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

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The Rapidly Oscillating Ap Star HR 1217 The Effect of a Magnetic Field on Pulsation by Christopher J. Cameron B.Sc, 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 OF T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R 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: / 4 • I U , \ A A ^ Degree: M c < U , o £ 5, Year: The University of British Columbia Vancouver, BC Canada Department of P lwcV. , „ , J £<\. Abstract i i Abstract T h e r a p i d l y o sc i l l a t i ng A p ( r o A p ) stars p rov ide a un ique o p p o r t u n i t y to observe a n d test processes t ha t u n t i l recent ly c o u l d o n l y be s tud ied i n d e t a i l for the S u n . A s a class, these s tars possess ext reme examples of chemica l inhomogenei t ies t h r o u g h the effects of g r a v i t a t i o n a l se t t l ing , m i x i n g a n d r ad ia t ive accelerat ions; magne t i c fields t ha t affect b o t h the m i c r o - a n d macroscop ic phys ics tha t influence s te l lar s t ruc ture ; a n d h i g h over tone pu l sa t ions . F o r the la t te r , we m a y use the too l s developed for asteroseismic ana lys is : the inference of the i n t e r n a l s t r u c t u r a l proper t ies of a s tar based o n i n f o r m a t i o n f r o m i t s p u l s a t i o n a l ins tab i l i t i e s . O n e of the mos t s t ud i ed of the r o A p stars is H R 1217. T h e r a p i d v a r i a b i l i t y of th i s s tar was first d iscovered b y K u r t z (1982). L a t e r , K u r t z et a l . (1989) ex tended p h o t o m e t r i c observat ions 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 . T h e y discovered s i x p r i m a r y frequencies w i t h pe r iods near 6 minu tes a n d a spac ing p a t t e r n t ha t is remin iscen t of pu l sa t ions observed i n the S u n . T h i s thesis presents a rev ised frequency analys is 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 Telescope ( W E T ) c a m p a i g n i n la te 2000. I n p a r t i c u l a r , we ident i fy a new frequency at 2788.94 / / H z w i t h a charac te r i s t i c second-order p - m o d e spac ing of 2.63 / / H z f r o m another frequency p rev ious ly r epo r t ed 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 mode l s of A stars are also presented w i t h a d i scuss ion o f h o w a magne t i c f ie ld c a n p e r t u r b the acous t ic frequencies. T o a c c o m p l i s h th is , the v a r i a t i o n a l m e t h o d used b y C u n h a & G o u g h (2000) is adap ted to ca lcu la te magne t i c p e r t u r b a t i o n s to the acous t ic modes ca l cu la t ed f rom our evo lu t i ona ry models . These p e r t u r b a t i o n s are used to e x p l a i n the unexpec t ed frequency spac ing observed i n H R 1217. Contents i i i Conten ts A b s t r a c t . i i C o n t e n t s i i i L i s t o f T a b l e s v L i s t o f F i g u r e s v i A c k n o w l e d g e m e n t s v i i i 1 I n t r o d u c t i o n 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 in 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 Mode l 13 1.5 The roAp Star H R 1217 15 1.6 A n Overview of this Thesis 18 2 D a t a & A n a l y s i s 20 2.1 The Whole Ea r th Telescope 20 Contents i v 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 C u r v e s 23 2.3.1 R u n Se lec t ion 26 2.4 F requency A n a l y s i s 28 2.5 T h e U n w e i g h t e d D a t a Resu l t s 31 2.5.1 E s t i m a t i n g Unce r t a in t i e s a n d Signif icance 35 2.6 T h e W e i g h t e d D a t a R e s u l t s 39 3 Stellar Evolution and Pulsation Models 48 3.1 S te l l a r 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 P r o p e r t i e s 52 3.2 P u l s a t i o n M o d e l s 55 3.2.1 M o d e l Frequencies 58 3.3 M a g n e t i c Effects 62 3.3.1 T h e F requency P e r t u r b a t i o n s 65 4 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 L i s t 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 fre-quency solutions explored 42 3.1 Ten models selected from the 569 that fall wi thin the Hipparcos 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 wi thin 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 . 105 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 List of Figures v i L i s t of F igures 1.1 A schemat ic d i a g r a m showing the ob l ique ro t a to r m o d e l geometry. . . . . 5 1.2 A schemat ic representa t ion of different spher ica l h a r m o n i c modes 10 1.3 T h e v a r i a t i o n i n the m e a n l igh t of H R 1217 t h r o u g h the B f i l ter as a f u n c t i o n of the r o t a t i o n phase of the s tar 14 1.4 A b o v e is a schemat ic d i a g r a m of the frequencies found f r o m the 1986 observat ions of H R 1217 ( K u r t z et a l . , 1989) 17 2.1 A m a p of the observator ies tha t p a r t i c i p a t e d i n the observa t ions of H R 1217 d u r i n g X C O V 2 0 23 2.2 T h e end resul t of the QED 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 2.3 A Four i e r s p e c t r u m of d a t a f rom the runs n o 2 9 0 0 q l ( top) a n d joy-002 ( b o t t o m ) 27 2.4 T h e f ina l l igh t curve o b t a i n e d after the Q E D r e d u c t i o n a n d the b a r y c e n t r i c cor rec t ions have been t a k e n in to account 29 2.5 T h e Fou r i e r s p e c t r u m of the entire r u n s h o w n i n F i g u r e 2.4 30 2.6 A schemat ic of the frequencies found f r o m the unweigh ted d a t a 33 2.7 T h e r a t i o be tween the c o m m o n ampl i t udes ( top) a n d differences i n fre-quencies (bo t tom) f r o m th i s d a t a set a n d the K u r t z et a l . (1989) d a t a set 34 2.8 Improvement of the s t a n d a r d dev ia t ions of the res iduals as each frequency is r emoved 36 2.9 T h e F W H M of the m a i n frequency i n the unweigh ted d a t a 37 List of Figures v i i 2.10 A c o m p a r i s o n between the weighted a n d unweigh ted mode l s u s ing E q u a -t i o n 2.8 44 2.11 Schema t i c a m p l i t u d e d i ag rams for mode l s 14 a n d 32 46 2.12 Schemat i c a m p l i t u d e d i ag rams for mode l s 18 a n d 27 47 3.1 A theore t i ca l H R d i a g r a m showing the extremes of the pa rame te r space . 53 3.2 T h e convect ive core mass ( top) a n d the convect ive envelope mass (below) as a func t i on of age for the mode ls s h o w n i n F i g u r e 3.1 . . . . 54 3.3 T h e v a r i a t i o n of A z / i n / / H z (top) a n d the m o d e l age i n G y r ( b o t t o m ) as a func t i on of the r a t i o X / Z for mode l s t ha t fa l l w i t h i n the Hipparcos er rorbars 56 3.4 A n echelle d i a g r a m for the 1.5M© mode l s w i t h X = 0.700, Z = 0.012 a n d Z = 0.014 59 3.5 O n the left, a n echelle d i a g r a m for 1 . 6 M 0 mode l s . T h e p lo t b e l o w shows the second order spac ing as a func t ion of frequency 60 3.6 Eche l l e d i ag rams for the mode l s l i s t ed i n T a b l e 3.1 62 3.7 Second order spac ing d iag rams for the mode ls l i s t ed i n T a b l e 3.1 63 3.8 T h e magne t i c pe r tu rba t ions ca l cu la t ed for mode l s 1 to 5 u s ing a v a r i a t i o n a l p r i n c i p l e 66 3.9 T h e magne t i c pe r tu rba t ions ca l cu la t ed for mode l s 6 to 10 u s ing a v a r i a -t i o n a l p r i n c i p l e 67 VUl Acknowledgements I ' d l ike to b e g i n b y t h a n k i n g m y supervisor J a y m i e M a t t h e w s . O v e r the past few years he has exposed me to 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 oppo r tun i t i e s . O f course there were o ther non-scient i f ic ; b u t equa l ly educa t iona l , experiences as w e l l . H i s gu idance a n d pa t ience has m a d e th i s 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 me unde r s t and some of the finer po in t s of s te l lar p u l s a t i o n theo ry a n d for b e i n g the second reader of th i s thesis. I don ' 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 used i n commen 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 ike t o t h a n k D a v i d G u e n t h e r for the use of h is s te l lar 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 to encourage a n d show interest i n m y work . 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 t h r o u g h o u t t h i s work . I w o u l d l ike to t h a n k Steve 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 cqua in t ed w i t h Q E D a n d to T h e W h o l e E a r t h Telescope c o l l a b o r a t i o n for co l l ec t ing , a n d g i v i n g m e access to , such great da t a . A l s o , t hanks to G e r a l d H a n d l e r for h is c o m m e n t s o n m y earl ier d a t a r e d u c t i o n . S p e c i a l 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 t h i s w o u l d have been poss ib le . Y o u r suppor t over the years has he lped me i n b o t h m y pe r sona l a n d academic life. T h a n k y o u . I ' d l ike to t h a n k m y friends at home a n d at U B C for h e l p i n g make th i s process a fun one. F i n a l l y , I can ' t over look D r . M a r k P . F . H u b e r for l i s t en ing to m y ran ts over the last year a n d for h a n d i n g i n th i s thesis wh i l e I was away. 1 Chapter 1 In t roduc t i on 1.1 Ap Stars and Chemical Abundances In the A p 1 nomenc la tu re , the A represents the spec t r a l class a n d the p s tands for spec t roscop ica l ly pecu l i a r . T h e A c lass i f ica t ion c a n be somewhat m i s l e a d i n g since the g roup also inc ludes spec t r a l types r a n g i n g f r o m late B to ea r ly F w i t h a t e m p e r a t u r e range of 14000 > Teff > 7000 K. Pecu l i a r i t i e s i n these s tars are observed as spec t r a l l ine s t r eng th anomal ies t ha t are s ign i f ican t ly different f r o m the m a j o r i t y of o ther s tars a n d are in te rp re ted as abundance enhancements or deple t ions o n the surface of the s tar . N e a r the m a i n sequence there exists a va r i e ty of c h e m i c a l l y p e c u l i a r stars. E a c h g roup is defined b y a specific t empera tu re range, magne t i c f ield s t reng th , spec t r a l l ine anomal ies , a n d the i r p u l s a t i o n a l proper t ies . A s a n example , the ho t t e r A p s tars w i t h t empera tu res i n the range of 10000 - 14000 K , c a n exh ib i t e i ther S i or H g a n d M n pecu l i a r i t i e s i n the i r spec t ra . T h e difference here is the A p S i s tars e x h i b i t a m a g n e t i c field w h i l e the A p H g M n stars genera l ly do not . W e c a n d i v i d e the A p stars i n two general categories based o n the i r t e m p e r a t u r e a n d the observed abundances . T h e cooler s tars w i t h Tejj < 10000 K are classif ied as A p S r C r E u stars a n d show ve ry s t rong l ine s t rengths for the rare ea r th e lements 2 S r , C r , a n d E u . In a d d i t i o n , these stars general ly have weak O l ines . T h e ho t t e r A p stars 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. 2elements La to Er Chapter 1. Introduction 2 Teff > 10000 K show strong Si lines and are known as A p Si stars. These stars sometimes show strong C a lines as well as some enhancement in 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 5 times the solar value and under abundances down to 1 0 - 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 main 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 wi th 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; bo th of which wi l l be discussed later in this chapter. A general discussion of magnetic fields in stars is given by Mathys (1989) and Landstreet (1992a; 1992b; and 1993). 1 . 2 . 1 Ap Spectra and the Zeeman Effect The perturbations to stellar spectra produced by line transitions of an atom in the pres-ence of a magnetic field are described by the Zeeman effect. If an atom 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 total angular momentum from the L S coupling is represented by J , then in the presence of a magnetic field there becomes 2 J + 1 available states for the electron. Each of these magnetic substates are 3elements Sc to Ni Chapter 1. Introduction 3 represented b y the integer q u a n t u m number m. If the difference i n magne t i c substates is A m = 0, t h e n t h e y are referred to as TT componen t s . If A m = ± 1 , t h e n the substates are k n o w n as a componen t s . T h e n componen 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 wave leng th , w h i l e the a componen t s have wavelengths tha t are o n ei ther side of the u n p e r t u r b e d wave leng th . T h e spac ing of the o componen t s is p r o p o r t i o n a l to the magne t i c f ie ld s t r eng th for fields w i t h s t rengths u p to a few tens of kiloGauss. T h e r e l a t i o n be tween the spac ing of the 7r a n d a componen t s is g iven b y (cf. Lands t ree t 1992b) A A = A 2 e 5 / 4 7 r m c 2 (1.1) where A is the u n s p l i t wave length , B is the magne t i c f ie ld , a n d e / m a n d c are the charge-to-mass r a t i o of the e lec t ron a n d the speed of l igh t , respect ively. T h i s r e l a t i o n assumes t ha t the L a n d e factor; ca l cu la t ed f r o m the s p i n q u a n t u m numbers , is equa l t o one. E v e n i f the Z e e m a n componen t s of a l ine are not fu l ly resolved, as is a lmos t a lways the case, the TT a n d a componen t s t end to b r o a d e n the spec t r a l l ine . If th i s b r o a d e n i n g c a n be measured the effective f ield s t r eng th m a y be es t imated . However , there are o ther effects t ha t compete to b r o a d e n a spec t ra l l ine such 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 macroscop ic tu rbu lence . T h e Z e e m a n b r o a d e n i n g domi na t e s for a s m a l l subset of stars w i t h B > 10 k G a n d pro jec ted veloci t ies v s in ( i ) < 10 k m / s (Lands t ree t , 1992b). 1.2.2 Some Observable Magnetic Quantities T h e sens i t iv i ty t o the magne t i c f ie ld varies f r o m l ine t o l ine . B y obse rv ing l ines f r o m different elements, one c a n deduce the magne t i c field s t r eng th b y c o m p a r i n g the w i d t h s of these l ines t o each other . I n th i s way, the Z e e m a n componen t s c a n p rov ide a measure of the magne t i c field s t r eng th in tegra ted over the s te l lar disc . T h i s q u a n t i t y is k n o w n as the m e a n field m o d u l u s a n d m a y be denoted as < B > . F o r example , M a t h y s et a l . (1997) present measures of the magne t i c field m o d u l u s for 40 A p stars. Chapter 1. Introduction 4 T h e spec t r a l l ines inf luenced b y a magne t i c f ie ld also possess in te res t ing p o l a r i z a t i o n proper t ies . I n f o r m a t i o n ga ined f o r m the p o l a r i z a t i o n of spec t r a l l ines c a n fur ther be used to set l i m i t s o n magne t i c field geometry. I f a spec t r a l l ine is observed 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 respect ive ly) l igh t , the c o n t r i b u t i o n to the spec t r a l l ine f r o m each of the o componen t s is different. T h e difference i n the p o s i t i o n of the m e a n wave leng th of the l ine i n R C P a n d L C P l igh t p rov ides a measurement of the l ine-of-sight componen t of the magne t i c f ie ld averaged over the s te l lar d isc . T h i s measurement is k n o w n as the m e a n l o n g i t u d i n a l field s t r eng th a n d is represented b y Bi. F o r mos t A p stars, 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 of the s tar are the o n l y magne t i c observa t ions ava i lab le because they are r e l a t ive ly easy to o b t a i n (Lands t ree t , 1993). T h i s is because the measurement is most sensit ive to modes t fields w i t h s imp le s t ruc tures a n d no a priori i n f o r m a t i o n abou t the l ine profi le is needed. I n fact, the v a r i a t i o n i n B\ w i t h r o t a t i o n phase (j) is p e r i o d i c a n d is g iven b y the s imple a n a l y t i c express ion Bi(4>) « OABp (cos/3cosi + s i n s i n i cos 0) (1.2) where Bp is the p o l a r field s t rength ; assumed to be a centred d ipo le , i is the angle be tween the r o t a t i o n ax i s a n d the l ine of sight, a n d the magne t i c pole is i n c l i n e d to the r o t a t i o n ax i s of the s tar b y the angle (3. T h i s express ion m a y be der ived f r o m ob l ique ro t a to r m o d e l of S t i b b s (1950) a n d assumes a l i m b d a r k e n i n g coefficient of un i ty . T h e geomet ry of the ob l ique ro t a to r is s h o w n i n F i g u r e 1.1. A s the s tar rotates, the angle f r o m the magne t i c field pole to the l ine of sight changes. T h u s , the change i n the aspect o f the magne t i c field leads to a m o d u l a t i o n of Bi w i t h the r o t a t i o n of the s tar . If there is a s t rong 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 abou t i t s geomet ry f r o m the t ransverse field componen t . T h e 7r componen t of a Z e e m a n t r ip l e t w i l l u sua l l y sa tura te before the o componen t s . T h e in tegra ted l ine profi le w i l l t h e n leave a net l inear 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 th i s net l inear p o l a r i z a t i o n one can , i n some cases, o b t a i n un ique values for i, (3 a n d Bp defined i n E q u a t i o n 1.2. T h i s charac ter i s t ic of the Z e e m a n componen t s is k n o w n as b r o a d b a n d l inear p o l a r i z a t i o n . Chapter 1. Introduction 5 Figure 1.1: A schemat ic d i a g r a m showing the ob l ique ro t a to r m o d e l geometry . T h e r o t a t i o n axis a n d d ipo le magne t i c ax i s are l abe l ed R a n d M respect ively. T h e l ine of sight to E a r t h is t o w a r d the r igh t . A n g l e s 4>, j3 a n d i are defined i n Sec t i on 1.2.2. T h e r e are o ther techniques for ex t r ac t i ng magne t i c f ie ld i n f o r m a t i o n f r o m spec t ra . T h e m o m e n t technique, for example , fits spec t r a l l ines measured w i t h different p o l a r i z a -t ions (i.e., different Stokes parameters) w i t h l ine momen t s defined b y (A — A o ) n + 1 . Here , A a n d Ao are the in t eg ra l over the l ine profile a n d the cen t ro id o f the wave leng th respec-t i v e l y (see M a t h y s 1989). T h e momen t s of different Stokes parameters y i e l d i n f o r m a t i o n abou t the f ield geometry. F o r ins tance, the first m o m e n t of the difference be tween the in t ens i ty i n L C P a n d R C P (Stokes V ) l ight recovers the m e a n l o n g i t u d i n a l f ie ld ( M a t h y s , 1988). Chapter 1. Introduction 1.3 Diffusion in Ap Stars 6 I n the contex t of A p stars, the separa t ion of elements as a means to e x p l a i n the observed abundance patches was first s t ud ied as ea r ly as the late 1960s a n d ea r ly 1970s (e.g., M i c h a u d , 1970). I n p r i n c i p l e , any element tha t is heavier t h a n the 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 ink under the influence of gravi ty . E x c e p t i o n s occu r for elements t ha t have a b s o r p t i o n l ines at the wavelengths near the l o c a l flux m a x i m u m . I n these cases, the elements m a y be l ev i t a t ed u p w a r d t o w a r d the s te l lar surface i f the r a d i a t i v e forces are greater t h a n the g r a v i t a t i o n a l force. O n c e the elements reach a n e q u i l i b r i u m p o s i t i o n be tween g r a v i t a t i o n a l a n d r ad ia t ive forces t hey m a y accumula t e i n sufficient a m o u n t s a n d cause abundance anomal ies at these loca t ions . R a r e E a r t h elements, such as Sr , m a y accumula t e at the surface of A p stars i n th i s way. G e n e r a l rev iews of dif fus ion t h e o r y m a y be found 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 & Proffitt (1993). T h e process of element sepa ra t ion descr ibed above is, however, a fragile one. If there are t u rbu l en t or convect ive veloci t ies i n the uppe r a tmospheres of these s tars t ha t exceed the diffusion veloci t ies of a few c m / s , the abundance anomal ies w i l l s i m p l y be m i x e d away ( M i c h a u d , 1976). W h i l e the m a i n sequence A stars do not have large convect ive zones, the A giants do . T h i s exp la ins w h y the A - t y p e g iants loose the i r abundance pecu l i a r i t i e s . It s h o u l d also be no ted tha t diffusion t i m e scales are o n the order o f « 1 0 6 yrs ; a s m a l l f r ac t ion of the evo lu t i ona ry t imescale of a near solar-mass s tar . T h u s , c h e m i c a l s epa ra t ion m a y occur ea r ly i n the evo lu t i on of these stars. D i f fus ion t h e o r y p red ic t s a the t h i n layer of abundance anomal ies at the surface of these stars. A s no ted 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 abundances of ~ 1 0 4 solar values) were representat ive of the in te r io r m e t a l content of these stars, t hey w o u l d c o n t a i n nea r ly a l l of these elements i n the Un ive r se . S ince A p stars represent a p p r o x i m a t e l y one i n a hund red t h o u s a n d stars, t h i s can ' t p o s s i b l y be the case. I n order for the diffusion m e c h a n i s m i n these s tars t o be efficient, there mus t be a s t a b i l i z i n g m e c h a n i s m against tu rbu lence as discussed above. It is be l i eved t ha t the Chapter 1. Introduction 7 magne t i c f ield p rov ides 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 stars. It also has the added effect o f caus ing h o r i z o n t a l l y d i s t r i b u t e d abundances o n the s te l la r surface near regions of h o r i z o n t a l magne t i c f ield ( M i c h a u d et a l . , 1981). T h i s occurs w h e n ions of a n element are l e v i t a t e d to depths where the magne t i c f ield becomes i m p o r t a n t c o m p a r e d to the r ad i a t i ve 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 at the s te l lar surface. M a g n e t i c fields c a n de-saturate some l ines of a n element t h r o u g h the Z e e m a n effect. T h i s resul ts f r o m the Z e e m a n sp l i t l ines b e i n g exc i t ed at s l i g h t l y different frequencies t h a n the same l ines p r o d u c e d w h e n no magne t i c f ield is present. W h e n a m a g n e t i c f ie ld is i n d u c e d , these l ines w i l l no longer have the o p t i m a l w i d t h s a n d energies for r ad i a t i ve forces t o overcome gravi ty . Therefore , some elements w i l l become less suscept ib le t o r a -d ia t ive accelera t ions w h e n there are magne t i c influences. T h e p o l a r i z a t i o n o f the Z e e m a n componen t s also affects the h o r i z o n t a l acce le ra t ion of elements; a d d i n g to the d i s t r i b u t i o n of e lements observed for the A p stars ( B a b e l & M i c h a u d , 1991, a n d M i c h a u d , 1996). 1.4 Stellar Seismology and the Ap Stars R a p i d v a r i a b i l i t y 4 of s tar l igh t was first observed i n the S u n . L e i g h t o n et a l . (1962) found t ha t s p a t i a l l y incoherent wave pa t te rns exist o n the surface of the S u n w i t h pe r iods near 5 minu tes . U l r i c h (1970), Le ibache r k, S t e i n (1971) were the first t o in te rpre t these osc i l l a t ions as s o u n d waves p r o d u c e d i n the solar in te r io r tha t resonate i n acous t ic cavi t ies genera ted b y changes i n the l o c a l sound speed. These sound waves p roduce s t a n d i n g wave pa t t e rns as they propaga te a r o u n d the in te r ior of the s u n w i t h i n these acous t ic cavi t ies . S ince these ea r ly beg inn ings , m i l l i o n s of p -modes have been observed i n the S u n . T h e name p — mode comes f r o m the fact tha t pressure is the res to r ing force for the s o u n d waves t ha t are responsible for the observed osc i l l a t ions . These p -modes have p roved to be a power fu l t o o l for s t u d y i n g the in t e rna l proper t ies of the S u n t h r o u g h helioseismology. 4The 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 of he l iose ismology is a k i n to s t u d y i n g i n t e rna l p roper t ies o f the E a r t h u s ing seismology. A few examples of successful inferences o n the solar in t e r io r m a d e f r o m he l iose i smology inc lude a n es t imate of the i n t e rna l r o t a t i o n rate , the d e p t h of the solar convect ive zone, cons t ra in t s o n element diffusion, a n d the r u n of s o u n d speed. A recent r ev iew of he l iose ismology is p r o v i d e d by 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 the late 1970s, observat ions of s tars h a d no t s h o w n p -modes w i t h pe r iods a n d frequency spac ing s i m i l a r t o those observed i n the S u n . T h i s changed w h e n K u r t z (1978, 1982) d iscovered the r a p i d l y o sc i l l a t i ng A p stars. These stars e x h i b i t r a p i d v a r i a b i l i t y i n the i r l igh t curves w i t h pe r iods between 5 a n d 15 minu tes a n d s emi -ampl i t udes under a p p r o x i m a t e l y 8 mi /Zzmagni tudes t h r o u g h a J o h n s o n B filter. T o date , there are 32 k n o w n r o A p stars: K u r t z & M a r t i n e z (2000) repor t 31 a n d the mos t recent, H D 12098, is r epo r t ed b y G i r i s h et a l . (2001). Some of these stars are m u l t i - p e r i o d i c a n d e x h i b i t f requency pa t t e rns s i m i l a r t o those observed i n the S u n . T h e best example of t h i s is the star of t h i s thesis, H R 1217 (see Sec t ion 1.5). Some de ta i l ed rev iews of r o A p stars 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 (2000). I n general , the A p stars show three types of va r i ab i l i t y . These i nc lude spec t r a l l ine s t rength , magne t i c f ield s t rength , a n d p h o t o m e t r i c ( in tegra ted l igh t ) va r i a t ions . T h e p h o t o m e t r i c va r i a t ions c a n further be s u b - d i v i d e d in to l o n g - t e r m ( L T ) va r i a t i ons a n d r a p i d osc i l l a t ions (the r o A p s tars) . E a c h of these forms of v a r i a b i l i t y seem to be i n -te rconnec ted t h r o u g h the c o m p l e x geomet ry of the magne t i c field, abundance pa t t e rns , s te l lar r o t a t i o n a n d i n c l i n a t i o n of the s tar to the observer. I w i l l b e g i n w i t h a d i scuss ion of n o n r a d i a l osc i l l a t ions a n d the i r relevance to the r o A p stars. T h e c o n n e c t i o n be tween the magne t i c , spec t ra l , L T a n d r a p i d va r i a t ions a n d the geomet ry tha t l i n k s t h e m w i l l t h e n be discussed. 1.4.1 Nonradial Oscillations N o n r a d i a l o sc i l l a t ions are discussed i n d e t a i l b y U n n o et a l . (1989). A b r i e f i n t r o d u c t i o n to areas of p a r t i c u l a r relevance to the r o A p stars is presented be low. Chapter 1. Introduction 9 T o first order , a s l owly ro t a t i ng , non-magne t i c s tar is sphe r i ca l ly s y m m e t r i c . P e r t u r -b a t i o n s t o th i s sphe r i ca l s y m m e t r y c a n be descr ibed b y sphe r i ca l h a r m o n i c s , Ylm{9,(j)). Here 9 a n d 0 are the u s u a l angu la r coord ina tes of the spher ica l coo rd ina t e sys tem. T h e s t u d y of osc i l l a t ions i n stars c a n be descr ibed as s m a l l pe r t u rba t i ons t o p h y s i c a l var iab les w i t h i n t he s ta r s u c h as pressure o r dens i ty o r i n r a d i a l d i sp lacements £ r . I n general , we m a y e x p a n d r a d i a l d isp lacements i n te rms of spher ica l h a r m o n i c s as where the n is the r a d i a l order of the osc i l l a t ions a n d corresponds to the n u m b e r of r a d i a l nodes (zeroes) i n the e igenfunct ion £r>ne, one is the eigenfrequency of the osc i l l a t ions , a n d t is t i m e . T h e sphe r i ca l h a r m o n i c provides the angula r de sc r ip t i on of the modes t h r o u g h the ind ices £ a n d m. T h e i n d e x £ is c o m m o n l y ca l l ed the degree of the m o d e a n d d iv ides the surface i n to regions osc i l l a t i ng i n oppos i te phase. If £ = 0, the o s c i l l a t i o n m o d e is r a d i a l ; or more precisely, sphe r i ca l ly s y m m e t r i c . T h e index m is the a z i m u t h a l order of the m o d e a n d represents the number of l o n g i t u d i n a l nodes o n a sphere. P h y s i c a l l y , the order m c a n be re la ted to the phase ve loc i ty of a wave t h r o u g h T h e s ign of m ind ica tes the d i r e c t i o n i n w h i c h the wave t ravels . A s t a n d i n g 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 the supe rpos i t i on of two waves t r ave l i ng i n oppos i te d i rec t ions since the b a c k g r o u n d state is , to first order , sphe r i ca l l y s y m m e t r i c . In t e rms of q u a n t i z a t i o n of the indices , m m a y take integer values f r o m — £ to + £ a n d £ is a nonnega t ive integer. I f m = 0 a l l n o d a l l ines o n a sphere are l ines of l a t i t ude , w h i l e i f m = £, t h e y are a l l l ines of long i tude . These are k n o w n as z o n a l a n d sec tora l modes respect ively . If m takes o n values be tween these extremes, the modes are k n o w n as tesseral modes a n d of the £ n o d a l l ines, £— \m\ of w h i c h are l ines of 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 5 i n F i g u r e 1.2 for the cases £ = 3 a n d m = 0 to 3. N o t e t ha t 5Image taken from the Delta Scuti Network homepage: http://www.deltascuti.net/DeltaScutiWeb/indexl .html (1.3) n,£,m (1.4) Chapter 1. Introduction |m|»2 H * f » 3 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 £r,n£ and one in Equation 1 .3 are independent of ra. This results in a degeneracy 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-Cn,t)n (1 .5) where il is the angular rotation frequency of the star measured in the rotating frame Chapter 1. Introduction 11 a n d CUte is k n o w n as the L e d o u x constant ( L e d o u x , 1951). T h e L e d o u x cons tan t is m o d e l dependent a n d is of the order l / £ r . N o w each frequency m a y be desc r ibed i n a n i n e r t i a l reference frame b y the u n p e r t u r b e d frequency o f r o m E q u a t i o n 1.3 p l u s or°^im f r o m E q u a t i o n 1.5. I n the case of the r o A p stars, the s t rong magne t i c f ield p roduces pressures t ha t are greater t h a n the l o c a l gas pressure near the s te l lar surface. A t r a d i t i o n a l p e r t u r b a t i o n a p p r o a c h i n desc r ib ing the effect of a magne t i c f ield o n a p u l s a t i o n m o d e is no longer v a l i d . I n d i v i d u a l o s c i l l a t i o n modes cannot be descr ibed b y a single set o f n , £, a n d m values. Ins tead, each m o d e must be expanded i n a n inf in i te s u m of sphe r i ca l h a r m o n i c s to account for the c o u p l i n g between the magne t i c f ield a n d p u l s a t i o n geomet ry (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 , 2000, a n d C u n h a & G o u g h , 2000) . T h e effects of the s t r ong magne t i c f ield o n the normal modes (i.e., those ca l cu l a t ed i n the case o f no magne t i c f ie ld or ro t a t ion ) w i l l be discussed i n Sec t i on 3.3 i n t e rms of a v a r i a t i o n a l a p p r o a c h used 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 of a few miZZzHer tz 6 ( m H z ) are consis tent w i t h p - m o d e frequencies i n non-degenerate s tars descr ibed 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 observa t ions of p u l s a t i n g stars y i e l d o n l y the con t r ibu t ions of I modes w i t h s m a l l values. T h i s is because the c o n t r i b u t i o n f r o m modes w i t h £ > 3 gets s m o o t h e d ou t w h e n the l igh t f r o m a s te l lar d isc is in tegra ted . T h e h i g h overtones of such low-degree pu l sa t ions have a s y m p t o t i c behav io r tha t m a y be exp lo i t ed for stars l ike the r o A p stars . If a sphe r i ca l l y s y m m e t r i c s tar pulsates a d i a b a t i c a l l y w i t h p -modes of n £, we m a y use the a s y m p t o t i c t heo ry of Tas sou l (1980, 1990) to descr ibe the frequencies b y i V « Au(n + £/2 + e) + T (1.6) where e is a cons tant , a n d T is a s t r u c t u r a l l y dependent q u a n t i t y t ha t is a n order of m a g n i t u d e smal le r t h a n the first t e r m . It is convenient to i n t roduce the frequency v 6 A period of 5 minutes corresponds to a frequency of 3.3 mHz Chapter 1. Introduction 12 since i t is u s u a l l y found i n the observa t iona l l i t e ra ture o n p u l s a t i n g stars. T h e r e l a t i o n be tween the angu la r frequency defined i n equa t i on 1.3 a n d tha t o f E q u a t i o n 1.6 is s i m p l y u = o/(2n). T h e factor Au i n E q u a t i o n 1.6 w i l l be used extens ive ly i n th i s thesis . It is r e l a t ed to the t i m e i t takes s o u n d to cross the d iameter of the s tar a n d is u s u a l l y referred to as the large spac ing . T h i s f requency is w r i t t e n i n t e r m s o f the s o u n d speed c t h r o u g h , - i Au = 2 [ dr/c . Jo (1.7) where R is the r ad ius of the star . Since th i s q u a n t i t y is a p p r o x i m a t e l y re la ted to the m e a n dens i ty of the star , i t m a y be w r i t t e n i n t e rms of the s te l lar mass M , a n d r ad iu s v i a the r e l a t i o n Au = (0.205 ± 0.011) f j H z (1.8) where G is the g r a v i t a t i o n a l constant . T h e n u m e r i c a l factor arises f r o m m o d e l c a l c u l a -t ions pe r fo rmed 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) rewrote E q u a t i o n 1.8 i n t e rms of the stars effective t empera tu re Teff, a n d l u m i n o s i t y L , t o give Au = (6.64 ± 0.36) x 10-ieM1/2T^fL~3/4 H z (1.9) w i t h L a n d M i n solar un i t s a n d T e / / i n degrees K e l v i n . It s h o u l d be no ted tha t there is a n a m b i g u i t y i n the i n t e r p r e t a t i o n of the observed frequency spac ing depend ing o n the £ values. If the modes have orders t h a t differ f r o m each other b y one a n d £ values tha t a l te rnate be tween even and o d d values, E q u a t i o n 1.6 y ie lds a f requency spac ing of a p p r o x i m a t e l y Au/2. I f however, consecut ive £ values are ei ther a l l even or o d d , the frequencies w i l l be spaced b y abou t Au; the large spac ing . T h e r e is also a second order spac ing 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 te rms of the s o u n d speed, t h i s spac ing y i e ld s S^ocAu fR-^-dr (1.10) Jo r dr T h e de ta i l s l e ad ing to th i s equa t i on m a y be found i n the w o r k of T a s s o u l (1990). Chapter 1. Introduction 13 T h i s spac ing is referred to as the s m a l l spac ing . C lose t o the center of a star , the l ead ing c o n t r i b u t i o n t o E q u a t i o n 1.10 comes f r o m the 1/r factor i n the in t eg rand . T h u s , t h i s spac ing m a y be used t o infer proper t ies of the in ter iors o f s tars (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 the r o A p stars, however, the d iagnos t i c power of m a y be l i m i t e d because the magne t i c p e r t u r b a t i o n s t o the frequencies m a y b e of t he same order , or larger t h a n , the s m a l l spac ing ( 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 Sec t i on 1.4 is shown for H R 1217 a long w i t h the v a r i a t i o n of i t s m a g n e t i c f ie ld i n F i g u r e 1.3. It is c lea r ly s h o w n tha t the m e a n l igh t v a r i a t i o n o f the s tar is i n phase w i t h the magne t i c f ie ld . T h e m o d u l a t i o n of the m a g n e t i c f ie ld is desc r ibed w e l l b y the ob l ique rotator m o d e l of S t i b b s (see Sec t i on 1.2.2). K u r t z (1982) suggested t ha t the L T v a r i a b i l i t y a n d r a p i d va r i a t ions m a y be 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 he ca l l ed the obl ique pulsator m o d e l . I n th i s m o d e l , the p u l s a t i o n ax i s o f the s tar is a l i gned w i t h the magne t i c ax i s o f the star; b o t h o f w h i c h are i n c l i n e d to the r o t a t i o n axis . A s the star rota tes the aspect of the p u l s a t i o n a n d m a g n e t i c ax i s va ry ; l e ad ing t o m o d u l a t i o n w i t h the r o t a t i o n of the star . T h e geomet ry is the same as i n F i g u r e 1.1, b u t n o w the magne t i c ax i s a n d p u l s a t i o n ax i s are one i n t he same. 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 frequency o. K u r t z (1982) showed tha t i n the geomet ry of the ob l ique pu lsa tor , the l u m i n o s i t y v a r i a t i o n w i t h ro t a -t i o n m a y be expressed as oc pp(cosa) cos [at + <pp] ( 1 - H ) Ii where P™ is the associa ted Legendre p o l y n o m i a l , <pp is a n a r b i t r a r y phase, a n d the angle a is the v a r y i n g angle be tween the magne t i c field axis a n d the l ine of s ight . T h i s angle has the same func t iona l dependence as i n the obl ique ro t a to r m o d e l of E q u a t i o n 1.2. I n the case of a d ipo le m o d e (£ = 1), the Legendre p o l y n o m i a l is equa l t o cos a a n d E q u a t i o n 1.11 m a y be expanded as ^ « A0 cos (at + <pp) + Ai [cos ({a + Q.}t + </?p) + cos ({a - £l}t + <pp)] (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 the m e a n l igh t of H R 1217 t h r o u g h the B f i l ter as a func t i on of the r o t a t i o n phase of the s tar (upper pane l ) . T h e m a g -ne t ic f ield v a r i a t i o n defined b y B\ i n Sec t i on 1.2.2 as a func t i on of r o t a t i o n phase is s h o w n i n the lower pane l . P u l s a t i o n d a t a a n d m a g -ne t ic 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) respect ively. T h i s figure was o b t a i n e d f r o m M a t t h e w s (1991). T h e a m p l i t u d e s are g iven 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 the above re la t ions , we see tha t a single d ipo le m o d e is sp l i t i n t o a t r i p l e t e x a c t l y spaced b y the r o t a t i o n p e r i o d of the star . I n general , t h i s m o d e l p red ic t s t ha t a m o d e w i t h degree £ is sp l i t in to 2 ^ + 1 frequencies. T h e r o A p s tars do e x h i b i t such fine s t ruc tu re i n the i r frequencies. Chapter 1. Introduction 15 Frequencies m a y also be sp 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. These 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 s tar because of the L e d o u x cons tant . C a l c u l a t i o n s of the L e d o u x constant for A s tar mode l s b y S h i b a h a s h i & T a k a t a (1993) suggest a value of Cn>( « 1 0 ~ 3 . A s t r ingent obse rva t iona l cons t ra in 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 the frequency spac ing of the r o A p s tar H R 3831. T h e y p lace a n uppe r l i m i t of 1 0 ~ 6 o n Cnf at the 3 a confidence level . T h e co inc idence o f magne t i c a n d p u l s a t i o n phase a l o n g w i t h the above cons t ra in t are s t r ong i n d i c a t i o n s t h a t the observed fine s t ruc tu re i n the frequency s p e c t r u m of the r o A p stars is not the resul t of r o t a t i o n a l l y p e r t u r b e d m - m o d e s . A s h o r t c o m i n g of th i s s imple m o d e l pu t fo r th b y K u r t z (1982) is i n the a m p l i t u d e a s y m m e t r y observed be tween sp l i t frequencies. F r o m E q u a t i o n 1.12, we see t h a t the s p l i t frequencies s h o u l d have the same a m p l i t u d e . T h i s is not the case for the A p stars. T h i s p r o b l e m is avo ided i f one takes i n to account the effects of b o t h r o t a t i o n a n d magne t i c f ie ld o n the frequencies t h r o u g h the C o r i o l i s a n d L o r e n t z forces. T h i s co r r ec t i on b y D z i e m b o w s k i & G o o d e (1985) cor rec t ly p red ic t s b o t h the r o t a t i o n a l l y sp l i t frequencies a n d the a m p l i t u d e asymmet r i e s . T h e mos t recent c o n t r i b u t i o n to the ob l ique ro t a to r m o d e l comes f r o m B i g o t & D z i e m -b o w s k i (2002). T h e y used a non-pe r tu rba t ive a p p r o a c h to show tha t the cent r i fuga 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 the frequency shifts w h i l e the C o r i o l i s force is d o m i n a n t i n d e t e r m i n i n g the a m p l i t u d e asymmet r ies . These au thors also show tha t the p u l s a t i o n , r o t a t i o n , a n d magne t i c axis are a l l i nc l i ned to each other . 1.5 The roAp Star HR 1217 T h e focus of thesis is the A p star H R 1217; a .k .a H D 24712 or D O E r i . T h i s was one of the first A p stars to be ident i f ied as a r o A p s tar ( K u r t z , 1982) a n d has s ince become one o f the mos t s tud ied . Recent ly , 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 abou t the abundance , magne t i c a n d p h o t o m e t r i c character is t ics of H R 1217. T h e mos t recent p h o t o m e t r i c d a t a is presented i n C h a p t e r 2 of th i s w o r k . A rev iew of some of the o ther 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 Bp respectively. Estimated uncertainties on the above angular measurements are « 2 — 3° and the uncertainly of the polar field strength is « 5%. The mean longitudinal field, 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-0.8 [/LtHz] 6 8 . 0 4 •§ 0.6 0.4 0 .2 hv/2 [ /xHz] 3 3 . 3 8 3 4 . 6 6 _i i ii 6 7 . 9 1 3 3 . 2 5 6 8 . 1 4 8 4 . 8 8 3 4 . 8 9 4 9 . 9 9 J i i _J i i i 2 . 6 5 2 .7 v ( m H z ) 2 .75 2.8 F i g u r e 1.4: A b o v e is a schemat ic d i a g r a m of the frequencies found f r o m the 1986 observat ions of H R 1217 ( K u r t z et a l , 1989). T h e ax i s are a m p l i t u d e i n rmZZzmagnitudes a n d frequencies i n m H z . T h e t w o poss ible values of ei ther Au of « 68 /xHz (blue) or Au/2 of « 34 /xHz (black) are shown. N o t e the strange spac ing of 49.99 a n d 84.88 /xHz i n d i c a t i n g a poss ible missing f requency at the r e d l a b e l ?. observed frequency spac ing . B o t h Au a n d Au/2 are presented i n F i g u r e 1.4. If the modes "are a l t e r n a t i n g between even a n d o d d £ values we w o u l d expect t o see a Au of « 68 ^xHz. In fact, the a l t e rna t i ng spac ing of 33 a n d 34 ^tHz a d d to th i s conc lu s ion . If the Chapter 1. Introduction 18 modes were a l l even or o d d , we w o u l d expect t ha t the spac ing be tween adjacent modes i n F i g u r e 1.4 w o u l d r e m a i n the same. T h e p o s s i b i l i t y tha t the modes are a l l o f the same degree cannot be r u l e d ou t . W h a t is not shown i n F i g u r e 1.4 is the fine s t ruc tu re spac ing a r o u n d a l l bu t the first f requency (read f r o m the left) . E a c h of the frequencies, 2 t h r o u g h 6, are a c t u a l l y t r ip le t s w i t h frequency spac ing of a p p r o x i m a t e l y 0.9 / / H z ; the r o t a t i o n a l frequency o f the s tar . T h e o r i g i n of these sp l i t t i ngs was discussed i n S e c t i o n 1.4.3. Suffice i t t o say, the unexpec t ed frequency spac ing between the last two frequencies i n F i g u r e 1.4 does not he lp de te rmine i f we are observ ing the large spac ing or h a l f of the large spac ing . It was o n l y i n the pas t few years; a lmos t a decade after K u r t z et a l . (1989) released the i r resul ts , t ha t the spac ing controversy seemed to be resolved. 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 able to ca lcula te a p a r a l l a x for H R 1217 based o n a n inferred large spac ing of 68 / i H z . T h e i r p r ed i c t ed p a r a l l a x of IT = 19.23 ± 0.54 mas was s h o w n to be consis tent w i t h the recent Hipparcos p a r a l l a x of IT = 20.41 ± 0.84 mas. T h u s , a Av/2 of 34 uHz seems to have been r u l e d ou t b y a n obse rva t ion m a d e i ndependen t l y f r o m asteroseismic analys is . A n o t h e r recent success i n the i n t e rp re t a t i on of the frequency spac ing of H R 1217 was p r o v i d e d b y C u n h a (2001). She p red ic t ed t ha t magne t i c d a m p i n g c o u l d be the cause o f the m i s s i n g frequency i n the 1986 d a t a set. She also showed t h a t some frequencies m a y be shif ted b y a p p r o x i m a t e l y 10-20 / / H z because of magne t i c field effects ( C u n h a & G o u g h 2000, a n d C u n h a 2001) . I n 2000, H R 1217 was selected to be observed i n ano the 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 frequency 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 fur ther evidence t ha t the magne t i c f ie ld a n d the osc i l l a t ions i n r o A p stars are in te rconnec ted . 1.6 An Overview of this Thesis Progress i n the s t u d y of the A p a n d r o A p stars has been s t ead i ly inc reas ing over the past few years w i t h the development of new obse rva t iona l a n d theore t i ca l too ls . I n th i s thesis I a t t e m p t to t ie together the mos t recent p h o t o m e t r i c observat ions of the r o A p s tar H R Chapter 1. Introduction 19 1217 w i t h the latest g r i d of A star evo lu t i ona ry a n d p u l s a t i o n mode l s . I n C h a p t e r 2 I ou t l ine the r e d u c t i o n a n d frequency ana lys i s 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 the W h o l e E a r t h Telescope c o l l a b o r a t i o n i n la te 2000. A n in t ro -d u c t i o n to the g l o b a l obse rva t ion concept a n d d a t a a c q u i s i t i o n is presented i n Sec t ions 2.1 a n d 2.2, respect ively. T h e r e d u c t i o n procedures a p p l i e d to the r a w t ime-series d a t a a n d the frequency ana lys i s fol lows i n Sec t ions 2.3 a n d 2.4. T h e resul ts o f t he ana lys i s is t h e n presented i n S e c t i o n 2.5 a n d compared to resul ts o b t a i n e d u s ing a va r i e ty of d a t a w e i g h t i n g schemes i n Sec t i on 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 mode l s are presented i n C h a p t e r 3. A l t h o u g h the m a i n focus of t h i s thesis is H R 1217, the evo lu t i ona ry mode l s c a l c u l a t e d for th i s w o r k cover a large enough pa ramete r space to be relevant to o ther r o A p stars . Specif ic improvemen t s to the p u l s a t i o n mode l s (see Sect ions 3.2 t h r o u g h 3.3.1) i nc lude b o t h n o n -a d i a b a t i c (energy gains a n d loses t h r o u g h r ad i a t ive processes) a n d magne t i c effects. A f ina l d i scuss ion c o m p a r i n g the results of the d a t a ana lys i s a n d the s te l lar mode l s is presented i n the f ina l chapter of th i s work . 20 Chapter 2 D a t a & A n a l y s i s 2.1 The Whole Earth Telescope T h e W h o l e E a r t h Telescope ( W E T ) is a c o l l a b o r a t i o n of as t ronomers a n d faci l i t ies f r o m a r o u n d the w o r l d whose col lec t ive goa l is t o o b t a i n h igh-qua l i ty , con t inuous , t ime-series p h o t o m e t r y of va r iab le stars. O r i g i n a l l y , W E T was o rgan ized to s t u d y va r i ab le , degen-erate stars. Its goals were expanded to inc lude types o f var iab les such as 5 S c u t i , r o A p , ca t ac lysmics , a n d sub -dwar f B stars. T h e m a i n advantage of a g l o b a l c a m p a i g n over s ingle-si te observa t ions i s t he r e d u c t i o n of d a t a gaps i n a g iven set of t i m e series measurements . S u c h gaps are c o m m o n w i t h s ingle-si te observat ions because any p a r t i c u l a r s tar is o n l y v i s ib l e for a c e r t a i n f r ac t ion of t he n igh t . I n a g l o b a l c a m p a i g n , observator ies are d i s t r i b u t e d i n l ong i tude , so one si te c a n s tar t observat ions w h e n another site has f in ished (see F i g u r e 2.1). I n p r i n c i p l e , con t inuous coverage of a chosen target c a n be ob ta ined . O f course, weather a n d / o r equ ipmen t p r o b l e m s m a y also l ead to a loss o f d a t a . T h e effect o f these m i s s i n g d a t a is seen as a l i a s ing i n frequency analys is of var iab le stars. T h e gaps p r o d u c e a d d i t i o n a l peaks i n the F o u r i e r s p e c t r u m of the da ta , m a k i n g iden t i f i ca t ion of a rea l o s c i l l a t i o n frequency dif f icul t . T h e more faci l i t ies t ha t pa r t i c ipa t e i n a W E T r u n , the greater the p o s s i b i l i t y for longer obse rv ing t i m e o n a target . T h i s increases the frequency r e s o l u t i o n o f the d a t a a n d helps ident i fy fine s t ruc ture i n the Four i e r s p e c t r u m . T h i s fine s t ruc tu re is c r u c i a l for m o d e iden t i f i ca t ion , r o t a t i o n a l i n fo rma t ion , a n d i n the case of the r o A p stars, i n f o r m a t i o n abou t the magne t i c field. F o r reviews of 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 HR 1217 with WET High-speed photometry is the preferred technique for observing r o A p stars. This tech-nique involves a continual monitoring of the star, wi th 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 in a loss of data on the rapid light variations exhibited by roAp stars. The 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 in 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 in 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; wi th a magnitude of B « 6.3, background light is not a major problem. Thus, isolating the star in a diaphragm wi th a diameter less than 20 arcseconds wi l l not significantly reduce the ratio of sky-light to star-light. The principal source of noise in the frequency range of interest is scintillation (short-period sky trans-parency variations). The rapid oscillations of roAp stars; wi th periods of approximately 5 minutes, are not affected by sky variations that occur wi th periods of a half-hour or more. A good review of the general technique for observing r o A p stars wi th W E T is provided by K u r t z k Mart inez (2000). In late 2000, the Whole Ea r th Telescope launched its twentieth extended coverage campaign, X C O V 2 0 1 . H R 1217 was included as one of the primary targets of this cam-paign, which ran from Nov. 6 to Dec. 11. During X C O V 2 0 , a variety of telescopes and instruments were used. The telescope apertures ranged from 0.6 m to 2.1 m and instru-ments 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 22 observations at C T I O 2 used a two-channel photometer, while observations at B A O 3 , Na in i Ta l 4 , Teide 5 , Mauna K e a 6 , and McDona ld 7 used three-channel photometers. The single-channel photometers were used at S A A O 8 , Per th 9 , and SSO 1 0 . Each of these sites are shown on the map in Figure 2.1. The three-channel photometers and sky monitoring procedures are described by Kle inman et al . (1996). Observations wi th one-and 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. Each site observed the target using a Johnson B filter wi th 10 sec integration times. Since H R 1217 is a bright star, larger telescopes required neutral density filters to keep count rates below 10 6 s - 1 in 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 photo-metric observations. The 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 Appendix A . This log includes the observatories that participated, the dates of participation and the telescope that was used. K u r t z et al . (2003) review the X C O V 2 0 observing run wi th emphasis on H R 1217. The need for high-precision data results i n some runs being selected for the final data analysis while others had to be discarded. Runs 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 and further detail on reduction procedures are outlined in the following sections of this chapter. 2Cerro Tololo Inter-American Observatory, Chile 3Beijing Astronomical Observatory, China 4Uttar Pradesh State Observatory, Naini Tal Manora Peak, India 5Observatorio del Teide (Tenerife), Spain 6Mauna Kea Observatory, Hawaii, U.S.A 7McDonald Observatory, Texas, U.S.A. 8 South African Astronomical Observatory, South Africa 9Perth Observatory, Australia 1 0 Sliding Spring Observatory, Australia 1 1 This data log was adapted from the one on the XCOV20 website 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, and i n Table A . l 2.3 Preparing the Light Curves Each individual run was reduced following the procedures outlined by Nather et al . (1990). The standard reduction software for W E T is called QED. This software performs various photometric reduction tasks such as: deadtime corrections, sky interpolation and subtraction for multiple channels, sky extinction corrections, and polynomial fitting of light curves. It is important to use a standard reduction package to reduce each run 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. The reference manual for QED was written by R . E . Nather in 2000 and can be found on the W o r l d Wide W e b 1 2 . A n overview of the reduction procedure for an individual run is described below. 12http://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 instru-ment used, the integration time used, default extinction coefficients and the raw counts as a function of time from the beginning of the run. The data is displayed on the screen so the user can identify bad data by eye. The selection of bad data is subjective, and the person performing this ini t ial step uses individual observing logs to help identify such points. Typically, bad data points include those collected while clouds pass i n front of the star, or points that have counts inconsistent wi th the rest of the run due to the star drifting in the diaphragm. The sky measurements are obtained by observing the sky; i.e., a measurement while there is no star is in the diaphragm. These sky measurements are marked where indicated in the observing logs. The deadtime corrections and the sky subtractions are then performed on the data. The 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. Each of the channels has a deadtime correction applied and the default deadtime constants for each instrument are used. The deadtime correction to the counts is given by Ct = C o / ( l — GVd) , where Co is the original count, Ct is the corrected count, and tj is the deadtime constant. These constants can be changed in QED 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. The airmass is simply the thickness of the atmosphere as a function of the zenith angle z. This change in brightness is known as the atmospheric extinction X and i t is related to the zenith angle by (Hiltner, 1962) X = sec{z) -0.0018167(sec(^)-1)-0.002875(sec( ,z)-1) 2 -0.0008083(sec(^) - 1 ) 3 (2.1) Once the atmospheric extinction is calculated, the counts are corrected using the equation logio(Cs/Ci) = kexX/2.5. In this relation, Cj and Cs correspond to the instru-mental counts and the corrected counts respectively. The extinction coefficient is given Chapter 2. Data & Analysis 25 b y kex a n d is adjus ted i n order t o remove as m u c h of the e x t i n c t i o n t r e n d as is poss i -ble . T h u s , i f t he e x t i n c t i o n co r rec t ion per fec t ly sub t rac t s b r igh tness v a r i a t i o n s caused b y a i rmass , the l ight curve w o u l d have no l o n g - t e r m t r end . F o r l igh t curves where the e x t i n c t i o n co r r ec t ion doesn ' t remove such l o n g t e r m t rends comple te ly , p o l y n o m i a l s of v a r y i n g degree were sub t r ac t ed f r o m the curve. T h e degree of the p o l y n o m i a l never exceeds 3, a n d i n most cases, a p a r a b o l i c or l inear fit was sufficient. /home/wet/People/chris/redb/MDR136.1 C Fa*: . : Jun 9 04:21 2002 *-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 K o L P I J l J - l rci.i?so B]ed_24B IBM.6061070 Baryc_412.2414.1.0602 #_Who;_mdr #_Wh trn :_ctio |_TeLtcp:.l.Bm #J>rogrmi.{JuiltO.U #_PhoU.muP3B '.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_-b.0015 I • ' ' I ' I • I ' I ' I ' I I I 0 2000 4000 6000 6000 10000 PrequencyOiHz) Figure 2.2: T h e end resul t of the QED r e d u c t i o n for r u n m d r l 3 6 at C T I O . T h e t o p panels show the channe l 1 ( top) a n d 2 d a t a a n d the lower panels show the Four i e r t ransforms o f each. A g a i n , channe l 1 is the top Fou r i e r p lo t . T h e r e d u c t i o n deta i l s f r o m the QED o u t p u t are p r i n t e d o n the F i g u r e . T h e resul ts of the above r e d u c t i o n procedures us ing QED are s h o w n i n F i g u r e 2.2 for the r u n m d r l 3 6 (see T a b l e A . l ) . A two-channe l pho tome te r was used a n d each of the reduced channels c a n be seen i n the t op two panels of the F i g u r e . T h e F o u r i e r t r ans forms Chapter 2. Data & Analysis 26 of each o f the channels is s h o w n i n the lower two panels . These Fou r i e r p lo t s show power at l o w frequencies f r o m r e m a i n i n g l ong - t e rm t rends i n the da t a . S m a l l he l iocent r ic cor rec t ions to the t imes caused b y changes i n the E a r t h ' s p o s i t i o n w i t h respect t o the s tar due to i t s o r b i t a l m o t i o n h a d to be t a k e n in to account . A s a f ina l s tep after the QED r educ t ion , the t i m e d a t a for each r u n are p l aced i n the same i n e r t i a l frame a n d the counts are n o r m a l i z e d abou t the m e a n count . A su i tab le reference frame for a l l of the t imes is the barycenter of the So la r S y s t e m . T h u s , each r u n has i t s t i m e s cor rec ted w i t h reference t o the barycenter of the So la r S y s t e m a n d l i s t e d s ince a t i m e T 0 . F o r X C O V 2 0 , T0 = 2451854.5 ba rycen t r i c J u l i a n D a t e ( B J E D ) . T h e f ina l resul t is t h e n b i n n e d i n 40 sec b ins to increase s igna l to noise a n d to decrease c o m p u t a t i o n a l t i m e d u r i n g the frequency analys is . T h i s d a t a t h e n goes t h r o u g h a se lec t ion p rocedure desc r ibed i n the fo l lowing sec t ion . 2.3.1 Run Selection Before frequency ana lys i s c o u l d beg in , runs w i t h the best s ignal - to-noise h a d to be se-lec ted . F i r s t , low-frequency noise was removed b y f i t t i ng a n d s u b t r a c t i n g s inusoids w i t h frequencies be low 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 selected u s ing the software Period98 1 3 (Sper l , M . 1998). M o r e deta i l s o n th i s software package c a n be found i n Sec t i on 2.4. Low-f requency noise i n the d a t a is caused b y l o n g - t e r m s k y va r i a t ions a n d other i n s t r u m e n t a l drif ts . S u c h frequencies s h o u l d have i dea l l y been r e m o v e d b y the e x t i n c t i o n co r rec t ion a n d a low-order p o l y n o m i a l fit; however , a m p l i -tudes w i t h pe r iods greater t h a n a ha l f -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 not affect the frequencies above 2 m H z tha t are i m p o r t a n t i n r o A p s tar s tudies . T h e Fou r i e r s p e c t r u m of each r u n was e x a m i n e d a n d those tha t h a d a s igna l - to -noise greater t h a n a p p r o x i m a t e l y three at the frequency w i t h the largest a m p l i t u d e were selected. F i g u r e 2.3 shows examples of a selected r u n a n d a rejected one. T h e noise level 1 3http://www. astro. univie.ac.at/~dsn/dsn/Period98/current Chapter 2. Data & Analysis 27 F i g u r e 2 . 3 : A Four i e r s p e c t r u m of d a t a f r o m the runs n o 2 9 0 0 q l ( top) a n d j o y -002 ( b o t t o m ) . It is clear tha t the a m p l i t u d e a n d noise levels for the n o 2 9 0 0 q l are m u c h bet ter t h a n those i n the joy-002 r u n . T h e r e is also a large frequency at a p p r o x i m a t e l y 720 c y c l e s / d a y (8.3 m H z ) i n the joy-002 r u n caused b y a p e r i o d i c d r ive error i n the telescope. Chapter 2. Data & Analysis 28 a n d a m p l i t u d e of the m a i n frequency at a p p r o x i m a t e l y 2.7 m H z is c l ea r ly different for b o t h runs . A l t h o u g h the a m p l i t u d e s of r o A p osc i l l a t ions c a n v a r y over the r o t a t i o n cycle of the star , the l ow s ignal- to-noise i n the joy-002 ( b o t t o m p lo t i n F i g u r e 2.3) r u n makes i t unaccep tab le for further frequency analys is . A f t e r b a d runs were e l im ina t ed , the r e m a i n i n g d a t a consis ts of 330.5 hrs o f pho -t o m e t r y over a p e r i o d of 35 days, as p l o t t e d at a compressed scale i n F i g u r e 2.4. T h i s covers s l i g h t l y less t h a n three r o t a t i o n cycles of H R 1217, w h i c h has a r o t a t i o n p e r i o d of a p p r o x i m a t e l y 12.5 days. T h e r e su l t ing d u t y cycle for the entire r u n is 3 3 % . 2.4 Frequency Analysis A l l f requency ana lys i s was pe r fo rmed us ing the software package Period98 (Spe r l , 1998). T h i s package uses a non l inear least squares fitting m e t h o d to ca lcu la te the amp l i t udes , frequencies, a n d phases tha t m i n i m i z e the res iduals of a g iven fit t o the da t a . T h e f i t t i n g f u n c t i o n is a s inuso id of the f o r m where Z is a zero-poin t offset t ha t takes i n to account any l inear t rends i n the d a t a . Ai a n d fa are the a m p l i t u d e a n d phase of the ith f requency t/j, respect ively . Period98 does not r e ly o n p r e w h i t e n i n g to ca lcu la te a frequency s o l u t i o n for the l igh t curve . P r e w h i t e n i n g is the process where b y a p e r i o d i c func t i on is r emoved f r o m the d a t a a n d the res iduals are used for the next s tep i n the r e d u c t i o n p rocedure . B y d o i n g th is , each frequency depends o n the p r ev ious ly de t e rmined frequencies t h r o u g h the res iduals . T h i s process is used to de termine w h i c h frequencies are i n c l u d e d i n the fit. T h e frequency w i t h the largest a m p l i t u d e i n the Fou r i e r s p e c t r u m is ident i f ied a n d used as a guess frequency for the least squares c a l c u l a t i o n . T h e res iduals f r o m th i s fit are t h e n used to ident i fy the next highest frequency. A t each step, a l l ident i f ied frequencies, a m p l i t u d e s a n d phases are i m p r o v e d v i a the least squares c a l c u l a t i o n . T h i s process of pa rame te r i den t i f i ca t ion a n d improvemen t cont inues u n t i l the a m p l i t u d e s i n the F o u r i e r (2.2) Chapter 2. Data & Analysis 29 HR 1217 330.5hrs of coverage 20 1 1 0 £ 0 a. 6 -10 20 10 ~i 1 1 r- T - i 1 r i ii i i (t in r p H mil -J 1 L_ 10 J — i — i — i — i i i i i • • 15 16 17 J — i i t i i i i . * . i 20 21 22 E _ i — 1 — i — i i l i i i i i i i i 24 26 28 30 32 34 BJED (2451854.5 +) [days] F i g u r e 2 . 4 : T h e final l ight curve o b t a i n e d after the Q E D r e d u c t i o n a n d the ba rycen t r i c correct ions have been t a k e n in to account . T h e l igh t curve shows 330.5 hrs of d a t a w h i c h cor responds to a d u t y cyc le o f a p p r o x i m a t e l y 33%. T h e a m p l i t u d e s are measured i n un i t s o f m i l l i m a g n i t u d e . s p e c t r u m are at the level of the noise. A m o v i n g average technique is used to es t imate the noise level i n the F o u r i e r s p e c t r u m . Period98 ca lcu la tes the average of the amp l i t udes i n the Fou r i e r s p e c t r u m i n a g iven frequency b i n a n d t h e n moves a long to the next frequency b i n . If the noise was w h i t e (frequency independen t ) , one c o u l d s i m p l y m o d e l i t 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 o b t a i n e d f r o m different ins t ruments , under v a r y i n g seeing condi t ions , con t r ibu te different ly to the noise of the Chapter 2. Data & Analysis 30 2550 2800 2650 2700 2750 2800 2850 2900 "1—' '. i/[MHz] i ' i i i i i i i i i i : Window 1 1 1 1 1 1 1 ..J I i I i . i V 1 220 230 240 250 i^cycles/day] F i g u r e 2 . 5 : T h e Four i e r s p e c t r u m of the entire 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 shows the w i n d o w func t ion for th i s da t a . T h e w i n d o w was ca l cu la t ed f rom the Four i e r t r a n s f o r m of a s inuso id w i t h the largest a m p l i t u d e frequency ident i f ied i n the d a t a a n d s a m p l e d at the same t imes as the d a t a set. d a t a set. Therefore , i f the b o x size is chosen appropr ia te ly , the average o f the frequencies i n each b o x represents the noise s p e c t r u m w e l l . T h e first search of Four i e r space was pe r fo rmed i n the range f r o m 0 m H z to the N y q u i s t f requency of 12.5 m H z . A l l power i n the Fou r i e r s p e c t r u m was ident i f ied be tween 2.5 m H z a n d 2.9 m H z as was found i n ear l ier observat ions (e.g., K u r t z & Seeman , 1983, a n d K u r t z et a l . , 1989). I n order to speed the Four i e r ca l cu la t ions t he l a t t e r f requency range was exp lo red a n d s a m p l e d at 1 x 1 0 ~ 5 m H z in tervals . T h e resul ts of the ana lys i s Chapter 2. Data & Analysis 31 w i l l be presented i n Sect ions 2.5 a n d 2.6. T h e noise of the s p e c t r u m was ca l cu l a t ed at in te rva ls of 3 x 1 0 ~ 3 m H z us ing a frequency b i n size of 23 / J H Z (2 c y c l e s / d a y ) . T h i s b i n size was chosen so tha t o n l y the noise near the frequency b e i n g cons idered was used i n the c a l c u l a t i o n . . 2.5 The Unweighted Data Results T h e resul ts o f the above r e d u c t i o n procedures for the unweigh ted d a t a set are s h o w n i n T a b l e 2 .1 . P resen ted are 21 frequencies cons i s t ing of 8 p r i m a r y frequencies a n d the i r r o t a t i o n a l sp l i t t i ngs . Frequencies l abe led V\ t h r o u g h v5 a n d u7 are consis tent w i t h those ident i f ied b y K u r t z et a l . (1989) i n the 1986 c a m p a i g n . T h i s ana lys i s also recovers the new f requency of 2791.48 / / H z t ha t was p red i c t ed 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 Sec t i on 1.5). T h i s frequency is l abe led ve(old) for th 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 frequency is r epor t ed i n th i s w o r k at 2788.94 / / H z . T h i s frequency fits the a l t e r n a t i n g spac ing p a t t e r n of 34.5 a n d 33.5 / / H z a n d is l abe l ed as u&(new) i n T a b l e 2.1. T h e spac ing between u6(new) a n d u^{old) is 2.63 / / H z a n d the spac ing be tween ue(old) a n d u7 is a p p r o x i m a t e l y 15 / / H z . T h i s new frequency was not p rev ious ly ident i f ied b y K u r t z et a l . (2002) because t h e y e x a m i n e d a smal le r subset of the da ta . I n c l u d i n g more d a t a (i.e., e x t e n d i n g the t i m e coverage) increases the frequency r e so lu t i on because t he discre te F o u r i e r t r a n s f o r m is we igh ted e x p o n e n t i a l l y b y the number of d a t a po in t s . T h u s , inc reas ing the l e n g t h o f a d a t a set p roduces shaper spec t ra l features. A n u m e r i c a l ana lys i s b y L o u m o s & D e e m i n g (1978) also showed the spec t r a l r e so lu t ion A / of a d a t a set of l e n g t h A T is g iven 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 of th i s r e l a t i o n for u n c e r t a i n t y es t imates is g iven i n Sec t i on 2.5.1. A schemat ic a m p l i t u d e spec t r a for these resul ts is g iven i n F i g u r e 2.6. Chapter 2. Data & Analysis 32 Table 2 . i: Resu l t s of the frequency ana lys i s of the u n -weigh ted da ta . T h e frequencies, a m p l i t u d e s a n d the phases are shown. T h e r o t a t i o n a l spacings 5v a n d the i n -fered large spacings, A ^ , are also shown. v§(old) was the s i x t h frequency found p rev ious ly b y K u r t z et a l . (2002) a n d ue(new) is a new frequency ident i f ied i n th i s s tudy. v (mHz) A m p . ( m m a ) * <Si/(mHz) A i / ( m H z ) 2.61953721 0.247873599 0.061262411 + 2.62052628 0.101942968 0.19455008 9.8907E-04 V2-V\ 0.03340692 - 2.6519788 0.239812675 0.006703652 9.6533E-04 ^2 2.65294413 0.793304662 0.145744578 + 2.65389542 0.181838918 0.300326255 9.5129E-04 0.03454836 - 2.68644247 0.332793541 0.200445295 1.0500E-03 "3 2.68749249 0.554168633 0.11110511 + 2.68841904 0.215781733 0.336983675 9.2655E-04 V4-V3 0.03343462 - 2.72006539 0.408958037 0.505309948 8.6172E-04 2.72092711 1.14018239 0.782933145 + 2.72182605 0.436426134 0.015917406 8.9894E-04 0.03440263 - 2.75432171 0.218422957 0.114667745 1.0080E-03 2.75532974 0.278092265 0.224817247 + 2.75623738 0.119315631 0.489636487 9.0764E-04 i/6 (new)-i/5 0.03360699 ug (new) 2.78893673 0.126677281 0.606594168 + 2.78996948 0.090747676 0.507306041 1.0327E-03 - 2.7906266 0.130919596 0.197095448 9.4253E-04 VQ (old)-i/6 (new) VQ (old) 2.79156913 0.193687427 0.37786162 0.0026324 + 2.79213576 0.070257486 0.092062776 5.6663E-04 V7-v& (old) - 2.80562143 0.190737069 0.450445087 0.0149975 u7 2.80656663 0.120611222 0.485536548 9.4520E-04 T h e average va lue of the fine frequency s t ruc ture is 9.27 x 1 0 - 4 m H z w i t h a s t a n d a r d Chapter 2. Data & Analysis 33 T 1 1 1 1 1 1 1 r 1 r 0.8 V, 0.6 0.4 0.2 0.8 E Jj, ¥ 0.6 3 E 0.4 0.2 llll . 2.65 2650 2.7 •/(mHz) 2700 2.75 "[^ Hz] 2.8" 2800 ' J L_Ld_ 230 235 i/[cycles/day] 240 F i g u r e 2.6: A schematic of the frequencies found from the unweighted data (top). The frequencies are listed i n Table 2.1. O n the bottom, the amplitudes of the frequencies and the noise level after al l frequencies are prewhitened. The ( ) line represent four times the noise level and the (. . .) line represents three times the noise level. Chapter 2. Data & Analysis 34 error o n the m e a n o f « 1%. A s s u m i n g H R 1217 is a n ob l ique ro ta to r , the fine s t ruc tu re 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 s tar . T h i s w o u l d i m p l y a r o t a t i o n p e r i o d of 12.5 days ± 1 % ; consistent w i t h values of the r o t a t i o n p e r i o d de r ived i n the l i t e ra tu re . F o r example , K u r t z & M a r a n g (1987) use p h o t o m e t r y to deduce a p e r i o d of 12.4572 ± 0.0003 days f r o m the 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 po la r i sed l ight d a t a a n d o b t a i n a p e r i o d of 12.4610 ± 0.0011 days f r o m the magne t i c field va r i a t ions . (mHz) , (mHz) F i g u r e 2 . 7 : T h e r a t i o be tween the c o m m o n a m p l i t u d e s ( top) a n d differences i n frequencies (bo t tom) f rom th i s d a t a set a n d the K u r t z et a l . (1989) d a t a set. T h e frequencies are consistent w i t h each o ther w h i l e the a m p l i t u d e s of the t h i r d a n d f o u r t h frequency ( read f r o m the left) are c l ea r ly different. T h e net a m p l i t u d e a n d frequency differences are s h o w n i n the uppe r r ight of each p lo t . W h i l e the de r ived frequencies are consistent w i t h the 1986 da ta , the a m p l i t u d e s differ Chapter 2. Data & Analysis 3 5 s igni f icant ly . F i g u r e 2.7 compares the difference be tween the p r i n c i p l e a m p l i t u d e s a n d frequencies t ha t were c o m m o n to b o t h th i s d a t a set a n d the 1986 da t a . T h e a m p l i t u d e s of frequencies u3 a n d v± show the largest difference be tween the two d a t a sets. T h e y differ b y a p p r o x i m a t e l y 6 0 % a n d 40%, respect ively. However , the net a m p l i t u d e difference be tween the 6 frequencies is s m a l l , at —0.070 ± 0.192 m m a g . A s each of the frequencies was ident i f ied a n d p rewhi t ened , the s t a n d a r d d e v i a t i o n of the res iduals was ca l cu la t ed . Frequencies were r emoved u n t i l the s t a n d a r d d e v i a t i o n of the res iduals approached a constant value. These resul ts m a y be found i n F i g u r e 2.8. A f t e r the twenty-f i rs t frequency was removed, the s t a n d a r d d e v i a t i o n of the res iduals is i m p r o v e d b y less t h a n a 0 .1%. E s t i m a t e s of the unce r t a in ty a n d s ignif icance of the der ived frequencies w i l l be discussed i n the next sec t ion . 2.5.1 Estimating Uncertainties and Significance U n c e r t a i n t y es t imates for the frequency analys is were ca r r i ed out i n three different ways . T h e first is t ha t used b y K u r t z & Wegner (1979) to es t imate the frequency r e so lu t i on o f two c losely spaced frequencies. T h e y state tha t the 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 each frequency unce r t a in ty 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 he f requency r e so lu t i on e s t ima ted b y L o u m o s h D e e m i n g (1978). T h i s m a y be ca l cu l a t ed u s ing A ' = s i r ( 2 ' 3 ) where A T is the l eng th of the obse rv ing r u n . U s i n g th i s m e t h o d , the f requency uncer-t a i n t y is e s t ima ted as A / « 8 x 1 0 - 5 m H z for A T = 35.12 days . T h e second m e t h o d is c losely re la ted to the first a n d es t imates the n u m b e r o f inde-pendent frequencies f r o m the 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 the m a i n peak i n the 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, the F W H M is e s t ima ted to be 6 x 1 0 - 4 m H z . T h i s is a p p r o x i m a t e l y a n order of m a g n i t u d e above the u n c e r t a i n t y es t imate f r o m E q u a t i o n 2.3. T h e t h i r d technique es t imates the unce r t a in ty 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 assumes the t imes are ce r t a in a n d the counts are subject 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. Each of the frequencies are listed i n Table 2.1. The frequencies are prewhitened i n order of decreasing amplitude. The 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 con-sidered 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, aa = (2/iV)1/V(m) (2.4) Chapter 2. Data & Analysis 37 2720 2721 2722 I 1__J I I I I I 1 I I I I I I I . . . I I I I 234.95 235 235.05 235.1 235.15 u[c/d] F i g u r e 2 . 9 : T h e F W H M of the m a i n frequency i n the unweigh ted d a t a . T h i s is used as a n es t imate for the unce r t a in ty of the frequencies de r ived f r o m the r e d u c t i o n of the unweighted da ta . 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 considered is represented b y a, the l eng th o f the r u n is T, the n u m b e r of d a t a po in t s N, a n d the root -mean-square of the res iduals is a(m). These equa t ions are a p p l i e d to the res iduals of the s inuso id fit as each frequency is p r ewh i t ened . T h e error es t imates for t h i s p rocedure are l i s t ed i n T a b l e 2.2. A c c o r d i n g to M o n t g o m e r y h O ' D o n o g h u e (1999), i f the r a n d o m noise is cor re la ted w i t h t ime , the uncer ta in t ies c o u l d be underes t ima ted . T h e y urge t ha t these values o n l y be t a k e n as a lower l i m i t of the unce r t a in ty a n d suggest tha t the a c t u a l resul t m a y be a n Chapter 2. Data & Analysis 38 order of magnitude higher. If this is the case, and all of the uncertainties in Table 2.2 are increased by an order of magnitude, they would be consistent wi th the previous uncertainty estimates for this data. K u r t z et al . (2002) derive uncertainties formally from their least squares analy-sis during a preliminary data reduction. They obtain a frequency uncertainty that is « 1 0 - 5 m H z . Given the above information, a conservative estimate on the frequency uncertainties would be 1 x 10~ 4 mHz . 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 l imits only. av (mHz) <7a(mma) CT^ (mHz) (TAi/(mHz) 2.49E-06 0.0141 0.0054 + 2.42E-06 0.0137 0.0052 4.91E-06 V2-V\ 5.16E-06 - 2.44E-06 0.0140 0.0053 5.11E-06 2.67E-06 0.0153 0.0058 + 2.40E-06 0.0138 0.0052 5.07E-06 VZ-V2 5.23E-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 4.95E-06 f 4 - f 3 5.29E-06 - 2.46E-06 0.0144 0.0053 5.18E-06 2.73E-06 0.0160 0.0059 + 2.48E-06 0.0146 0.0054 5.21E-06 1/5-1/4 5.13E-06 - 2.34E-06 0.0139 0.0051 4.74E-06 "5 2.40E-06 0.0143 0.0052 + 2.30E-06 0.0137 0.0050 4.70E-06 1/6 (new)-f5 4.68E-06 UQ (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 4.61E-06 1/6 (old)-i/6 (new) (old) 2.33E-06 0.0140 0.0050 4.61E-06 + 2.27E-06 0.0137 0.0049 4.60E-06 continued on next page Chapter 2. Data & Analysis 39 Table 2.2: continued av (mHz) <7 a(mma) 04, iT fo (mHz) ( T A „ ( m H z ) 1/7 2.28E-06 2.26E-06 0.0138 0.0137 0.0049 0.0049 4.54E-06 1/7-1/6 (old) 4.59E-06 E s t i m a t e s for the s igna l t o noise at w h i c h a frequency m a y conf ident ly be ident i f ied 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 used a m p l i t u d e spec 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 t he H u b b l e Space Telescope 's F i n e G u i d a n c e Sensor gu ide s tar d a t a to show tha t a s igna l t o noise of 3.6 w o u l d p roduce a 9 9 % confidence leve l for f requency iden t i f i ca t ion , w h i l e a s igna l to noise of 4.0 w o u l d p r o d u c e a 99 .9% confidence level . These levels are consistent w i t h those suggested b y Brege r et a l . (1993) a n d those discussed b y A l v a r e z et a l . (1998), a n d Breger et a l . (1999). T h o s e au thors also s tate t ha t th i s m e t h o d is equivalent to the Scargle false a l a r m p r o b a b i l i t y test (Scargle , 1982) w h i c h assigns a confidence level for a n ident i f ied frequency a s suming w h i t e noise. F o l l o w i n g these au thors , a s ignif icant confidence level for a frequency iden t i f i ca t ion is o b t a i n e d where the a m p l i t u d e of the frequency is at least 3.5 t imes the noise (see F i g u r e 2.6). 2.6 The Weighted Data Results T h e d a t a co l lec ted d u r i n g a W E T c a m p a i g n has noise charac ter i s t ics t ha t are a c o m b i -n a t i o n of noise f r o m each i n d i v i d u a l obse rv ing r u n . I n order to o b t a i n the best s igna l t o noise, one mus t consider e i ther r e m o v i n g some of the da t a , or we igh t ing the da t a . R e m o v i n g d a t a m a y not be the most desirable a l te rna t ive since gaps i n the l igh t curve p r o d u c e aliases i n the Fou r i e r spec t rum. T h i s , i n t u r n , confuses frequency iden t i f i ca t ion . It is also i m p o r t a n t t o note t ha t the Four i e r t r a n s f o r m is weigh ted b y the n u m b e r of p o i n t s cons idered . T h u s , l ower ing the n u m b e r of p o i n t s b y r e m o v i n g d a t a w i l l a lso i n -crease the noise i n the ca l cu la t ed s p e c t r u m . In th i s thesis, different w e i g h t i n g schemes w i l l be cons idered i n a n a t t empt to p roduce the best frequency s o l u t i o n for the H R 1217 Chapter 2. Data & Analysis 40 da t a . T w o we igh t ing schemes discussed i n H a n d l e r (2003) w i l l be descr ibed , as w e l l as the mod i f i ca t i ons t o these m e t h o d s used i n th i s w o r k to ana lyze the H R 1217 da t a . T h e first we igh t ing scheme is s igma-cutoff we igh t ing . I n th i s m e t h o d , each d a t a po in t is we igh ted based o n the res iduals f rom a g iven fit. E a c h weight is ass igned b y the re la t ions Wi = 1 if G{ < Kares Wi = (Kares/ai)x if a > Kares (2.7) where K a n d x are free parameters , Oi is the r m s res idua l of the ith p o i n t a n d ares is the average s t a n d a r d d e v i a t i o n of the res iduals . F o r example , R o d r i g u e z et a l . (2003) choose values oi K — 1.0 a n d x = 2.0. Period98 uses th i s m e t h o d to ass ign weights to da ta ; however, K a n d x are fixed at 1 a n d 2, respect ively, a n d i t is u p to the user to choose the cut-off a m p l i t u d e ares. F r a n d s e n et a l . (2001) use th i s f unc t i on of Period98 as one of the we igh t ing schemes i n the i r pape r . B o t h t h e y a n d H a n d l e r (2003) c a u t i o n tha t the value of crre3 s h o u l d be chosen wisely . I f i t is not , b a d d a t a t ha t falls w i t h i n the cut-off c o u l d be g iven fu l l weight , or g o o d d a t a c o u l d be g iven l ow weight . T h i s m e t h o d is also l i m i t e d b y i t s dependence o n a p r ede t e rmined s o l u t i o n to ca lcu la te res iduals . A n o t h e r m e t h o d of d a t a we igh t ing descr ibed i n H a n d l e r (2003) is l igh t curve var iance we igh t ing . I n th i s m e t h o d , the d a t a are b o x c a r s m o o t h e d to remove the p o i n t - t o - p o i n t differences i n intensi ty . T h e inverse of the s m o o t h i n g func t i on is t h e n used to ass ign weights to the da t a . B y d o i n g th is , n igh t - to-n ight va r i a t ions i n the d a t a c a n be t a k e n in to account . T h e free parameters i n th i s m e t h o d w o u l d be the size of the t i m e b i n b e i n g cons idered a n d the exponent b y w h i c h y o u weight the da ta ; i .e., the x i n E q u a t i o n 2.7. T h e m a i n d isadvantage of th i s m e t h o d is tha t the s m o o t h i n g f u n c t i o n c a n be g rea t ly affected b y o u t l y i n g po in t s . T h i s may, i n t u r n , resul t i n a n i m p r o p e r weight ass ignment . I n t h i s s tudy, four we igh t ing me thods based o n the above schemes were tested. E a c h w e i g h t i n g p rocedure is s u m m a r i z e d as: m e t h o d 1 : T h e Period98 we igh t ing scheme is used w i t h a va r i e ty of cutoff Chapter 2. Data & Analysis 41 a m p l i t u d e s a r e s . m e t h o d 2 : T h e d a t a weights are assigned us ing E q u a t i o n 2.7 w i t h the average s t a n d a r d error of the res iduals f rom a fit . T h i s fit is o b t a i n e d f r o m the resul ts of the unweigh ted d a t a ana lys i s (see Sec t ion 2.5). W h e n E q u a t i o n 2.7 is a p p l i e d to da t a , the free parameters are va r i ed to explore the i r effect o n the f ina 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 error o f t he res idua ls t o a fit are c a l c u l a t e d at va r ious t i m e b ins before E q u a t i o n 2.7 is app l i ed . 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 weights f r o m a s m o o t h i n g fi l ter i n tha t the change i n the noise f r o m t i m e b i n to t i m e b i n is t a k e n in to account . m e t h o d 4 : T h e weights are ca l cu la t ed a n d a p p l i e d us ing E q u a t i o n 2.7 as each new frequency is added . B y d o i n g th is , we don ' t r e ly o n res iduals f r o m a f ina l fit t o the da t a . T h i s m e t h o d a t t emp t s to overcome the dependence o n a p rede t e rmined s o l u t i o n b y cons ide r ing the weight to be a pa ramete r t ha t changes w i t h each s tep i n the frequency analys is . E a c h of the resul ts 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 resul ts f r o m the unweigh ted d a t a t h r o u g h a s ta t i s t i c xlomp defined b y where values w i t h the subscr ip t 0 refers to those o b t a i n e d f r o m the unweigh ted da t a , / represents a n observed t ime-series d a t a po in t , ffit is a p o i n t c a l c u l a t e d u s ing E q u a t i o n 2.2, a n d the s t a n d a r d er ror of the res iduals f rom the fit for a g iven d a t a b i n are represented b y a. M o d e l s w i t h var ious t i m e b ins a n d cut-off a m p l i t u d e s were cons idered . S o m e of the mode l s cons idered modes to be t r ip le t s , quinte ts , 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 of b o t h . O t h e r mode l s were cons t ruc ted a s suming t ha t the frequencies are sp l i t b y e x a c t l y the r o t a t i o n p e r i o d of the star . I n the la t te r case, frequencies were ad jus ted after each least squares i t e r a t i o n to be sp l i t w i t h a frequency of 9.2897 x 1 0 - 4 m H z ; the average of the Xcomp £ £ = _ ( / . - / / i t o ) 2 / ( c 7 j o ) (2.8) Chapter 2. Data & Analysis 42 measurements 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 phases were t h e n i m p r o v e d us ing Period^. A l l mode l s are s u m m a r i z e d i n T a b l e 2.3. Table 2.3: A s u m m a r y of the we igh t ing parameters for the 34 different frequency solu t ions exp lored . T h e p a r a m -eters are defined i n E q u a t i o n 2.7. T h e c o l u m n l abe l ed method refers to the we igh t ing me thods descr ibed i n Sec-t i o n 2.6. T h e points /bin c o l u m n is the number of d a t a po in t s per t i m e b i n . If method 1 is used, the cutoff a m -p l i t u d e is presented ins tead of the number of d a t a po in t s per b i n . See the comments c o l u m n for specific r e d u c t i o n deta i ls . model # K X points /b in or crrea{mrnag] method method = 1 comment 1 1.0 1.0 500 2 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 12 no weight 1 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 /b in or ores[mmag] method method = 1 comment 15 2.0 1.0 0.8 1 See model 12 comment (New Weight). 16 - - no weight 1 Calculate a solution where all frequencies are triplets. 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 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 5 1 See model 25 comment (New Weight). 27 2.0 1.0 1 1 See model 25 comment (New Weight). 28 2.0 1.0 0.8 1 See model 25 comment (New Weight). 29 - - 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 Adjust frequencies to be exactly rotationally split. 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 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 mode l s are c o m p a r e d to the unweighted d a t a t h r o u g h E q u a t i o n 2.8. T h e resul ts Chapter 2. Data & Analysis 44 are s h o w n i n F i g u r e 2.10. A lower xlomp s ta t i s t ic impl i e s lower res iduals for the fit . I n a l l cases exp lo red , the improvemen t i n the res iduals was less t h a n 1%. A n a l t e red ve rs ion of the unweigh ted d a t a was used to test the sens i t iv i ty of the xlomp s t a t i s t i c t o a p o o r f i t . In t h i s case; m o d e l n u m b e r 33, the unweighted d a t a h a d i t s fine s t ruc tu re spaced e x a c t l y b y the r o t a t i o n p e r i o d of the star wh i l e the amp l i t udes a n d phases were not a l te red . T h e resul ts i nd ica t e a xlomp t ha t is a p p r o x i m a t e l y 5% worse t h a n i n the other mode l s . M o d e l number s 25 a n d 34 s h o u l d be c o m p a r e d to m o d e l 33. B o t h of these mode l s force the frequencies of the fine s p l i t t i n g for the unweighted d a t a t o be the r o t a t i o n a l f requency of the s tar . T h e excep t ion i n m o d e l 25 is tha t the frequencies, a m p l i t u d e s a n d phases were t h e n i m p r o v e d u s ing Period98. F o r m o d e l 34, o n l y the a m p l i t u d e s a n d phases were t h e n i m p r o v e d u s i n g Period98. T h e lowest peaks i n F i g u r e 2.10 cor respond to mode l s 14, 18, 27 a n d 32. Schema t i c a m p l i t u d e d i a g r a m s showing the weighted noise are presented i n F i g u r e s 2.11 a n d 2.12. I n a l l cases considered, v7 shows o n l y a signif icant double t s t ruc ture . E a c h o f the o ther frequencies show, at least, s ignif icant t r ip le t s t ruc ture . C o m p l e t e d a t a tables for these 4 mode l s are g iven i n A p p e n d i x C . I n the other s t a n d a r d error was c o m p u t e d for mode l s w i t h the same frequen-cies, b u t w i t h different weight parameters . T h e resul ts show tha t the 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 ~ 6 w h i l e the a m p l i t u d e s differ b y less t h a n a h a l f of a percent . C h a n g i n g the pa ramete r space for the d a t a weights has l i t t l e effect o n t h e o u t c o m e o f the da t a . T h e c o m p u t e d s t a n d a r d errors for a l l cases are also presented i n A p p e n d i x C . Chapter 2.. Data & Analysis 45 -I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1—I 0.898 - - J i i i i I i i i , I , , , i 1_ 0 10 20 30 0 10 20 30 Model Number n Model Number n 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 unweigh ted mode l s u s i n g E q u a t i o n 2.8. T h e p lo t at the t op shows the s t a t i s t i c for the 34 mode l s descr ibed i n T a b l e 2.3. T h e r ight p lo t is a n e x p a n d e d v i e w of the first 32 models . In a l l cases, the mode l s v a r y f r o m the unweighted m o d e l b y no more t h a n 5%. Chapter 2. Data & Analysis 1.2 1 h 0.8 0.6 0.4 0.2 2700 - i 1 L _ J i i i I i i L V \ 4 230 235 240 i / [ cyc le s /day ] '0 2700 27S0 230 235 i / [oyc les /day] 240 Figure 2.11: Schematic amplitude diagrams for models 14 (top) and 32 (bot-tom). The ( ) line represent four times the noise level and the (. . .) line represents three times the noise level. The noise is calculated from the weighted residuals. Chapter 2. Data & Analysis 47 230 235 y [ c y c l e s / d a y ] 230 235 u [ c y c l e s / d a y ] 240 Figure 2.12: Schematic amplitude diagrams for models 18 (top) and 27 (bot-tom). The ( ) line represent four times the noise level and the (. . .) line represents three times the noise level. The noise is calculated from the weighted residuals. 48 Chapter 3 Stel lar E v o l u t i o n and P u l s a t i o n M o d e l s A g r i d of e v o l u t i o n a r y a n d p u l s a t i o n mode l s of A type stars is c a l cu l a t ed e x p l o r i n g a va r i e ty of parameters . T h e effect of a magne t i c f ie ld o n the r e su l t i ng p u l s a t i o n frequencies is also e s t ima ted for these mode l s . In th i s chapter , the procedures for c a l c u l a t i n g the mode l s are discussed a n d the resul ts of the ca lcu la t ions are i n t r o d u c e d . 3.1 Stellar Evolution Models T h e s te l lar e v o l u t i o n mode l s were ca lcu la t ed us ing the Y a l e S te l l a r E v o l u t i o n C o d e w i t h R o t a t i o n ( Y R E C 7 ) i n i t s n o n - r o t a t i n g conf igura t ion (Guen the r et a l . , 1992). Y R E C 7 solves the mechan i ca l , conserva t ion , a n d energy t r anspo r t equat ions of s te l la r s t ruc tu re u s ing the H e n y e y r e l a x a t i o n scheme (Henyey et a l , 1964). A de ta i l ed d i scuss ion of the equat ions of s te l lar s t ruc ture c a n be found i n the text of K i p p e n h a h n & Weige r t (1994). T h e dens i ty of a s te l lar m o d e l is re la ted to the o ther m a t e r i a l funct ions; i.e., the t e m p e r a t u r e a n d pressure o b t a i n e d f r o m the s te l lar s t ruc tu re equat ions , t h r o u g h the e q u a t i o n of s tate ( E O S ) . Y R E C 7 in terpola tes be tween O P A L E O S tab les (Rogers 1986, a n d Rogers , Swenson, &; Iglesias, 1996) w i t h different compos i t i ons i n order to o b t a i n the app rop r i a t e densi t ies 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 rou t ines needed to solve the energy t r a n s p o r t equa t ions o f s te l la r s t ruc tu re are i n t e r p o l a t e d f r o m t w o separate tables . If the t empera tu re o f a mass she l l i n a g iven m o d e l is greater t h a n 6000 K , the O P A L o p a c i t y tables are u t i l i z e d (Iglesias & Rogers , 1996). If the t empera tu re of a she l l is less t h a n 15000 K , the low- tempera tu re (molecular ) o p a c i t y tables of A l e x a n d e r & Fe rguson Chapter 3. Stellar Evolution and Pulsation Models 49 (1994) are used to o b t a i n the opaci ty . In regions of the m o d e l where these t empera tu res over lap , a l inear r a m p func t ion is used to l i n k the two o p a c i t y tables . A l l tables are ca l cu l a t ed for a solar e lementa l abundances (Grevesse et a l . , 1996). F r o m th i s s t r u c t u r a l i n f o r m a t i o n , the nuclear r eac t i on network; e.g., the p r o t o n - p r o t o n or ca rbon-n i t rogen-oxygen chains , is ca l cu la t ed for each m o d e l she l l t o de te rmine the energy generated v i a nuclear b u r n i n g us ing the nuclear cross sect ions o f B a h c a l l et a l . (2001). T h e new in te r io r e lementa l abundances are ca l cu l a t ed f r o m th i s process a n d another m o d e l is evo lved f r o m th i s comple te m o d e l t o a l a te r t i m e s tep. In t h i s s tudy, mode l s are evolved f rom the zero age m a i n sequence ( Z A M S ) to the a p p r o a c h of the base of the r ed giant b r a n c h (see F i g u r e 3.1). E a c h m o d e l genera ted has a p p r o x i m a t e l y 3000 shells evenly d i s t r i b u t e d a m o n g the in te r ior , envelope a n d the a tmosphere . T h e m o d e l in te r ior represents the inner ~ 9 9 % of the m o d e l b y mass w h i l e the envelope makes u p the o ther 1%. T h e ove r ly ing a tmosphere is c a l c u l a t e d assum-i n g a f requency independent t empera tu re -op t i ca l d e p t h ( T — r ) r e l a t ion ; k n o w n as the E d d i n g t o n gray a p p r o x i m a t i o n . 3.1.1 The Parameter Space T o b e g i n to specify a s te l lar m o d e l , the mass, l uminos i ty , 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 of the s tar are requi red . I n mos t cases, there are spec t roscopic observa t ions tha t y i e l d i n f o r m a t i o n abou t the effective t empera tu re a n d the heavy m e t a l (elements w i t h a t o m i c n u m b e r s greater t h a n 2) content . T h e l u m i n o s i t y is also e s t i m a t e d f r o m p h o t o m e t r i c observat ions , or apparent m a g n i t u d e a n d p a r a l l a x measurements ; w h i l e the mass is o n l y t i g h t l y cons t ra ined i f the s tar be longs to a b i n a r y f r o m w h i c h a confident o r b i t a l geomet ry m a y be der ived . T h e r e is also the p o t e n t i a l t o use as teroseismic ob-servat ions to fur ther c o n s t r a i n the evo lu t i ona ry s ta tus of a s tar t h r o u g h paramete r s l ike the large spac ing A i / (see E q u a t i o n 1.7). E v e n i f a s te l lar mode le r h a d i n f o r m a t i o n o n a l l these observable quant i t ies , the s t a n d a r d t r ea tment of convect ive energy t r an spo r t adds ano ther free pa ramete r k n o w n as the m i x i n g l eng th pa ramete r a ( B o h m - V i t e n s e , Chapter 3. Stellar Evolution and Pulsation Models 50 1958). T h i s pa ramete r sets the number of pressure scale-heights a convect ive element rises before re leas ing i t s heat t o the s u r r o u n d i n g p l a s m a . Recen t ly , C u n h a et a l . (2003) e x a m i n e d a set of s te l lar mode l s for H R 1217 i n hopes 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 tatus . T h e i r resul ts are c o m p l i m e n t a r y t o those of th i s thesis a n d there is a n over lap i n the choice of parameters to explore . T h e effective t empera tu re of H R 1217 is es t ima ted f r o m two sources. R y a b c h i k o v a et a l . (1997) use the i r spec t r a l synthesis code to es t imate a n effective t empera tu re of Teff = 7250; w i t h o u t a quo ted uncer ta in ty . T h e o ther es t imate o f the Teff comes f r o m the S t r o m g r e n p h o t o m e t r y of 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) used th i s p h o t o m e t r y t o es t imate a n effective t empera tu re of 7400 ± 100 K . I n th i s work , we combine these to resul ts t o o b t a i n Teff = 7400^200 K . T h i s is i d e n t i c a l t o the cons t ra in t s o n effective t empera tu re used b y C u n h a et a l . (2003). T h e mass of H R 1217 is e s t ima ted f rom prev ious A s tar mode l s t o be 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 Sa io , 1985, a n d M a t t h e w s et a l , 1999). A t igh te r cons t ra in t for the mass of H R 1217 of 1.8 ± O . 3 M 0 is der ived b y W a d e (1997) a s s u m i n g i t is a r i g i d ro ta to r . I n th i s s tudy, the masses were va r i ed f r o m 1.3 to 1 . 8 M 0 i n 0 . 5 M © steps. T h e l u m i n o s i t y of H R 1217 is also es t ima ted us ing two different approaches . T h e first is f r o m the observed large spac ing of w 68 fxEz. U s i n g E q u a t i o n 1.9 w i t h the above es t imates o n Teff, the mass, a n d the large spac ing , a as teroseismic l u m i n o s i t y is c a l c u l a t e d to be 8.2tl'^LQ. I f we observe h a l f the large spac ing of « 34 / . H z , the l u m i n o s i t y is e s t ima ted to be 2 O . 7 l 4 ; g . L 0 . T h e second m e t h o d is a d i rec t c a l c u l a t i o n of the stars l u m i n o s i t y f r o m the Hipparcos p a r a l l ax . M a t t h e w s et a l . (1999) derive a l u m i n o s i t y of 7.8 ± O . 7 L 0 f r o m the Hipparcos pa ra l l ax ; consistent w i t h a Au « 68 / i H z . T h e m e t a l l i c i t y of H R 1217 is es t imated f r o m the spec t roscopy of R y a b c h i k o v a et a l . (1997), w h i c h ind ica tes [Fe/H] « 0.32 ± 16%. A s s u m i n g Fe is a t racer of the in te r io r m e t a l content o f a star , a heavy m e t a l mass f rac t ion of Z — 0.008 c a n be e s t ima ted f r o m the re la t ions X + Y + Z=l, (3.1) Chapter 3. Stellar Evolution and Pulsation Models 51 (3.2) a n d AY _ (n - Yp) (3.3) AZ (Z* - Ze) where X a n d Y are the mass fract ions of hyd rogen a n d h e l i u m a n d AY I AZ is the G a l a c t i c enr ichment pa ramete r . T h e subscr ip t s p , * a n d © denote p r i m o r d i a l , s te l lar a n d solar values, respect ively . T o comple te the ca l cu l a t i on , values of (Z/X)® = 0.0244 (Grevesse et a l , 1996), Yp = 2.232 ± 0.003 ( O l i v e & S t e igman , 1995), a n d AY j AZ = 2.5 (Bressan et a l . , 1994) were adop ted . A d i f f icul ty arises for the A p stars i n tha t the i r inferred m e t a l content f r o m spec t r a d r a m a t i c a l l y changes over the magne t i c phase of the s tar . R y a b c h i k o v a et a l . (1997) give values 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 magne t i c phases. If i t is (na ive ly) a ssumed tha t the average of a l l of the meta l s for each of the 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 the in te r ior heavy m e t a l content of H R 1217, one c a n ca lcu la te f r o m E q u a t i o n s 3.1 - 3.3, a Z tha t varies be tween 0.019 a n d 0.024. T h u s , the infer red in te r io r Z for H R 1217 c a n v a r y over a large range d e p e n d i n g o n h o w the surface m e t a l l i c i t y is used. It s h o u l d be no t ed tha t there is no reason to bel ieve t ha t the surface m e t a l content of the A p stars is representat ive of the in te r io r m e t a l content . However , for completeness , Z = 0.008 to 0.022 i n steps of 0.002 is used for the e v o l u t i o n a r y mode ls ; a l o n g w i t h a n e s t ima ted h y d r o g e n mass f r ac t ion of X = 0.700 to 0.740 i n steps of 0.020. T h e large extent of the pa ramete r space i n c o m p o s i t i o n encompasses the uncer ta in t ies f r o m the observed (Z/X)Q, Yp, AY/AZ, 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 parameters of a = 1.4, 1.6, a n d 1.8 are used i n the m o d e l g r i d . If a s tar possesses a convect ive envelope, a m a y be used to set the a d i a b a t i c t e m p e r a t u r e gradient at the base of the convect ive zone. S ince the r ad iu 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 Teff, v a r y i n g a w i l l s l i gh t ly change the r ad ius of the m o d e l ( G u e n t h e r et a l . , 1992). S ince the envelopes of A stars are essent ial ly r ad ia t ive , any s t r u c t u r a l changes f r o m the different values of a are s m a l l w h e n c o m p a r e d to the t o t a l pa rame te r space b e i n g cons idered . Chapter 3. Stellar Evolution and Pulsation Models 52 3.1.2 Model Properties T o get a n i n d i c a t i o n of the pa ramete r space exp lo red , the ex t remes o f mass a n d c o m -p o s i t i o n are p l o t t e d o n the theore t i ca 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 the error boxes ca l cu la t ed f rom the Hipparcos l uminos i t y , a n d b o t h the as-te rose ismic luminos i t i e s . It is clear tha t the er rorbars for the as teroseismic l u m i n o s i t y ca l cu l a t ed f r o m a Av/2 ~ 34 fiRz are o n l y crossed b y h i g h mass m o d e l s w i t h Z values t ha t are greater t h a n the solar value of Z « 0.02. It is also evident t ha t the change i n the heavy m e t a l content of the mode l s p roduces the largest change i n the p o s i t i o n of the m o d e l o n the 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 5 s te l lar mode l s genera ted for the m o d e l g r i d . D e t a i l e d mode l s ( con ta in ing i n f o r m a t i o n necessary for the p u l s a t i o n analys is) were o u t p u t i n age steps of 0.05 G y r f r o m the Z A M S , decreas ing the n u m b e r of mode l s b y a n order of magn i tude . O f those models , 569 fa l l w i t h i n the Hipparcos l u m i n o s i t y error b o x ( H L E B ) . T h e m a j o r i t y of these mode l s have a mass of « 1 . 6 M 0 a n d a m e t a l l i c i t y range f r o m Z « 0.014 to Z « 0.020. A fu l l l i s t i n g of a l l of these mode l s c a n be found i n T a b l e B . l of A p p e n d i x B . It is also in te res t ing to l ook at the convect ive proper t ies of the c a l c u l a t e d mode l s since convec t ion has a n influence o n b o t h the evo lu t i ona ry s ta tus a n d p u l s a t i o n a l p roper t i e s of a s tar , as w e l l as i m p l i c a t i o n s for the surface chemica l inhomogenei t i es of A p stars. F i g u r e 3.2 shows the e v o l u t i o n of b o t h the convect ive core mass a n d the convect ive envelope mass for the mode l s s h o w n i n the H R d i a g r a m (F igu re 3.1). M o d e l s w i t h h igher Z a n d X have convect ive cores tha t last longer t h a n those w i t h the lower Z a n d X mass f ract ions . I n general , a m o d e l w i t h a g iven mass tha t has a h igher m e t a l content w i l l t ake longer to reach the same e v o l u t i o n a r y state as a m o d e l w i t h a lower m e t a l content . T h e same is t rue for mode l s w i t h a larger hyd rogen mass f rac t ion ; a l t h o u g h to a lesser extent . I n essence, the mode l s w i t h the lower Z content c o n t a i n more nuc lear fuel; i nc reas ing energy p r o d u c t i o n i n the nuclear b u r n i n g core. F o r a g iven Z , the mode l s w i t h a h igher X have less efficient energy t r anspor t i n the in te r ior because of the i r sma l l e r convect ive cores. Chapter 3. Stellar Evolution and Pulsation Models 53 T ' • 1 ' 1 ' ' ' ' 1 ' ' ' r Log(T e f f ) F i g u r e 3 . 1 : A theore t i ca l H R d i a g r a m showing the ext remes o f the pa rame te r space. T h e inner red error b o x a r o u n d <g) represents the 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 * repre-sent the luminos i t i e s ca lcu la t ed f r o m a Au of 68 / / H z a n d Au/2 34 / / H z . A l s o s h o w n are two l ines of constant Au. T h e m e t a l l i c i t y has the greatest effect of the e v o l u t i o n a r y s ta tus of the s te l la r m o d e l : sh i f t ing the t racks t o w a r d the lower r igh t for lower Z values. T h e mass of the convect ive envelope is ins ignif icant i n these mode l s . I n fact, i t is o n l y a few of the mode l s s h o w n i n F i g u r e 3.2 tha t envelope convec t ion is a large f r ac t ion of the envelope mass ( « 1% of the m o d e l mass) . These mode l s m a y be m o r e efficient at m i x i n g away patches of c h e m i c a l inhomogenei t ies ; r u l i n g ou t l o w mass a n d h i g h Z mode l s as cand ida tes A p stars. F o r e v o l u t i o n a r y mode l s tha t fa l l w i t h i n the H L E B , the spread i n Au is s m a l l . T h i s Chapter 3. Stellar Evolution and Pulsation Models 54 33 \ S o 33 o o o > a o C J 0.2 h -=1.311.-U-l.SM... U-l .BM.-Z-0.008 Z-0.022 2 3 age [Gyr.] 20.008 \ s <o "^0.006 V a. > c "0.004 a> _> *J o 0) C 00.002 u 1 1 • ' ' 1 i i i i I i ' • 1 ' ' T _ r _ H-1.3M. _ U-l.SU. Z = 0.022 X = 0.700 - X = 0.740 -1 'X '' ; L / ! i 1 i i i i 1 • • i! i i 2 3 age [Gyr.] Figure 3.2: T h e convect ive core mass ( top) a n d the convect ive envelope mass (below) as a func t ion of age for the mode 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 to the mass of the s tar . T h e envelope represents a p p r o x i m a t e l y the outer 5% of the m o d e l r ad ius . 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 error bars i n F i g u r e 3.3 represent the m a x i m u m a n d m i n i m u m values of Au a n d X / Z for a g iven mass. T h e o n l y mode l s t ha t do no t r ep roduce Chapter 3. Stellar Evolution and Pulsation Models 55 the observed A z / of 68 /xHz are the l ow Z , 1 . 3 M 0 mode l s . O n c e aga in , the mode l s tha t seem to best reproduce the observed large spac ing are those w i t h a mass of 1.5 to 1 .6M© a n d a c o m p o s i t i o n of ( X , Z) « (0.700, 0.017). It s h o u l d be no t ed t ha t the large spacings presented i n th i s p lo t are ca l cu la t ed d i r ec t l y f r o m the in t eg ra l de f in i t i on i n E q u a t i o n 1.7, a n d no t f r o m the spac ing of the ca lcu la t ed o s c i l l a t i o n frequencies. Spac ings ca l cu l a t ed u s ing these two m e t h o d s differ f rom each other b y a few /xHz; due to the i n t eg ra l averag ing of the s o u n d speed 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 the r igh t i n F i g u r e 3.3 is the spread i n age for each of the mode l s t ha t fa l l i n the H L E B . A s is expec ted , the mode ls w i t h the lowest mass a n d the highest X / Z r a t i o are the oldest . 3.2 Pulsation Models T h e p u l s a t i o n ca lcu la t ions are ca r r i ed out us ing the n o n a d i a b a t i c p u l s a t i o n package of D a v i d G u e n t h e r ; J I G 8 (Guen the r , 1994). J I G 8 solves the s ix l i nea r i zed , n o n a d i a b a t i c equat ions of n o n r a d i a l s te l lar osc i l l a t ions u s ing the H e n y e y r e l a x a t i o n m e t h o d (Henyey et a l . , 1964). These s ix c o m p l e x p a r t i a l different ial equat ions descr ibe the r a d i a l depen-dence of the v e r t i c a l a n d h o r i z o n t a l d isp lacement 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 to the en t ropy a n d the r ad i a t ive luminos i ty , as w e l l as the E u l e r i a n p e r t u r b a t i o n s to the 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 i ts r a d i a l der iva t ive . T h e t i m e dependence of the p e r t u r b e d quant i t i es is p e r i o d i c t h r o u g h the func t ion exp(iut); where the u is the c o m p l e x eigen-frequency. A comple te i n t r o d u c t i o n to b o t h the ad i aba t i c 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 equat ions c a n be found 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 equat ions are solved us ing the E d d i n g t o n a p p r o x i m a t i o n (Saio &; C o x , 1980). R a d i a t i v e losses a n d gains are t a k e n in to account t h r o u g h the r a d i a t i v e f lux i n the en t ropy equa t i on where T , s, t, a n d p are the t empera tu re , entropy, t ime , a n d the densi ty, respect ively . T h e vec tor F is the s u m of the r ad ia t ive a n d convect ive fluxes. If the 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: The variation of Az/ in / / H z (top) and the model age i n G y r (bot-tom) as a function of the ratio X / Z for models that fall wi th in the Hipparcos errorbars. The errorbars on the plot represent the maximum and minimum values for each mass. Chapter 3. Stellar Evolution and Pulsation Models 57 the divergence of the convect ive f lux are ignored , F becomes; i n the E d d i n g t o n a p p r o x i -m a t i o n , P r - _ < L V f e « + _ ! . T £ l (3.5) 3 « p [47r A-KK at] where c is the speed of l igh t , a is the r a d i a t i o n dens i ty constant , a n d K is the opac i ty . T h e process b y w h i c h J I G 8 ob ta ins the eigenfunct ions a n d frequencies involves five steps o u t l i n e d b y G u e n t h e r (1994). A f t e r a n i n i t i a l guess of the o s c i l l a t i o n frequency a n d a degree £ of the mode , the inner t u r n i n g po in t rt is a p p r o x i m a t e d f r o m the r u n of s o u n d speed cs i n the m o d e l u s ing the r e l a t i on cs(rt)/rt — u/y/£(£ + 1). T h e inne rmos t she l l is set at r ad ius where the wave number is 1 0 - 8 t imes smal le r t h e n the va lue of inner t u r n i n g p o i n t . T h e ad iaba t i c o sc i l l a t i on equat ions are so lved w i t h the outer 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 removed; 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 the ad i aba t i c e igenfunct ions (see U n n o et a l . , 1989). O n c e th i s i n i t i a l guess is ob t a ined , the comple te l i nea r i zed a d i a b a t i c equat ions are so lved t h r o u g h the r e l a x a t i o n m e t h o d . N e x t , the n o n a d i a b a t i c o s c i l l a t i o n equat ions are so lved w i t h the outer 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 emoved us ing the rea l eigenfrequency o b t a i n e d f r o m the a d i a b a t i c ca l cu la t ions . A n i n i t i a l guess for the i m a g i n a r y par t of the n o n a d i a b a t i c f requency is t h e n ca l cu l a t ed f r o m the w o r k a n d k ine t i c energy in tegra ls b y use of the r e l a t i o n W(Re)/wk = —4nr). Here , 77 is k n o w n as the g r o w t h rate of the m o d e a n d i t is the r a t i o be tween the i m a g i n a r y a n d rea l pa r t s of the osc i l l a t i on frequency. T h e w o r k a n d k i n e t i c energy of the m o d e are g iven , respec t ive ly b y W(r) = -47r 2 r 2 Im ( £ p * £ r ) (3.6) a n d 1 fM wk = 2Uld Jo I 5 v I2 d m (3-7) where 8p* is the L a g r a n g i a n pressure va r i a t i on , a n d £ r is the v e r t i c a l pa r t of the r a d i a l d i sp lacement , Sr. F i n a l l y , the eigenfunct ions a n d eigenfrequencies o b t a i n e d i n the l a t t e r s tep are used to ca lcu la te the s o l u t i o n to the fu l l l i nea r i zed 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 equat ions u s ing the H e n y e y m e t h o d . Chapter 3. Stellar Evolution and Pulsation Models 58 3.2.1 Model Frequencies Since p h o t o m e t r i c s te l lar p u l s a t i o n observat ions do not give disc-resolved i n f o r m a t i o n , the mos t i m p o r t a n t d iagnos t i c of s te l lar s t ruc ture comes f r o m the f requency s p a c i n g desc r ibed i n Sec t ions 1.4.2 a n d 1.5. I n pa r t i cu l a r , the large spac ing Au gives i n f o r m a t i o n a b o u t the m e a n dens i ty of the s tar a n d the second order spac ing 8^ p rov ides i n f o r m a t i o n a b o u t the m e a n m o l e c u l a r weight near the s tel lar core; i.e., age i n f o r m a t i o n . I n t h i s sec t ion , we present the p u l s a t i o n resul ts for a l l s te l lar mode l s t ha t l ie w i t h i n the H L E B . T h e m a t c h i n g of m o d e l frequencies to observed frequencies is a subjec t ive process since there are u s u a l l y more modes exc i t ed i n mode l s t h a n are observed. T h e m o d e l t ha t reproduces the observed o s c i l l a t i o n s p e c t r u m of a s tar is not un ique . Recen t ly , however, G u e n t h e r & B r o w n (2004) have developed a m e t h o d to improve the m a t c h i n g of mode l s to observat ions . T h e i r m e t h o d involves m i n i m i z i n g a x2 s t a t i s t i c be tween a large g r i d of m o d e l frequencies a n d a n observed s p e c t r u m . O n c e the i r m o d e l g r i d is e x p a n d e d to a more diverse pa ramete r space, t h i s m e t h o d promises to be a m u c h more quan t i t a t i ve m e t h o d of m o d e iden t i f i ca t ion . T h e c o m p l e x i t y r e su l t i ng i n the large n u m b e r of o s c i l l a t i o n modes exc i t ed i n a set of s te l lar mode l s is i l l u s t r a t ed i n F i g u r e 3.4. T h e p lo t of 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 shows r idges of power 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 , l ines of c o m m o n £ r u n v e r t i c a l l y o n the d i a g r a m , w h i l e the h igher order p - m o d e frequencies (larger n values) are sor ted ho r i zon ta l ly . I n F i g u r e 3.4, the n o n a d i a b a t i c frequencies a n d large spac ing are p l o t t e d for a l l of the 1 .5M© mode l s w i t h h y d r o g e n mass f rac t ion of 0.700 a n d m e t a l mass f rac t ions of 0.012 a n d 0.014. T h e ca l cu l a t ed frequencies a l te rnate be tween even a n d o d d £ values as the order of the modes is increased. T h e v e r t i c a l do t t ed l ines are frequencies ident i f ied i n th i s s t u d y for H R 1217 (see Sec t i on 2.5). Se lec t ing the p rope r s te l lar m o d e l , a n d even the correct degree o f the mode , is ve ry diff icul t . I n fact, even t h o u g h there are more modes exc i t ed for the Z = 0.014 models , there are a s ignif icant n u m b e r o f modes ca l cu l a t ed f r o m the Z = 0.012 mode l s near Au « 68 / i H z . Chapter 3. Stellar Evolution and Pulsation Models 59 90 1 = 0 M = 1.5 M 0 , X = 0.700 1 = 1 Z = 0.013 o 1 = 2 Z = 0.014 60 2400 2600 2800 3000 i/[AtHz] Figure 3.4: A n echelle d i a g r a m for the 1.5M© mode 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 ack l ines are the frequencies ident i f ied for H R 1217 i n Sec t i on 2.5. N o t e the modes a l te rna te be tween b o t h even a n d o d d degrees. T h e dependence of the frequency spac ing 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 th i s figure show 1 . 6 M 0 mode l s w i t h different c o m p o s i t i o n s 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 of 1.05 G y r comes closest Chapter 3. Stellar Evolution and Pulsation Models 60 1 1 1 1 1 1 1 • • 1 . i i | , • • • J5.BO Gyr • • "» "* • • 0.90 7. = 0.014 •a •* * * * Z = 0.016 .• • " " 0.80 Z = 0.020 -A X =700 X =720 - 8 ° " ° ° * . » X =740 . _ # = Model Age ° ° ° ° ° ° »! 0 5 M = 160 M e - a = l.B -1.10 -- o 0 ° ° ° ° o „ -° a - 0 ° 0 ° ° ° 0 { 1 ° ° „ 1.20 ° ° = > o o o o 8 1 , . . 1 . 2600 2B00 3000 v(MHz) 2600 2800 "(MHZ) Figure 3 .5 : O n the left, a n echelle d i a g r a m for 1 . 6 M Q mode l s . These mode l s were selected based o n the d ive r s i ty of the c o m p o s i t i o n a n d age of the mode l s for th is mass. T h e p lo t be low shows the second or-der spac ing 5(2) as a func t ion o f frequency. T h e n u m b e r labe ls o n the p lo t i nd ica t e the mode ls age; where the o p e n s y m b o l s are the younger mode l s . T h e legend for the p l o t o n the left app l ies t o b o t h p lo t s . t o the 68 / i H z ; however, the younger mode ls w i t h a solar Z of 0.020 a n d a lower X = 0.700 are a lso close. T h e o lder mode l s show a decrease i n the large spac ing ; i n d i c a t i n g a n inc reas ing s te l lar r ad ius (decrease i n m e a n densi ty) as the s tar evolves t h r o u g h the error box . T h e p lo t o n the r igh t i n F i g u r e 3.5 shows the second order spac ing as a f u n c t i o n o f frequency. It s h o u l d be 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 the p u l s a t i o n mode l s ca l cu l a t ed for th i s thesis . T h i s d iagnos t ic d i a g r a m shows the same genera l features Chapter 3. Stellar Evolution and Pulsation Models 61 as those i n a n echelle d i a g r a m ; however, the mode l s w i t h a s m a l l spac ing of « 3 / / H z m a y be i m p o r t a n t for the i n t e rp re t a t i on of the frequency analys is of Sec t ions 2.5 a n d 2.6. Table 3.1-. T e n mode l s selected f r o m the 569 t h a t f a l l w i t h i n the Hipparcos l u m i n o s i t y error bars . These mode l s s a m -ple a range of age, c o m p o s i t i o n , mass a n d m i x i n g l eng th . T h e fu l l m o d e l l i s t i n g m a y be found i n A p p e n d i x A . del # MQ L o g ( 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 mode l s w i t h a range i n mass, 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 pa rame te r f r o m the 569 mode l s tha t fa l l w i t h i n the 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 ag rams are s h o w n for each of these mode l s i n F igu re s 3.6 a n d 3.7, respect ively . T h e echelle d i ag rams of F i g u r e 3.6 show the same genera l t rends as discussed ear l ier . T h e in te res t ing t h i n g to note is tha t mode l s 6 - 1 0 s lowly increase i n Au. T h i s is caused b y b o t h the younger age of the more massive models , as w e l l as, t he i r lower m e a n densi t ies . T h e mode l s w i t h the second order spac ing of 3 / / H z are also e i ther o lder , or have a mass less t h a n s~ 1 . 5 M 0 . Chapter 3. Stellar Evolution and Pulsation Models 62 2600 2800 i/OiHz) 2600 2800 K(MHZ) Figure 3.6: Echelle diagrams for the models listed i n Table 3.1. The model number is listed at the top of the diagram. 3.3 Magnetic Effects The effects of the magnetic field on both the oscillation frequencies and the eigenfunc-tions have been described by a number of authors (e.g., Shibahashi & Takata, 1993, Dziembowski & Goode, 1996, Bigot et al., 2000, and Cunha h Gough, 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. The usual perturbation techniques for calculating the magnetic effects on stellar oscillations are no longer valid. However, there has been recent suc-cess in the interpretation of r o A p oscillations in terms of a variational principle (Cunha, 2001). B y only calculating the perturbed eigenmodes and not the eigenfunctions, the method of Cunha k. Gough (2000) is an attractive alternative to a perturbation analysis of magneto-acoustic modes. The work of Cunha & Gough (2000) follows from that of Campbel l & Papaloizou Chapter 3. Stellar Evolution and Pulsation Models 63 2600 2800 2600 2800 I/(MHZ) Figure 3.7: Second order spac ing d i ag rams for the mode l s l i s t ed i n T a b l e 3.1. T h e m o d e l number is l i s t ed at the t op of the d i a g r a m . (1986). These au thors d iv ide the star i n to a t h i n outer b o u n d a r y layer a n d the in te r io r . I n the b o u n d a r y layer, the L o r e n t z forces are comparab le to; or larger t h a n , the gas pressure, w h i l e i n the in te r ior the field is essent ial ly force free. S ince the magne t i c b o u n d a r y layer is t h i n , a p l ane -pa ra l l e l a p p r o x i m a t i o n m a y be used. It is also assumed tha t the effects of the magne t i c f ield c a n be c a l c u l a t e d l o c a l l y at each l a t i t ude . I n th i s case, the field o n l y varies i n the v e r t i c a l d i r e c t i o n a n d has c o m -ponents B = (Bx,0,Bz). T h e magneto-acous t ic waves are desc r ibed u s i n g a h o r i z o n t a l wavenumber k = (kx,ky,0). W i t h th i s i n f o r m a t i o n , the ad i aba t i c , m a g n e t i c a l l y n o n -diffusive, p u l s a t i o n equat ions m a y t h e n be 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) -u2pu = i | k | W + (B • V ) 2 — - A ? _ L (B • V) ( V - 0 (3.8) Mo Mo k | where a n d Chapter 3. Stellar Evolution and Pulsation Models 64 - W > = (B • V) 2 f + (B • V) (V-0 (3-9) Mo Mo I K- I - w 2 t f , = ^ - <?V- ( t f ) - — [(B • V) (V-0] + (B • V ) 2 ^ (3.10) MO PO W = i.Vp+U+*) ( V . O - ( B - V ) < B ' ^ (3.11) \ M u / Mo 9 = i ^ (3.12) I n the above re la t ions , the d isp lacement vector £ is decomposed in to a v e r t i c a l c o m p o n e n t £z = £ • ez, a componen t tha t is pe rpend icu la r to the wavenumber v = £-(ez x k) / | k |, a n d a c o m p o n e n t t ha t is p a r a l l e l to the wavenumber u — (£ • k) / | k |. T h e l o c a l grav-i t a t i o n a l acce le ra t ion is denoted b y g a n d the first ad i aba t i c exponent is 7. A l l o ther s y m b o l s have the i r u s u a l meanings . I n the deep in te r ior , a magneto-acous t ic m o d e c o m p l e t e l y decouples i n t o a pure A l f v e n i c m o d e a n d a pure acous t ic mode . T h i s was verif ied 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 the J W K B a p p r o x i m a t i o n , the magne t i c modes i n the in te r ior m a y be descr ibed b y the func t ions (cf. C u n h a & G o u g h 2000) K», umz) ^ P 1 / 4 (C, D)exp p-l'\C+,D+)exp np0pu2\2 .kxBxz sr) iz~i 2 dz — i-. fz (popu>2\ .kxBxz + (3.13) where C , D , C + , a n d D+ 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 expec ted t o diss ipate before t hey are reflected back t o w a r d the surface o f the s tar ( R o b e r t s & S o w a r d , 1983). W i t h th i s i n m i n d , the constants C+ a n d D+ i n E q u a t i o n 3.13 m a y t h e n be set to zero t o assure tha t no 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 waves occur i n the in te r io r . In the in te r io r , the v e r t i c a l componen t of the u n c o u p l e d modes is essent ia l ly acous t ic . T h e a m p l i t u d e of th i s v e r t i c a l m o d e m a y be represented 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) 6* ^ " 7^/2 c o s (^Jz x d z + SP ) ( 3 - 1 4 ) Ax1'2 P1 were 5P is a phase a n d K is the v e r t i c a l acoust ic wavenumber . T h e coo rd ina t e z a n d z* represent the d e p t h i n the b o u n d a r y layer a n d the p o s i t i o n of the base of the b o u n d a r y layer, respect ively . T h e n u m e r i c a l so lu t ions of the sys tem of E q u a t i o n s 3.8 to 3.10 are m a t c h e d to the a s y m p t o t i c r e l a t ions 3.13 a n d 3.14 a t each l a t i t u d e to o b t a i n va lues for t he m a g n e t i c phases 5P. T h e p u r e l y acous t ic case, i.e, B = 0 i n E q u a t i o n s 3.8 to 3.10, is also m a t c h e d on to the a s y m p t o t i c re la t ions to o b t a i n the u n p e r t u r b e d phases 5pQ. Phase shifts A5P are t h e n o b t a i n e d f r o m the difference between these 2 phases at each l a t i t ude . W i t h the above i n f o r m a t i o n , the v a r i a t i o n a l m e t h o d of C u n h a k G o u g h (2000) c a n be used to es t imate the first-order frequency shifts of the e igenmodes caused b y a magne t i c f ie ld . T h e frequency shifts m a y be ca lcu la t ed f r o m where the average of the phase shifts A5P over a sphere is f* A6p{Yem)2sm6d9 Au A5P p (3.15) ( 3 - 1 6 ) T h e quant i t i es w i t h a subscr ip t of zero are those expec ted i n the case o f no magne t i c field. A comple te de sc r i p t i on of the n u m e r i c a l p rocedure used to ca lcu la te the frequency shifts is o u t l i n e d i n the appendices of 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 to the acous t ic osc i l l a t ions are ca l cu l a t ed u s ing a n a d a p t e d ve r s ion of the code used b y C u n h a k G o u g h (2000). T h i s new vers ion of the code was u p d a t e d b y C u n h a (pr iva te c o m m u n i c a t i o n , 2003) to read the o u t p u t f r o m the e v o l u t i o n a r y mode l s presented i n S e c t i o n 3.1. F requency shifts were 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 ing E q u a t i o n 3.15. T h e magne t i c pe r t u rba t i ons t o frequencies f r o m 900 to 3100 Chapter 3. Stellar Evolution and Pulsation Models 66 40 20 h 4 < V -20 05 -40 \ -I ' ' ' ' I 1 1 1 1 I ' 1 model: 1 2 3 4 5 1=1 Models 1 - 5 40 I . . . . I 1000 2000 3000 f(yuHz) 1000 2000 3000 20 _ 15 N X ^ 10 < ^ 5 o h 1 1 1 1 -_ ^ \ \ \ / - V \ N \ / / " 1 , , , , 1 , , , 1000 2000 3000 y(/iHz) 1000 2000 3000 i/(/itHz) Figure 3.8: T h e magne t i c pe r tu rba t ions ca l cu la t ed for mode l s 1 to 5 u s i n g a v a r i a t i o n a l p r inc ip l e . T h e uppe r p lo t s show the rea l f requency shift for modes w i t h degrees £ = 1 (left) a n d £ = 2 ( r igh t ) . T h e lower p lo t s show the i m a g i n a r y pa r t of the frequency shifts for the same degrees. uRz are s h o w n i n F igu re s 3.8 a n d 3.9 a s suming a d i p o l a r magne t i c f ie ld w i t h a p o l a r s t r eng th o f 4.0 k G . T h i s frequency range a n d magne t i c f ie ld s t r eng th is consis tent w i t h measurements of H R 1217 (see Sec t i on 1.5). Chapter 3. Stellar Evolution and Pulsation Models 67 Models 6 - 1 0 1000 2000 3000 1000 2000 Figure 3 .9: The magnetic perturbations calculated for models 6 to 10 using a variational principle. The upper plots show the real frequency shift for modes wi th degrees £ — 1 (left) and £ = 2 (right). The lower plots show the imaginary part of the frequency shifts for the same degrees. Each of Figures 3.8 and 3.9 show the real and the imaginary frequency shifts for modes wi th a degree of £ — 1 and 2. Note that there is very litt le difference in the real frequency shifts for modes of different degree. The imaginary part of the £ = 2 modes Chapter 3. Stellar Evolution and Pulsation Models 68 are; however, c l ea r ly lower. T h i s was also s h o w n for the case o f a p o l y t r o p e s te l la 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 p a r t o f the frequency i s pos i t ive , t he magne to -acous t i c m o d e loses energy. T h e smal le r a m p l i t u d e of the i m a g i n a r y pa r t for the I = 2 modes suggest t ha t t hey are m u c h less suscept ible to d a m p i n g f r o m the magne t i c field. T h i s effect depends o n the geomet ry of the m o d e a n d the magne t i c f ie ld . C u n h a Sz G o u g h (2000) show t h a t t he m a x i m u m energy loss o f a magne to -acous t i c m o d e occu r s app rox-i m a t e l y at the l a t i t u d e where BZBX is a m a x i m u m . T h i s occurs at 4 5 ° for a d i p o l a r magne t i c field. M o d e s of h igher 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 the equator more q u i c k l y t h a n modes of lower degree. W h e n the phase shifts are averaged over a sphere, t he net effect is a smal le r i m a g i n a r y frequency shif t for m o d e s of larger t. S i m p l y pu t , these modes have lower amp l i t udes near the l a t i t ude where the m a x i m u m of energy loss occurs . It is be l i eved tha t the r e su l t ing frequency shifts are caused b y energy loss t h r o u g h the c o u p l i n g of i n w a r d l y p r o p a g a t i n g magne t i c s low waves a n d the acous t ic m o d e s ( C u n h a & G o u g h , 2000) . T h e frequency j u m p s change ve ry l i t t l e be tween the mode l s . T h i s makes i t dif f icul t t o use the magne t i c pe r t u rba t i ons as a d i s c r iminan t be tween different e v o l u t i o n a r y mode l s . It is i m p o r t a n t t o no t ice the frequency shifts be tween modes of different degrees are o f t he same order as t he second-order spac ing desc r ibed 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 observed 8^ as a d iagnos t ic of s te l lar s t ruc ture ( D z i e m b o w s k i & G o o d e , 1996). 69 Chapter 4 Discuss ion and Conc lus ions E a r l y p h o t o m e t r i c observat ions o n the r o A p s tar H R 1217 (e.g., K u r t z et a l . , 1989) showed a p u l s a t i o n p a t t e r n tha t was charac ter i s t ic of h igh-order p -modes observed i n the S u n (see F i g u r e 1.4). T h e m a i n differences between the solar o s c i l l a t i o n s p e c t r u m a n d tha t of H R 1217 were the observed a m p l i t u d e s a n d frequency spacings. I n general , the a m p l i t u d e s of the r o A p s tars are greater t h a n the solar o s c i l l a t i o n amp l i t udes . F r equency spacings suggest a A i / o f a p p r o x i m a t e l y 135 / / H z for the s u n (e.g., A i n d o w et a l . , 1988), w h i l e the infer red large spac ing for H R 1217 is ha l f th i s va lue at 68 / / H z , consis tent w i t h a lower m e a n dens i ty of the A p star . T h e d a t a r e d u c t i o n presented i n C h a p t e r 2 for the X C O V 2 0 c a m p a i g n o n H R 1217 y ie lds resul ts tha 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 the 1986 c a m p a i g n . T h e two no tab le except ions are the a m p l i t u d e s o f the o s c i l l a t i o n modes (see F i g u r e 2.7) a n d the new frequencies at 2788.94 a n d 2791.57 / / H z (see T a b l e 2.1). T h e la t te r of these new frequencies was also ident i f ied 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 he X C O V 2 0 d a t a b y K u r t z et a l . (2002). T h e a m p l i t u d e s of the t h i r d a n d fou r th frequencies ident i f ied i n the 1986 d a t a differ f r o m those ident i f ied i n th i s s t u d y (see Sec t i on 2.5) b y & 6 0 % a n d 4 0 % respect ively . A l t h o u g h the a m p l i t u d e s be tween the X C O V 2 0 d a t a a n d the 1986 d a t a are c l ea r ly dif-ferent, the net a m p l i t u d e difference (defined b y V J ^ o o o — ^1986]) is o n l y —0.07 ± 0.192 m m a g . T h i s suggests there is a n exchange of power ( a m p l i t u d e squared) be tween the observed modes w h i l e the net power is conserved. T h e exact m e c h a n i s m govern ing the exchange of power be tween i n d i v i d u a l modes is not k n o w n . It is also in te res t ing t ha t the net f requency difference between the c o m m o n frequencies ident i f ied i n b o t h d a t a sets is o n l y [^ 2000 — 1^986] = 0.56 ± 0.96 / / H z . T h i s corresponds to a n average frequency Chapter 4- Discussion and Conclusions 70 difference of 0.09 ± 0 . 9 6 / / H z . T h e persistence of these observed frequencies suggests t ha t the m o d e l i fe t imes i n H R 1217 are s table over the 14 year t i m e gap be tween observa t ions . M o d e se lec t ion a n d l i fe t ime are p a r t i c u l a r l y in te res t ing i n r o A p stars. F o r example , the r o A p star H R 3831 exh ib i t s o s c i l l a t i o n modes tha t are s table over 20 years of obser-va t ions ( K u r t z et a l . , 1997) w h i l e the r o A p star H D 60435 shows some m o d e l i fe t imes of less t h a n 7 days ( M a t t h e w s et a l . , 1987). A l o n g basel ine for observa t ions o n other r o A p s tars w o u l d be inva luab le to our unde r s t and ing of 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 . , 2001). T h e frequency ana lys i s presented i n Sect ions 2.5 a n d 2.6 suggest t ha t the a m p l i t u d e l i m i t at w h i c h frequencies c a n conf ident ly be ident i f ied f r o m the current t ime-series re-d u c t i o n has been reached. O n c e each of the 21 frequencies h a d been r emoved b y the unwe igh ted d a t a analys is , the res iduals 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 improvemen t be tween the unweighted frequency ana lys i s a n d each of the we igh ted frequency reduc t ions presented i n Sec t ion 2.6 is also less t h a n 5% (see F i g u r e 2.10). W h i l e the we igh ted noise levels i n t he Fou r i e r spec t r a of F i g u r e s 2.11 a n d 2.12 are c l ea r ly lower t h a n the noise of F i g u r e 2.6, the general resul ts are the same. It is no t c lear h o w the we igh t ing of the frequencies affects the d e t e r m i n a t i o n of the confidence levels discussed i n S e c t i o n 2.5.1. Regardless , a lmos t a l l of the frequencies c a n be desc r ibed b y a t r i p l e t fine s t ruc tu re . T h e excep t ion is the frequency u7. E v e n w h e n v7 is forced to be fitted as a r o t a t i o n a l l y sp l i t t r ip l e t , the a m p l i t u d e of one of the componen t s is a lways less t h a n 0.035 m m a g (see Tab le s O l a n d C . 2 ) . T h i s is a lways be low the noise; even w h e n i t is we igh ted . A doub le t s t ruc ture is not p red ic t ed b y the ob l ique pu l sa to r m o d e l . It s h o u l d be no t ed t ha t least-squares fits ca l cu la t ed u s ing Period98 a lways converged close to the same ampl i t udes , frequencies a n d phases. T h i s is i l l u s t r a t e d i n Tab le s C . 5 t h r o u g h C . 1 0 of A p p e n d i x C . T h e frequencies, ampl i tudes , a n d phases were averaged a n d a s t a n d a r d error of the m e a n was ca l cu la t ed for frequency so lu t ions f r o m T a b l e 2.3 w i t h the same ident i f ied frequencies. T h e s t a n d a r d error o n the frequencies, a m p l i t u d e s a n d phases are o f the order 1 0 - 6 m H z , 1 0 - 2 m m a g , a n d 1 0 - 2 r ad ians . T h i s is a tes tament to the robustness of Period98. Chapter 4- Discussion and Conclusions 71 T h e absence of the frequency u(old) (see T a b l e 2.1) i n the 1986 d a t a is e x p l a i n e d b y C u n h a (2001) as s t rong magne t i c d a m p i n g of the o s c i l l a t i o n mode . T h e m o d e l s presented 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 agreement w i t h her resul ts for a p o l y t r o p e s te l lar m o d e l . These mode l s show a n increase of d a m p i n g t o w a r d the seventh frequency 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 the observed spac ing be tween frequencies 6 a n d 7 of ~ 15 / / H z ; -20 / / H z away f r o m the expec ted value . A s d iscussed i n S e c t i o n 3.3.1, the m a g n e t i c d a m p i n g of m o d e s decreases w i t h i n -creas ing degree (larger £ values) . T h i s is a consequence of the m o d e a n d magne t i c f ie ld geomet ry as w e l l as the we igh t ing of the modes b y the v a r i a t i o n a l m e t h o d . T h e modes w i t h the larger £ values have smal le r amp l i t udes near l a t i tudes of m a x i m u m magne t i c d a m p i n g . T h e net d a m p i n g of t he h igher degree m o d e s is less w h e n the average o f t he magne t i c effects is ca l cu la t ed for a l l l a t i tudes . D a m p i n g increases w i t h inc reas ing i m a g i -n a r y frequency 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 are less for the £ = 2 modes t h a n the co r re spond ing £ — 1 modes . T h e s p a c i n g be tween b o t h o f the new modes (u(old) a n d u(new)) i s a p p r o x i m a t e l y 2.63 / / H z . T h i s is consistent w i t h the second-order frequency spacings c a l c u l a t e d for the s te l lar mode l s i n S e c t i o n 3.2.1 (see F igu re s 3.5 a n d 3.7). If t h i s is a s m a l l spac ing , E q u a t i o n 1.10 shows; b y def in i t ion , tha t the modes differ i n degree b y 2. C o n s i d e r a s ingle 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 is s t a b i l i z e d b e l o w observable levels t h r o u g h magne t i c d a m p i n g . If th i s m o d e exchanges power be tween i t se l f a n d a m o d e of larger degree, the a m p l i t u d e of the new m o d e m a y g row to observable levels because i t is less d a m p e d b y magneto-acous t ic in te rac t ions . T h i s conjecture is s u p p o r t e d b y the obse rva t i on o f two c lose ly spaced modes i n the X C O V 2 0 d a t a w h i l e there is no evidence for s i m i l a r modes i n the 1986 d a t a set. However , th i s is diff icul t to p red ic t e m p i r i c a l l y since the mechan i sms 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 are s t i l l u n c e r t a i n . W h i l e the ob l ique pu l sa to r mode l s discussed i n Sec t i on 1.4.3 are no t used to p red ic t the a m p l i t u d e a symmet r i e s observed i n t h i s w o r k , a p r e d i c t i o n o f the m o d e l s is tes ted u s i n g the fine frequency spac ing of the X C O V 2 0 da ta . T h e ob l ique pu l sa to r m o d e l , i n a l l i t s forms, p red ic t s tha t the frequencies are e x a c t l y sp l i t b y the r o t a t i o n frequency of Chapter 4- Discussion and Conclusions 72 the s tar . If the modes are descr ibed ins tead b y t r ave l ing waves ( m ^ 0) discussed i n Sec t i on 1.4.1, one w o u l d expect depar tures f rom the exact spac ing because o f the L e d o u x cons tan t of E q u a t i o n 1.5. B y averaging the difference be tween the fine spacings 5u of T a b l e 2.1 a n d the r o t a t i o n frequency of 9.2897 x 1 0 - 4 m H z (the 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 (1987); see S e c t i o n 2.6), t he L e d o u x cons tant is e s t ima ted to be 7.2 x 1 0 ~ 5 ± 11%. T h i s is s t i l l 2 orders of m a g n i t u d e lower t h a n the value o f Cnj « 1 0 ~ 3 c a l cu l a t ed b y S h i b a h a s h i k T a k a t a (1993). It is also i n agreement w i t h the value of Cn%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 the fine s t ruc tu re observed i n the 1986 d a t a set. T h u s , i t is u n l i k e l y t ha t the observed r o t a t i o n a l s p l i t t i n g is descr ibed b y E q u a t i o n 1.5. T h e d iagnos t i c power of a second-order spac ing for the r o A p s tars has been ques t ioned b y some au thors (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 . , 2001) . T h e reason is t ha t the magne t i c pe r tu rba t ions be tween modes of differ ing degree is of the same order as the s m a l l spacings . However , i t is s t i l l in te res t ing to d r a w some conc lus ions f r o m the c a l c u l a t e d s m a l l spacings o f S e c t i o n 3.2.1 i n l i gh t o f t he poss ib le observed s m a l l spac ing of 2.63 pRz be tween frequencies v{old) a n d v{new). I n p a r t i c u l a r , the e v o l u t i o n a r y mode l s tha t reproduce the observed second-order spac ing of 2.63 / / H z have a mass of « 1 . 6 M 0 a n d a c o m p o s i t i o n of ( X , Z) ~ (0.720, 0.017) (see F i g u r e s 3.5 a n d 3.7). If the ca l cu la t ed large spac ing is also c o m p a r e d to the observed va lue of 68 / / H z , the e v o l u t i o n a r y m o d e l p roper t ies w o u l d be consistent w i t h those f r o m the s m a l l spac ing est imates; i.e, a sub-solar Z , solar X , a n d a mass of 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). These resul ts are also conf i rmed b y the w o r k of C u n h a et a l . (2003). However , no one 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 the observed 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 lower g l o b a l m e t a l l i c i t y con f i rm the resul ts of L e b r e t o n et a l . (1999). It is u s u a l l y assumed tha t s tars i n the solar ne ighbo rhood , l ike H R 1217, have a solar me ta l l i c i t y . L e b r e t o n et a l . (1999) ca lcula te a m e t a l l i c i t y range of —1 < [Fe/H] < 0.3 for a sample of stars w i t h dis tances less t h a n 30 p c . W h i l e there is no w a y to abso lu te ly de te rmine the in te r io r m e t a l content of any s tar f r o m the observed surface abundances , Chapter 4- Discussion and Conclusions 73 it is espec ia l ly diff icul t for the A p stars because of there pecu l i a r spec t r a l features. T h e o l d a s s u m p t i o n of solar m e t a l l i c i t y mode ls for nearby stars m a y have to be rev ised . T h e mode l s t ha t fa l l w i t h i n the l u m i n o s i t y e r ro rbox defined by the Hipparcos p a r a l l a x w i t h a mass less t h a n 1 . 5 M Q 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 convec t ion . I n the case of a 1 . 3 M 0 m o d e l w i t h c o m p o s i t i o n ( X , Z ) = (0.740, 0.022), the envelope is a lmos t comple t e ly convect ive b y the t i m e i t enters the defined e r ro rbox . These mode l s have left the m a i n sequence a n d are a p p r o a c h i n g the base of the r ed g ian t b r a n c h . W h i l e s m a l l convect ive zones (<C 0 .1% M * ) exist i n o ther mode l s , the convect ive envelope mode l s presented i n F i g u r e 3.2 c a n c lea r ly be r u l e d out as A p m o d e l candida tes . T h e reason b e i n g t ha t any surface inhomogenei t ies w o u l d be m i x e d away b y the on-set of such deep surface convect ive zones. M o d e l s presented i n t h i s s t u d y are a va luab le asset for c o n s t r a i n i n g the phys ics of A p a n d r o A p stars . F u t u r e improvement s w i l l i nc lude be t te r a p p r o x i m a t i o n s t o the m o d e l a tmospheres . Specif ica l ly , the use of a n E d d i n g t o n grey a tmosphere i n the m o d e l ca l cu l a t i ons p red ic t s a n i s o t h e r m a l acous t ic cut-off f requency t h a t i s 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 to the T - r re la t ions for r o A p s tars have been used 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 the first case, a T - r was a r t i f i c i a l l y mod i f i ed to cause a steeper t empera tu re gradient near the s te l lar surface; m i m i c k i n g the effect of a chromosphere . However , A u d a r d et a l . (1998) use m o d e l a tmospheres w i t h a surface c o m p o s i t i o n specific to H R 1217 bu t w i t h a g l o b a l Z = 0.02. B o t h groups were successful i n r a i s i n g the acous t ic cut-off so tha t the ca l cu l a t ed frequencies a n d the observed frequencies were i n agreement. M o d e l a tmospheres have been ca lcu la t ed b y W e r n e r Weiss of the U n i v e r s i t y o f V i e n n a (pr iva te 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 we are c u r r e n t l y a d a p t i n g t h e m so they c a n be used w i t h Y R E C 7 . T h e r e are t w o improvemen t s over the ear l ier w o r k of A u d a r d et a l . (1998). F i r s t , the e v o l u t i o n a r y mode l s w i l l no t be res t r i c ted to a constant surface gravi ty . T h e effect th i s w i l l have is expec ted to be s m a l l ; however, i t is consistent w i t h a n e v o l v i n g sequence of s te l lar mode l s . T h e second, a n d more i m p o r t a n t , is the a d d i t i o n of m u l t i p l e c o m p o s i t i o n s t ha t Chapter 4- Discussion and Conclusions 74 reflect the observed change i n abundance pa t te rns as H R 1217 rotates . T h u s , a n u m b e r of independent e v o l u t i o n a r y t racks w i l l be ca lcu la t ed a l l o w i n g us to m o d e l the frequencies w i t h a n a z i m u t h a l dependence. These new mode l s w i l l t h e n also be subject t o the magne t i c frequency p e r t u r b a t i o n analys is presented i n Sec t i on 3.3. T h e q u a l i t y of p h o t o m e t r i c d a t a of H R 1217 is also expec ted to i m p r o v e i n the near fu ture s ince the M O S T 1 ( 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) space se i smology mic ro-sa te l l i t e (aper ture = 15 cm) was successfully l aunched o n J u n e 30, 2003. T h i s pro jec 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 funct ions are as terose ismology a n d the de tec t ion of reflected l igh t f rom exoplanets . M O S T is a l r eady surpass ing i t s expec ted per formance a n d v i e w i n g targets w i t h a p h o t o m e t r i c p rec i s ion of ~ a few par t s -p e r - m i l l i o n i n the frequency range relevant for r o A p s tars a n d w i t h d u t y cycles of 9 9 % ( M a t t h e w s , p r iva t e c o m m u n i c a t i o n ) . A s a compar i son , the X C O V 2 0 d a t a covered a d u t y cycle of « 33%; m a k i n g i t a h i g h l y successful g r o u n d based s tudy, a n d reached a noise level of « 0.1 m m a g ; a lmos t 2 orders of m a g n i t u d e greater t h a n M O S T . I n la te 2004, M O S T w i l l observe H R 1217 for jus t over 3 weeks. T h i s s h o u l d be sufficient t o cover 3 r o t a t i o n pe r iods of the s ta r p r o v i d i n g a n adequate t ime-base for h i g h frequency re so lu t ion . S ince H R 1217 w i l l be observed o n l y 4 years after the X C O V 2 0 observat ions , there c o u l d be in teres t ing resul ts o n the m o d e l i fe t imes of the s tar . T h e new mode l s presented i n th i s thesis c o m b i n e d w i t h i m p r o v e d observa t ions f r o m space s h o u l d shed new l ight o n the phys ics govern ing the s t ruc ture a n d e v o l u t i o n of A p a n d r o A p stars. 1Microvariability & Oscillations of STars or Microvariabilite et Oscillations STellaire 75 B i b l i o g r a p h y 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 der R a a y , H . B . 1988, i n Se i smology of the S u n a n d Sun- l ike Stars , E S A S P - 2 8 6 , 157 A l e x a n d e r , D . R . & Ferguson , J . W . 1994, A p J , 437, 879 A l v a r e z , M . , He rnandez , 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 . , Massac r i e r , G . , L i u , Y . Y . , L i , Z . P . , G o u p i l , M . J . , Cor t e s , T . R . , M a n g e n e y , A . , D o l e z , N . , V a l t i e r , J . C , V i d a l , I., Spe 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 rovos t , J . , k Weiss , W . W . 1998, A & A , 335, 954 B o h m - V i t e n s e , E . 1958, Zei tschr i f t fur A s t r o p h y s i c s , 46, 108 B a b e l , 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 of the A s t r o n o m i c a l O b s e r v a t o r y Ska lna te P le so , 27, 431 B a g n u l o , S., L a n d i deg l ' Innocen 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 B a h c a l l , J . N . , P i n s o n n e a u l t , M . H . , k B a s u , S. 2001, 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. 2001, M N R A S , 323, 362 B i g o t , L . k D z i e m b o w s k i , W . 2002, A & A , 391, 235 Chapter 4- Discussion and Conclusions 76 B i g o t , L . , P rovos 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 . , H a n d l e r , G . , G a r r i d o , R . , A u d a r d , N . , Z i m a , W . , P a p a r o , M . , Be ichbuchne r , F . , Z h i - P i n g , L . , S h i - Y a n g , J . , Z o n g - L i , L . , A i - Y i n g , Z . , P i k a l l , H . , S t a n k o v , 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 . , Pa jdosz , G . , Z o l a , S., T h o m a s s e n , T . , S o l -h e i m , J . - E . , Se rkowi t sch , E . , Reegen, P . , R u m p f , T . , Schmalwieser , A . , k M o n t g o m e r y , M . H . 1999, A & A , 349, 225 Breger , 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 . , O s t e r m a n n , W . , P a p a r o , M . , k Scheck, M . 1993, A & A , 271, 482 Bres san , A . , C h i o s i , C . , k Fago t to , F . 1994, A p J S , 94, 63 C a m p b e l l , C . G . k P a p a l o i z o u , J . C . B . 1986, M N R A S , 220, 577 C h r i s t e n s e n - D a l s g a a r d , J . 2002, R e v i e w s of M o d e r n P h y s i c s , 74, 1073 C u n h a , M . S. 2001, M N R A S , 325, 373 C u n h a , M . S., Fernandes , J . M . M . B . , k M o n t e i r o , M . J . P . F . G . 2003, M N R A S , 343, 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 D z i e m b o w s k i , W . A . k G o o d e , P . R . 1996, A p J , 458, 338 F randsen , S., P i g u l s k i , A . , N u s p l , J . , Breger , M . , B e l m o n t e , J . A . , D a l l , T . H . , A r -entoft, T . , S te rken , C . , M e d u p e , T . , G u p t a , S. K . , P i n h e i r o , F . J . G . , M o n t e i r o , M . J . P . F . G . , B a r b a n , C , C h e v r e t o n , M . , M i c h e l , E . , B e n k o , J . M . , B a r c z a , S., Szabo , 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 . 2001 , A & A , 376, 175 G a b r i e l , K . , Noe l s , A . , Scuflaire , R . , k M a t h y s , G . 1985, A & A , 143, 206 Chapter 4- Discussion and Conclusions 77 G a u t s c h y , A . , Sa io , H . , k Harzenmoser , H . 1998, M N R A S , 301, 31 G i r i s h , V . , Seetha, S., M a r t i n e z , R , Josh 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 Sagar , R . 2001, A & A , 380, 142 Grevesse, N . , Noe l s , A . , k Sauva 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 G u e n t h e r , 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, 937 G u e n t h e r , D . B . , D e m a r q u e , P . , K i m , Y . - C , k P i n s o n n e a u l t , M . H . 1992, A p J , 387, 372 H a n d l e r , G . 2003, 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 techniques. (Ch icago , U n i v e r s i t y P re s s [1962]) Iglesias, C . A . k Rogers , 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 emens , 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 . , Me 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., Pa jdosz , G . , K r z e s i n s k i , J . , S o l h e i m , J . - E . , B r u v o l d , A . , O ' D o n o g h u e , D . , K a t z , M . , V a u c l a i r , 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 . , P r o v e n c a l , J . L . , k Hansen , C . J . 1995, A p J , 450, 350 K i p p e n h a h n , R . k Weige r t , A . 1994, S te l la r S t ruc tu re a n d E v o l u t i o n (S te l la 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 A s t r o n o m y a n d A s t r o p h y s i c s L i b r a r y ) Chapter 4- Discussion and Conclusions 78 K l e i n m a n , S. J . , N a t h e r , R . E . , k P h i l l i p s , T . 1996, P A S P , 108, 356 K u r t z , D . k Wegner , G . 1979, A p J , 232, 510 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 Stars , 1436, 1 — . 1982, M N R A S , 200, 807 — . 1990, A R A k A , 28, 607 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 ve 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 . , Me tca l f e , T . S., M u k a d a m , A . , Na the 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 . , Sek iguch i , K . , J i a n g , 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 Perez , J . M . , S o l h e i m , J . E . , Johannessen, 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 . , G i o v a n n i n i , O . , M a t t h e w s , J . M . , C a m e r o n , C , V a u c l a i r , G . , N i t t a , 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 ve 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 . , Me tca l f e , T . S., M u k a d a m , A . S., Na the 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 . , Sek iguch i , K . , J i a n g , X . , Shobbrook , R . , A s h o k a , B . N . , Seetha , S., J o s h i , S., 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 Perez , J . M . , S o l h e i m , J . -E . , Johannessen , 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 . , G i o v a n n i n i , O . , k M a t t h e w s , J . M . 2002, M N R A S , 330, L 5 7 K u r t z , D . W . k 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 . , M a r t i n e z , P . , Seeman, J . , C r o p p e r , M . , C l e m e n s , J . C , K r e i d l , T . J . , S te rken , C , Schneider , H . , Weiss , 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 Kur t z , D . W . k Seeman, J . 1983, M N R A S , 205, 11 Kur 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 Modern Astronomy, 5, 105 Landstreet, J . D . 1993, in A S P Conf. Ser. 44: I A U Col loq . 138: Peculiar versus Normal Phenomena in A-type and Related Stars, 218 Lebreton, Y . , Perrin, M . N . , Cayrel, R. , Bagl in , 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 Col loq . 138: Peculiar versus Normal Phenomena i n A- type and Related Stars, 274 Loumos, G . L . k Deeming, T . J . 1978, Astrophys. and Space Science, 56, 285 Mathys, 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 . , Kur 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, in A S P Conf. Ser. 203: I A U Col loq . 176: The Impact of Large-Scale Surveys on Pulsating Star Research, 74-75 Matthews, J . M . , Wehlau, W . H . , k Kur t z , D . W . 1987, A p J , 313, 782 Michaud, G . 1970, A p J , 160, 641 Michaud, G . 1976, in I A U Col loq. 32: Physics of A p Stars, 81 Michaud, G . 1996, in I A U Symp. 176: Stellar Surface Structure, 321 Michaud, G . , Charland, Y . , k Megessier, C . 1981, A & A , 103, 244 Michaud, G . J . k Proffitt, C . R . 1993, in A S P Conf. Ser. 44: I A U Col loq . 138: Peculiar versus Normal Phenomena in A-type and Related Stars, 439 Montgomery, M . - H . k O'Donoghue, D . 1999, D S S N (Vienna), 13 Moon , 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, in I A U Symp. 105: Observational Tests of the Stellar Evolut ion Theory, 47 Roberts, P. H . k Soward, A . M . 1983, M N R A S , 205, 1171 Chapter 4- Discussion and Conclusions 81 R o d r i g u e z , E . , C o s t a , V . , H a n d l e r , G . , k G a r c i a , J . M . 2003, A & A , 399, 253 Rogers , F . J . 1986, A p J , 310, 723 Rogers , F . J . , Swenson, F . J . , k Iglesias, C . A . 1996, A p J , 456, 902 R y a b c h i k o v a , T . A . , Lands t r ee t , J . D . , G e l b m a n n , M . J . , B o l g o v a , G . T . , T s y m b a l , V . V . , k Weiss , W . W . 1997, A & A , 327, 1137 Sa io , H . k C o x , J . P . 1980, A p J , 236, 549 Scargle , J . D . 1982, A p J , 263, 835 S h i b a h a s h i , H . k Sa io , H . 1985, P A S J , 37, 245 S h i b a h a s h i , H . k T a k a t a , M . 1993, P A S J , 45, 617 S t i bbs , D . W . N . 1950, M N R A S , 110, 395 Tassou l , M . 1980, A p J S , 43 , 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 . , Sa io , H . , & Sh ibahash i , H . 1989, N o n r a d i a l osc i l l a t ions of s tars ( N o n r a d i a l osc i l l a t ions of stars, T o k y o : U n i v e r s i t y of T o k y o Press , 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 . , Johnson , 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 . , Skaret , 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 tu rgeon , 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, in A S P Conf. Ser. 42: G O N G 1992. Seismic Investigation of the Sun and Stars, 331 Wolff, S. C . 1983, The A-stars: Problems and perspectives. Monograph series on. nonthermal phenomena in stellar atmospheres (The A-stars: Problems and perspec-tives. Monograph series on nonthermal phenomena in stellar atmospheres) Appendix A. XCOV20 Observing Log 83 Appendix A X C O V 2 0 Obse rv ing L o g T h i s a p p e n d i x conta ins the obse rv ing log for H R 1217 d u r i n g X C O V 2 0 . See the tab le c a p t i o n for deta i ls . Table A . i : A d a t a l og ob t a ined d u r i n g X C O V 2 0 . T h e log conta ins r u n names, telescopes used, the n u m b e r of p o i n t s col lec ted , a n d comment s f r o m observers. T h e r u n s m a r k e d w i t h a * are runs t ha t were used i n the frequency ana lys i s . See Sec t i on 2.3.1 for se lec t ion c r i t e r i a . R u n N a m e Telescope D a t e ( U T ) Start T i m e ( U T ) # of Points Observer C o m m e n t s m d r l 3 6 * C T I O 1.5m 6-Nov-00 2:23:30 1760 2001 year in original data! m d r l 3 7 * C T I O 1.5m 9-Nov-OO 1:55:20 2524 vignetting present, re-reduced to correct (somewhat) m d r l 3 8 * C T I O 1.5m 10-Nov-OO 1:59:50 1756 sky in channel 2 m d r l 3 9 * C T I O 1.5m 12-Nov-OO 2:16:00 2422 sky in CH2, dome problems in curve m d r l 4 0 * C T I O 1.5m 13-Nov-OO 1:34:30 2580 moon! m d r l 4 1 * 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! m d r l 4 2 * 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 t e i d e O l * Teide 0.8m 16-Nov-00 0:42:10 1342 nice chl m d r l 4 3 * 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 m d r l 4 4 * C T I O 1.5m 17-Nov-OO 1:18:20 2730 good run n o l 7 0 0 q 2 * Hawaii 0.6m 17-Nov-OO 7:28:00 1375 good night for first half n o l 7 0 0 q 3 * Hawaii 0.6m 17-Nov-OO 12:34:20 196 good second half of night! t e i d e N 0 4 * 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 Log 84 Table A . l : continued R u n N a m e Telescope D a t e ( U T ) Start T i m e ( U T ) # of Points Observer C o m m e n t s n o l 8 0 0 q l * Hawai i 0.6m 18-Nov-00 7:22:30 2360 nice run - some clouds sa-od047* S A A O 1.9m 18-Nov-00 23:29:00 798 short but sweet teiden05* Teide 0.8m 18-Nov-00 22:53:20 3892/2 reduction problems n o l 9 0 0 q 2 * Hawai i 0.6m 19-Nov-OO 10:14:20 1423 Nice run! subtracted 8066 cts from ch3 sa-od048* S A A O 1.9m 19-Nov-00 18:55:00 2611 some dome glitches and 1% drop in counts -reason unknown t e i d e N 0 6 * 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 n o 2 0 0 0 q l * Hawai i 0.6m 20-Nov-00 7:37:00 2249 low amp part of rotat ion cycle p v b l l l 8 Bick ley 0.6m 18-Nov-00 12:28:31 2220 Single channel da ta from Pe r th p v b l l l 9 Bick ley 0.6m 19-Nov-OO 12:15:13 1495 sa-od049* S A A O 1.9m 20-Nov-OO 18:51:00 2651 Darragh's Last Stand - G o o d night - no fit applied and 17pt. sky chunk so wrote in two chunks p v b l l 2 0 Bick ley 0.6m 20-Nov-OO 12:19:12 2454 asm-0081* M c D o n a l d 2.1m 21-Nov-OO 9:06:10 349 not used p v b l l 2 1 Bick ley 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 mi l l ion at 10 sec int. t imes sa-m0003* S A A O 0.75m 21-Nov-OO 19:26:50 4857 5sec integrations - counts 4 mi l l ion per point j x j -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 intermittent clouds joy-002 M c D o n a l d 2.1m 23-Nov-OO 3:59:50 1958 Transparency variations (?) t e i d e N 0 8 * Teide 0.8m 22-Nov-OO 22:07:20 1364 5 sec integrations, humidi ty at end n o 2 3 0 0 q l * Hawai 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 s s o l l 2 3 a * sso 23-Nov-OO 12:42:28 467 not used: clouds, t im ing i n question s s o l l 2 3 b * sso 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 t e i d e n l O * tenerife 80cm 23-Nov-OO 22:05:40 3943 p v b l l 2 3 * Bick ley 0.6m 23-Nov-OO 12:15:41 1298 saOm0006* S A A O 0.75m 24-Nov-00 18:18:00 5646 Per iodic i ty in the sky at t e i d e n l l * Tenerife 25-Nov-OO 2:03:30 1229 Huge seeing but good data nonetheless continued on next page Appendix A. XCOV20 Observing Log 8 5 Table A . l : continued R u n N a m e T e l e s c o p e D a t e S t a r t T i m e # o f O b s e r v e r C o m m e n t s ( U T ) ( U T ) P o i n t s p v b l l 2 4 * Per th 24-Nov-00 12:18:40 2494 good first part j o y - 0 0 9 M c D o n a l d 2.1m 25-Nov-00 4:22:50 1567 clouds - not used j x j - 0 1 2 2 B A O 0.85m 25-Nov-OO 15:41:20 563 clouds - not used n o 2 5 0 0 q l * Hawai i 0.6m 25-Nov-00 7:03:10 2457 very good run t e i d e n l 2 * tenerife 80cm 25-Nov-OO 22:09:20 4040 long, good run j o y - 0 1 2 * M c D o n a l d 2.1m 26-Nov-00 3:55:50 1497 good da ta j x j - 0 1 2 3 B A O 0.85m 26-Nov-00 13:00:00 329 n o 2 6 0 0 q 2 * Hawai i 0.6m 26-Nov-00 6:59:30 2370 another nice run j x j - 0 1 2 4 * B A O 0.85m 26-Nov-00 14:14:50 1536 strong transparency variations s a - m 0 0 0 7 * S A A O 0.75m 26-Nov-00 18:28:40 5408 5 sec integrations, a nice run! j o y - 0 1 6 * M c D o n a l d 82" 27-Nov-OO 4:03:00 1956 clouds, c h l / c h 2 div is ion n o 2 7 0 0 q l * Hawai i 0.6m 27-Nov-00 6:38:00 2460 another nice run s a - m 0 0 0 8 * S A A O 0.75m 27-Nov-OO 18:27:50 5464 scattered on ends, but good signal t e i d e n l 3 * Teide 0.8m 26-Nov-OO 21:50:30 316 a very short run, 5 sec integrations j x j - 0 1 2 7 * B A O 0.85m 27-Nov-OO 13:44:10 1733 beautiful! t e i d e n l 4 * Teide 0.8m 27-Nov-OO 22:28:20 2910 Steve d i d this one...isn't it nice? :) j o y - 0 2 0 * M c D o n a l d 82" 28-Nov-00 4:04:20 2315 clouds, non-standard reduction to br ing out signal n o 2 8 0 0 q l Hawai i 0.6m 28-Nov-00 6:45:00 886 most ly clouds s a - h - 0 4 6 * S A A O 74" 28-Nov-00 18:54:30 2674 good run, about an hour t r immed from end s s o l l 2 7 S S O 0.6m 27-Nov-OO 10:03:13 859 there's signal here! j x j - 0 1 3 0 * B A O 0.85m 28-Nov-00 16:23:50 768 lots of clouds, but d iv ided by channel 2 to get a signal t e i d e n l 5 * Tenerife 0.8 m 28-Nov-00 22:01:50 3976 large seeing but signal O K j o y - 0 2 5 * M c D o n a l d 2.1 m 29-Nov-00 4:00:40 1895 middle part on clouds n o 2 9 0 0 q l * Hawai i 0.6m 29-Nov-00 6:41:00 2479 single channel but beautifull ! s a - g h 4 6 5 * S A A O 74" 29-Nov-OO 20:30:30 1942 t e i d e N 1 6 * Tenerife 0.8m 29-Nov-00 21:18:50 4092 first half clouds, second half gorgeous! j o y - 0 2 8 * M c D o n a l d 2.1 m 30-Nov-00 3:54:20 1899 a dataset so good even a theorist can reduce it! :) n o 3 0 0 0 q l * M a u n a K e a 24" 30-Nov-00 6:40:50 2461 single-channel j x j - 0 1 3 1 B A O 0.85m 29-Nov-OO 12:58:40 801 run through clouds - unusable j x j - 0 1 3 4 B A O 0.85m 30-Nov-OO 13:22:20 833 Too many clouds. Rejected. s a - g h 4 6 6 * S A A O 74" 30-Nov-00 19:30:20 618 first of four pieces of the night s a - g h 4 6 7 * S A A O 74" 30-Nov-OO 21:19:50 311 second piece s a - g h 4 6 8 * S A A O 74" 30-Nov-OO 22:15:20 848 th i rd piece s a - g h 4 6 9 * S A A O 74" l-Dec-00 0:39:00 560 last piece t e i d e n l 7 * Teide 0.8m 30-Nov-OO 21:17:50 4276 middle part missing because continued on next page Appendix A. XCOV20 Observing Log 86 Table A . l : continued R u n N a m e T e l e s c o p e D a t e S t a r t T i m e # o f O b s e r v e r C o m m e n t s ( U T ) ( U T ) P o i n t s mirror loose ssc-1201 S S O 0.6m l-Dec-00 9:47:57 1938 needed lots of low-f filtering s a - g h 4 7 0 * S A A O 74" l -Dec-00 18:57:20 1259 first part of night s a - g h 4 7 1 * S A A O 74" l -Dec-00 22:30:00 1348 second part of night j o y - 0 3 1 M c D o n a l d 82" l -Dec-00 3:48:50 1591 too many clouds dangit! t e i d e n l 8 * Tenerife l -Dec-00 22:04:00 2780 target not centered s a - g h 4 7 2 * S A A O 74" 2-Dec-00 18:46:50 1020 s a - g h 4 7 3 * S A A O 74" 2-Dec 21:38:30 1600 t e i d e n l Q * Teide 0.8m 2-Dec 21:46:50 3631 not useable s a - g h 4 7 4 * S A A O 74" 3-Dec 18:35:20 2713 good run, ampli tude is decreaseing s a - g h 4 7 5 * S A A O 74" 4-Dec-00 21:51:00 1530 clouds, transparency variations t s m - 0 0 8 7 * M c D o n a l d 82" 5-Dec-00 5:50:40 1100 electronics problems, no signal :( t s m - 0 0 8 9 * M c D o n a l d 2.1m 6-Dec-00 3:47:00 1448 observer never d i d sky i n c h l at a l l r O O - 0 2 2 * Na in i t a l 40in. 6-Dec-00 18:55:00 859 r O O - 0 2 5 * U P S O l m 7-Dec-00 16:53:20 842 short run of reasonable qual i ty t e i d e n 2 2 * Teide 0.8m 8-Dec-00 21:37:40 3788 poor conditions to start, then cloud t e i d e n 2 4 * Teide 0.8m 9-Dec-00 21:01:50 1386 5 sec. good short run t e i d e n 2 7 * Teide 0.8m 1 l-Dec-00 0:20:40 1741 5 sec. good short run 87 Appendix B A L i s t i n g of the E v o l u t i o n a r y M o d e l s T h i s a p p e n d i x prov ides a l i s t i n g of parameters for the 569 mode l s t ha t fel l w i t h i n the er ror b o x defined b y the H i p p a r c o s l u m i n o s i t y of 7.8 ± 0.7 LQ a n d the effective t empera tu re of Teff = 7400^200 K . A d i scuss ion of these pa ramete r s m a y be found i n S e c t i o n 3.1.1. Table B . i : A l i s t i n g of the mode l s t ha t fa l l w i t h i n the H i p -parcos l u m i n o s i t y errorbars . T h e tab le l i s ts the M a s s i n So la r un i t s , the effective t empera tu re i n degrees K e l v i n , the l o g a r i t h m of the l u m i n o s i t y i n So la r un i t s , the age i n u n i t s of 1 0 9 years, the large spac ing i n / zHz , the h y d r o g e n a n d heavy m e t a l mass fract ions, a n d the m i x i n g l e n g t h pa ramete r for each m o d e l . MQ L o g ( T e / / ) L o g ( L / L Q ) A g e ( G y r s ) 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 L o g ( T e / / ) L o g ( L / L 0 ) Age(Gyrs ) Av(nHz) X Z a 1.4 3.87098 0.859405 1.4 72.6281 0.72 0.008 1.8 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 L o g ( T e / / ) L o g ( 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 L o g ( T e / / ) L o g ( L / L 0 ) Age (Gy rs ) 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 L o g ( T e / / ) L o g ( L / L 0 ) A g e ( G y r s ) 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 1 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 1 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 1.5 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 1 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 1 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 1 70.899 0.7 0.014 1.4 1.5 3.86952 0.874471 0.95 72.5136 0.7 0.014 1.4 continued on next page Appendix B. A Listing of the Evolutionary Models Table B . l : continued MQ L o g ( T e / / ) L o g ( L / L 0 ) Age (Gy rs ) X Z a 1.5 3.87179 0.870235 0.9 74.1147 0.7 0.014 1.4 1.5 3.8739 0.865959 0.85 75.6791 0.7 0.014 1.4 1.5 3.87091 0.926943 1.1 67.065 0.7 0.012 1.8 1.5 3.87416 0.922573 1.05 68.986 0.7 0.012 1.8 1.5 3.87081 0.926943 1.1 66.9671 0.7 0.012 1.6 1.5 3.87408 0.922573 1.05 68.9015 0.7 0.012 1.6 1.5 3.87074 0.926943 1.1 66.8963 0.7 0.012 1.4 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 L o g ( T e / / ) L o g ( L / X , Q ) Age(Gyrs ) 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 L o g ( T e / / ) L o g ( L / L 0 ) Age(Gyrs ) AV(IJ,HZ) X Z a 1.55 3.86546 0.902089 0.95 68.5729 0.7 0.016 1.6 1.55 3.86808 0.89794 0.9 70.226 0.7 0.016 1.6 1.55 3.8705 0.893779 0.85 71.8318 0.7 0.016 1.6 1.55 3.87276 0.889562 0.8 73.4064 0.7 0.016 1.6 1.55 3.8595 0.909967 1.05 65.0596 0.7 0.016 1.4 1.55 3.86254 0.906075 1 66.784 0.7 0.016 1.4 1.55 3.86537 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 L o g ( T e / / ) L o g ( L / L Q ) Age(Gyrs ) X Z a 1.6 3.87146 0.904496 0.95 72.0366 0.74 0.014 1.6 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 L o g ( T e / / ) L o g ( L / L 0 ) A g e ( G y r s ) Av'pHz) X Z a 1.6 3.8712 0.907276 0.85 71.5616 0.72 0.016 1.4 1.6 3.87338 0.903129 0.8 73.0904 0.72 0.016 1.4 1.6 3.85828 0.883976 0.85 68.841 0.7 0.02 1.8 1.6 3.86042 0.880244 0.8 70.1876 0.7 0.02 1.8 1.6 3.85783 0.883993 0.85 68.4121 0.7 0.02 1.6 1.6 3.86007 0.880261 0.8 69.8307 0.7 0.02 1.6 1.6 3.85767 0.883976 0.85 68.2481 0.7 0.02 1.4 1.6 3.85993 0.880244 0.8 69.6879 0.7 0.02 1.4 1.6 3.86179 0.928187 0.95 65.2113 0.7 0.018 1.8 1.6 3.86466 0.924254 0.9 66.8359 0.7 0.018 1.8 1.6 3.86734 0.920222 0.85 68.4449 0.7 0.018 1.8 1.6 3.86986 0.916113 0.8 70.0399 0.7 0.018 1.8 1.6 3.86158 0.928187 0.95 65.0071 0.7 0.018 1.6 1.6 3.8645 0.924254 0.9 66.6799 0.7 0.018 1.6 1.6 3.86721 0.920222 0.85 68.3156 0.7 0.018 1.6 1.6 3.86975 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 L o g ( 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 L o g ( T e / / ) L o g ( L / L Q ) Age(Gyrs ) Av(fiHz) X Z a 1.65 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 L o g ( T e / / ) L o g ( 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 MQ L o g ( T e / / ) Log (L /L©) Age(Gyrs ) Ai/(nHz) X Z a 1.7 3.87435 0.893294 0.4 77.1483 0.72 0.022 1.8 1.7 3.8715 0.90147 0.5 74.71 0.72 0.022 1.8 1.7 3.87309 0.897583 0.45 75.9758 0.72 0.022 1.8 1.7 3.8698 0.905362 0.55 73.4184 0.72 0.022 1.8 1.7 3.868 0.909232 0.6 72.1076 0.72 0.022 1.8 1.7 3.86609 0.913075 0.65 70.7766 0.72 0.022 1.8 1.7 3.86193 0.920658 0.75 68.0785 0.72 0.022 1.8 1.7 3.86406 0.916886 0.7 69.4326 0.72 0.022 1.8 1.7 3.85969 0.924385 0.8 66.721 0.72 0.022 1.8 1.7 3.85732 0.928007 0.85 65.3619 0.72 0.022 1.8 1.7 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 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 MQ Log(re//) L o g ( L / L 0 ) A g e ( G y r s ) AufaHz) X Z a 1.7 3.86313 0.852096 0.45 76.7749 0.74 0.022 1. 1.7 3.86192 0.855628 0.5 75.7297 0.74 0.022 1. 1.7 3.8606 0.859189 0.55 74.637 0.74 0.022 1. 1.7 3.85931 0.862978 0.6 73.5504 0.74 0.022 1. 1.7 3.85782 0.866448 0.65 72.4199 0.74 0.022 1. 1.7 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. 102 Appendix C Tables of the Weigh t ed D a t a Resu l t s T h e resu l t s f r o m the we igh ted d a t a ana lys i s are presented i n Tab l e s C . l , C . 2 , C . 3 , a n d C . 4 for the four mode l s w i t h the lowest xLmp s ta t i s t i c . E a c h of the m o d e l are ident i f ied b y n u m b e r i n the tab le c a p t i o n wh i l e a desc r ip t ion of h o w each m o d e l was reduced is desc r ibed i n S e c t i o n 2.6 a n d T a b l e 2.3. Tab le s C . 5 , C . 6 , C . 7 , C . 8 , C . 9 , a n d C . 1 0 present the average frequency, a m p l i t u d e a n d phase for m o d e l s w i t h t he same ident i f ied frequencies, b u t w i t h different w e i g h t i n g parameters . T h e s t a n d a r d error of the m e a n for each of the frequencies, a m p l i t u d e s a n d phases are deno ted as a. M o d e l s are ident i f ied b y n u m b e r i n the t ab le c a p t i o n a n d r e d u c t i o n descr ip t ions m a y be found i n Sec t ion 2.6 a n d T a b l e 2.3. T a b l e d : W e i g h t e d m o d e l n u m b e r 14. v ( m H z ) A m p . ( m m a ) 4> <5i/(mHz) Ai/ ( m H z ) - 2.61858507 0.070913352 0.910846 0.00094979 2.61953486 0.24294598 0.0675517 + 2.62054495 0.095604089 0.192343 0.00101009 2.65118266 0.049705656 0.499794 0.00078301 0.03341159 - 2.65196567 0.227132022 0.0337749 0.00098078 1/2 2.65294645 0.785205032 0.136531 + 2.65389001 0.198684201 0.305039 0.00094356 ++ 2.65491013 0.036362386 0.295139 0.00102012 2.68645653 0.316776759 0.182504 0.00103671 V3-L>2 0.03454679 2.68749324 0.554052254 0.100974 + 2.68840557 0.202862201 0.373698 0.00091233 2.71912802 0.050704868 0.24473 0.0009345 0.03343382 continued on next page Appendix C. Tables of the Weighted Data Results Table C . l : continued v ( m H z ) A m p . ( m m a ) 4> dV(mHz) Ai/ ( m H z ) - 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 I /5-f4 0.03440303 "5 2.75533009 0.273609593 0.2087 + 2.75623487 0.122168294 0.499628 0.00090478 2.78801304 0.06013802 0.351141 0.00091362 i/6 (new)-i/5 0.03359657 1/6 (new) 2.78892666 0.103978465 0.648353 + 2.79001282 0.096200533 0.417312 0.00108616 2.79059069 0.131995516 0.25712 0.00093511 i/6 (old)-i/6 (new) 1/6 (old) 2.7915258 0.210137403 0.42461 0.00259914 + 2.79250869 0.085426427 0.612155 0.00098289 2.80563375 0.18459977 0.430217 0.00094644 1/7-1/6 (old) 0.01505439 2.80658019 0.129121204 0.486159 + 2.80739686 0.034644142 0.762129 0.00081667 Table c.2: Weighted model number 18. v ( m H z ) A m p . ( m m a ) 4> <Si/(mHz) Ai/ ( m H z ) - 2.61856429 0.071942551 0.964752 0.0009714 2.61953569 0.229725434 0.06479 + 2.62053759 0.093576944 0.185547 0.0010019 1/2-1/1 0.03341124 - 2.65197721 0.237377855 0.006794 0.00096972 2.65,294693 0.778877511 0.138822 + 2.65388537 0.194361604 0.316978 0.00093844 1/3- V2 0.03454832 - 2.68644101 0.326210402 0.208817 0.00105424 2.68749525 0.55849757 0.100642 + 2.68839972 0.207372074 0.384754 0.00090447 1/4-^3 0.03343165 - 2.72006234 0.383081687 0.513257 0.00086456 continued on next page ix C. Tables of the Weighted Data Results Table C.2: continued v ( m H z ) A m p . ( m m a ) 4> <5i/(mHz) Ai/ ( m H z ) 1/4 2.7209269 1.16752688 0.781976 + 2.72182919 0.421655594 0.014391 0.00090229 1/5-1/4 0.03439753 - 2.75431064 0.208573433 0.117343 0.00101379 2.75532443 0.272340984 0.225663 + 2.75623806 0.113454944 0.478726 0.00091363 i/6 (new)-i/5 - 2.78800841 0.051362814 0.372008 0.00092015 0.03360413 1/6 (new) 2.78892856 0.116984091 0.642873 + 2.79002184 0.098770964 0.396236 0.00109328 1/6 (old)-i/6 (new) - 2.79063517 0.123071617 0.17611 0.00089349 0.0026001 1/6 (old) 2.79152866 0.213303151 0.423763 + 2.79251626 0.081582989 0.606488 0.0009876 1/7-1/6 (old) - 2.80563794 0.187002768 0.423267 0.00092045 0.01502973 1/7 2.80655839 0.127268129 0.526702 + 2.807416 0.028527146 0.740054 0.00085761 Table c.3: W e i g h t e d m o d e l n u m b e r 27. v ( m H z ) A m p . ( m m a ) * <5i/(mHz) Ai/ ( m H z ) 1/1 2.61952756 0.250389276 0.075044 0.001022 + 2.62055003 0.097967802 0.188145 1/2-1/1 0.03341767 - 2.65197412 0.236102849 0.010989 0.000971 1/2 2.65294523 0.779581638 0.13877 + 2.65389634 0.198998402 0.293861 0.000951 0.03454714 - 2.6864492 0.318197961 0.196758 0.001043 "3 2.68749237 0.554835233 0.105008 + 2.68840277 0.204518648 0.377414 0.00091 1/4-1/3 0.03343572 - 2.72005924 0.387951466 0.52163 0.000869 1/4 2.72092809 1.16502003 0.780566 + 2.72182907 0.425400013 0.014838 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 ( m H z ) A m p . ( m m a ) 4> <5t/(mHz) Ai/ ( m H z ) - 2.75431424 0.202921595 0.121033 0.00102 2.75533394 0.27371851 0.207484 + 2.75623869 0.120983186 0.480159 0.000905 i/6 (new)-i/5 0.03360725 1/6 (new) 2.78894119 0.110023328 0.610227 0.000952 + 2.78989309 0.089420136 0.629435 - 2.79062019 0.133141748 0.20376 0.000905 i/6 (old)-i/6 (new) 1/6 (old) 2.79152475 0.206709227 0.420454 0.00258356 + 2.79244265 0.09004229 0.718838 0.000918 1/7-1/6 (old) - 2.80563819 0.182137724 0.428567 0.00094 0.0150535 2.80657825 0.135530035 0.482141 Table c.4: W e i g h t e d m o d e l n u m b e r 32. 1/ ( m H z ) A m p . ( m m a ) 4> <5i/(mHz) Ai/ ( m H z ) - 2.61862362 0.073309533 0.832223 0.00091106 2.61953468 0.239755041 0.069636 + 2.62048027 0.095932955 0.245718 0.00094559 ++ 2.62118363 0.080067584 0.011993 0.00070336 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 1/3-1/2 0.03454477 - 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 0.0009255 1/4-^3 0.03343682 - 2.72005692 0.378040442 0.523409 0.00087041 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 ix C. Tables of the Weighted Data Results Table C.4: continued 1/ ( m H z ) A m p . ( m m a ) <t> <5i/(mHz) Ai/ ( m H z ) 0.03441038 - 2.75431941 0.205991964 0.113945 0.0010183 2.75533771 0.270062018 0.19682 + 2.75623916 0.121934717 0.491413 0.00090145 1/6 (new)-i/5 0.03360677 v% (new) 2.78894448 0.110591288 0.599474 + 2.78997949 0.095712189 0.48221 0.00103501 1/6 ( o l d ) - f 6 (new) 0.00258473 - 2.79058457 0.134939722 0.261878 0.00094464 1/6 (old) 2.79152921 0.212994084 0.416727 + 2.79247997 0.083587537 0.67673 0.00095076 1/7-1/6 (old) 0.01505025 - 2.80563634 0.183135677 0.431134 0.00094312 1/7 2.80657946 0.13235342 0.483335 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 ( m H z ) 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 ( m H z ) 0 V A m p l i t u d e ( m m a ) °~Amp <t> 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 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 values for weighted r e d u c t i o n mode l s 12 t h r o u g h 15. A l s o S h o w n are the s t a n d a r d errors o n the average, o, for each of the frequencies, a m p l i t u d e s a n d phases. v ( m H z ) Ov A m p l i t u d e ( m m a ) OAmp 0~<f, 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 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 2.7562337 1.61E-06 0.120785 0.001266 0.50185174 0.0029707 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 Table C.6: continued 108 v ( m H z ) A m p l i t u d e (mma) QAmp 4> 0 > 2.8056314 2.8065777 2.807390 3.1E-06 3.34E-06 1.03E-05 0.1854348 0.1242794 0.0377643 0.001802 0.003883 0.002778 0.43582401 0.48549101 0.77594125 0.0061031 0.0051416 0.0130536 Table c.7: A v e r a g e values for weighted r e d u c t i o n mode l s 16 t h r o u g h 19. A l s o S h o w n are the s t a n d a r d errors o n the average, cf, for each of the frequencies, a m p l i t u d e s a n d phases. v ( m H z ) <7„ A m p l i t u d e (mma) 4> 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 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 1 030E-06 0.11496046 3 801E-03 0.63687551 4 427E-03 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 values for weighted r e d u c t i o n mode l s 21 t h r o u g h 24. A l s o S h o w n are the s t a n d a r d errors o n the average, o, for each of the frequencies, a m p l i t u d e s a n d phases. 1/ ( m H z ) av 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 2 720927 4E-07 1.148638 0 005753 0.782802 0.000868 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 values for weighted r e d u c t i o n mode l s 25 t h r o u g h 28. A l s o S h o w n are the s t a n d a r d errors o n the average, o, for each of the frequencies, a m p l i t u d e s a n d phases. v ( m H z ) 0 V A m p l i t u d e (mma) °~Amp 4> c<t> 2.619531 2.620545 2.651977 4.11E-06 5.16E-06 2.73E-06 0.249794 0.099087 0.23728 0.002486 0.001556 0.002204 0.070423 0.184552 0.007028 0.005858 0.001297 0.002996 ix C. Tables of the Weighted Data Results Table C.9: continued v ( m H z ) Ov A m p l i t u d e (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 values for weigh ted r e d u c t i o n mode l s 30 t h r o u g h 32. A l s o S h o w n are the s t a n d a r d errors o n the average, o, for each of the frequencies, a m p l i t u d e s a n d phases. v ( m H z ) ov A m p l i t u d e (mma) ""Amp 0(f, 2.618615 4.53E-06 0.072739 0 000822 0 842523 0.007433 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 ( m H z ) o-v A m p l i t u d e ( m m a ) 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 

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