8.5E-07  0.211407  0 003885  0.114906  0 001496  2 755333  2.55E-06  0.27421  0 002233  0.212606  0 008042  2 756236  1.83E-06  0.120356  0 001713  0.498439  0 003515  2 788935  4.66E-06  0.120502  0 004973  0.612645  0 006893  2 789984  4.04E-06  0.096761  0 001691  0.471863  0 011191  2 790598  6.85E-06  0.133643  0 001462  0.246016  0 008433  2 791538  4.43E-06  0.202389  0 006484  0.410531  0 003102  2 792518  1.99E-05  0.069908  0 008051  0.61472  0 032057  2 805631  2.67E-06  0.185083  0 001732  0.438911  0 004142  2 806572  3.75E-06  0.124083  0 004466  0.487065  0 004595  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0085153/manifest

Comment

Related Items