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The oscillation modes of Delta Scuti stars Kennelly, Edward James 1994

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THE OSCILLATION MODES OF DELTA SCUTI STARS By Edward J. Kennelly B. Sc. (Astronomy) University of Western Ontario M. Sc. (Astronomy) University of British Columbia  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  in THE FACULTY OF GRADUATE STUDIES GEOPHYSICS AND ASTRONOMY  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  September 1994  ©  Edward J. Kennelly, 1994  __  In presenting this thesis in  partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  cc’d Aroror  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  SpI.  O 199L(  Abstract  Many S Scuti stars exhibit multiperiodic oscillations and are therefore considered to be good candidates for studies in stellar seismology; the modes of oscillations act as probes of the internal stellar structure. For S Scuti seismology to be successful, both the oscifiation modes and frequencies must be determined observationaily, then compared with theoreti cal models. Standard photometric observations can provide direct information only about the frequencies of those low-degree oscillations which give rise to large disk-integrated variations. In this thesis, oscillations of both low- and high-degree were studied by ana lyzing the variations introduced in the Doppler-broadened profiles of rapidly rotating S Scuti stars. The incidence of high-degree variability within the S Scuti instability strip was inves tigated with an spectroscopic survey of  ‘  50 bright stars, carried out at the Dominion  Astrophysical Observatory. Evidence for high-degree variations was discovered in 14 stars which were located both on the main sequence and in more evolved stages of evolution. A few stars (e.g., r Peg and  92  Tau) were investigated further with high-resolution  (2.4 A/mm) spectra obtained at the Canada-France-Hawaii telescope.  Using a two-  dimensional Fourier technique to transform the variations within the line profiles in both time and Doppler space, a representation of the variations was produced from which the apparent frequency and the apparent azimuthal order (or degree) could be directly determined. The technique was especially successful at uncovering multiple modes of oscillation. The observed mode spectra were found to be consistent with prograde, sec toral modes which oscillate with nearly equal frequency in the corotating frame of the star. However, the frequencies of the high-degree modes were lower than expected if they U  result from pressure waves trapped near the stellar surface. To study the oscillations in detail, observations lasting several days are required in order to provide adequate frequency resolution. As part of the 1992 MUSICOS campaign, 4 days of nearly continuous observations of 82 Tau were obtained from sites in China, France, the Canary Islands, and Kitt Peak. Analysis of the MUSICOS data has revealed several low- and high-degree modes (1  8) at frequencies between 11 and 17 cycles day . 1  in  Table of Contents  Abstract  ii  List of Tables  vii  List of Figures  ix  Acknowledgements 1  2  xii  The Oscillations of S Scuti Stars  1  1.1  Introduction  1  1.2  A Description of Stellar Oscillations  2  1.3  The Properties of S Scuti Stars  7  1.4  Stability Analysis of S Scuti Stars  10  1.5  The Problem of Mode Selection  14  1.6  Seismology of S Scuti Stars  1.7  Seismology of S Scuti Stars  1.8  Goals for this Thesis  -  -  .  A Test for Convective Overshooting The Internal Rotation Proffle  .  .  15 16 18  The Identification of Oscillation Modes  19  2.1  Methods of Mode Identification: A Summary  19  2.2  Fourier Analysis  22  2.3  Analysis of Radial Velocity Variations  25  2.4  2-D Fourier Representation of Line-Proffle Variations  26  2.5  Demonstrations of the 2-D Fourier Technique  30  iv  3  2.6  2-D Fourier Prewhitening  49  2.7  Discussion  50  Line-Profile Variations in the S Scuti Instability Strip 3.1  Introduction  52  3.2  Organization of the Survey  54  3.3  Observations  55  3.4  Analysis of the Survey  70  3.5  Discussion  79  4 The Line-Profile Variations of r Pegasi  5  6  52  82  4.1  Introduction  82  4.2  The 1990 CFH Observations  83  4.3  The 1992 DAO Observations  91  4.4  Discussion  95  The Line-Profile Variations of 62 Tauri  99  5.1  Introduction  5.2  The 1990 CFHT Observations  103  5.3  Observations from DAO  111  5.4  Discussion  112  99  Four Line-Profile Variables Re-Visited 6.1  Introduction  114 114  1  6.1.1  HR 1298:  6.1.2  HR 2707: 21 Mon  115  6.1.3  HR 3888: v UMa  115  6.1.4  HR 5329:  2  En  114  Boo  116 V  6.2  Fourier Analysis of the 1987 CFHT Observations  116  6.3  Discussion  123  7 The MUSICOS Observations of 62 Tauri  8  126  7.1  Introduction  126  7.2  The Goals of MUSICOS  126  7.3  Organization of the MUSICOS 92 Campaign  127  7.4  The MUSICOS Observations  129  7.5  Reduction of the MUSICOS Spectra  131  7.5.1  Reduction of the Xinglong Data  131  7.5.2  Reduction of the OHP Data  133  7.5.3  Reduction of the WilT Data  134  7.5.4  Reduction of the Kitt Peak Data  136  7.5.5  Corrections to the Line-of-Sight Velocity  138  7.5.6  Continuum Normalization  138  7.5.7  Spectral and Temporal Resolution  141  7.6  Radial-Velocity Variations  142  7.7  Line-Profile Variations  150  7.8  Discussion  167  Summary and Conclusions  176  Bibliography  180  vi  List of Tables  2.1  Methods of Mode Identification  2.2  Numerical Simulations of High-Degree Modes  2.3  Simulations of Low-Degree Modes  2.4  Fourier Representation of Multiple Modes  3.1  Survey Selection Criteria  3.2  Survey of S Scuti Candidates  56  3.3  Survey Observations  59  3.4  Detection of Line Profile Variations  65  4.1  Fourier Analysis of the Line-Proffle Variations of r Peg (CFHT)  87  4.2  Fourier Analysis of the Velocity and Light Variations of r Peg (CFHT &  .  46 •  .  .  55  UH)  88  4.3  Fourier Analysis of the Line-Profile Variations of T Peg (DAO)  94  5.1  Summarizing  5.2  Fourier Analysis of the Line-Profile Variations of  5.3  Fourier Analysis of the Velocity Variations of  6.1  Characteristics of the Observed Stars  124  7.1  MUSICOS 92: Requirements  128  7.2  MUSICOS 92: Sites and Instruments  129  7.3  MUSICOS 92: Observations of 92 Tau  130  92  Tauri  102  vii  92  92  Tau  Tau  110 110  7.4  MUSICOS 92: Spectral Resolution  7.5  MUSICOS 92: Analysis of the Radial Velocity Variations  7.6  MUSICOS 92: Further Analysis of the Radial Velocity Variations  vu’  .  141 147  .  .  .  .  149  List of Figures  1.1  Velocity Maps of Surface Oscillations  2.1  Simulated High-Degree Line-Profile Variations  33  2.2  Fourier Representation of High-Degree Line-Profile Variations  34  2.3  Fourier Representation of Large Amplitude Variations  38  2.4  Fourier Representation of Inclined stars with 0  2.5  Fourier Representation of Radial Oscillations.  2.6  Fourier Representation of £  =  1 Oscillations  2.7  Fourier Representation of £  =  2 Oscillations  2.8  Fourier Representation of £  =  3 Oscillations  2.9  Line-Profile Simulations with Multiple Modes  4  —m  £  40  2.10 Fourier Representation of Multiple Modes 3.1  Detection of Line-Profile Variations  3.2  Detection of Line-Profile Variations  3.3  Mean Spectra and Their Variations  68  3.4  Mean Spectra and Their Variations  69  3.5  Line-Profile Variations in the Instability Strip  3.6  The Period-Luminosity-Colour Relation for S Scuti Stars.  3.7  The Distribution of Projected Rotational Velocities  4.1  The Line-Profile Variations of r Peg (CFHT)  84  4.2  Fourier Representation of the Line-Profile Variations in r Peg (CFHT).  85  ix  -  -  Example One Example Two  •  62 63  4.3  Simultaneous Radial Velocity and Light Variations of r Peg (CFHT & UH). 89  4.4  The Line-Proffle Variations of r Peg (DAO)  4.5  Fourier Representation of the Line-Proffle Variations in r Peg (DAO).  4.6  Radial Velocity Variations of r Peg (DAO)  95  4.7  Identification of the High-Degree Modes  97  5.1  The Orbital Radial Velocity Variations of  5.2  Mean Spectra of  5.3  The Line-Proffle Variations of  5.4  The Fourier Representation of the Line-Profile Variations of  5.5  The Radial Velocity Variations of  6.1  Fourier Representation of the Line-Profile Variations in  o1  En (I)  118  6.2  Fourier Representation of the Line-Proffle Variations in  1  Eri (II)  119  6.3  Fourier Representation of the Line-Profile Variations in 21 Mon  120  6.4  Fourier Representation of the Line-Proffle Variations in v UMa  121  6.5  Fourier Representation of the Line-Proffle Variations in ic 2 Boo  122  7.1  Spectral Coverage for the Individual MUSICOS Sites.  131  7.2  The Observational Window Functions  132  7.3  Stability at the MUSICOS Sites  137  7.4  The Radial Velocity Variations of  7.5  The Spectral Window Function and Amplitude Spectra of  7.6  Residual Velocities after Removing Five Frequencies  146  7.7  Amplitude Spectrum Before and After Prewhitening  148  7.8  Comparison of the MUSICOS Frequencies with Breger (1989)  151  7.9  Time Series Observations from Xinglong (HJD 2448964)  152  92  92  92 .  Tau  104  Tau at Different Phases of the Orbit 92  106  Tau  92  92  x  93  108 92  Tau.  .  109  .  Tau.  111  Tau  144 92  Tau  .  .  .  .  145  7.10 Time Series Observations from OHP (HJD 2448964).  153  7.11 Time Series Observations from Kitt Peak (HJD 2448964)  154  7.12 Time Series Observations from Xinglong (HJD 2448965).  155  7.13 Time Series Observations from OHP (HJD 2448965)  156  7.14 Time Series Observations from WilT (HJD 2448965) 7.15 Time Series Observations from Kitt Peak (HJD 2448965). 7.16 Time Series Observations from Xinglong (HJD 2448966)  159  7.17 Time Series Observations from OHP (HJD 2448966) 7.18 Time Series Observations from WilT (HJD 2448966)  161  7.19 Time Series Observations from Kitt Peak (HJD 2448966)  162  7.20 Time Series Observations from Xinglong (HJD 2448967)  163  7.21 Time Series Observations from OHP (HJD 2448967)  164  7.22 Time Series Observations from WHT (HJD 2448967)  165  7.23 Time Series Observations from Kitt Peak (HJD 2448967)  166  7.24 The Two-Dimensional Fourier Representation of 82 Tau.  168  7.25 A Detailed Two-Dimensional Fourier Map of 62 Tau. 7.26 Slices of the Two-Dimensional Fourier Map of 7.27 The Frequency Spectrum of  92  Tau  xi  92  Tau.  .  .  .  .  170 174  Acknowledgements  I’d like to thank Gordon Walker, Stephenson Yang, Bill Merryfield, Jaymie Matthews, Eiji Kambe, David Bohiender, the MUSICOS team, the staff at DAO, the UBC astron omy grad students (esp. David Woods), my family and friends, the staff at the University Village Restaurant, and the magic of black bean sauce.  xli  Chapter 1  The Oscillations of S Scuti Stars  1.1  Introduction  The vibrational properties of a star are largely determined by its size, mass and structure. The aim of stellar seismology is to use the natural oscillations of a star to probe internal structure. Any star for which multiple modes are detected (or suspected) is a good candidate for seismology. During the last decade, seismology of the Sun has proven to be very successful. However, studies of stars other than the Sun are required to investigate the properties of stellar structure as a. function of mass and age. The S Scuti stars are 1.5 to 2.5M, stars which lie within an extension of the Cepheid instability strip near the main sequence. Many S Scuti stars are known to undergo  miii  tiperiodic oscillations. As many as 5 to 10 modes have been identified by observers. The prospect of S Scuti seismology is therefore particularly attractive. If detailed descriptions of the oscillations for stars at different stages of evolution can be acquired then the obser vations could be used to test theories of stellar structure and evolution. With this goal in mind, the identification of modes and frequencies in S Scuti stars has become an extensive area of research in recent years. Especially successful have been photometric campaigns lasting several weeks which have provided the necessary frequency resolution to reveal multiple modes of oscillation (e.g.,  62  Tau: Breger et al. 1989; GX Peg: Michel et al.  1992). However, the theoretical frequency spectra for S Scuti stars are extremely rich and the identification of the relatively few observed oscillation modes from photometry alone  1  Chapter 1. The Oscillations of S Scuti Stars  2  is difficult. Fortunately, spectroscopic observations of S Scuti stars can complement re sults from photometry. For rapidly rotating stars, the variations on the stellar surface are mapped into the Doppler broadened absorption proffles of the star. By analyzing this “Doppler image”, information about the oscillation modes can be obtained, including those which would otherwise go undetected in photometric studies. 1.2  A Description of Stellar Oscillations  It is useful at this point to introduce some of the terminology which is used to describe the general oscillations of a star. Additional details about the theoretical aspects of stellar oscillations can be found in the monographs by Cox (1980) and Unno et al. (1989). Given a model of stellar structure, the equations of hydrodynamics (i.e., the equations of mass, momentum, and energy conservation, and the equation of radiative diffusion) which describe the deformations and oscillations of a star can be formulated as an eigen value problem. (Usually, the star is assumed to be spherically symmetric; that is, the effects due to rotation and magnetic fields which distort the star are neglected.) Solutions to the eigenvalue problem can be found regardless of whether or not the mechanism for sustaining the oscillations is known. However, treatment of the driving mechanism is required to predict the amplitudes of the modes which do propagate. The eigensolutions of the equations of motion can be separated into components which describe the variations in the radial and angular directions in a. system of spherical coordinates (r, 6, çb). The dependence of the oscillations on 0 and  4’  can be written in  terms of spherical harmonics. Two orthogonal families of solutions are possible which are referred to as the spheroidal and toroidal modes. The displacements are described by:  phfr, 3  4 )1m(6, 4’)e’ 6,4’, t) = A(r)(1, k, k_ t  [1.1]  Chapter 1. The Oscillations of S Scuti Stars  Et.(r, 8, 4, t)  =  A(r)(O,  3  _L.,  _)ym(9,  qf)e”.  Here k is the ratio of the horizontal to radial velocity amplitude,  =  is the frequency  ii  g5) are the spherical harmonic functions. The magnitude of  of oscillation, and k  [1.21  2 (where M and R are the mass and radius of the star) is determined by v 3 GM/R  the boundary conditions on the pressure at the surface of the star. It is related to the pulsation constant  Q  =  Pocif7 by k  =  74.4  2 Q  where P 03 is the period of oscillation  and 3 is the mean density of the star. For the high-frequency oscillations k << 1. A mode of oscillation is characterized by its frequency and three quantum numbers. The order n is usually taken to represent the number of radial nodes between the center of the star and the surface. The angular dependence of the oscillations is described by the degree £ and the azimuthal order m. The quantity, £ nodal lines in planes parallel to the equator and  21m1  longitude. The azimuthal order can take values m  =  —  ml  specifies the number of  specifies the number of nodes in —I, —t + 1,  ...,  L In a spherically  symmetric star, there is no preferred axis and oscillations of a given degree are said to be degenerate with 2t + 1 values. Examples illustrating the appearance of the stellar surface for different modes of oscillation are provided in Figure 1.1. In the absence of rotation, the eigenfrequency of a toroidal mode is v  =  0. In this case,  the modes represent steady eddy motions and not oscillatory motions. In a rotating star, toroida.l modes correspond to Rossby waves (r-modes) with frequency, z’ where  =  2m/e(t +1)  is the frequency of rotation. Since the radial component of toroidal oscillations  is zero, these kinds of modes provide no information about the internal structure of the star. On the other hand, a radial component to the spheroidal modes does exist and these modes may be envisioned as waves which propagate within the star. The general name for oscillations for which there is a dependence on angular position is nonradial  Chapter 1. The Oscillations of S Scuti Stars  4  a  Figure 1.1: Velocity Maps of Surface Oscillations. The line-of-sight component of the surface velocity variations resulting from nonradial oscillations is illustrated for modeswith: a)t=3,m=O,b)t=3, ImI=2,c)t=3, mI=3,d)t=1O, ImI=O, e) £ = 10, ImI = 6, and f) £ = 10, ImI = 10. Dark areas represent motion away from the observer; light areas represent motion towards the observer.  Chapter 1. The Oscillations of S Scuti Stars  5  pulsations (nrp). In the presence of rotation, the degeneracy of frequencies of nonradial oscillations is removed. In the inertial frame, the frequency of nonracial oscillation in a uniformly rotating star is given as  Vnjm =  v , 0 .  —  mIZ(1  —  + O(V),  where v , is the intrinsic frequency of oscillation of a nonrotating star and 0 frequency of rotation and  [1.3] is the  is a (small) dimensionless quantity which depends on  the eigenfunctions of the nonrotating star and on its structure (reflecting the fact that the star is no longer spherically symmetric). Normally, the convention is chosen such that negative values of m represent modes which are prograde (i.e., they propagate in the same direction as rotation) while positive values of m represent retrograde modes. Moreover, if the ratio of the rotation-to-pulsation frequencies is large (f’/v > 0.2) then it no longer appropriate to describe any given mode of oscillation in terms of a single spherical harmonic. Instead the oscillation modes must be described in terms of linear combinations of spheroidal and toroidal components (Lee & Sa.io 1990 and Aerts & Waelkens 1993). Both pressure and gravity can act as restoring forces for the pulsations giving rise to two classes of oscillations, generally referred to as acoustic (or pressure) and gravity waves (also called p- and g-modes). The frequencies of these modes are determined by two characteristic frequencies. The Lamb frequency is related to the velocity of sound within the star:  ” 2 L  The sound speed is defined as c(r)  ‘—  =  £(t+1)cfr) 2 r  14  Fipofr)/pofr), where p0(r) and po(r) are the  Chapter 1. The Oscillations of S Scuti Stars  pressure and density of the unperturbed star, and  6  1 r  =  (ctlnp(r)/dln p(r))ad is the  adiabatic exponent. The sound speed increases with decreasing radius. The Brunt VaisaTh frequency describes the oscillation frequency of a bubble of gas due to buoyancy forces: GM(r)(ldlnQ,o(r)/Po(r))) 2 r Jr 1 F  [1.5]  and is sensitive to evolutionary changes in stellar structure.  Generally, the acoustic  N f 2 r)  waves are characterized by high frequencies (v 2 > L, N ) and the gravity waves by low 2 frequencies  (i?  <  ). 2 L, N  A third group of modes, called the fundamental (or f-modes), correspond to surface waves because they have no radial nodes within the star. These modes oscillate with frequencies (approximately) described by 2 G M21(t—1) 2f+1  1 [.6]  which are intermediate to those of the p- and g-modes. The p- and g-mode oscillations propagate in regions or cavities located within the star, their location depending on the structure of the star and the mode of oscillation. Pressure waves propagate near the surface of a star. The depth to which these waves penetrate the star is determined by the sound speed within the star and by the frequency and nonradial degree of oscillation (see Equation 1.4). Radial modes are a special case of pressure waves for which £  =  0. With no dependence on angular position, radial modes  of oscillation penetrate all the way to the center of a star. Modes of oscillation with nonzero values of the nonradial degree propagate nearly vertically at the surface of the star but are eventually refracted by the increase of sound speed with increasing depth. Since all waves propagate similarly near the surface of the star, the information they  Chapter 1. The Oscillations of S Scuti Stars  7  carry then reflects the conditions in the regions which they penetrate. It is this property which makes it theoretically possible to determine the run of sound speed with depth within the star (for example) from a detailed description of the observed oscillations. 1.3  The Properties of S Scuti Stars  There are now more than 250 known S Scuti stars. Catalogues of these objects (pri marily based on photometric observations) have been compiled by L6pez de Coca et al. (1990), Garcia et al. (1993), and Rodrfguez et al. (1994). The most thorough reviews written about the S Scuti stars were prepared by Breger (1979) and Wolff (1983). More recent reviews have appeared in conference proceedings (e.g., Breger (1990), Yang (1991), Matthews (1993), and Breger (1993)). Theoretical reviews have been written by Dppen (1993) and Dziembowski (1990a). Progress in S Scuti research during the years from 1990 to 1993 has been reported by Kurtz (1994). In this section, a brief summary of the observational properties of S Scuti stars is presented. Then, in the following sections, theoretical aspects of S Scuti stars are reviewed. The S Scuti stars are late A- and early F-type stars. Their periods of oscillation are typically less than a third of a day and the amplitudes of their light curves are typically a few milli-magnitudes. The amplitude of their radial velocities are usually less than 5 km s . It is now generally accepted that two groups of S Scuti stars exist 1 (Breger 1991): the nonradial pulsators and the radial pulsators (referred to in the past as dwarf Cepheids, RRs stars, and Al Vel stars). The radial pulsators oscillate with one or two large amplitude radial modes. They are situated above the main sequence and are characterized by low projected rotational velocities (v sin i < 30 km s , where i is 1 the inclination of the star to the line of sight). By contrast, the nonradial pulsators can exhibit many (low-degree) modes, including radial modes, but the amplitudes of these  Chapter 1. The Oscillations of 5 Scuti Stars  8  modes are small and are commonly seen to vary from one observing run to the next. The stars in this group can be either in the main sequence or post-main sequence stages of evolution. Their projected rotational velocities are usually large (typically v sin i  =  150  km s ) but S Scuti stars are not generally observed with rotational velocities above 200 1 km s 1 (Wolff 1983). In addition, a third group of stars which are undoubtably related to the S Scuti stars have population II abundances. These objects exhibit primarily radial modes and are referred to as SX Phe stars. The low-amplitude, multiperiodic S Scuti stars are particularly interesting for seis mology. Until recently, the explanation for the quasi-periodic behaviour of some S Scuti light curves in terms of beating between two or more frequencies had been questioned. In many cases, the time coverage of the observations was insufficient to establish whether multiple modes actually existed or whether the periods and amplitudes themselves were variable. Observations obtained from a single site were often insufficient to provide an swers because of the uncertainties introduced by one-day aliases in the periodograms of the variations. However, with the introduction of multi-site campaigns, this question has been resolved; both possibilities occur. Some stars have been found to be variable with the same frequency spectrum even after several years. For example, the S Scuti star 62 Tau was twice observed as part of an international photometric campaign by Breger et  al. (1987) and Breger et al. (1989) and both times the same five frequencies were uncov ered. For other stars, variations in amplitude have been clearly documented. Such was the case for the monoperiodic star r Peg (Breger 1991). In addition, very slow changes in period P can be expected owing to evolution. Both positive and negative values of dP/dt have been reported by Breger (1990), possibly suggesting that evolution occurs in both the redward and blueward directions on the HR-diagram. Multi-site photometric campaigns have had amazing success in the last decade. Fre quency spectra for numerous stars have come from two camps: the STEPHI network  Chapter 1. The Oscillations of S Scuti Stars  9  (Stellar Photometry International) and the Delta Scuti Network (DSN). In a recent cam paign, the Delta Scuti Network joined forces with the Whole Earth Telescope (WET, a project dedicated to the observation of rapid variations in white dwarf stars) to form the GLOBUS network and to search for low- and high-order oscillations in the star FG Vir (Breger 1993). A project to obtain multi-site CCD photometry, STACC (a Small Tele scope Array with CCD Cameras) has also been organized Frandsen (1992). The results from many of these campaigns have been summarized by Belmonte et al. (1992). How ever, Matthews (1993) and Kurtz (1994) warn that observers may have overinterpreted their data and that many of the published frequencies could be false. By comparison with photometry, spectroscopic time series observations of S Scuti stars have been few. Yang et ai. (1982), Yang & Walker (1986a), and Yang, Walker, & Bennett (1987) obtained precise radial velocity measurements of three S Scuti stars using a technique which superimposed a calibration spectrum on the stellar spectrum. Campos & Smith (1980) and Smith (1982) studied the asymmetries of line profiles introduced by low-degree modes of oscillation and attempted to fit model profiles to the observations to identify the modes of oscillation. Line-profile variations were also discovered in the rapidly rotating S Scuti star UMa,  o1  En by Yang & Walker (1986b) and in o En, 21 Mon, v  Boo by Walker, Yang, & Fahlman (1987). In these stars, variations appear as a  series of bumps which travel through the proffles from blue to red. Similar features have been observed in 0- and B- type stars and are usually attributed to high-degree nonradial oscillations. Using a method of trial and error to fit the observations with synthetic line proffles generated with a geometrical model, Kennelly et al. (1991) confirmed that the variations observed in  Boo could be reproduced by high-degree oscillations. However,  the fact that less satisfactory results were obtained for the other, more complex stars (Kennelly 1990) reflected the shortcomings of the trial-and-error approach to line-proffle fitting as a method of mode identification.  Chapter 1. The Oscillations of S Scuti Stars  10  Although progress in. S Scuti research has been significant in the last several years, many puzzling aspects remain. Not all stars lying within the bounds of the S Scuti instability region appear as photometric variables. Only one-third of the stars are variable with amplitudes greater than 0.01 magnitude (Breger 1979). It is usually assumed that most or all stars within the strip are variable but with amplitudes below the sensitivity of present day techniques. This idea is supported by the almost exponential increase in the number of variable stars with decreasing amplitude. The identification of the factors which determine the amplitudes of the variations is one of the greatest problems confronting S Scuti research. Statistical studies indicate that large amplitude radial pulsations are present only in slowly rotating stars and that rapid rotation tends to support nonradial pulsations especially in main sequence stars. The nonvariabi]ity of some chemically peculiar A stars is believed to be caused by the depletion of He from the ionization zone by gravitationally induced diffusion (and a corresponding increase in metals supported by radiation pressure) resulting in stability against pulsation. However, this theory does not account for variations in other chemically peculiar stars. Other factors including age, binarity, and atmospheric structure have also been suggested as factors which might influence the stability of the S Scuti stars. 1.4  Stability Analysis of S Scuti Stars  The stability of S Scuti Stars has been studied by several authors during the past two decades. In the seventies, the variable stars lying in the lower instability strip were thought to belong to two distinct classes which at that time were identified as S Scuti and dwarf Cepheids (sometimes called Al Velorum stars or RRs stars). The S Scuti stars were those which exhibited low amplitudes and seemed to be Population I stars undergoing normal evolution ofF the main sequence. The dwarf Cepheids were those  Chapter 1. The Oscillations of S Scnti Stars  11  stars which exhibited one or two frequencies of oscillation with large amplitudes. It was suspected (at that time) that the dwarf Cepheid stars were Population II objects but the masses and evolutionary stage of these objects were very uncertain. In the case of doubly periodic stars, it was observed that the ratio of the oscillation frequencies could be used to identify the modes of oscillation. For example, the period ratio between the first harmonic and the fundamental mode was determined observationaily to be 0 /P 1 P while the period ratio for the next two harmonics was 1 /P 2 P  =  =  0.77  0.81. Early theoretical  studies attempted to explain the mechanism of pulsation in these stars and to determine the evolutionary stage(s) of these objects by matching the observed period ratios in dwarf Cepheids. The excitation of radial oscillations was studied by Chevalier (1971), Petersen & Jorgensen (1972), and Stellingwerf (1979). (Stellingwerf gives a more complete list of references.) The evolutionary models constructed by Chevalier (1971) with masses of 1.8 and 2.0 Me and a composition consistent with Population I stars were found to be unstable for both the fundamental and first overtone pulsations.  Driving for the  oscillations was produced by the ,-mechanism acting at the lower edge of the second Helium ionization zone. Upon maximum compression the flow of radiation is limited by the increase in opacity  (,)  resulting from the ionization of Helium. The period ratio of  /P 1 P the oscillations was found to be 0  =  0.756, although this value was not recognized  to be in disagreement with the observations. Petersen & Jorgensen (1972) found that the fundamental mode and at least three overtones were unstable to oscillations and they argued that the period ratios of the dwarf Cepheids were consistent with population II compositions. Cox et al. (1978) suggested that the depletion of helium due to diffusion in the outer layers could explain the observed period ratios. Stellingwerf (1979) showed that modes as high as the fifth overtone are unstable. In fact the growth rate for these high overtone pulsations was found to be larger than for the fundamental mode even in regions  Chapter 1. The Oscillations of S Scuti Stars  12  of the instability strip where only fundamental oscillations are observed. Three zones of driving were identified in Stellingwerf’s models, associated with the hydrogen and the two helium ionization zones. The period ratios derived by Stellingwerf eliminated the need to invoke population II compositions to explain the observations (though a few stars, e.g., SX Phe, were still recognized as population II). About a decade later, the discrepancy between the observed and theoretical period ratios had still not been resolved. Andreasen & Petersen (1988) calculated that theoretical period ratios of both the S Scuti stars and the classical Cepheids could be brought into agreement with observations with an overall enhancement of the standard Los Alamos opacities. New opacities were calculated which more or less confirmed Andreasen and Petersen’s predictions. Meanwhile, more and more frequencies were being detected in the low-amplitude S Scuti stars which could only be explained by the existence of nonradial oscillations. The stability of S Scuti stars against nonradial pulsations was examined by Dziembowski (1977), Fitch (1981), and Lee (1985). However, the calculations by Fitch resulted in oversimplified frequency spectra due to an erroneous application of interior boundary conditions (Dziembowski & Królikowska 1990). For those S Scuti stars at the beginning of their evolution on the main sequence, adiabatic calculations are sufficient to determine the frequencies of nonracial oscillation. Slightly evolved stars have a centrally condensed structure and nonadiabatic effects should be considered.  However, the behaviour of  evolved S Scuti stars differs from that of the Cepheid variables where the degree of central condensation in the core is far greater and radiative dissipation strongly damps gravity modes in the core. For S Scuti stars, radiative dissipation is not enough to inhibit the existence of gravity modes with short wavelengths in the core (Dziembowski 1977, Lee 1985). Dziembowski (1977) found that the ic-mechanism can drive nonradial oscillations in a great variety of stellar models. An evolutionary sequence of three S Scuti models was  Chapter 1. The Oscillations of S Scuti Stars  calculated for a. star with mass M  =  13  . For less evolved models, pressure modes 0 1.5M  with a wide range of 1-values were shown to be unstable with frequencies bounded by those of the fundamental radial mode and first three overtones. Some high-degree modes (1  =  8 and 16) were shown to be partially trapped in the interior of the star. For more  evolved models, even the low-degree modes are observed to have a g-mode like behaviour; that is, the oscillations are free to propagate in the interior as well as near the surface. Among the low-degree modes, only those modes with £  =  1 and £  =  4 with frequencies  near those of the radial overtones are found to be well-trapped in the outer regions of the star. For all models, the most unstable modes are those with very high 1-values (1> 128) which are driven by the H I ionization zone. Oscillations with low 1-values (1 < 32) are driven by the He II ionization zone. However, since the theoretical growth rates for many of the radial modes are calculated to be small, whereas observations indicate that these modes are excited with large amplitudes, Dziembowski suggested that linear theory is insufficient to estimate oscillation amplitudes. Lee (1985) expanded on the work of Dziembowski, generating a model of a 2M star in the hydrogen shell burning stage. In agreement with Dziembowski, Lee finds that a number of radial and nonradial modes can be excited by the ic-mechanism. The most unstable radial mode is the fourth overtone. Low-degree nonradial modes with frequencies that are similar to the those of the radial modes correspond to pseudo p-modes which are partially trapped in the envelope of the star. These modes have growth rates which are comparable to the radial modes. However, for the most part, the oscillation spectrum of low-degree modes is composed of pseudo g-modes (which are mostly trapped in the interior) and modes with mixed character. The g-mode nature of the oscillations was linked to the gradient in chemical composition left behind by the shrinking convective core. The Brunt-Vä.isä1. frequency increases substantially in the ji-gradient zone and g-modes can be partially trapped there. On the other hand, high-degree modes can be  Chapter 1. The Oscillations of S Scuti Stars  14  distinguished as either pseudo p-modes or pseudo g-modes because the evanescent zone separating the two propagation zones becomes sufficiently thick that modes are effectively trapped in one region or the other. (For illustrations, see Section 15 of Unno et al. 1989.) 1.5  The Problem of Mode Selection  If so many modes are theoreticafly unstable in S Scuti stars while relatively few are ob  served, what mechanism determines the amplitudes of oscillation in these stars? In more luminous stars (RR Lyraes and Cepheids), the amplitudes of oscillation are limited by the saturation of the ,-mechanism. Ste]lingwerf (1980) showed that if low-degree modes were to saturate the driving mechanism in S Scuti stars then variations with amplitudes larger than one magnitude would be observed. Clearly this never occurs. It is also unlikely that driving is saturated by a large number of high-t modes since the existence of these modes would be observed as substantial macrotubulent broadening in the absorption proffles of the stars. The correct mechanism seems to be that suggested by Dziembowski (1980). Nonlinear coupling of the acoustic modes with gravity modes can limit the growth of the acoustic modes in the outer envelope. Only those modes which are well trapped in the envelope are observed. The details of this problem were studied in a series of papers by Dziembowski (1982), Dziembowski & Królikowska (1985), Dziembowski, Królikowska  & Kosovitchev (1988), and Dziembowski & Kr6likowska (1990) and has been reviewed recently by Dziembowski (1988) and Dziembowski (1990b). If Dziembowski’s theory is supported by observational tests, the task of mode identi fication could be greatly simplified for the evolved S Scuti stars. The implications if the theory is not affirmed are dim. Some stochastic process may be involved in the selection of modes. The situation for the main sequence variables is not so formidable because these stars have simpler theoretical frequency spectra and are therefore more favourable  Chapter 1. The Oscillations of S Scuti Stars  15  for mode identification.  1.6  Seismology of S Scuti Stars  -  A Test for Convective Overshooting  As an illustration of the seismic potential of S Scuti stars consider the problem of convec tive overshooting. The cores of stars more massive than 1M 0 are convective (Kippenhahn & Weigert 1990). Complete mixing of elements is assumed to occur in this region. Buoy ancy forces accelerate elements of gas up to the boundary of the core. Braking occurs only after the boundary is crossed and therefore inertia can carry elements past the border of the core. Simple estimates of this process indicate that the acceleration within the core is considerably smaller than the magnitude of the braking beyond it. In this case, the core boundary is a “hard wall” and overshooting elements are stopped within a negligible fraction of a pressure scale height. However, the situation changes if the effects of the overshooting on the thermal structure of the star are considered. Rising elements of gas which overshoot the core boundary are cooler than their surroundings and will increase the temperature gradient. This increase in temperature gradient decreases the braking force allowing elements to penetrate farther into the region of stability. The amplitude of this effect is highly uncertain since the calculations depend on arbitrary parameters. 0 model by Maeder (1975) indicate that elements are accelerated Calculations for a 2M to 30 or 40 m s 1 in the core and overshooting can reach amplitudes of up to 14% of a pressure scale height. This corresponds to a 30% increase in the mass of the core. As the star evolves, mixing introduced from beyond the core boundary increases the luminosity and the lifetime of stars. Recent calculations by Mowlavi & Forestini (1994) suggest that convective overshooting in 2.5M stars extends main sequence lifetimes by 20% and widens the main sequence of the HR diagram by Slog Tef ,e  =  0.04. Although present day  wisdom suggests that convective overshooting is important, a better theory of convection  Chapter 1. The Oscillations of S Scuti Stars  16  is needed to completely resolve the problem. The seismology of S Scuti stars could be used to measure the extent to which convec tive overshooting affects stellar evolution. Gravity modes can be trapped in the chem ically inhomogeneous region of evolved stars left behind by the shrinking core. Dziem bowski & Pamjatnikh (1991) have shown that the determination of the frequency of a particular low-degree gravity mode can provide strong constraints on the amplitude of the overshooting. 1.7  Seismology of S Scuti Stars  -  The Internal Rotation Profile  As a second example of the applicability of S Scuti seismology, the effect of frequency splitting by stellar rotation is revisited. Recall that modes with different values of £ probe the star to different depths. The dependence of the amplitude of rotational splitting is also 1-dependent (see Equation [1.3]). Therefore, observations of rotational splitting for modes of different degrees can be used to determine the structure of the internal rotation (e.g., Brown 1991). In practice, the identification of rotational splitting in stellar frequency spectra is rarely used for anything more than a method of mode identification. For example, the frequencies of oscillation observed in the S Scuti star 1 Mon have been interpreted as a rotationally split £  =  1 mode. In this case, the frequency spacing between the modes  are nearly equal, which would be the case if only first order effects were important. However, Saio (1981) showed that for rapidly rotating stars, the second order effects of rotational splitting can be significant. (The effects of tidal splitting due to a close binary companion were also shown to be non-negligible.) Saio applied his calculations to a polytropic model of a star with  ii =  3 and ‘y  =  5/3, appropriate for massive stars in  which radiation pressure dominates. A relation of the form  Chapter 1. The Oscillations of S Scuti Stars  v  =  —  (1  —  17  vo(..)2 2 Ci)m1 + C  [1.7]  was derived. In this equation, the coefficient C 2 is dimensionless and includes both rotational and tidal terms which are proportional to m 0 and m . Values of C 2 1 and those coefficients making up C 2 were calculated for the polytropic model for P1- to p -modes 6 with £  =  1,2, and 3. The results were tabulated against values of the dimensionless  frequency w 2  =  v / 3 R 2 GM where M and 1? are the mass and radius of the star. Saio  applied his calculations to the observations of three early-type stars and attempted to identify the modes of oscillations from the observed frequency spectrum. The asymmetry introduced by the second order effects on frequency splitting were shown to be significant in these stars and moderate success at the identification of oscillations modes was achieved for two of the three stars. Although Saio recognized the existence of frequency splitting in S Scuti stars, he did not apply his calculations to any stars of this type. From multisite observations, five frequencies were observed in the S Scuti star, GX Peg (Michel et al. 1992). Goupil et al. (1993) showed that the frequencies could be matched with theoretical models which included overshooting and the new OPAL opacities if the frequencies corresponded to n I  =  1, n  =  =  2 and n  =  3 radial modes, plus a rotationally split  3 nonradial mode. The identification of £  =  0 and 1 modes is consistent  with the suggestion of Dziembowski (1990b) that only those modes effectively trapped in the envelope are observed. For those models which included convective overshooting, this is true of the n  =  3, £  =  1 mode, while the n  =  1 and 2, £  =  1 modes which have  frequencies within the observed range are not well trapped in the envelope. OX Peg is a member of a close binary system and tidal synchronization, at least in the outer layers has likely occurred. The rotational splitting between the £  =  1 modes  provides a measure of the rotation rate within the star. Using the results of Saio (1981),  Chapter 1. The Oscillations of S Scuti Stars  18  Goupil et al. find that GX Peg cannot be rotating as a solid body. Instead, the rate of rotation inside the star is more rapid than at the surface. 1.8  Goals for this Thesis  The primary goal of this thesis was to exploit the information contained within Doppler broadened line profiles to study both the frequencies and modes of oscillation in S Scuti stars by direct observation. The possibility of identifying both low- and high-degree modes from line-profile variations will be investigated. Although models for high-degree oscillations are not generally available, the predictions made by Dziembowski and Lee concerning the trapping of nonradial modes in the envelope of the stars could be tested with observations of high-degree oscillations. So far relatively few stars have been ob served for the presence of high-degree variations. A comparative study of the properties of the line-profile variable S Scuti stars could provide additional insights into the physics of this class of stars. In Chapter 2, a method of analysis for the detection of high-degree oscillations from line-proffle variations is introduced and tested using numerical simulations. In Chapter 3, the results from a survey of line-profile variability among rapidly rotating stars in the S Scuti instability strip is presented. The variations in a few of these stars are examined in detail in Chapters 4, 5, and 6. The observations obtained for this survey lack the frequency resolution to detect all the modes of oscillation which may be present in these stars. In Chapter 7, observations of  92  Tauri obtained during a multisite spectroscopic  campaign are presented and analyzed together with previously published frequencies  derived from two extensive photometric campaigns on this star.  Chapter 2  The Identification of Oscillation Modes  2.1  Methods of Mode Identification: A Summary  The identification of frequencies of oscillation and the modes that give rise to them is the key to progress in stellar seismology. Quite often the realization of this goal requires more data than can be easily obtained by a single observer. To resolve closely spaced frequencies, the observations must be of sufficient duration to sample the beat period of the modes. Many nights may be needed to resolve all the frequencies. However, observations obtained from a single site are often insufficient to study multiperiodic stars because daily gaps in the data introduce a one-cycle per day ambiguity in the choice of the frequencies. Once a frequency is misidentified, the determination of additional frequencies becomes more and more unreliable. For example, Jones et al. (1993) determined six frequencies in  Boo based on photometric observations during 13 nights at a single site.  However, from a more careful analysis of this data, Kurtz (1994) verified the presence of only three frequencies. The problem with aliases can be combated with the organization of international campaigns. With telescopes around the world following the same star, nearly continuous observations can be obtained if the weather cooperates at all sites. Campaigns to obtain simultaneous photometry have been very successful because large amounts of observing time can be obtained with relative ease on smaller telescopes. Being a measure of the disk-integrated light, photometry is capable of measuring the frequencies of low-degree  19  Chapter 2. The Identification of Oscillation Modes  modes (0  £  20  3). Indirect methods of mode identification must be used to determine  the degree £, azimuthal order m, and radial order n of the oscillations from the pho tometry. Some of these methods have been discussed in Chapter 1. A summary of the various methods of mode identification is presented in Table 2.1. Often unambiguous identifications of oscillation modes can not be made because of three complications in herent to the S Scuti stars themselves: the frequencies of modes resulting from different radial orders may overlap; not all theoretically unstable modes appear with observable amplitudes; and in evolved stars, both pressure- and gravity-modes may be excited with similar frequencies. Spectroscopic observations can provide additional information about the modes of oscillation. For example, a precision in radial velocity of ±50 m s 1 is equivalent to a precision of ±0.mOOl in photometry (Yang 1991). Highly accurate differential velocities, down to  .—‘  10 m s , have been measured in stars with high-resolution spectra and using 1  methods of precise calibration (e.g., Campbell, Walker, & Yang 1988). If simultaneous spectroscopy and photometry is obtained, then empirical results suggest that the ratio of the amplitudes between velocity and light indicates whether a given mode is radial: 2K/11m  80 km s mag 1 or nonradial: 2K/Im,,  55 km  1 (Yang 1991). mag  A direct measure of the modes of oscillation can be obtained from studies of spectro scopic line-profile variations in rotating stars. Velocity variations on the stellar surface perturb the rotation proffle of the star and introduce traveling bumps into the rota tionaily broadened absorption proffles. Several methods have been developed to extract information about the modes from the observations of variable profiles in 0-B and S Scuti stars. The method of fitting line-profiles with synthetic data can produce convinc ing results but the method usually fails if the variations are very complicated (Kennelly 1990). Balona (1986a, 1986b, 1987) showed that the variations in a line profile due to  Chapter 2. The Identification of Oscillation Modes  21  Table 2.1: Methods of Mode Identification  LIGHT & RADIAL VELOCITY VARIATIONS: LIMITATIONS: (0 £ 3); Only those modes which produce large disk integrated variations. Q-values & Model Calculations Period Ratios Colour Phase Lags Velocity-to-Light Ratios Frequency Splitting  -  -  -  -  -  the radial order of low-degree modes radial modes low-degree modes can distinguish radial from nonradial modes the 2t + 1 values of m  LINE-PROFILE VARIATIONS: LIMITATIONS: (0 £ 4,); Where £moe depends on the rotational broadening and the intrinsic width of the proffle. Typically, £ma can be as high as 10 to 20. Proffle Fitting The Moment Method Fourier Doppler Imaging  -  -  -  simple spectra of low- and high-degree modes low-degree modes low- and high-degree modes  low-degree oscillations can be described in terms of the variations in the first few mo ments of the line profile. The zeroth moment is the equivalent width (a normalization factor). The first moment is the radial velocity. The second moment reveals variations in the width of the line. This method is useful for studying low-degree modes of oscillation but the amplitudes of the moments decrease rapidly for higher degrees. Observations cannot be obtained with a sufficient signal-to-noise ratio to make the method practical for high-degree modes. The method most commonly used for the study of 0 and B stars was introduced by Gies & Kullavanijaya (1988). In their method, a Fourier analysis is  Chapter 2. The Identification of Oscillation Modes  22  performed on the temporal dependence of the variations to determine the frequencies of oscillation and the modes are calculated from the phase variations of the individ ual Fourier components across the line profile. This type of analysis has been dubbed “Fourier Doppler Imaging”. An extension of this technique is introduced in Section 2.4. To be effective, investigations require both high-time and high-spectral resolution with good signal-to-noise. For S Scuti stars, exposures times are limited to about 10 minutes in order to provide sufficient phase coverage of the variations. The instrumental resolution must be comparable to the intrinsic resolution of the star (i.e., about 5 km s FWHM for metailic lines) or at least be much smaller than the projected rotational velocity of the star under investigation. The amplitude of line-profile variations is generally less than one percent of the continuum which means that the spectra must have signal-to-noise S/N> 100 in order to “see” the variations. For these reasons, only the brightest S Scuti stars have been examined for spectroscopic variations and very extensive campaigns to obtain spectroscopic observations are not often attempted. The observing constraints for the detection of line-profile variations in 0 and B stars are slightly more relaxed since these stars rotate more rapidly and have longer periods of variation. Multisite spectroscopic campaigns lasting about a week have been successfully organized for the study of these objects (e.g., Reid et al. 1993). 2.2  Fourier Analysis  Many methods for extracting frequencies from time-variable data have been devised for astronomical purposes (e.g., see Fullerton 1986). In this thesis, only two techniques are of immediate interest: Least Squares Fitting and Fourier Periodogram Analysis. For both techniques, the observations are assumed to be described by sinusoidal functions specified only by a period, amplitude and phase. The fact that the two methods are  Chapter 2. The Identification of Oscillation Modes  23  mathematically equivalent has been shown by Deeming (1975). The Fourier transform f(v) of a function F(t) is defined as  f  f(v)  F(t) 2 e d 4 ” t.  [2.1]  Now, if F(t) is a measure of some time-variable quantity from a star, N  F(t)  A sin(2irv (t 2  —  ti))  [2.2]  with N frequencies v, amplitudes A, and phases 2 27rv t , then the Fourier transform of F(t) is given by N  f(v)  AS(v  —  v)  [2.3]  =  where Sj is the Dirac delta function which has the property that Sj  =  0 for v  . In 2 v  practice, real observations are obtained during a finite interval of time. In this case, the data D(t) can be described as  D(t)  =  F(t) W(t). .  [2.4]  For example, if W(t) is the box function described as  W(t)=1  —<t<.  [2.5]  Tsinc(wTv)  [2.6]  W(t)=O then the Fourier transform of this function,  w(v) =  is called the window function and  TsinirTv irTzi  =  Chapter 2. The Identification of Oscillation Modes  d(v)  =  f(v) * w(v) =  L  f(t’)w(v  24  z))di”  [2.7]  will be the Fourier representation of the data. This correspondence is called the convo lution theorem. The convolution of f(v) with tv(v) means in effect that every frequency z’ will have associated with it, the pattern described by w(v). Therefore, the ability to resolve frequencies is determined by the duration of the observations. Astronomical observations 1 t  (j  =  are  usually obtained at a finite number of discrete times  1,2, .M). In this case, by analogy with the continuous Fourier transform, the . .  discrete Fourier transform is defined as M  f(v)  ) 1 F(t  e2TI/tj.  [2.8]  :1=1  Note that this equation is dimensionally different than Equation 2.1 since the dt term has been dropped. If At  =  t  —  t_ is a constant, a special algorithm known as the FFT  (Fast Fourier Transform) can be used to calculate f(v). Otherwise f(v) can be evaluated directly by writing M  f(v)  =  )[cos(2rvt) + i sin(2rvtj)] 1 E F(t  [2.9]  :1=1  and treating the real and imaginary parts separately. It is common to finely sample  f(v)  in order to better identify the frequencies. For equally spaced data  VNyq  1/2At  [2.10]  defines the upper limiting frequency, called the Nyquist frequency, above which no infor mation about the variations can be recovered. For data which isn’t equally spaced the generalized Nyquist frequency is usually adopted (Scargie 1982) where At is the mean spacing between observations.  Chapter 2. The Identification of Oscillation Modes  25  Variations in the data appear as peaks in the Fourier periodogram which is defined as  P(v)  =  [(Re[f(v)])2 + 2 (Im[ffr)]) ] .  [2.11]  and has been normalized to account for power at both positive and negative frequencies. The amplitude spectrum is defined as the square root of the periodogram and is preferred in this thesis because it does not downplay the noise with respect to weak signals and translates immediately into an empirical quantity (the amplitude of variation). 2.3  Analysis of Radial Velocity Variations  The velocity variations associated with low-degree modes produce Doppler shifts and dis tort the absorption profiles of a star. Radial velocity variations correspond to variations in the first moment of the line profiles, i.e., variations in the positions of the lines. For a given velocity amplitude on the surface of the star, radial modes of oscillation produce the largest variations in radial velocity. The amplitudes of variation arising from nonra  dial modes are smaller because of cancellation between oppositely phased variations on the surface. Modes with .( > 3 produce no detectable radial velocity variations. To a good approximation, the radial velocity variations of low-amplitude S Scuti stars may be described as sinusoidal. Accurate measurements of radial velocity variations can be made by cross-correlating the position of a given line profile in the target spectrum with that of a template spec trum (e.g., Tonry & Davis 1979, Fahiman & Gla.spy 1973). With this type of method, the relative shift between the two proffles is determined by the maximum in the corre lation function plotted against the relative velocity separation. For variable star work, the mean spectrum generated from a series of observations may be used as the template  Chapter 2. The Identification of Oscillation Modes  26  spectrum. Very precise measurements of radial velocity may be made with this technique by increasing the number of spectral lines in the calculation. The uncertainty in deter mining the position of a line is inversely proportional to the strength of the line profile but in the calculation, the lines are weighted by the square of their strength (Tonry & Davis 1979). To determine the frequencies, amplitudes and phases of the oscillations, the method of Least Squares Fitting has proved very effective. For example, the program PERDET, written by Breger (1980a), is based on the method of nonlinear least squares by Bevington (1969) and can be used to fit up to seven frequencies simultaneously. Using a method of prewhitening, frequencies can be determined one by one by subtracting successive multisinusoidal fits from the data. Initial guesses for the frequencies, etc. may be determined from peaks in the Fourier amplitude spectrum of the data (or residuals). Alternative methods carry out the prewhitening in the Fourier domain by making a. sliding fit of the window function to the peaks in the periodogram (Wehlau & Leung 1964, Gray & Desikachary 1973). 2.4  2-D Fourier Representation of Line-Profile Variations  In the case of the Sun, disk-resolved measurements of the intensity or velocity variations can be obtained and translated directly into the desired spectrum of frequencies and spherical quantum numbers  (ii,  £, m) by decomposing the pattern of surface variations  into spherical harmonics and making a Fourier transformation in time. Stars cannot be resolved in the same way as the Sun but through techniques of Doppler imaging some of the spatial information about surface variations can be recovered. On the apparent disk of a rotating star, strips parallel to the projected rotation axis will have the same line-of-sight velocity and produce the same Doppler shift. The maximum shift occurs at  Chapter 2. The Identification of Oscillation Modes  27  the limb of the star and equais ±RC sin i, where R is the radius of the star, 11 is the rotational frequency of the star, and i is the inclination of the rotation axis from the line of sight. For rapidly rotating stars, a one-dimensional image of the stellar surface is mapped into each absorption line-profile. The ability to resolve features on the surface of stars is determined by the rate of rotation, the inclination, and the intrinsic width of the absorption line. Features confined to the surface of the star (such as star spots or chemical peculiar ities) appear as bumps or dips in the line-profiles which travel at a rate determined by the period of rotation of the star. The same pattern of line-profile variation is repeated after one revolution of the star. Knowing this period and with an estimate of the radius of the star, the inclination can be determined. Since the rate of variation is dependent on the latitude of the feature, inversion techniques may be used to convert the line-profile variations to maps of the stellar surface (e.g., Rice, Wehlau, & Khokhlova 1989, Donati et al. 1992). Nonracial pulsations are not necessarily fixed on the surface of the star the way spots are but move at a rate different from that of rotation. The measurement of the rate of travel of features in the line profiles can provide only a measure of the combined rotation period and wave period in the equatorial direction. If more than one mode is present then waves will travel at different rates and an inversion similar to that applied to surface features is no longer possible. The surface velocity field generated by spheroidal nonradial pulsations described in terms of spherical coordinates (r, 8,  V(6, 4’, t)  =  4’)  and time t is  9 j.p 4 0 , kV 0 (cos 8)ei(m+) m , kv 0 1v  [2.12]  where V 0 is the velocity amplitude of the oscillations in the radial direction, k is the  Chapter 2. The Identification of Oscillation Modes  ratio of the horizontal to radial velocity amplitude,  28  ii  is the frequency of oscillation, and  Pj”(cos 0) are associated Legendre polynomials of degree £ and azimuthal order m. The latitudinal dependence of the oscillation amplitude is described by the associated Legendre functions Fe (cos 9). If (for the moment) the assumption is made that for the m stars observed, the rotation axis is nearly perpendicular to the line of sight, then the largest contribution to the line-profile variations may be assumed to result from sectoral modes (1  =  ml).  Imi  (Modes with £  suffer from cancellation from oppositely directed  variations on either side of the equator.) The amplitude of a sectorai mode is largest at the equator and will decrease to zero at the poles. The observed variations, therefore originate from the region located close to the equator of the star. (For non-magnetic, rotating stars the axis of oscillation is assumed to be aligned with the axis of rotation.) Therefore, to a good approximation the radial velocity field of an oscillating star may be represented by  V  =  [2.13]  This equation is of a form which naturaily lends itself to a Fourier analysis in both the temporal and spatial dimensions, analogous to the analysis of the solar oscillations. Assuming again that the star is observed equator-on (sin i  =  1), the time-varying  component of the rotationally broadened profiles (i.e., the residuals created by subtract ing an average profile) can be interpreted as Doppler images of (t  =  Im?)  equatorial  oscillations. In velocity coordinates, the edge of the star corresponds to ±v sin i which can be determined, for example, by fitting rotationally broadened synthetic profiles to the observations. To express these variations in a form consistent with Equation [2.13], the residuals within the proffle can be mapped onto a coordinate system corresponding to stellar longitude  cbj  using the relation  Chapter 2. The Identification of Oscillation Modes  =  29  sin_1(3.)  [2.14]  where v corresponds to the velocity of a given wavelength with respect to the center of the line. The window for the observations can be described by W(c, t)  =  S(4) T(t). With  respect to time, T(t) is the box function described by Equation 2.5. However, in the spatial direction, the amplitude and resolution of the variations decrease in the wings of the line proffles due to foreshortening at the limb of the star. Although the exact functional form of S(c) is not specified, its affect on the Fourier amplitude spectrum can be anticipated. Power within the sidelobes of the window function will be decreased at the expense of increasing the width of the central peak. In the spatial domain, the  resolution is already restricted by the fact that only half of the star can be observed. Therefore, excluding the effects due to foreshortening, the resolution (i.e., the full width at half maximum of the central peak) is at best Im  =  1/0.5  =  2. Using simulated  line-proffle variations to account for the spatial window function, the effective resolution will be estimated numerically in Section 2.5. Based on these principles a method of line-profile analysis was devised: the TwoDimensional Fourier Analysis wherein the line-profile variations are transformed in time and ‘Doppler space’ to Fourier space, where an amplitude spectrum expressed as a func tion of apparent frequency z) and apparent azimuthal order  is calculated from the real  and imaginary parts of the two-dimensional transform. The calculation is analogous to that of Equation 2.9 with the transforms in the temporal and spatial domains calculated in succession. Thus, if the time-variable component of the line proffles is written as G(#, t) with M and M observations in space and time then the 2-D Fourier transform g(ii, b) is calculated in two steps as follows:  Chapter 2. The Identification of Oscillation Modes  30  Mt  g’(, i))  ) [cos(2irbt 1 ) + i sin(2irC’tj)] 3 G(cj, t  [2.15]  g’(4, i)) [cos(2Trth4) + i sin(2irriiq!)].  [2.16]  =  j=1 M,  g(th, =  The order of the calculation is not important. The two-dimensional periodogram be comes:  =  2 + (Im[g(th, b)]) [(Re[g(th, b)]) ] 2  [2.17]  from which both the temporal and spatial frequencies of the modes can be determined. The Two-Dimensional Fourier technique is especially attractive because it simplifies the otherwise complex patterns in the proffle variations by decomposing them into Fourier components and enables the possibility of directly determining both frequencies and the modes of oscillation simultaneously. The advantages and limitations of the method are explored in the following section. 2.5  Demonstrations of the 2-D Fourier Technique  The line-profile variations resulting from a rotating, oscillating star were simulated using a numerical model. The NRP-program (written at UBC prior to this thesis) describes the oscillations of a star in terms of the spherical harmonics defined in Equation 2.12. In previous work, a version of this code was used to model the observed line-profile variations of  2 ic  Bootis (Kennelly et al. 1991) by a method of trial and error. A more  detailed description of the program can be found therein. Extensive testing of the code showed it to be consistent with the simulations of Kambe and Osaki (1988). The program was used to calculate synthetic profiles on which to demonstrate the Fourier technique.  Chapter 2. The Identification of Oscillation Modes  A model was calculated with mass, M velocity at the equator was (initially) to be i  = 900.  prograde, sectoral mode (t  =  =  0 and radius, R 2M  31  =  3R. The rotational  120 km s and the inclination of the star was taken  The star was made to oscillate with a single, high-degree, =  10, m  in the radial direction was low, V  =  —10). The velocity amplitude of the oscillations  =  2.5 km s and the amplitude in the horizontal  directions (V 9 and Vqs) were set to zero. The frequency of oscillation was v 0 day’ in the co-rotating frame (ii  =  =  17.0 cycles  24.9 cycles day 1 in the frame of the observer if second  order effects are neglected, see Equation 1.3). A series of snapshots was calculated with intervals in time chosen to reflect those of typical observations of S Scuti stars. (The intervals of time actually correspond to those of the observations of r Pegasi presented in Chapter 4). The simulated observations consisted of 38 profiles, lasted for more than 7 hours, and had a mean separation of about 12 minutes. Noise was not included in the initial simulations. The line-proffle variations produced with the NRP-model are presented in Figure 2.1. On the left-hand-side of this figure, the line proffles at each time interval are illustrated. In this diagram, time increases upwards and is indicated in days beside each proffle. On the right-hand-side, the residual variations after subtracting the mean proffle for the entire series are shown. The proffles are sampled by 51 points across the full width of the line, spaced at equal wavelength intervals. The data were transformed first to a velocity coordinate system, truncated at ±v sin i, then mapped onto the independent variable ç! according to Equation 2.14. With the data tabulated as a function of space and time  (,  t), the two-dimensional Fourier transform was calculated for frequencies between 0  1 and azimuthal orders between 0 and 50. The sampling in Fourier and 100 cycles day space was 101 points by 101 points. The results from these simulations are summarized in Figure 2.2 where, for the pur pose of display, the residual line-proffle variations and the resulting Fourier transform  Chapter 2. The Identification of Oscillation Modes  32  have been interpolated onto 128x128 grids. The line-profile variations are illustrated as a grey-scale plot in the top central diagram. To the left of this, a grey scale map of the line-of-sight velocity variations (excluding rotation) over the surface of the star is shown at an arbitrary phase. Note that the largest variations are indeed confined to the equatorial region for this sectorai mode. The top right diagram is the two-dimensional Fourier representation (amplitude spectrum) of the line-profile variations which was most effectively displayed as a contour plot. The figure has been normaJized to the maximum intensity, and levels are plotted at constant intervals from zero to maximum. As expected, the power is concentrated at an apparent azimuthal order near 10 and an apparent fre quency near 25 cycles day . The series of peaks which extend to higher and lower 1 frequencies are a reflection of the temporal window function of the simulations. No such pattern is evident in the direction of apparent azimuthal order. This is a result of the natural tapering of the variations towards the wings of the line profile (at the expense of broadening the central peak). In addition, a harmonic of the primary peak can be identified at approximately twice the frequency and azimuthal order of the largest peak. This secondary peak appears because the shape of the line-proffle variations is not ex actly sinusoidal. The lower two diagrams of Figure 2.2 illustrate as grey-scale maps, the one-dimensional Fourier representations of the line-profile variations calculated for both frequency (using Equation 2.15) and apparent azimuthal order. In Table 2.2, the observed position (ifl, 1’) and amplitude (a) of the primary peak is listed for this model and the ones that follow. The amplitudes are expressed as a fraction of the continuum intensity (but are related to the velocity amplitude of the input model). The frequencies have been determined with an accuracy of  0.1 cycles day 1  of the input value. The apparent azimuthal orders are measured to within  0.5 of the  input value. The full-width at half maximum of the central peak, in apparent frequency and azimuthal order, is zI’  =  3.3 cycles day’ (as expected) and  tiz. =  4. The large  Chapter 2. The Identification of Oscillation Modes  I  0297 r—  •  33  I••••I., I A  —1LlLV—————o.so5 —WW-—&297  —.--—0.29Q —-—0.290 /1—0276 —.\ —‘ l 26 j—O. __0.268 .253  —\  —J\flf.——-—-o 261  /,—o.z8  —\r’J,—-—0.2Z1 .207  —‘1\’LV--—o.228 —‘ifl---—0.221 —\\)‘--———o.214  185  —.—‘  ‘l V  j—O. ? 13 J30  \J  —‘I’iLLV———-—°•°  —AflY  .049  .030  _\f___0.038 .011  4505  4510  Wavelength  4515  4505  4510  4515  Wavelength  Figure 2.1: Simulated High-Degree Line-Profile Variations. The line-profile vari ations arising from a high-degree (1 = 10), sectoral oscillation mode were simulated using a numerical model. On the left, the series of line profiles is shown. The time of calcula tion (in days) is indicated beside each profile. The time-variable components of the line proffles were calculated by subtracting the average profile and are shown on the right.  Chapter 2. The Identification of Oscillation Modes  34  100  I  80  80 40 20 0 ••  ‘0  4’O  60  Apparent Azimuthal Order  ° 0 c 80  60 40  20 C..  tEjjE1w ..  10  ‘  .  •-0• )•0• Velocity (km  )  10  10  Apparent Azimuthal Order  Figure 2.2: Fourier Representation of High-Degree Line-Profile Variations. The line-proffle variations arising from a high-degree (( = 10), sectoral oscillation mode were simulated using a numerical model. The time-variable component of the profile variations are shown in the top middle diagram. The velocity map of the star at an arbitrary phase is shown on the top left. The Fourier representation of the line-profile variations is shown on the top right. The lower diagrams show the amplitude spectrum resulting from one-dimensional Fourier transforms in time and in space.  Chapter 2. The Identification of Oscillation Modes  35  value of th is a result of the limited information available in the spatial domain and the foreshortening which occurs in the wings of the absorption lines. The precision to which the oscillation modes can be identified will ultimately be limited by the width of Ath. (If the form of the spatial window function were known, it might be possible to correct for this projection effect. However such a correction would put increased weight on the information obtained from the line wings, where the signal-to-noise of the variations is very low. For these reasons, it was decided to accept the poor mode resolution.) The position of the primary peak in apparent azimuthal order, th larger than the expected value of  i =  =  10.5, is slightly  10. A bias towards large values of th was also  observed for other models with sectoral modes (not presented here). The difference between the observed and expected values of  ‘i  is usually larger than S  =  0.5 and  increases as I decreases. The origin of this discrepancy seems to be related to the way in which the surface variations are mapped into the line profiles. If the variations were due to temperature variations fixed to the surface of the star (e.g., spots), then only the resolution of the spots would be affected by the projection effects near the limb of the star. However, the line-proffle variations are introduced as perturbations of the rotational profile and result from a redistribution of light in accordance with the modified Doppler shift. In this case, the projection of the pulsation velocity along the line of sight is important and this also decreases towards the limb of the star (assuming oscillations in the radial direction only). Therefore, projection effects can affect not only the resolution but also the spacing of the bumps within the line profiles. The consequence of this is a measured value of  iz.  which is a little bit too large. It is because of this not quite  exact mapping of the modes that the observed values of i. are referred to as ‘apparent’ azimuthal orders. The effect of large oscillation amplitudes on the Fourier representation is demon strated in Figure 2.3. The top diagram of this figure shows the results from a model  Chapter 2. The Identification of Oscillation Modes  36  Table 2.2: Numerical Simulations of High-Degree Modes  Model Input  tm (c d’)  (c d—’)  V. (km s’)  ii  k  i (deg.)  Observed Output 1’ a (c d’)  10  -10  17.0  24.9  2.4  0.0  90  10.5  25.0  0.0053  10 10  -10 -10  17.0 17.0  24.9 24.9  10.0 2.4  0.0 1.0  90 90  10.5 8.8  25.0 25.0  0.0199 0.0178  10 10 10 10 10 10 10 10 10 10 10  -10 -09 -08 -07 -06 -05 -04 -03 -02 -01 00  17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0 17.0  28.2 27.1 25.9 24.8 23.7 22.6 21.5 20.3 19.2 18.1 17.0  2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4  0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0  45 45 45 45 45 45 45 45 45 45 45  10.8 10.5 10.6 10.5 10.5 10.5 10.6 10.6 10.8 10.8 10.8  28.0 27.0 26.0 25.0 23.5 22.5 21.5 20.5 19.0 18.0 17.0  0.0025 0.0024 0.0025 0.0024 0.0021 0.0019 0.0016 0.0013 0.0010 0.0008 0.0004  similar to the one in Figure 2.2 but with the velocity amplitude in the radial direction . The pattern of line-profile variations (center) and the map of 1 increased to 10 km s the stellar surface (left) look similar to those of the previous model, except the pattern of variation is now very non-sinusoidal. This sort of behaviour introduces a series of harmonics at integer multiples of the apparent frequency and apparent azimuthal order in the Fourier representation. The position and amplitude of the highest peak in this figure is listed in Table 2.2. Increasing the amplitude of the horizontal velocities (Vç, V ) has a more dramatic 9  Chapter 2. The Identification of Oscillation Modes  37  effect on the Fourier representation. The lower series of diagrams in Figure 2.3 shows the results for a model similar to the original one but with amplitudes: V  =  9 V  =  Vr  =  2.5  km s . In this case, a new pattern of line-profile variations emerges which give rise 1 to numerous peaks in the Fourier domain. Two peaks of comparable amplitude and identical frequency are observed on either side of the true value of the azimuthal order. In the next example showing the Fourier representation of high-degree modes, the inclination of the original model is changed to 45 degrees and a series of models with prograde modes and —10  m  0 were generated. (The value of the projected rotational  velocity was maintained at 120 km sl.) A few of these models are displayed in Figure 2.4. The results for all models are listed in Table 2.2. Apart from a shift in frequency due to the increased rotational velocity of the model and a diminished amplitude, there is very little difference between the Fourier representation of the first model of Figure 2.4 (with £  =  —m  =  10) and that of the original equator-on model. For the other models with  Imi, the results are more surprising; the value of the apparent azimuthal order more closely represents £ than it does Imi. (This effect was first pointed out by Dr. Douglas  £  Gough (1992).) With the help of the stellar velocity maps, it can be understood why this is so. With inclinations different from 90 degrees, a significant portion of the lower half of the star is hidden from view and complete cancellation between oppositely directed variations on either side of the equator (for ImI  =  £  —  2j + 1, where  j  is an integer)  never occurs. Instead, there is a net velocity perturbation along each line parallel to the axis of rotation. These perturbations introduce bumps in the rotationally broadened profile of the star, the number of which corresponds to the value of 1/2. Although the amplitude of the variations becomes smaller with decreasing  mI, for inclined  stars one  would expect to observe in Fourier space, a series of rotationally split peaks at roughly constant apparent azimuthal orders corresponding to modes with differing rn-values and equivalent degrees. This affect was seen in simulations with inclinations as large as  Chapter 2. The Identification of Oscillation Modes  38  1uIJ  B0  —  O 60 C? C?  = -  C  () 40  20 0  0  to  0  10 20 30 40 Apparent Azimuthal Order  6C  C 4? 0  r..  U  l=1O,m=-1O  L.  Velocity (km )  Apparent Azimuthal Order  Figure 2.3: Fourier Representation of Large Amplitude Variations. Two models were calculated to test the effects of amplitude on the representation in Fourier space. The top series of diagrams show the stellar velocity map (left), the radial line-profile variations (center), and the Fourier representation (right) for a model with surface radial velocity variations equal to 10 km s_L. The bottom series of diagrams show the same representations as above but for a model with horizontal velocity variations equal to the radial velocity variations at 2.5 km s . 1  Chapter 2. The Identification of Oscillation Modes  i  =  39  80. At even larger inclinations, power is transferred to the harmonic located at  (2, 2ii) as the resulting profile variations become more and more nonsinusoidal. Even at i  =  90 complete cancellation may not occur between out-of-phase variations on the  two hemispheres because the pattern of the line-profile variations may not be exactly sinusoidal. In theory, the ratio of power between (2th, 2i) and  (, 1’) provides a measure  of the inclination. However, the shape of the variations (and therefore this ratio) is also sensitive to the velocity amplitude on the stellar surface. In the next series of models, the possibility of identifying low-degree modes, those that can been seen photometricaily, from the Fourier representation is considered. Figures 2.5, 2.6, 2.7, and 2.8 illustrate the results of models generated with £  =  0, 1,2, and 3. Only  the prograde modes are shown but the results for all degenerate modes are provided in Table 2.3. Again, the apparent azimuthal order remains nearly constant for all values of m. However, for these modes the measured value of the degree is also very much larger than is the input value. For example, a radial mode of oscillation produces a peak at =  1.76. This can be understood in the following way. Radial oscillations cause no  distortion of the line profile but rather introduce a shift in wavelength. As line-proffle variations, they introduce S-shaped variations since the amplitude of the variations must decrease to zero where the edge of the proffle meets the continuum. In Fourier space, this produces a peak near  =  1.76. The values of the degree for other modes are similarly  determined to be too large but the difference between the expected and measured values decreases with increasing degree. As a result, the difference between the measured values of the apparent azimuthal order resulting from modes separated by one degree is less than one. Therefore distinguishing between low-degree modes with the Fourier method is intrinsically more difficult than identifying high-degree modes. However, it should be possible to at least distinguish say £  =  3 modes from £  =  0 or 1. The identification of low  degree modes might be aided further by the fact that the amplitude of the line-proffle  Chapter 2. The Identification of Oscillation Modes  40  80 60  g.  40  0  10 20 30 40 Apparei Azimuthal Order  0  10 20 30 40 Apparent Azimuthal Order  6  80 60  i  40 20  lOG •a  80 60  20  .0  1..  10 20 30 40 Apparent Azimuthal Order  -0 )) Velocity (km 4’)  6  a. ‘U  •  0.0  I  =  10, m =0  I:  4 * 4 0 Velocity (km s’)  Apparent Azimuthal Order  Figure 2.4: Fourier Representation of Inclined stars with —t m 0. The same series of diagrams as in the previous figures is presented for stars with and inclination of 45 degrees and for m = —10, —9, —6,0.  Chapter 2. The Identification of Oscillation Modes  41  50  I  40 U  30  I 20 10  b.i Velocity (km )  1=0  io  n  0  5  10  15  20  25  Apparent Azimuthal Order  Figure 2.5: Fourier Representation of Radial Oscillations. The surface velocity map, residual line-profile variations, and the Fourier representation are illustrated for an £ = 0 mode of oscillation. variations resulting from sectoral modes increases with increasing degree for the same surface velocity amplitude, whereas the opposite behaviour is true of the radial velocity variations. Finally, a model with five modes of oscillation present was constructed and its Fourier representation was calculated from the time series of line profiles. All five modes were sectora.l with £  =  0,3,6, 10, and 16 and had the same frequency of 17 cycles day 1  in the corotating frame of reference. The inclination of the star was i  =  45°. All other  parameters were the same as above. The line-proffle variations resulting from this model, shown in Figure 2.9, produce a very irregular pattern. It is not possible to identify the modes of oscillation by counting the bumps present in any given profile. However, with the 2-D Fourier technique it is possible to recover all the modes of oscillation. Each mode is represented by one of the large peaks in the Fourier representation shown in Figure 2.10. These results are summarized in Table 2.4. The apparent frequencies agree well with the input values and the apparent modes agree well with their expected values. Although the amplitudes of the surface oscillations are the same for all modes, the measured amplitude  Chapter 2. The Identification of Oscillation Modes  42  50  30 20  0  0  0 o  10 I  •  .1  5 10 15 20 Apparent Azimuthal Order 50 40 C,  30 C  a  20  .110  1=1, m  =  0  Velocity (km )  5 10 15 20 Apparent Azimuthal Order  25  Figure 2.6: Fourier Representation of £ = 1 Oscillations. The surface velocity, residual line-proffle variations, and the Fourier representation are illustrated for £ = 1, m = —1, and 0 modes of oscillation.  Chapter 2. The Identification of Oscillation Modes  43  40 30 C  2O  to  5 15 20 10 Apparent Aztmuthal Order  440  j. 30 C  a  g.  20  C C  a. a. • •1 ••• Apparent Aztmuthal Order  • 40  30  20  0000  I -  0  1=2, m =0  Velocity (km )  -  -  I  -  -  -  -  I.  -  -  -  I  -  5 10 15 20 Apparent Azimuthal Order  -  -  -  25  Figure 2.7: Fourier Representation of £ = 2 Oscillations. The surface velocity, residual line-profile variations, and the Fourier representation are illustrated for £ = 2, m = —2, —1, and 0 modes of oscillation.  Chapter 2. The Identification of Oscillation Modes  44  I.  440  I  a  110 5 10 15 00 Ap.r.M A.h..th.1 0,d. K  440  6 10 15 20 App.,oL A.Im,th.I 0d IC 140  r  0 P 110 0  5  10  15  20  25  App.ront A.th.I 0rd.  0  0  0  App.ea1 .Lh.1 0r.r  Figure 2.8: Fourier Representation of £ = 3 Oscillations. The surface velocity, residual line-profile variations, and the Fourier representation are illustrated for £ = 3, m = —3, —2, —1, and 0 modes of oscillation.  Chapter 2. The Identification of Oscillation Modes  45  Table 2.3: Simulations of Low-Degree Modes  Model Input  £  m  i.’o  (c d’)  v  (c  ) 1 a—  Observed Output  Vr ) 1 (km s  k  i (deg.)  Li  a  (c d ) 1  00  00  17.0  17.0  2.5  0.0  45  1.8  17.0  0.0012  01 01 01  -01 00 +01  17.0 17.0 17.0  17.9 17.0 16.1  2.5 2.5 2.5  0.0 0.0 0.0  45 45 45  2.5 2.0 2.5  18.0 17.0 16.0  0.0015 0.0008 0.0015  02 02 02 02 02  -02 -01 00 +01 +02  17.0 17.0 17.0 17.0 17.0  18.5 17.9 17.0 16.1 15.4  2.5 2.5 2.5 2.5 2.5  0.0 0.0 0.0 0.0 0.0  45 45 45 45 45  3.2 2.9 3.2 2.9 3.2  19.0 18.0 17.0 18.0 15.0  0.0017 0.0015 0.0005 0.0015 0.0017  03 03 03 03 03 03 03  -03 -02 -01 00 +01 +02 +03  17.0 17.0 17.0 17.0 17.0 17.0 17.0  19.4 18.5 17.9 17.0 16.1 15.4 14.6  2.5 2.5 2.5 2.5 2.5 2.5 2.5  0.0 0.0 0.0 0.0 0.0 0.0 0.0  45 45 45 45 45 45 45  4.1 3.8 4.0 4.3 4.0 3.8 4.1  20.5 19.0 18.0 17.0 16.0 15.0 13.5  0.0018 0.0018 0.0011 0.0005 0.0016 0.0018 0.0018  Chapter 2. The Identification of Oscillation Modes  46  Table 2.4: Fourier Representation of Multiple Modes  Model Input £  16 10 06 03 00  m  -16 -10 -06 -03 00  (c d— ) 1  (c d— ) 1  V (km s1)  17.0 17.0 17.0 17.0 17.0  34.8 28.2 23.7 20.4 17.0  2.5 2.5 2.5 2.5 2.5  V  is greatest for the mode with £  =  k  i (deg.)  0.0 0.0 0.0 0.0 0.0  45 45 45 45 45  Observed Output a ) 1 (c d—  16.5 10.7 6.7 4.0 2.0  35.0 28.0 24.0 20.5 16.5  0.0025 0.0024 0.0022 0.0018 0.0012  16 and smailest for the radial mode.  What are the optimal conditions for observing line-profile variations? A limited num ber of simulations were performed to shed some light on this question. Two models similar to the one presented in Figures 2.9 and 2.10 were generated with values of v sin i  =  60  and 150 km s. The resulting proffles were processed to include instrumental broad ening (described by a Gaussian function) and noise. The instrumental broadening was chosen to match the values that would be obtained from high-resolution observations at the Canada-France-Hawaii telescope (T’,  =  4 km s’) and from lower-resolution ob  servations at the Dominion Astrophysical Observatory (V 3  16 km  s1).  added with a normal distribution to simulate both good observations (S/N relatively poor observations (S/N  =  Noise was =  500) and  100). The quality of the results were judged from  the relative amplitudes of the detected peaks to the noise level in Fourier space. For the model with v sin i  =  150 km s , all the modes could be easily resolved in the high 1  resolution observations at noise levels down to S/N  =  100. Comparable results could also  be obtained at low-resolution if the signal-to-noise was 500. The results from the models  Chapter 2. The Identification of Oscillation Modes  I _____.\\•/f____0.305 ’? 29 j—O. \ —‘ J,—0.290 —..——-.‘./,r———0.283 .278 .288 l 26 j—O. 253  47  I’’’•I’’’, I .305 .297 .290 .283 .278 .268 .261 .253 .247  .235 .228 .221 .214 .207 199 185 .178 .171 .184 157 :149 f0. 137 .130 .123 .115 .109  ,,—o.ioi  .228 .221 .214 .207 199 . 185 .178 .171 .164 157 :149 . 137 .130 .123 .115 .109 101 .090 082  .062 .075 .068  .068 .057 .049  0.O49 .038 .030 .022 .011  Eooo  .038 .030 .022 .075 011 000  4505  4510  Wavelength  4515  4505  4510  4515  Wavelength  Figure 2.9: Line-Profile Simulations with Multiple Modes. A sequence of line profiles (left) and residual variations (right) resulting from a model with five sectora.l modes with .t = 0,3,6, 10, and 16 are shown.  I.  U)  ‘-  U)  NO  i-iCD  CD  O  CD  rj  0 CD?  %: CD  I-SI  CDCD U) —  oo  OO  CD  +  OO•  aq CD  CD ‘1  ‘1  0  (-..  .  0  0  ‘1 CD :ic’’  0  0  0  0  l’3 0  C) 0 C) 0  ApparenI Frequency (cyc daf’) 0 0  00  0.  0  0  L  Chapter 2. The Identification of Oscillation Modes  with v sin i  =  49  1 indicate that high-resolution observations are preferable for the 60 km s  study of more slowly rotating stars. However, even at this resolution, very high-degree modes of oscillation (1> 10) could not be recovered. Information only about the modes with very low degrees could be recovered from the low-resolution simulations. 2.6  2-D Fourier Prewhitening  The previous simulations demonstrate that with the Two-Dimensional Fourier technique even complex line-profile variations can be represented in such a way as to provide a straightforward interpretation of stellar oscillation spectra. However, to fully interpret the Fourier maps of stars, a method of two-dimensional prewhitening is desired which can take into account the window function of the observations and beating between different modes of oscillation. Two possible schemes come to mind. A least squares fit to the residual variations could be made using the positions of the peaks in Fourier space as the initial guesses for the frequencies and modes of oscillations. In this way, the data would be prewhitened in a manner which is analogous to the analysis of radial velocity variations in Section 2.3. Alternatively, the Fourier representation could be analyzed by fitting two-dimensional window functions to the peaks in Fourier space, analogous to the method of Gray & Desikachary (1973). The problem with these methods is that there is no good analytical representation for the Doppler (and temporal) component of the line-proffle variations. An approximation to the two-dimensional window function was generated by sampling the data as observed and multiplying the result by a window of the form:  S(’)  =  0.5(1.0 + cos(2#)).  [2.18]  but this function did not provide a very good fit to the simulated profile variations nor  Chapter 2. The Identification of Oscillation Modes  50  did the resulting Fourier transform have exactly the right full-width at half maximum. Alternatively, the NRP-model itself can be used to generate a kind of window function which would even reproduce the non-sinusoidal properties of the stellar oscillations and the nonlinearities of mode interaction. The possibility of prewhitening of using synthetic spectra generated by the NRP-program was tested with a simulation involving several modes which produced a complex pattern of peaks in Fourier space. The peaks were identified one by one. With each identification, a model was generated that reproduced all the modes so far identified. Subtracting the Fourier representation of that model from the original simulations effectively removed all the signal due to the previously identified modes. The remaining modes could be identified in the difference plot. 2.7  Discussion  The Two-dimensional Fourier method can provide a more or less direct measurement of both the frequency and mode (degree) of oscillation. Nonlinear effects introduced by the projection of the velocity variations within the absorption proffles conspire to make the identification of modes more difficult than one would like. However, with the help of numerical simulations, these projection effects can be anticipated. In practice, even greater uncertainty in mode identification is likely to be introduced by erroneous determinations of the projected rotational broadening. The spectra of S Scuti stars are expected to be rich with spectral features. For rapidly rotating stars, blending between neighbouring lines can be a problem. Uncertainties in the estimate of the continuum level and in the projected rotational velocity are unavoidable. Since the derived values of the apparent azimuthal order scale approximately with adopted value of the projected rotational velocity the observed modes are directly affected by this uncertainty. Still  Chapter 2. The Identification of Oscillation Modes  51  the 2-D technique offers a tremendous advantage over traditional methods of trial-anderror proffle fitting because it allows complex line-proffle variations to be resolved into multiple modes. In particular, it will prove extremely useful for the interpretation of the line-proffle variations observed in several S Scuti stars which are presented in Chapters 4 through 7 of this thesis.  Chapter 3  Line-Profile Variations in the S Scuti Instability Strip  3.1  Introduction  Prior to this thesis, high-degree modes of oscillation had been detected only in the four 6 Scuti stars observed by Walker et al. (1987):  o  En, 21 Mon,  v  2 Boo. These UMa, ,c  stars were intentionally chosen to be quite similar in their observational properties. Lineprofile variations were discovered first in  1 o  En by Yang & Walker (1986) while attempting  to measure its radial velocity variations. To increase the chance of finding other stars which also show similar behaviour, stars resembling  o1  Eri were chosen for observation.  All four stars were known S Scuti stars, reported to display hourly photometric variations. The temperatures and luminosities of these objects were not too different from each other with spectral types quoted as being F211-III, A8Vn, F2IV, and A8IV. Finally, the projected rotational velocities of these objects were identified as 96, 121, 110, 127 km s. All four stars displayed patterns of line-profile and radial velocity variations which varied in complexity. This thesis began with an observing run at the Canada-France-Hawaii 3.6 m telescope (CFHT) in October 1990. The objective of the observations was to increase the sample of known S Scuti line-profile variables by securing high-resolution spectra (2.4 A/mm) of two stars during successive nights of observation. As an added discniminant to the identification of modes, simultaneous photometric observations were also obtained. Can didates for observation were chosen from Breger’s (1979) catalog of known S Scuti stars.  52  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  53  One month prior to the CFHT observations, the variability of several candidates was investigated during four nights of observation on the 1.2 m telescope at the Dominion Astrophysical Observatory (DAO). In order to achieve signal-to-noise ratios comparable to those expected at CFHT, the observations at DAO were made at lower resolution (10 A/mm). For this reason, it was expected that only the low-degree modes would be de tected. However, variations in the line profiles (as well as radial-velocity variations) were seen in about half the observed stars. Afterwards, numerical simulations demonstrated that high-degree modes could be studied in rapidly rotating stars with observations made at the DAO 1.2 m telescope even at low resolution. The observations at CFHT were successful. Time series observations of two new line-proffle variables,  82  Tau and r Peg, were obtained. However, questions about just  how common high-degree oscillations are among stars in the instability strip remained. For example, it is necessary to observe rapidly rotating stars in order to see line-profile variations. All stars observed to show line-proffle variations have 70 < v sin i < 160 km s . Does the existence of high-degree modes require rapid rotation or is this merely 1 a selection effect? Is there a connection between the modes which are excited and the temperature or luminosity of the star? Certainly one might expect to see differences in the pulsational properties of main sequence and post main sequence S Scuti stars since the theoretical pulsation spectra for these groups are very different. Finally, could many of the photometrically constant stars in the S Scuti instability strip actually be variable with high-degree modes? To answer some of these questions, additional observations were made over a period of two years at the Dominion Astrophysical Observatory. These observations, combined with previous data collected at CFHT, comprised a new survey of line-profile variations in the S Scuti instability strip. The objectives of the spectroscopic survey were: 1) To discover new line-proffle variables and study their variations in detail.  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  54  2) To examine the relationship between high- and low-degree modes. 3) To investigate the dependence of pulsation on the stage of evolution. 4) To determine the boundaries of the instability strip for high-degree modes. 5) To examine the dependence of pulsation on rotation (v sin i). In this chapter, the observation, reduction, and analysis of the survey data is dis cussed. In many cases, the duration of the observations was sufficient only to determine if variations existed but was not suitable for measuring the periods and modes of os cillation. Therefore, in this chapter, the observations are analyzed without reference to the particular modes which may be present. In the following chapters, those stars for which good coverage was obtained are analyzed in more detail. Several days of nearly continuous coverage was obtained for the line-profile variable,  62  Tau during the 1992  MUSICOS campaign; these observations are presented separately in Chapter 7. 3.2  Organization of the Survey  Survey stars were selected based on criteria concerning the brightness, projected rota tional velocity, spectral type, and declination of all objects falling within or near the (photometrically determined) boundaries of the S Scuti instability strip. Table 3.1 sum marizes the criteria. Using the Bright Star Catalog (Hoffleit 1982) as the source, 55 stars were found which satisfied the selection criteria. Of these, 46 were successfully observed. In addition, 8 stars which failed to satisfy the brightness criteria and 1 star which exceeded v sin i  =  200  km s were included in the survey. The characteristics of the observed stars are presented in Table 3.2. The stars are identified by their HR number and the coordinates, magnitude, spectral type, radial velocity (in km s ), and projected rotational broadening (in km s1) 1 are listed for each star. The abbreviations SB1, SB2, SBO, etc. indicate that the systemic  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  55  Table 3.1: Survey Selection Criteria  Spectral Type  A2  Projected Rotational Broadening  60  Brightness  m  Declination  5> —10  F5  -  -  200 km s  4.5  radial velocities are variable because the stars are spectroscopic binaries with one or two components detected or for which orbital elements are available. In the final column of this table, the periods of those stars which are known to be variable based on photometric observations are listed. The periods were obtained from the catalogue of S Scuti stars recently compiled by Rodrfquez et al. (1994). The magnitude of the light variations in these stars is typically Am  =  0.m03. Those stars which did not satisfy all the selection  criteria are listed at the end of the table. Note that most of these were observed because they were known to be S Scuti stars. The only stars which were missed by the survey include HR 1387, 1473, 2714, 3799, 5107, 5478, 5867, 6149, and 6771, none of which are known to be variable. 3.3  Observations  All observations were made using the coudé spectrographs at 1.2m telescope at the Do minion Astrophysical Observatory and the 3.6m Canada-France-Hawaii telescope. At the DAO, low resolution spectra were obtained (10  A  mm’) in order to maintain a  high signal-to-noise level without jeopardizing the time resolution. However, on a few  Chapter 3. Line-Profile Variations in the S Scuti Instability  Strip  56  Table 3.2: Survey of S Scuti Candidates  HR  21 269 343 403 553 580 622 804 840 1298 1392 1394 1412 1560 1666 2298 2484 2540 2763 2777 2852 3569 3572 3690 3757 3888 3974 4031 4357  Right Ascension  0 0 1 1 1 2 2 2 2 4 4 4 4 4 5 6 6 6 7 7 7 8 8 9 9 9 10 10 11  9 56 11 25 54 3 9 43 50 11 26 26 28 52 7 23 45 52 18 20 29 59 58 18 31 51 7 16 14  7.2 44.6 4.8 46.9 38.0 26.5 32.0 18.6 34.1 51.9 18.1 20.4 39.3 53.7 51.3 46.2 17.8 47.4 5.7 7.5 6.1 14.7 29.2 50.8 30.9 1.3 25.5 41.3 6.0  Declination  59 38 55 60 20 72 34 3 38 -6 22 15 15 -5 -5 4 12 33 16 21 31 48 11 36 63 59 35 23 20  9 29 9 14 48 25 59 14 19 50 48 37 52 27 5 35 53 57 32 58 46 2 51 48 3 2 14 25 31  8 56 0 9 34 16 16 16 12 19 51 7 17 10 6 34 54 43 27 57 55 42 30 16 41 27 41 3 32  Vmag  Spectral Type  Radial Velocity  2.27 3.87 4.33 2.68 2.64 3.98 3.00 3.47 4.23 4.04 4.28 4.49 3.40 4.39 2.79 4.44 3.36 3.60 3.58 3.53 4.18 3.14 4.25 3.82 3.67 3.80 4.48 3.44 2.56  F2111-IV A5V A7V A5111-IVv A5V A2V A5111 A3V F2111 F211-III A8Vn FOV A7111 F4111+A6111 A3111 A5IV F5111 A3111 A3V F2IV FOV A7IV Mm A3V FOTV F2IV A7V FOul A4V  +11 +08 +09 +07 —02 —14 +10 —05 +14 +11 +35 +38 +40 —06 —09 +15 +25 +02 —09 +04 —04 +09 —14 +04 —10 +27 —18 —16 —20  SB SB SB SBO SB2 SB2O V V SB1 SBO SB1O SBO SB V? SB SB SBO SB SBO SB SB V V? V SB V  v sin i  Period (days)  70 72 102 113 79 84 76 183 149 96 196 192 78 153 179 124 70 128 154 111 68 151 68 165 140 110 148 84 181  0.1009  0.0747 0.1484 0.163 0.0756  0.1327  Chapter 3. Line-Profile Variations in the 6 Scuti Instability Strip  Survey of 6 Scuti Candidates  Declination  HR  Right Ascension  4534 4660 5264 5435 5570 5733 5735 5788 5789 6095 7069 7377 8130 8450 8454 8494 8984  11 12 14 14 14 15 15 15 15 16 18 19 21 22 22 22 23  49 15 1 32 57 24 20 34 34 21 47 25 14 10 9 14 42  5.3 25.0 38.7 5.2 11.4 30.1 44.0 48.4 48.4 55.4 1.0 29.1 46.8 11.1 59.3 59.3 3.3  14 57 1 38 -4 37 71 10 10 19 18 3 38 6 33 57 1  34 1 32 18 20 22 50 32 32 9 10 6 2 11 10 2 46  623 1351 1444 1547 2707 4368 5329 5960 8880  2 4 4 4 7 11 14 15 23  9 19 33 51 11 16 13 57 20  25.0 57.6 50.6 22.4 23.5 39.6 28.7 48.3 38.1  25 14 14 18 -0 -3 51 54 23  56 2 50 50 18 39 47 44 44  -  57  continued  v sin i  Vmag  Spectral Type  Radial Velocity  25 57 41 22 38 33 1 15 20 9 47 49 22 51 43 34 56  2.14 3.31 4.26 3.03 4.49 4.31 3.05 3.80 3.80 3.75 4.36 3.36 3.72 3.53 4.29 4.19 4.50  A3V A3V A3V A7111 FOV FOV A311-III FOIV FOIV A9111 A5111 F3IV F2IV A2V F5111 FOIV A7V  —00 —13 —02 —37 +22 —13 —04 —38 —42 —35 —45 —30 —21 —06 +02 —01 +12  V V SB V  SB SB  121 177 150 139 117 84 171 70 80 141 79 85 89 117 139 86 63  25 7 41 23 7 6 24 54 26  4.98 5.59 4.65 5.10 5.45 4.47 4.54 4.95 4.60  F2111 FOIV A8V A7IV-V A8Vn-F3Vn A7IVn A8IV FOIV A5V  +01 +42 +40 +37 +30 —03 —17 —11 +16  V SB1? SB2O V V V? SB SB V  154 109 117 141 121 225 127 140 143  SB V V? V SB SB? SB SB SB2  Period (days)  0.073  0.1557  0.0412  0.054 0.067 0.042 0.10 0.071 0.0763 0.0543  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  occasions high-resolution observations (2.4  A  58  ) were obtained when made in con 1 mm  junction with other observing projects. Three detectors were used at the DAO. The Reticon is a one-dimensional detector with 1872 pixels of dimension ‘750p x 15g. The Reticon is sensitive in the blue (with a quantum efficiency of  70%) but the readout  noise of this detector is high (350 e per pixel). The 512 x 512 Ford CCD has 2 0p pixels and low readout noise (3 e per pixel) but this device is red sensitive with a quantum efficiency of only 20% at 4500A. The 620 x 1024 RCA CCD has 15jL pixels and is blue sensitive. The quantum efficiency of the RCA is 80% at 4500A but the detector also suffers from high readout noise (52 ej. Initially, observations were made with the Reti con detector but this was later replaced by the RCA CCD in an attempt to improve the signal-to-noise of the observations. However, it was found that the two detectors produced comparable results. Spectra obtained at CFHT had a reciprocal dispersion of 2.4  A  1 and an 1872 Reticon was used as detector. The spectral region of the ob mm  servations was dependent on the resolution and the detector which was used but always included three absorption lines at 4501 A(Ti II), 4508 A(Fe II), and 4515 (Fell)  A.  Table 3.2 summarizes the characteristics of the observing runs. The resolution was calculated from the full width at half maximum of the emission lines in the calibra tion spectra. The numbers quoted represent the average over all nights during a given observing run. The duration of the time series and the length of the exposures varied from star to star. Calibration spectra were obtained about every two hours during the night. Several flat field spectra were usually obtained after each series of stellar spectra with signal levels which matched those of the observations. Several zero-second bias frames were obtained and averaged throughout the night during observations involving the CCDs. The data were processed using the routines within IRAF . 1 IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the 1  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  59  Table 3.3: Survey Observations  Site  UT Date  CFH DAO CFH DAO DAO DAO DAO DAO DAO DAO DAO  1987 Feb 12-13 1990 Aug 01-04 1990 Oct 03-04 1990 Dec 28 1991 Apr 17 1991 Aug 22-25 1991 Dec 05-06 1992 Mar 10-14 1992 Apr 10-14 1992 Oct 05 1992 Dec 17  Dispersion  2.4 10.1 2.4 10.1 10.1 10.1 10.1 10.1 2.4 10.1 2.4  A/mm A/mm A/mm A/mm A/mm A/mm A/mm A/mm A/mm A/mm A/mm  Detector  Resolution  Reticon Reticon Reticon Reticon Ford CCD Reticon RCA CCD RCA CCD RCA CCD RCA CCD Reticon  38000 19000 40000 15000 17000 18000 16000 12000 52000 14000 45000  Reticon data were preprocessed by first subtracting the baseline or dark exposure. Next an additive correction was applied to the data to correct for the 4-point noise pattern introduced by the four separate readout circuits or video lines. The corrected spectra were divided by flat field spectra to correct for differences in the responses of the individual pixels. Finally a 4-point multiplicative correction (gain normalization) and an 8-point additive correction were applied to the data. The preprocessing of the CCD data began with the removal of the overscan level from each exposure. Then the bias frame or, in some instances, a dark frame was subtracted from each of the spectra. (The exposed area of the CCD encompassed about 40 pixels but in order to shorten the time spent reading out the detector, pixels were binned on-chip in the direction perpendicular to the dispersion. In this case, one superpixel represented Association of Universities for Research in Astronomy (AURA), Inc., under cooperative agreement with the National Science Foundation.  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  60  8 physical pixels on the device.) Each exposed column of the flat field spectra was normalized to an intensity of unity so that when the stellar frames were divided by the flat fields, the relative intensity in each column was preserved. The spectra were then extracted by adding the signal in each of the exposed columns. For both Reticon and CCD observations, the wavelength calibration of the spectra was performed using a time-weighted average of the dispersion relations derived from the arc spectra obtained preceding and succeeding the stellar exposures. The stability of the CFHT and DAO coudé spectrograplis are known to be good, though systematic drifts usually occur during the night. Corrections for the heliocentric velocity were made to the spectra (without interpolating the data) and cosmic ray features were removed by examining each spectrum and assigning corrected intensities to the affected pixels. The continuum is poorly defined in the blue-violet spectra of rapidly rotating A- and F-stars. Blending between neighbouring lines in afl spectral regions leaves little evidence of the true continuum. In the reductions, a somewhat arbitrary choice of the continuum level was made for each time series. With the mean spectrum normalized to this level, the normalization of individual spectra in a series was performed by dividing each spectrum  by the mean, then fitting a low-degree polynomial through the residuals, and dividing each of the original spectra by the fitted functions. Line-proffle variations can be detected if the variations exceed the noise level in the continuum. The mean absolute deviation of each time series was calculated by summing the absolute values of the differences between each spectrum and the mean. The am plitude of this function provides a measure of the amplitude of the noise in an average spectrum in the series. Since the signal level of the continuum is normalized to unity, the signal-to-noise of the observations can be determined by taking the inverse of the noise amplitude. If line-profile variations are present in the data, the amplitude of the ‘noise’ will increase at the positions of the absorption lines. (There will also be a small increase  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  61  in noise within the proffles due to photon statistics amounting to less than 12% of the  continuum noise for profiles with depths less than 20% of the continuum level.) There fore, if the amplitude of the noise correlates with the depth of the profiles, line-proffle variations are likely present in the data. The null hypothesis, i.e., no correlation between noise amplitude and line depth (in the wavelength region from 4463A to 4524A), was tested for each set of observations by using the F-test criterion (Press et al. 1992) to compare the variance from a least-squares best fit through the data (i.e., non-zero slope) to the variance from the mean (i.e., zero slope). The F-test returns the probability that the two variances are consistent. Figures 3.1 and 3.2 illustrate the procedure in the case of a positive detection of line-profile variations (HR 1412) and in the case of a null de tection (HR 6095). The figures show the mean spectrum calculated from the time series of each star (top), the absolute mean deviation of the intensity variations (middle) , and the amplitude of the ‘noise’ as a function of line depth (bottom). In each case, the line of best fit has been drawn though the noise amplitude versus line depth relation and the slope of the curve as well as the probability that the observations could be fit by a straight line with zero slope are indicated. This method of detecting variations provides a measure of the variability of a star independent of the mode of oscillation. The results from the variability analysis are summarized in Table 3.4. The observa tions are grouped according to the run during which the data were collected and listed in the order that observations were made. The columns of this table reveal the number of spectra in the time series, a representative exposure time, the heliocentric Julian date corresponding to the start of the observations, the duration of the time series in hours, the noise level of the continuum, the amplitude of the line-profile variations (LPVs), and the probability that line-proffle variability is present (i.e., one minus the probability derived from the F-test). Note that the LPV probability is negative if the least-squares fit returned a negative slope (indicating a decrease in variability amplitude with line  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  I  I  I  HR 1412:  I  I  I  62  I  I  —  JD 2448596  4480  4460  4500  4520  Wavelength (A)  4460  4480  4500 Wavelength  I  I  I  I  I  I  I 0.0192 +/  SLOPE = PROABIUTY  40.01  I 0  0  I  =  I  I  I  I  I  I  I  I  I  I  I  4520  (A) I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  I  0.0011  0.000  I  0.1  I  I  0.2 Line Depth  I  0.3  I  0.4  Figure 3.1: Detection of Line-Profile Variations Example One. Line-proffle variations in HR 1412 were detected as enhanced variations relative to the noise level of the continuum. The top figure shows the mean spectrum of HR 1412 for the time series obtained on HJD 2448598; the middle figure shows the amplitude of the absolute mean deviation as a function of wavelength; and the bottom figure shows the amplitude of the variations plotted as a function of line depth. A line of best fit (with slope 0.019 ± 0.001) has been drawn through the data. The F-test statistic indicates that there is a zero percent chance that this data could be equally well described by a line with zero slope. -  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  I  I  I  HR 6095:  I  I  I  I  JD 2448694  4460  4480  4500 Wavelength  I  I  63  I  I  I  4520  (A)  I  I  I  0.01 0  M4 4460  4480  4500 Wavelength  I  ..  0.01  -  I  I  I  I  I  I  I  I  I  I  I  I  I  I  4520  (A)  I  SLOPE = 0.0003 +/— 0.0006 PRODABIUTY = 0.985  0  ci) ..  0  I  0  I  I  0.1  I  I  •.  .  I  I  0.2 Line Depth  Figure 3.2: Detection of Line-Profile Variations  I  0.3  -  0.4  Example Two. Same as Figure  3.1 but showing the null detection of variations in HR 6095.  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  64  depth). The amplitude of the noise and of the line-profile variations were determined from the least-squares fit. The LPV amplitude represents the amplitude of the variations at the maximum line depth within the spectral region under consideration (i.e., within the MgII-FeI blend at )4482  A).  The non-LPV contribution to this quantity (i.e., the  noise) has been removed assuming that the two sources of variations add in quadrature. The observations of HR 5435 during the run starting on HJD 2448723 were treated as a single set, therefore the noise and LPV amplitudes and the LPV probability is listed for the entire run after the first night. LPV probabilities greater than 0.99 correspond (effectively) to a 3 sigma detections of variability. Stars in this catagory are: HR 0021, HR 1298, HR 1412, HR 1547, HR 2707, HR 3888, HR 5329, HR 5435, HR 8494, and HR 8880. Variations were detected in HR 1444 at the about 2 sigma level (i.e., with a probability of 0.95). One sigma detections (with a probability greater than 0.68) were made for HR 0403, HR 0580, HR 5960, and HR 8984. For the remainder of the stars, no variations could be detected above the level of the noise. In Figures 3.3 and 3.4, representative illustrations of the mean spectra and absolute mean deviations are shown for the spectral region between 4460A and 4525A for the stars in which variations were detected. During the reductions, care was taken to identify sources of error which might intro duce false detections of variability. Instabilities in the spectrograph introduce wavelength shifts between one exposure and the next. During the course of a night the wavelength zero-point of the spectrograph may drift by up to one pixel or more. If changes in the spectrograph occur slowly over time this effect can be minimized with frequent wave length caiibration of the data. Large effects generally occurred at the beginning of the night and after the detector has been refilled with liquid nitrogen. Variations in the zero point of the spectra were detected during the 1991 August observations. This effect was observed to occur during the last series of the night and  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  65  Table 3.4: Detection of Line Profile Variations  HR  No. of Spectra  Exposure Time (sec.)  Heliocentric Julian Date  Duration of Series (hours)  Noise Level  Amplitude of LPVs  LPV Probability  1298 3888 5329 1298 1351 2707  28 17 16 14 09 12  500 800 525 500 400 850  6838.70227 6838.90852 6839.07838 6839.69930 6839.80234 6839.87177  4.33 3.83 2.37 2.09 1.10 2.73  0.0025 0.0017 0.0018 0.0019 0.0059 0.0036  0.0075 0.0064 0.0094 0.0066 0.0029 0.0083  1.000 1.000 1.000 1.000 —0.050 1.000  0021 8494 1412 8494 8880 1392 5435 5735 1412 5849 5960 0840 1444  30 15 13 16 10 11 12 13 14 10 08 11 09  300 600 660 660 900 1200 500 750 900 700 1200 900 900  8136.67809 8136.81495 8136.94554 8137.64461 8137.78687 8137.89596 8138.63304 8138.74338 8138.90238 8139.63217 8139.72028 8139.84014 8139.96301  2.93 2.86 2.30 2.93 2.34 3.46 1.83 3.09 3.41 1.74 2.34 2.59 2.00  0.0011 0.0013 0.0010 0.0012 0.0010 0.0013 0.0007 0.0005 0.0011 0.0009 0.0011 0.0007 0.0011  0.0063 0.0028 0.0041 0.0026 0.0040 0.0011 0.0020 0.0001 0.0051 0.0014 0.0015 0.0002 0.0017  1.000 1.000 1.000 1.000 1.000 0.450 1.000 —0.012 1.000 0.509 0.866 0.015 0.942  8494 1412 8880 0623 1547  14 52 38 06 12  600 300 600 600 450  8168.76448 8168.94164 8169.73227 8170.05391 8170.10315  3.70 5.41 7.32 0.92 1.48  0.0040 0.0012 0.0017 0.0011 0.0020  0.0043 0.0101 0.0048 0.0004 0.0051  0.849 1.000 1.000 0.005 1.000  0021 3888 3757 5435  08 09 13 17  500 1200 900 600  8254.62392 8254.72033 8254.86444 8255.02406  0.98 3.00 3.46 3.08  0.0012 0.0011 0.0012 0.0008  0.0051 0.0039 0.0008 0.0031  1.000 1.000 0.170 1.000  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  66  Line-Proffle Variability- continued  HR  No. of Spectra  Exposure Time (sec.)  Heliocentric Julian Date  Duration of Series (hours)  Noise Level  Amplitude of LPVs  LPV Probability  4368 5435  10 28  300 500  8364.62563 8364.66788  0.68 5.32  0.0030 0.0017  0.0025 0.0019  0.336 0.848  8494 0553 0403 8130 8450 0343 7377 0269 0021 1394 5435 7069 0021 0622  13 11 10 10 08 06 09 11 10 08 10 04 16 05  900 600 660 800 780 900 600 800 300 900 600 1200 500 900  8491.68854 8491.84782 8491.94067 8492.66901 8492.77287 8492.90233 8493.66972 8493.76067 8493.89350 8493.94243 8494.64784 8494.73436 8494.79715 8494.90912  3.51 1.85 1.49 2.12 1.62 1.39 1.54 2.68 0.75 1.99 1.50 0.96 2.39 1.14  0.0016 0.0009 0.0009 0.0011 0.0010 0.0019 0.0010 0.0013 0.0013 0.0026 0.0010 0.0016 0.0013 0.0022  0.0031 0.0004 0.0012 0.0003 0.0006 0.0008 0.0012 0.0015 0.0042 0.0024 0.0022 0.0017 0.0050 0.0007  0.998 0.003 0.702 0.017 —0.111 —0.006 0.514 0.533 0.990 0.379 1.000 0.207 1.000 0.015  1412 3569 8984 2484 4031  18 13 04 13 07  400 400 600 400 420  8596.80703 8596.89613 8597.59743 8597.92869 8598.02034  2.12 1.53 0.73 2.05 0.56  0.0011 0.0012 0.0014 0.0019 0.0023  0.0072 0.0013 0.0037 0.0007 0.0004  1.000 0.625 0.896 —0.006 —0.019  2484 4357 5570 5733 2298  22 17 10 10 17  300 300 600 400 300  8692.68487 8692.84503 8692.92081 8692.99949 8693.62403  2.05 1.57 1.44 1.14 1.72  0.0020 0.0013 0.0029 0.0025 0.0028  0.0006 0.0001 0.0012 0.0013 0.0028  0.007 0.020 —0.020 0.036 0.553  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  67  Line-Profile Variability- continued  HR  No. of Spectra  Exposure Time (sec.)  Heliocentric Julian Date  Duration of Series (hours)  Noise Level  Amplitude of LPVs  LPV Probability  2777 4357 4534 5788 5789 2852 3572 4660 5435 6095 1560 2763 4031 5264 5435 1666 3569  17 16 15 11 11 19 13 10 19 17 05 16 16 06 16 07 31  300 350 400 300 300 300 400 350 300 300 350 350 300 500 300 500 350  8693.71144 8693.79246 8693.88127 8693.97450 8693.96665 8694.65600 8694.74184 8694.81703 8694.89550 8694.98501 8695.62609 8695.69909 8695.79283 8695.87570 8695.96218 8696.62043 8696.67824  1.56 1.65 1.65 1.80 1.81 1.75 1.44 0.89 1.72 1.48 1.19 1.76 1.52 0.68 1.62 0.89 3.42  0.0017 0.0012 0.0012 0.0048 0.0028 0.0051 0.0028 0.0021 0.0020 0.0032 0.0041 0.0022 0.0027 0.0063 0.0025 0.0024 0.0023  0.0004 0.0001 0.0007 0.0042 0.0029 0.0040 0.0008 0.0013 0.0032 0.0007 0.0013 0.0008 0.0021 0.0023 0.0010 0.0013 0.0014  0.015 0.020 0.087 0.338 0.455 —0.490 —0.012 0.064 0.997 0.015 —0.015 —0.000 0.287 —0.012 0.007 0.024 0.191  5435 5435 5435 5435  06 32 02 11  300 900 700 700  8723.02246 8723.69336 8725.67783 8727.66315  0.53 8.32 0.21 2.63  0.0021  0.0017  0.989  8880 2540  31 09  900 600  8901.61977 8902.02023  9.39 1.41  0.0018 0.0013  0.0040 0.0004  1.000 0.012  8454 0580 3690  11 13 10  900 900 1200  8974.52733 8974.67727 8975.01048  2.97 3.09 2.88  0.0055 0.0032 0.0045  0.0030 0.0030 0.0009  0.278 0.853 0.005  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  I  1  1  I  I  I  I  I  I  HR 0021  J1 2448494  HR 0580  JD 2448974  HR 1298:  JD 2448836  HR 1412  ID 2448138  HR 1444:  JD 2448139  I  I  68  I,  0.01  0.01  0.8 0  1  IIIIIIIIIII:  0.01  0.8 0 • .—  0  1.  0.01  0.8 0 1  0.01  0.8  IIIIIiIIIIIii.  HR 1547:  1 0.8  0  JD2448170  0.01  -  -  0 1  HR 2707  JD 2446839  0.01  0.8 I  4460  I  I  I  I  I  4480 4500 4520 Wavelength (A)  0  4460  4480 4500 4520 Wavelength (A)  Figure 3.3: Mean Spectra and Their Variations. On the left, the mean spectra of a few selected stars, calculated from time series observations and normalized continuum of unity are shown. On the right, the absolute mean deviation for the series observations provides a measure of the amplitude of the line-proffle variations. signal-to-noise of an average spectrum in each series is determined from the noise in the continuum.  to a time The level  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  I I I I HR 3888:  I  I  I  I  I  JD 2448254  HR 5329:  JD 2446839  HR 5435:  JD 2448138  HR 5960:  JD 2448139  I  I  I  I  I  I  I  I  I  69  I  I  I  I  I  I  I,  O.Oi  :  oil  0IIIIIIIIIIIII.  HR 8494: JD 2448137 ,AJ\ jY\ J’/\ flJThj- ,  -  0.01  -  0.8-”’J1  HR 8880:  JD 2448137  0.01  :  0.8 lIlJllIIlIl HR 8984: JD 2448597  4460  4480 4500 4520 Wavelength (A)  04III4lI,  4460  4480 4500 4520 Wavelength (A)  Figure 3.4: Mean Spectra and Their Variations. Same as Figure 3.3  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  70  likely results from the warming of the detector. Changes in zero-point produce variations in the data which are related to the depth of the absorption line. The effect was easily detected in the 1991 August data because unusually large variations were introduced in the proffle of H’)’. Normally H 7 is not sensitive to line proffle variations because the intrinsic line proffle is broad compared to the Doppler broadening due to rotation. For most of the observations, the effect is small. However, in the case of HR 403, zero-point variations have lead to a false detection of line-profile variations at the 1 sigma level. During the 1992 March observations a more severe problem occurred. Condensation on the window of the CCD camera introduced variations throughout the entire spectrum. By blowing a weak stream of nitrogen gas across the surface of the the detector the prob lem was believed to have been eliminated. However, after reducing the data, evidence of patches of condensation could still be identified as regions of isolated variation not associated with absorption lines. The instrumental resolution during these nights was extremely poor, suggesting that some amount of condensation remained during all ob servations. However, among all of the stars observed, evidence for line-profile variability was apparent only in HR 5435 (which was already known to be variable). 3.4  Analysis of the Survey  Photometric indices for all stars except HR 403, HR 580, and HR 5788 were obtained either from the Hauck & Mermilliod (1980) uvbyf3 photoelectric photometric catalogue or, in some cases, from the photometry of variable stars provided by Breger (1979). The ubvy photometric system was introduced by Strömgren (1966) with wavelengths of the four bands specifically chosen to account for the effects of line blanketing and metaflicity. The (5  —  y) colour index is relatively unaffected by line blanketing and correlates reasonably  well with temperature. The colour index, m 1  =  (v  —  5)  —  (5  —  y) is intended to measure  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  the amount of line blanketing in the 4100  1 c  =  (u  —  v)  —  (v  —  A  71  region and correlates well with [Fe/H], while  b) is designed to measure the strength of the Baimer discontinuity and  correlates well with surface gravity. Quite often a fifth ifiter is introduced which provides a measure of the strength of the 11/3 line. The /3 index correlates well with temperature, is only weakly gravity sensitive and is also unaffected by line blanketing and interstellar reddening. The ubvy/3 photometric system was calibrated for F-, and A-stars by Crawford (1975) and Crawford (1979). F-stars were defined as those objects with 2.59 < /3 < 2.73. A-stars were defined as those objects with 2.72 < /3 < 2.89. Since F-type stars are numerous, nearby stars which are unaffected by reddening could be used in Crawford’s calibration. Accurate trigonometric parallaxes were used to set the zero point for the absolute magnitudes of these objects. For the A-stars, the zero point was determined using observations of bright stars in several open clusters. The calibration relation for absolute magnitude derived by Crawford took /3 as the independent variable and had the form:  M  =  M(ZAMS)  —  f  .  1 Sc  [3.3]  where Sc 1 is the magnitude difference between the observed c 1 index and the standard ci(/3) derived from Crawford’s calibration and Mv(ZAMS) is the absolute magnitude of a star on the zero-age main sequence with photometric index /3. The relationship between /3, the standard value of c 1 and the absolute magnitude can be determined from Table I of Crawford (1975) for F-stars and from Table I of Crawford (1979) for A-stars. For A-stars,  f=  9 and for F-stars,  f=  10. The uncertainty in the absolute magnitude  as determined by Crawford is ±0.25 mag. for F-stars and ±0.30 mag. for A-stars. The Crawford calibrations were used to derive absolute magnitudes for the stars in  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  72  the survey. Complete ubvy/3 photometry was unavailable for HR 5329, so its absolute magnitude was calculated from the relation between period-luminosity-colour relation for S Scuti stars (see below) and from the star’s parallax and apparent magnitude. Six stars, HR 1351, HR 1392, HR 1394 HR 1412, HR 1444, and HR 1547 are members of the Hyades cluster and their absolute magnitudes can be calculated from the distance modulus of the cluster (Gunn et al. 1988):  m  —  M  =  3.28 ± 0.26.  [3.4]  HR 1392, HR1394, HR 1412 and HR 1444 are spectroscopic binaries and the absolute magnitudes derived using this method do not agree with those determined from the photometric indices (by up to 1 magnitude). Two systems, HR 1412 and HR 1394 have been resolved during lunar occultations by Peterson et al. (1981) and the absolute magnitude of the primary stars were derived from the magnitude difference between the components. The stars are plotted in an HR-diagram of absolute magnitude versus (b  —  y) colour  in Figure 3.5. In this diagram, stars which exhibit line-proffle variability are indicated by filled circles. The size of the circle indicates the amplitude of the line profile variations. If multiple observations were made of the same star then the average amplitude is shown. The LPV amplitudes measured from observations obtained at CFHT (high resolution) were found to be larger by a factor of about 1.5 than those of the same star derived from observations obtained at DAO (low resolution). Therefore, the amplitudes derived from CFHT data were corrected by a factor of 0.67 in order to compensate for the differences in resolution. Stars which were detected to be variable at the one and two sigma level (i.e., HR 1444, HR 5960, and HR 8984) are included in the figure. Photometric indices were not available for HR 580. Stars in which variations were not detected are identified  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  73  by open circles in Figure 3.5 and in this case, the size of the circle represents the upper limit for line-profile variation amplitudes. For these stars the possible existence of lineprofile variations with amplitudes below the indicated level can not be excluded. The zero-age main sequence and the boundaries of the instability strip determined by Breger (1979) have been reproduced in this figure. The relative luminosity of the stars observed in the survey is not necessarily a good indication of the relative ages of these objects. The Crawford calibration scheme overlooks the effect that rotation has on the derived absolute magnitudes. The surface gravity of a star which is observed to rotate rapidly will be reduced relative to the value that would be observed if the star did not rotate. The Sc 1 index measures the reduced gravity. As a result, the absolute magnitudes of rapidly rotating stars could be lower than those of slowly rotating stars (with equal mass and age) by as much as 0.52 magnitudes for main sequence objects and by as much as 0.82 magnitudes for evolved stars (Crawford 1979). Therefore, stars which appear to be evolved because they are more luminous than main sequence stars may in fact also be main sequence stars. In addition, rotation may also affect the observed colours such that rapidly rotating stars will appear slightly cooler relative to nonrotating stars of equal mass and age. All of the stars observed in. the LPV survey rotate rapidly (v sin i > 60 km s’) and the effect of this can be illustrated by plotting the observed stars in a period-luminosity diagram. The pulsation periods of S Scuti stars obey a relationship between period, luminosity, and colour (the PLC relation) which can be used as an independent measure of the absolute magnitude. Many derivations of this relation exist in the literature. For example, Breger (1979) finds that the absolute magnitude M is related to the b  —  colour and the period of oscillation P in accordance with the relation:  M  =  —3.052 log P + 8.456(b  —  y)  —  3.121.  [3.5]  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  —1  i  I  I  I  I  I  I  I  o  undetected • variable  I  0  0.006  g  00.004  0  I•1412  /  00.003  o 0.002 1  °  I  1  Q0.o08  o  I  The 6 Scuti Instability Strip 0  0  74  0  0.001  ‘  !ir.8880 5329o  I  cP/  0  /  •270721  ,  00  o  i  \ \ 2  ZAMS  \  .  0  i  1298•  I I  •1547 8494 5960 • 1444 0  I  \,  3888• /  I i I I  8984  °“  /  3  I  0  I  I  0.1 (b—y)  I  I  0.2  I  I  0.3  Figure 3.5: Line-Profile Variations in the Instability Strip. The stars included in the survey are plotted in a HR-diagram of absolute magnitude (Me) versus (b—v) colour. Those stars found to be variable are represented by the solid symbols. The size of the circle indicates the amplitude of the variations. Those stars for which no variations were detected are plotted as open circles and in this case the size of the circle indicates the  upper limit to the variation amplitudes. The legend relates the size of the symbols to the amplitude in units where the continuum equals one. The zero-age main sequence and the boundaries of the photometrically determined instability strip are also shown.  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  75  (More recently, Fernie (1992) derived a PLC relation by assuming that the S Scuti stars lie in an extension of the Cepheid instability strip.) In Figure 3.6, the periods of those stars known to be variable are plotted against the colour-corrected absolute magnitudes (i.e.,  M  —  8.456(b  —  y)) determined from the photometry. The photometrically determined  periods compiled by Breger (1979) were adopted for all stars with the exception of HR 1444 and HR 5435. The period of HR 1444 was obtained from the catalog of Lopez de Coca et al. (1990). For HR 5435, the photometric period is claimed to be 0.25 cycles 1 (Auvergne et al. 1979), however the period adopted for this star in Figure 3.6 was day the one derived from its line-profile variations (P  =  0.047 days; Kennelly et al. 1992).  HR 5329 was excluded from the figure because its absolute magnitude was determined using Equation 3.5 since complete photometry was not available. HR 8984 is not known as a variable so a period was not available for this star. The solid line in Figure 3.6 is the best-fit PLC-relation derived by Breger (1979) using a complete sample of S Scuti stars. Almost all stars observed in the survey fall to the right of this line. This shift can be attributed to the effects of rotation on the derived magnitudes (and colours). Stars within one magnitude of the zero-age main sequence may be considered to be hydrogen-core burning objects (Crawford 1979). At least 5 variable stars seem to be in this stage of evolution. The amplitudes of the variations in these stars are small with a mean value of 0.0026 ± 0.0010 relative to a continuum value of unity. By contrast, the remaining 8 variable stars seem to be evolved objects and have somewhat larger ampli tudes of variation (0.0047 ± 0.0012 relative to the continuum). However, the amplitude of variation is proportional to the depth of the rotationally-broadened line proffles but even after scaJing the amplitudes, the correlation between the amplitudes of variations and the (assumed) evolutionary status remains. This correlation could result from the fact that the number of modes excited in evolved stars is expected to be significantly in creased over that of the main sequence stars. However, the systematic effects introduced  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  76  I  I  I  I  I  I  I  0.2  3888  ///“  0.1 0.08  /1298•  •21  .1412  /  a)  .  O.06  eo 5435• •8880  0.04  0.02  1547•  I  2  I  I  •8494  I 1 —  I I I 0 8.456 (b—y)  I  I —1  I  I  I —2  Figure 3.6: The Period-Luminosity-Colour Relation for S Scuti Stars. The pub lished photometric periods of the variables stars are plotted against their colour-corrected absolute magnitudes and compared to the PLC-relation derived by Breger (1979). Be cause the magnitudes of these objects have not been corrected for the effects of rotation, most of the stars fall to the right of the curve.  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  77  by rotation could mean that some of these seemingly evolved stars are still on the main sequences. Convective overshooting may also play a role in prolonging the main sequence lifetimes of these stars. The identification of oscillation modes could help to determine of the evolutionary stages of these objects. The discovery of main sequence line-proffle variables is important since main sequence stars are expected to exhibit less complicated eigenmode spectra so that the task of mode identification could be greatly simplified for these objects. The S Scuti stars are known to share the same distribution in a HR diagram as do normal A- and F-stars (Breger 1979). The distribution of survey stars in Figure 3.5 does not appear to be very different from distribution of normal stars although no rigorous statistical test was performed to support this claim. All of the stars found to be variable lie within (or very near to the edge of) the established boundaries of the instability strip. Forty-three percent of the stars lying within the strip were found to be variable. However, this result is likely biased by the intentional observation of S Scuti stars with apparent magnitudes greater than the cutoff value for the DAO survey. After excluding these objects, the incidence of variability becomes about 35 percent which agrees fairly well with the fraction of stars found to be variable in photometric surveys (Breger 1979). In fact, only those stars already known (or suspected) to be S Scuti stars were detected as line-profile variables in this survey. Notable exceptions are HR 580 and HR 8984 which gave some hint of line-profile variability at the one sigma level. However, the signalto-noise of the HR 580 observations was very low and coverage of HR 8984 was very limited. The analysis of the variations presented in this Chapter can not be used to discrimi nate between the modes of oscillation. This can only be achieved by analyzing the time series of observations using a method such as the Two Dimensional Fourier Technique. However, the simulations presented in Chapter 2 indicated that low-degree modes do not  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  78  give rise to substantial line-proffle variations in rapidly rotating stars. (Small shifts in radial velocity do not give rise to large intensity variations in broad absorption lines.) Thus, the observation of line-profile variability may be interpreted at least as an mclication that high-degree modes might be present. The observed variation of HR 21 is good example of the contrary. The line-profile variations of this star have a double peak structure (Figure 3.3) which would be expected to originate from the rather large radial velocity shifts resulting from the low-degree oscillations in this star. However, the vari ations observed in the other stars do not show this same double-peaked structure and may be produced by variable features which travel through the absorption lines. That no line-profile variations were detected (with certainty) in stars identified as photometrically ‘constant’ (to a certain precision) is significant. It means that highdegree modes of oscillation are not generally observed with large amplitudes in stars which do not exhibit low-degree variations. Therefore, whatever mechanism is operating to limit the amplitudes of oscillation in two-thirds of the stars lying within the S Scuti instability strip may also influence the amplitudes of high-degree modes. Finally, it is well established that the amplitude of the light variations in S Scuti stars is strongly correlated with the projected rotational velocity (e.g., Rodruez et al. 1994). Although, the mean value of v sin i for S Scuti stars is about 100 km s’, only the slowly rotating S Scuti stars (with v sin i < 20 km  s1)  exhibit large amplitude light variations.  For these objects, the variations (usually) result from only one or two radial modes. The oscillation spectrum of rapidly rotating stars is different, often exhibiting multiple radial  and nonracial modes of oscillation. Since rotation appears to be closely tied with the oscillation spectra of S Scuti stars, it is interesting to investigate whether or not rotation can also be correlated with the presence of high-degree oscillations. The distribution of projected rotational velocities for stars in the survey is illustrated in Figure 3.7 and can be compared to that expected for A-stars (Wolff 1983). The distribution of stars  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  79  in which line-proffle variations were detected is also plotted against v sin i in the figure. Unfortunately, it is difficult to comment on the similarity or differences between these distributions because the detection of line-profile variations suffers from several selection effects that are related to the value of v sin i. High-degree modes can not be observed in slowly rotating stars because the line profiles are too narrow. Also, line-proffle variations resulting from high-degree oscillations will be more difficult to detect in very rapidly rotating stars because the profiles are too shallow (and therefore the amplitude of the variations are smaller.) As a result, there is likely a tendency to detect line profile variations in stars if v sin i is near 100 to 120 km s . (The exact value will depend on 1 the mode of oscillation.) 3.5  Discussion  The goals of the spectroscopic survey were: to discover new line-profile variables, to examine the incidence of line-profile variability in the HR diagram, and to determine the dependence of the variations on the presence of low-degree modes, the evolutionary stage, and on the rotation velocity. The analysis presented in this chapter revealed 10-13 variable stars out of the 55 observed. (Four of these stars were identified prior to this thesis.) Only those stars which were known photometric variables exhibited variations in their line profiles. Thus, it was concluded that the amplitudes of high- and low-degree modes depend similarly on some unspecified selection mechanism. The amplitudes of the observed line-proffle variations tend to be larger in the apparently evolved stars than they are in the main sequence objects. However, the identification of the evolutionary stage of a given star is complicated by effect that rapid rotation has on the derived luminosities. Finally, the dependence of the line-profile variations on the rotation velocity of the star could not be determined because the observations suffer from selection effects which  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  I I I I I I I I I  I  I  80  I I I I I I  20-  -  IS_  5__  I 0  0  I  I  I 50 100 150 Projection Rotational Velocity (km s’) I  200  Figure 3.7: The Distribution of Projected Rotational Velocities. The distribution of rotational velocities for all the observed stars is plotted in the upper histogram and was found to resemble that expected for A-stars. The lower histogram shows the distribution of rotational velocities for the stars identified as variables.  Chapter 3. Line-Profile Variations in the S Scuti Instability Strip  81  are related to the projected rotational velocity of the stars (i.e., the ability to detect variations is sensitive to the width and depth of the line profiles). In addition to effects related to rotation, other selection effects may also have some bearing on the results of this survey. The observations are sensitive only to variations which occur on a time scale of one to two hours. The highest signal-to-noise was achieved for the brightest and therefore the most evolved stars (if all stars are at the same distance). Unresolved binaries could introduce errors in the adopted colours and magnitudes. The strength of the absorption lines being studied decreases with temperature. Therefore, line-profile variations will be difficult to detect in the spectra of early A stars. Weak lines in the spectra become stronger with decreasing temperature. Therefore, uncertainties introduced by blending with these lines will be greatest for late A and early F stars. In summary, line profile variations are most easily detected in bright stars with intermediate rotational velocities and spectra types. Line-profile variations in several S Scuti stars have been discovered with this spectro scopic survey. These variations (in most cases) are interpreted as evidence for high-degree oscillations. The variability is certain in 10 stars: HR 21, HR 1298, HR 1412, HR 1547, HR 2707, HR 3888, HR 5329, HR 5435, HR 8494, and HR 8880. For the remaining 4 stars (HR 580, HR 1444, HR 5960, and HR 8984) confirmation of the variations is needed. A Fourier analysis of the intensity variations within the profiles might be sufficient for this purpose; real variations would be periodic. However, the following chapters of this thesis will focus on the analysis of those stars for which the existence of variations is certain.  Chapter 4  The Line-Profile Variations of 1 Pegasi  4.1  Introduction  The S Scuti star r Pegasi (HR 8880) is classified as an A5 IV star with projected rotational broadening v sin i  =  150 km s . The variability of this star was discovered by Millis 1  & Thompson (1970) and confirmed by Fesen (1973) who proposed a single frequency of variation at 17.52 cycles day . Later, Michael & Seeds (1974) obtained 7 nights of 1 photometric observations and concluded that two frequencies were present at 18.406 and 20.429 cycles day . 1 Photometric observations obtained between 1966 and 1979 (including those cited above) were compiled and analyzed by Breger (1991). He found that afl the observations could be described by oscillations with a single frequency at 18.4052 cycles day 1 with variable amplitude and phase. A mathematical model with three very closely spaced frequencies was shown to reproduce these variations.  However, Breger considered it  more likely that the variations in amplitude and phase were due to the Blazhko Effect operating over the long term. (The characteristic of modulated oscillation amplitudes derives its name from S. Blazhko (1907) who first discovered the effect in RR Lyrae stars.) Based on estimates of the Q-value and the amplitude ratio and phase shift between light and colour variations, Breger concluded that the mode of oscillation responsible for these variations is a nonradial p or p4 mode with t  =  2.  In Chapter 3, observations of r Peg were presented that suggested the presence of  82  Chapter 4. The Line-Profile Variations of ‘r Pegasi  83  line-profile variations due to high-degree oscillations. In this section, the 1990 observa tions from CFHT are analyzed using the Fourier method and compared to observations obtained at DAO two years later. 4.2  The 1990 CFH Observations  On 1990 October 5 (UT), 38 high-resolution spectra (2.4 A mm’) were obtained of r Peg at the Canada-France-Hawaii telescope (CFHT) during 7.3 hours. The upper limiting frequency for unevenly sampled data is specified by the generalized Nyquist frequency, ’Nyq = 1  1/(2zt), where /.t is the mean time spacing between observations. For the  observations of r Peg,  . The upper limit of 1 60 cycles day  Iri is determined by  the intrinsic width of a stellar absorption profile and the projected rotational broadening. Resolution in frequency is proportional to the inverse of the total time coverage of the observations. For the CFHT observations of r Peg, AE  =  3.3 cycles day . In 1  the spatial domain, resolution is restricted because only half the stellar circumference can be observed and because of the foreshortening which occurs in the line wings. The simulations of Chapter 2 indicated that the full-width at half maximum of a peak resulting from line-profile variations is approximately ith  =  4.  A portion of the observed spectral region is presented in Figure 4.1 to illustrate the variations in the line profiles. In this figure, the profiles of A4501.278 Ti II, M508.289 Fe II, and )4515.342 Fe II have been combined in order to increase the signal to noise. Averaged over sufficiently long times, the line profile variations cancel leaving only the rotationally broadened spectrum. The mean was subtracted from each proffle to produce the time series of residuals shown on the right-hand-side. A steady progression of strong features moving from blue to red appear early in the data but suddenly die out mid-way. This effect could be caused by beating between high-degree modes of oscillation.  Chapter 4. The Line-Profile Variations of r Pegasi  11111111  IllilIllIll  1.037  84  liii.  1.030  fl 022 1.022  p... 1  1.000 4gfPP”1.000 994  \/° /°  .988 .979 968 .980 .954  .946 .939 .932  .948 .939  17 .910 .904  .917 .910 .904 .896 .889 .881  .932  .889 .881 .670 v.898 ‘\ t0.862 -  .870  .//-o  .855  —VWro:a48  .848 .841 .834  834 .822 .8  /°  .808 .800 789 .782 .763  .754 743  f\f0.732  743  ::::732  10.005 —200 0 200 400 —200 0 200 400 Velocity Velocity  Figure 4.1: The Line-Profile Variations of T Peg (CFHT). The line-proffle varia tions of of i- Peg are shown in this time series of spectra (left). On the right, residuals calculated by subtracting the mean profile from each spectrum help to clarify the vari ations. The times of observation are indicated to the right of each spectrum in days relative to HJD 2448169. The vertical scales are in units of the continuum intensity.  Chapter 4. The Line-Profile Variations of 1- Pegasi  5c  I  I  I  I  I  I  I  I  I  I  I  IFWHM  1  40  I  85  I  2Osini  -  1o>c3  0  0510152025 Apparent Azimuthal Order  Figure 4.2: Fourier Representation of the Line-Profile Variations in r Peg (CFHT). The variations were transformed in both space and time to produce this con tour map of the apparent azimuthal orders and frequencies of oscillation. The maximum of the transform is 0.00440 relative to a continuum of unity. The contours above the 3 sigma confidence level (at 0.00061) are plotted as thick curves. Contours below 3 sigma are thin. Theoretical limits to the pulsation frequencies of prograde sectoral modes are plotted for the standard He abundance model (thick solid lines ) and the He-rich model (thin solid lines ). The full width at half maximum resolution is indicated by the error bar labeled “FWHM”. The magnitude of first-order rotational splitting is indicated by the error bar labeled “2 sin i”.  Chapter 4. The Line-Profile Variations of r Pegasi  86  Variations in the residuals (of the averaged profiles) were analyzed using the twodimensional Fourier method outlined in Chapter 2. For these calculations, v sin i  =  150 km s was adopted. A contour map of the resulting Fourier amplitude spectrum is presented in Figure 4.2. The amplitudes are expressed as a fraction of the continuum level. The maximum of the Fourier transform is 0.0044 relative to a continuum of unity and the contours have been drawn at intervals of 0.0003. Confidence limits for the 2D transform were calculated using the method of Bootstrap Resampling (Press et al. 1992). The observed residuals were randomized in both time and space, then analyzed using the 2D Fourier transform method. The maximum of the resulting transform was recorded. After repeating this procedure one hundred times, the value recorded for the second highest maximum gives an indication of the level above which there is a 99% chance (about three sigma) that a peak observed in the true data could not be a result of spurious noise. For r Peg, the three sigma level occurs at 0.00061 relative to a continuum of unity. (Two sigma (95%) occurs at 0.00059 and one sigma (68%) at 0.00054.) In Figure 4.2, contours below 3 sigma are plotted as thin lines and contours above 3 sigma are plotted as thick lines. A series of peaks can be seen running diagonafly across the center of the  i—ii  diagram  with apparent frequencies £ between 18 and 32 cycles day 1 and apparent azimuthal orders  5i  between 3 and 18. These peaks are interpreted as likely modes of oscillation.  The error bar labeled ‘FWHM’ indicates the expected full width at half maximum of the peaks. The positions and amplitudes of several (prominent) peaks were measured and are tabulated in Table 4.1. The errors in the measured quantities: Ath cycles day , and iA 1 small peak at th  0.2, Az’  0.1  0.0002 were estimated using Monte Carlo simulations. The  2.6 and 1)  1 may not seem very significant but it is 18.5 cycles day  consistent with the frequency expected for the low-degree variations (i.e., at 18.4 cycles , Breger 1991). The remaining peaks in Figure 4.2 could be caused by beating 1 day  Chapter 4. The Line-Profile Variations of r Pegasi  87  Table 4.1: Fourier Analysis of the Line-Profile Variations of r Peg (CFHT)  Apparent Azimuthal Order i  Apparent Frequency (cycles day’)  Amplitude (continuum = 1)  2.6 6.6 11.1 14.9  18.5 22.3 21.0 29.8  0.0008 0.0027 0.0030 0.0044  between modes. The full spectrum was used to measure the radial velocity variations of r Peg. The spectra were interpolated to a common dispersion and the mean spectrum was calculated. The cross-correlation algorithm within IRAF was used to measure the shift in pixels between each spectrum in the series and the mean spectrum. The pixel shifts were converted to velocities (at 4500  A) and  are plotted in Figure 4.3. The error in each point  has not been determined. The cross-correlation program does provide an estimate of the error based on the width of the correlation peak but these are unrealistically large; the average error is 1.83 km s . Clearly, the accuracy of the measurements is better than 1 this value. Confidence limits for the radial velocity data were derived using Bootstrap Resam pling. The Fourier analysis of 100 random arrangements of the radial velocity data indicated that there is a 99% chance that peaks observed with amplitudes greater than 1 are not due to noise. 0.68 km s The observed radial-velocity variations were analyzed using the period analysis pro gram (PERDET) by Breger (1980a). With this program, the period, amplitude, and  Chapter 4. The Line-Profile Variations of ‘r Pegasi  88  Table 4.2: Fourier Analysis of the Velocity and Light Variations of r Peg (CFHT & UH)  Measurement  Frequency (fixed) (cycles day ) 1  Amplitude  Phase (at HJD 2448169)  RV  18.4  1 0.93 km s  0.64  IIU tB t.V  18.4 18.4 18.4  0.017 mag. 0.019 mag. 0.016 mag.  0.15 0.18 0.14  phase of the variations are determined by making a least-squares fit to the data with a sinusoidal function. Up to seven frequencies can be fit with the program. In the case of r Peg, an adequate fit was obtained with a single frequency of 18.4 cycles day , with 1 amplitude 0.93 km s . This frequency agrees extremely well with Breger’s published 1 value. The solid curve drawn through the data points in Figure 4.3 shows the fit. UBV photometry of r Peg, simultaneous with the CFHT observations, was obtained by Jaymie Matthews at the University of Hawaii 0.6 m telescope. His reduced light curves are plotted in Figure 4.3. Using Breger’s code once again, the frequency of oscillation was calculated for each wavelength band. The mean frequency was 18.2 cycles day 1 in agreement with the radial velocity data. The light curves of r Peg were fitted with sinusoids with the frequency fixed at 18.4 cycles day’. The solid curves in Figure 4.3 illustrate the fit and the results are summarized in Table 4.9. Notice that the radial velocity variations are about 180° out of phase with the variations in the V-band. The velocity-to-light ratio is 57 km s 1 mag 1 (in the V-band). This low number indicates that the mode responsible for the variations is nonradial, likely I  =  1 or 2.  Velocity-to-light ratios for radial modes are closer to 100 (Yang 1991). Potentially, the  Chapter 4. The Line-Profile Variations of r Pegasi  I •‘  1  . 0  0  I  I  I  I  I  I  I  89  I  I  I  I  I  I  I  I  I  II)  I  8169.7  8169.8 8169.9 8170 Heliocentric Julian Date (—244000)  I  0.02  8170.1  I  :1  0 I  —0.02  Ii  III  iii  liii  Ii:  AJ./.A/H  0.02  —0.02  Iii  liii  I  111111  1  0.02 0 —0.02 I  8169.7  I  8169.8 8169.9 8170 Heliocentric Julian Date (—244000)  8170.1  Figure 4.3: Simultaneous Radial Velocity and Light Variations of ‘r Peg (CFHT & UH). The radial velocity variations measured from the CFHT spectra are illustrated together with simultaneous photometry in three colours. Sinusoidal variations with a period of 18.4 cycles day’ provide a reasonable fit to all observations.  Chapter 4. The Line-Profile Variations of i Pegasi  90  ratio of the amplitude of the line-profile variations to that of radial velocity could also be used as an indication of the degree of oscillation. This quantity will be small for £  =  3  modes and large for radial modes. However, the value also depends on the projected rotational broadening. For r Peg, the RV-to-LP ratio is 930 km s . 1 The amplitude ratio and the phase shift between the colour and light variations can be used to estimate the mode of oscillation (Watson 1988), however, these quantities could not be accurately measured from the 1990 data. Breger (1991) derived AB_v/Av 0.39 ± 0.04, and cBv  —  qv  =  48 ± 100 for  T  =  Peg.  The projected rotational velocity of i Peg is quite large. It is therefore reasonable to assume that the inclination of this star can not be very different from i is true then tessoral modes with  Imi  =  £  —  =  900.  If this  1 are not likely to be observed because the  variations cancel across the equator. The remaining possibilities for the low-degree mode are: £  =  1 with  Imi  =  1 and £  =  2, with  Imi  =  in the 2-D representation is matched best with £  2, or 0. However, the peak observed =  1 mode (after taking into account  the projection effects discussed in Chapter 2). Theoretical calculations of massive stars (M  =  ) indicate that in the presence of rotation, prograde modes are more likely 0 12M  to be excited to large amplitudes than are retrograde modes (Carroll & Hansen 1982). Assuming that rotation has a similar effect of the oscillations of the less massive S Scuti stars (no calculations have ever been made), then a mode with (1, m) most likely source of the low-degree variations. If the mode is £ the light variations observed in  T  =  =  (1, —1) is the  1 it then follows that  Peg must be due entirely to temperature variations  since no light variations from geometrical distortions are introduced by this mode. The apparent azimuthal order of the peaks listed in Table 4.1 depend on the adopted value of the projection rotational broadening. By measuring the width of the proffle variations, an independent estimate of v sin i Adopting this value instead of v sin i  =  =  136 km s 1 was obtained for ‘r Peg.  150 km s causes the identified orders to be  Chapter 4. The Line-Profile Variations of ‘i- Pegasi  91  shifted to 2.6, 5.9, 10.0, and 13.7. The effect is largest for the high-degree modes but the identification of the low-degree mode is unchanged in this re-analysis. 4.3  The 1992 DAO Observations  On 1992 October 5 (UT) a second lengthy series of observations was obtained of  i-  Peg  from the Dominion Astrophysical Observatory. The observations were made at lower resolution (10.1 A/mm) but because the absorption lines of this star are broad, the lineprofile variations can still be reasonably sampled even at this resolution (with 30 pixels per profile). There were 31 spectra in the series obtained over 9.4 hours of observation. The Nyquist frequency of the observations is about 40 cycles day . The resolution in 1 1 FWHM. frequency is 2.56 cycles day The combined line-proffle variations of the M501, X4508, and )4515  A  lines are  illustrated in Figure 4.4. The pattern of variation is similar to that seen in the CFHT spectra. The two-dimensional Fourier spectrum resulting from the averaged variations is presented in Figure 4.5. The 99% confidence level for peaks resulting from this data was calculated to be 0.0010 relative to a continuum of unity. Figure 4.5 shows contours plotted at intervals of 0.0002. Those above the 99% level are plotted as thick lines. The maximum intensity of the highest peak is 0.0025. The error bar labeled “FWHM” illustrates the calculated resolution in frequency and azimuthal order. Like the 2D transform derived from the CFHT data, most of the power in this Fourier spectrum is associated with apparent frequencies between 18 and 32 cycles day 1 and with apparent modes between 5 and 20. The positions and amplitude of some of the largest peaks in this figure are listed  in Table 4.2. The uncertainties in these quantities are typicafly: , and EiA 1 cycles day  iz  0.5, t’  0.3  0.0005. The structure of the peaks in this plot hints that  additional unresolved modes may be present.  Chapter 4. The Line-Profile Variations of r Pegasi  92  1u1I1•  I  Zt.017 1.004  A  .986  ‘\  .956  958  O.930 4915  .901 \i\’  .888  ‘l\J’(VVV0.866 .884 .854 !!.874 .843  .831 .801 .790  _‘•__\  • /‘“—0.  9 9 9  .779  745  .745  4 4 3  .724  .769 .759 .734 .713  700 .690 .680 669  .889  646  .657 .646  .700 .690 .680  10.005 •  I,IIIIIIIIIII•II,II1I,,  —200  111.1•  0 200 400 —200 0 200 400 Velocity Velocity  Figure 4.4: The Line-Profile Variations of r Peg (DAO). Same as Figure 4.1 but for observations obtained at DAO (HJD 2448169).  CD  CD  I—i  aq  CD  -  Qo• C  —‘  o  0  CD  I  .  CD  00  ‘-÷ .  CD  -4.  CD  I—.  I-  Q CD  ‘  0  CJ1  0  z’)  .—cii  (4•  I-,  I-I  CD  I-  ‘  .  C;’  BCD0  Ci)C.”  OCD  oq  0  0  0 0  Apparent Frequency (cycles day ) 1 Cii 0  0  0  CD  0  CD  CD  Chapter 4. The Line-Profile Variations of r Pegasi  94  Table 4.3: Fourier Analysis of the Line-Profile Variations of ‘r Peg (DAO)  Apparent Azimuthal Order th  11.6 13.5 16.6  Apparent Frequency Amplitude (cycles day’) (continuum = 1)  21.4 27.0 30.6  0.0025 0.0022 0.0011  Radial velocity variations of ‘r Peg were measured from the DAO spectra using the cross-correlation technique. The results are illustrated in Figure 4.6. The variation at 18.4 cycles day’ was not recovered. The fluctuation which occurs a little past half way in the series is a result of instabilities in the spectrograph after refilling the CCD with liquid nitrogen. Ordinarily these data would have to be discarded. However, they have been retained here in order to demonstrate the effect of the error. The full amplitude of the shift introduced by refilling was actually 11.9 km s (1.2 pixels). Most of this has been eliminated by using a weighted average of the dispersion relations from the preceding and succeeding calibration spectra. The remaining error has an amplitude of 0.6 km s (0.06 pixels) and is not eliminated until the next set of calibration spectra are introduced (after  i-.’  1.2 hours). The spectra affected by this problem were included  in the analysis of the line-proffle variations and give rise to the peak observed at low apparent frequency and apparent azimuthal order in Figure 4.5. The corrupted spectra will not affect the variations observed at high degree and therefore still contain useful information.  Chapter 4. The Line-Profile Variations of ‘r Pegasi  ç -  o  2  i  I  I  95  I  I  I  I  1-  0  -  -  •  •....•••••.••  . • •  • .  • •  •  -  —2  -  -  I  I  I  I  I  8901.7  8901.8  8901.9  8902  Heliocentric Julian Date (—2440000) Figure 4.6: Radial Velocity Variations of i- Peg (DAO). Measurements of the radial velocity variations from DAO (HJD 2448901) give no indication of variations arising in the star. 4.4  Discussion  The Fourier maps generated from the CFHT and DAO observations, although taken two years apart, qualitatively agree very well. High-degree modes seem to occupy the same region of the Fourier spectrum and at least one mode near  =  11 appears in  both data sets. The most significant difference between the observations, however, is that no evidence of the low-degree variation can be found in the DAO data. The nondetection of variations in radial velocity is equivalent to photometric variations below 3 milli-magnitudes. Unfortunately, simultaneous photometric observations which might have supported this point were not obtained during the DAO observations. The disap pearance of the low-degree mode is not surprising given the history of this star. The photometric amplitude of r Peg has been known to vary from as low as 5 to as high as 12 milli-magnitudes (Breger 1991). It is unlikely, that the low-degree mode in  ‘,-  Peg  has disappeared altogether akin to the effect seen in early-type variable Spica (Smith 1985). However, the observations demonstrate that for a time r Peg could be mistaken as photometrically constant while high-degree modes persist with large amplitudes.  Chapter 4. The Line-Profile Variations of ‘r Pegasi  96  To derive the stellar characteristics of r Peg, the Strömgren photometry published in Hauck & Mermilloid (1980) have been adopted and the calibrations and suggested uncertainties of Crawford (1979) and Philip & Relyea (1979) were used. It was found that M  =  0.66 ± 0.3, Tef e  =  8000 ± 200 K and logg = 3.7 ± 0.1, and from these values  L = 43L , R = 3.4R and M = 2 0 .2Me. Note that these results differ from those of Breger (1991) who finds  T  Peg to be less luminous and hotter. The peak-to-peak  photometric variations imply temperature variations for this star of 1Tef f = 60 K which is much less than the uncertainty in the effective temperature. Dziembowski (1990b) examined pulsational instabilities for envelope models of S Scuti stars having three different luminosities and masses. The most luminous of these which has L = 40L 0 and M = 2M , closely resembles the inferred characteristics for r Peg. 0 While Dziembowski computed the range of unstable frequencies for radial modes, these can be adopted for nonradial modes of sufficiently small degrees  (  3) because the  radial variation of the eigenfunctions in the region where opacity mechanisms excite the mode is then only weakly f-dependent. The theoretical limits, calculated for prograde sectoral modes and corrected for rotation using Equation 1.3, are superimposed on the observed Fourier maps of r Peg in Figure 4.2 and 4.5. The thick solid lines denote the boundaries for a model of standard He abundance (Y = 0.28) and the thin solid lines denote the boundaries for a He-rich model (Y = 0.38). Note that the frequency of the low-degree mode falls within the boundaries of the theoretical model. When the relations are extrapolated to include high-degree modes, the limiting frequencies no longer have physical significance. Never the less, even though the region is less than 7 cycles day 1 in width, most of the inferred modes fall within the band of unstable frequencies. Thus, if the observed modes are prograde and sectoral, then each mode would have nearly the same oscillation frequency in the co-rotating frame of the star, about 17 cycles day . 1 Specific theoretical models of high-degree oscillations are not available to compare  Chapter 4. The Line-Profile Variations of r Pegasi  50  I  I  I  I  I  97  I  I  T  1  40  I  I  I  I  I  I  I  F—MODES  IFWHM  -  P—MODES 30  20-  G—MODES  -  0  0  I  I 5  I  I  I  I  I  10 15 Apparent Azimuthal Order  I  20  25  Figure 4.7: Identification of the High-Degree Modes. The high-degree modes observed in r Peg appear with frequencies which are lower than expected for p-mode oscillations. The spectrum of f-modes (solid line ) divides the spectra of p- and g-modes in an unevolved (nonrotating) star.  Chapter 4. The Line-Profile Variations of r Pegasi  98  with the observed oscillation spectrum of r Peg. However, the calculations of Dziem bowski (1977) and Lee (1985) predict that high-degree modes should be trapped either in the outer envelope as p-modes or in the interior as g-modes. Since modes trapped in the interior are unlikely to reach measurable amplitudes on the stellar surface, the observed high-degree variations are more reaiistically explained as p-modes (or f-modes). However, the modes observed in r Peg afl seem to have nearly the same frequency (in the corotating frame). Since frequency is an increasing function oft, the oscillation spectrum of ‘r Peg can conform to the interpretation of p-modes only if the the radial orders of the high-degree modes are as low as or lower than the radial orders of the low-degree mode (i.e., n  =  3 or 4). To demonstrate, the (approximate) theoretical f-mode spec  trum (for a nonrotating star) was calculated using Equation 1.6 and is superimposed on the observed Fourier spectrum of r Peg in Figure 4.7. (Only those contours above the 3 sigma significance level are illustrated.) The frequencies of the observed high-degree modes are shown to be lower than that which is expected for f-modes (even before mak ing the correction for rotation) whereas p-modes have characteristic frequencies that are greater than the f-mode spectrum. In the absence of specific theoretical predictions, this discrepancy remains unresolved.  Chapter 5  The Line-Profile Variations of 62 Tauri  5.1  Introduction  The S Scuti star  92  Tauri (HR 1412) is a bright spectroscopic binary  ( o 4 t =  3.41) and  a member of the Hyades cluster. The star was first discovered as a spectroscopic binary by Moore (1908) and Frost (1909). The first spectroscopic orbit was derived by Plaskett (1914). These initial observations were followed by the investigations of Petrie (1940) and Ebbighausen (1959). The orbit was found to be highly eccentric (e orbital period of P  =  =  0.750) with  140.728 days (Ebbighausen 1959). The spectrum of the secondary  star was not detected in these early studies. The system was finally resolved during lunar occultation observations by Peterson et al. (1981) which indicated that the components are separated by lxV  =  1.10 ± 0.05 and  that the secondary star is marginally bluer than the primary. Peterson et al. concluded that the primary is an A7IV star with V  =  3.75 and B—V  =  0.17 and that the secondary  is an ASV star with. V  =  4.85 and B  estimated to be M 1 / 2 M  =  0.8. Breger et al. (1987) modeled the photometric properties  —  V  =  0.16. The mass ratio of the system was  of components using the ubvyf3 system and concluded that magnitude differences between the components from V  =  0 to 1.5 magnitudes could be permitted.  Kr6likowska. (1992) employed evolutionary calculations of this system to further con strain the models of the primary and secondary star. The distance modulus to the Hyades was used to estimate the luminosity of the components. The membership of the  99  92  Tau  Chapter 5. The Line-Profile Variations of 92 Tauri  100  to the Hyades allowed further constraints to be placed on the chemical composition of the components. Kr6likowska concluded that the primary could be identified as an evolved star in the thick hydrogen-shell burning stage with mass M = 2.63 ± 0 0.01M and that the secondary was a main sequence star with mass M = 2.23 ± 0.01M . However, these 0 conclusions were subject to the assumption that overshooting in the convective core can be neglected. Significant overshooting might delay the evolution of the primary enough that the possibility that it too is a main sequence star can not be excluded. Recently, the secondary spectrum of 02 Tau was identified in the investigations of the spectroscopic binary orbit by Peterson (1991) and Peterson et al. (1993). In the first paper, a spectrum of Procyon was rotationaily broadened and used as a template in a cross-correlation calculation which took into account the signatures of both the primary and secondary star. A projected rotational velocity of v sin i  1 was derived 80 km s  for the primary whereas the secondary was found to rotate with v sin i = 150 km The mass ratio of the components was determined to be 1 /M = 0.876 in reasonable 2 M agreement with earlier estimates and with the calculations of Królikowska (1991). In a second study by Peterson et al. (1993) synthetic spectra were used as templates for the cross-correlation calculation.  A model which satisfied both the photometry and  spectroscopy was derived with Te 11 = 8400 K, log g = 3.9, and v sin i = 65 km s for the primary and Te 1 for the secondary. 1 = 8400 K, log g = 4.3, and v sin i = 170 km s The orbital parameters obtained from this investigation did not deviate much from the established values. The variabffity of 92 Tau due to stellar oscillations was first reported by loran (1977, 1979) who identified a frequency near 14 cycles day’ based on photometric observations. Further studies by Duerbeck (1978) confirmed the variability. Antonello & Mantegazza. (1983) later found frequencies at 13.26 and 13.22 cycles day 1 among others. In order to fully resolve all the frequencies of oscillations, a campaign to obtain multi-site photometric  Chapter 5. The Line-Profile Variations of 92 Tauri  101  observations of 62 Tau was orchestrated by Breger et aL (1987). Unprecedented coverage of the star was obtained during several weeks of observation to produce a frequency spectrum that was relatively free of day-to-day aliases. Four frequencies of oscifiation were identified in this study at 13.22970, 13.48090, 13.69362, and 14.31756 cycles day . 1 However, the authors were not able to unambiguously identify the oscillation modes from these frequencies alone. A second multi-site campaign was initiated by Breger et al. (1989) in order to inves tigate the persistence of the previously identified frequencies and to search for additional variations. Once again superb coverage was obtained, this time with multi-colour obser vations being made at one site. Frequencies consistent with those previously identified were derived from the data (at 13.229653, 13.480733, 13.693596, and 14.317637 cycles ) with one additional variation discovered at 14.614537 cycles day. However, 1 day— again reliable mode identifications could not be made without the availability of theo retical models to describe the expected frequency spectrum for 82 Tau. Meanwhile, an independent study by Kovacs S Paparó (1989) confirmed the frequencies identified by Breger et al. (1989). Based on calculations of the light-time delay due to the binary orbit, both studies concluded that at least most of the variations seen in 62 Tau must originate in the primary. Table 5.1 summarizes the known properties of the 62 Tau system. In this chapter, spectroscopic observations of 82 Tau obtained at the Canada-France-Hawaii 3.6m tele scope (CFHT) are analyzed in connection with the problem of mode identification in this star. Observations made at the 1.2 m telescope of the Dominion Astrophysical Ob servatory (DAO) are also discussed. In 1992, 82 Tau was made part of an international campaign (MUSICOS) to obtain high-resolution spectroscopic observations. These ob servations are discussed separately in Chapter 7.  Chapter 5. The Line-Profile Variations of 62 Tauri  102  Table 5.1: Summarizing 62 Tauri  • Orbital Parameters (This study) P = 140.738 ± 0.008 days K = 29.4 ± 0.1 km s  7 = 37.93 ± 0.08 km s e = 0.750 ± 0.002  asini = 0.377 ± 0.002 au 0 f = 0.361 ± 0.006M  • Spectral Type (Peterson et al. 1981, Peterson 1991, Peterson et al. 1993) Primary: Secondary:  A7IV A5V  v sin i = 65 km s 1 v sin i = 160 km s  • Temperature of the system from photometry (Breger et al. 1989) 1111 = Te  8200 ± 100 K  1 = 100 K Te  • Magnitude of the system from lunar occultations (Peterson et al. 1981) 0 = 3.41 V  =  1.10 ± 0.05  11/12 = 2.75  • Distance Modulus to the Hyades (Gunn et al. 1988) m  —  M = 3.28 ± 0.26  • Composition of the Hyades (VandenBerg & Poll 1989) 0.70  X  0.75  0.02  Z  0.03  • Evolutionary Stage (Królikowska 1992) Primary: Secondary:  H-Shell Burning Main Sequence  M = 2.63 ± 0.01M 0 M = 2.23 ± 0.O1M®  • Photometric Variations (Breger et al 1987, Breger et al. 1989) Freq. (c/d): Amp. (mag):  13.229653 0.0066  13.480733 0.0026  13.693597 0.0045  14.317637 0.0027  14.614537 0.0012  Chapter 5. The Line-Profile Variations of 92 Tauri  5.2  103  The 1990 CFHT Observations  A 5.41 hour time series of 82 Tau was obtained at CFHT on 1990 October 4 UT (HJD 2448168). The data consisted of 52 high-resolution (2.4 A/mm) spectra obtained with 300 second integration times. This translates to a Nyquist frequency of 230 cycles 1 and a resolution of 4.44 cycles day day . 1 Observations at CFHT were obtained during the orbital phase when the system was near perihelion allowing the greatest possible velocity separation between the two com ponents. The orbital radial velocity curve for  92  Tau is illustrated in Figure 5.1. In this  figure, measurements based on spectra obtained at CFHT and DAO have been included with published data from the radial velocity studies of Petrie (1950), Ebbighausen (1959), and Peterson et al. (1993)). The orbital parameters listed in Table 5.1 were calculated from the combined data using the programs of Andrew Walker (1993) and differ little from those published by Peterson et al. The data have been phased to HJD 2424216.632 with a period of P  =  140.738 days and the curve of best fit is shown. Note that because  the system is highly eccentric (e  =  0.762) the window during which the components are  at their greatest velocity separation is only a couple of days. The pair of dashed lines are separated by four days and indicate the phase during which the 1992 MUSICOS observations were obtained (Chapter 7). In Figure 5.2, the mean spectrum obtained during the 1990 CFHT observations is plotted together with mean spectra obtained from DAO at two different epochs (HJD 2448136 and HJD 2448596). The presence of the secondary spectrum on the blue side of the line profiles of the primary can be identified in the CFHT spectra by comparing the observations and is especially evident in the strong MgH-FeI blend at .X4482. The positions of the Doppler-shifted A4508 Fell line of the primary relative to the systemic ve locity of the system is indicated by the symbol ‘P’ in each of these spectra. The projected  Chapter 5. The Line-Profile Variations of 62 Tauri  100  104  I  I  x  x  0  0  >s  80  CFH9O x *  *  xo  xo  ::  lIx 0  -  0011  0  x  20  IOI  X  00  O*  0  *  ° X  00  0  MUSICOS  *  00  00  0 Petrie, 1935—39 x Ebbighausen, 1957—59 *Peterson, 1990—91 • Kennelly, 1990—91  P=140.738 days 0  I  0  -  I  I  0.5  I  I  1  I  I  1.5  Phase  Figure 5.1: The Orbital Radial Velocity Variations of 62 Tau. The radial velocity of the primary star of this binary system was measured from observations obtained at CFHT and DAO and added to those published in the literature to determine the orbital elements for the system. These data were phased to HJD 2424216.632 with the derived period of P = 140.738 days to produce this illustration. Observations obtained at CFHT when the Doppler shift between the components was greatest are indicated. Also shown is the phase during which the 1992 MUSICOS observations were obtained.  Chapter 5. The Line-Profile Variations of 62 Tauri  105  rotational broadening of the primary was measured by fitting rotationally broadened syn thetic proffles to the )4501 (Till) and M508 (Fell) lines to give vsini  75km s . The 1  position and projected rotational broadening of the secondary is not so easily determined. An independent measure of Doppler shift and broadening of the secondary was not de termined from the CFHT data, however the estimates derived by Peterson (1991) were at least consistent with the observations. The top panel of Figure 5.2 shows the absolute mean deviation of the time series of spectra obtained at CFHT (see Chapter 3 for a discussion of this quantity). Notice that the region of line-profile variability is confined within the limits of profiles of the primary star. There is no evidence of variability on the blueward side of these lines where the secondary spectrum is located. It can be concluded, therefore, that the oscillations  giving rise to the line-profile variations must originate from oscillations of the primary star as suggested by Breger et al. (1987). The width of the deviations due to line-proffle variability was used as an independent measure of the projected rotational broadening to give v sin i  =  65 km s based on the X4501 and A4508 lines. This value is in better  agreement with that obtained by Peterson et al. (1993) than is the value derived from line-proffle fitting. The variations in the line profiles of 62 Tau are illustrated in the time series of CFHT spectra shown in Figure 5.3. The profile displayed in this figure is a composite of the )4501, )i4508, and .X4515 line profiles. The residual variations formed after subtracting the mean profile are plotted on the right-hand-side of the figure. The labels beside each spectrum represent the central time of observation in days relative to HJD 2448168. The appearance of the variations in 62 Ta.u is very different from those seen in r Peg where there was an obvious progression of features which traveled from blue to red through the line proffles. The CFHT data of 62 Tau show a pattern of rising and falling residuals with very little motion through the profile. This type of behaviour could be caused by  Chapter 5. The Line-Profile Variations of 02 Tauri  I  I  I  HR 1412  I  106  I  (HiD 2448168)  I  I  I  0.0: 4460  4480  I  .1  • HR 1412  I  I I HR 1412  4460  I  I  4520  I  I  I  I  I  I  I  I  I  I  I  I  I  I  (HJD 2448168)  I  HR 1412  .1  4500 Wavelength (A)  I (HJD 2448136)  I  I  I  (HiD 2448596)  4480  4500 Wavelength (A)  4520  Figure 5.2: Mean Spectra of 02 Tau at Different Phases of the Orbit. The bottom three illustrations in this figure show the mean spectra obtained at CFHT (HJD 2448168), DAO (HJD 2448136), and DAO (HJD 2448596) at three different phases of the binary orbit. The position of the primary spectrum is indicated by the label ‘P’ relative to the systemic velocity for the star in each of these spectra. The presence of the secondary spectrum can be identified by comparing the observations. The top figure shows the absolute mean deviation plot for the observations at CFHT. Note that increase amplitude due to line-proffle variations is confined within the width of the rotationally broadened profiles of the primary.  Chapter 5. The Line-Profile Variations of 82 Tauri  107  beating between different modes. The observations were analyzed using the two-dimensional Fourier technique. Once again, in order to boost the signal-to-noise of the observations, the information in the three isolated proffles was combined. The lower limit to the projected rotational velocity (v sin i  =  ) was adopted for the calculation. The resulting Fourier map is 1 65 km s  illustrated in Figure 5.4. The amplitudes are measured as a fraction of the continuum level. Using the method of Bootstrap Resampling, the three sigma confidence level was calculated to be 0.0006 relative to a continuum of unity. Above this level, there is a 99% chance that an observed peak could not be due to noise. Contours in the Figure 5.4 are plotted at intervals of 0.0003. Thin lines indicate amplitudes below the confidence level and thick line indicate amplitudes above the confidence level. Two obvious peaks can be identified in the Fourier spectrum. The largest peak occurs at apparent azimuthal order ii  =  3.6 with frequency 1’  =  13.8 cycles day 1 and is likely  related to the photometric variations. A second smaller peak is also apparent at th with frequency 1>  =  =  9.3  16.0 cycles day 1 demonstrating that high-degree variations are also  present in this star. Most of the smaller peaks can be attributed to aliases of these two peaks. However, the third largest peak which appears at low frequency and high apparent azimuthal order may also be real. Table 5.2 summarizes the results. The errors in the measured quantities were estimated using Monte Carlo simulations to be: zth Az)  =  0.1, and AA  =  0.0003. Note that if v sin i  =  =  0.2,  1 were chosen for this star 75 km s  then the positions of the two largest peaks become shifted in apparent azimuthal order to 4.3 and 11.1 respectively. The radial velocity variations calculated using the cross-correlation routines in IRAF with the mean spectrum as a template (and evaluated at 4500  A)  confirm the presence  of low-degree variations. The radial velocity curve for 82 Tau is illustrated in Figure 5.5 and has been fitted by a sinusoidal variation with a single frequency of 13.7 cycles  Chapter 5. The Line-Profile Variations of 92 Tauri  PP  pip  108  iiiiiii  1.167 1.156 1.152  1:t iiII  —j,.--———1 .117 .113 —.—--—--..J,N’-——.—1 .109 f1 .106 .099 .—.‘‘1.099 1.087  072  1.O76 1.072 — ‘I’1 .069 .085  I:8 t:88 1.031 —.-A—.——-—1.031 1 027 .—J\.t..—-—-—L.027 1:O22 1.022 .017 I’’....-....-...—1.017 1.013  v1  1.005 1.001 .998  FZ  fTh —-_9g I 10.005 Ii  .1.  ._.  I  —200 0 200 400 —200 0 200 400 Velocity Velocity  Figure 5.3: The Line-Profile Variations of 92 Tau. Time series of spectra obtained at CFHT show variations within the line profiles of this star (left). The variations are more clearly seen in the series of residuals calculated by subtracting the mean profile from each observed spectrum (right). The numbers on the right-hand-side represent times of observation relative to HJD 2448168. The error bars indicate intensity relative to a continuum of unity.  Chapter 5. The Line-Profile Variations of 92 Tauri  5 c:  I  I  I  I  I  I  I  I  109  I  I  I  I  I  I  I  FWHMI  I  I  1 J  4f]sin  I  40  • 10  0  I  0  5  I  I  15 10 Apparent Azimuthal Order  I  20  25  Figure 5.4: The Fourier Representation of the Line-Profile Variations of 82 Tau. The line-profile variations were transformed in both time and space to produce this map of apparent azimuthal order and apparent frequency of oscillation. The error bar labeled ‘FWHM’ indicates the full width at half maximum resolution in frequency and azimuthal order. The diagonal lines indicate the frequencies theoreticafly predicted to be unstable for prograde sectoral modes. Limiting values for models calculated with normal Helium abundance (thick lines) and for Helium-rich models (thin lines) are shown. The error bar labeled ‘4 sin i’ gives an indication of the size of the rotational splitting. Contours have been drawn at intervals of intensity of 0.0003 relative to a continuum of unity and beginning at two standard deviations above the mean noise.  Chapter 5. The Line-Profile Variations of 82 Tauri  110  Table 5.2: Fourier Analysis of the Line-Profile Variations of 82 Tau  Apparent Azimuthal Order I’I  Apparent Frequency (cycles day— ) 1  Amplitude (continuum = 1)  3.7 9.2  13.8 16.1  0.0113 0.0044  Table 5.3: Fourier Analysis of the Velocity Variations of  92  Tau  Measurement  Frequency (fixed) (cycles day)  Amplitude  Phase (at HJD 2448168)  RV  13.66  1.67 km s  0.013  day. Residuals after subtracting variations at this frequency are non-negligible and reflect the fact that more than one mode is present. Better agreement is obtained by fitting the data with two sinusoids with frequencies 13.4 cycles day and 10.6 cycles day but the series is probably too short to warrant serious consideration of these frequencies. The ratio of amplitudes determined from the radial velocity and line-profile variations (RV-to-LP) is 150 km s. This value is smaller than that obtained for  ,-  Peg  but a comparison between stars is not meaningful without taking into consideration the amount of rotational broadening. In addition, the RV-to-LP calculation is meaningful only if the modes which have given rise to the radial velocity and line-proffle variations are the same. In the case of 92 Tau which is known to oscillate with several closely spaced  Chapter 5. The Line-Profile Variations of 92 Tauri  111  frequencies, this assumption may not be correct.  -  I  I  I  I  I  8169  Heliocentric Julian Date  I  I  I  8169.1 (—244000)  Figure 5.5: The Radial Velocity Variations of 92 Tau. Radial velocity variations were measured using a cross-correlation technique with the mean spectrum as a template. A sinusoidal fit with a single frequency of 13.7 cycles day 1 was made to the data and curve Non-negligible as the solid residuals obtained after through the points. is shown removing the variation at this frequency suggest that multiple frequencies are present.  5.3  Observations from DAO  Observations of  92  Tau were obtained at the Dominion Astrophysical Observatory as  part of the S Scuti survey. These spectra were obtained at lower resolution and did not have the same time coverage as the data from CFHT. The radial velocity and line-proffle variations measured from the DAO spectra support the results obtained with the CFHT data. Measurements of the projected rotational velocity based on line proffle fitting (77 km  _1)  and from the extent of line proffle variability in a plot of the absolute mean  ) indicate values which are somewhat larger than that measured 1 deviation (73 km s from the CFHT data. The DAO spectra also helped to define the orbital radial velocity curves for the  92  Tau system.  Chapter 5. The Line-Profile Variations of 92 Tauri  5.4  112  Discussion  Using the apparent magnitude determined by Peterson et al (1981) and the distance modulus to the Hyades cluster, the absolute magnitude of the primary star of  92  Tau  was calculated to be M = 0.47. This corresponds to a luminosity of L = 56L. The effective temperature of the primary was estimated to be Tejj = 8200 ± 100 K. From these values a radius of R = 3.75R 0 was derived. Taking v sin i = 65 km s , the amplitude 1 of frequency splitting owing to first-order effects of rotation can be calculated. In Figure 5.4, the frequency separation between two modes with m = 4 is indicated by the error bar labeled “4f sin i”. For this calculation, the inclination is assumed to be i = 90. Dziembowski (1990b) has calculated the boundaries of unstable frequencies for os cillating stars with luminosities up to L = 40L. The range of frequencies spanned by unstable radial modes of oscillation were calculated for  92  Tau by adopting the values  from Dziembowski’s most luminous model and then scaJing these frequencies using the period-luminosity-colour relation for S Scuti stars (Equation 3.5). The mass of the pri mary star M = 2.63M, derived by Królikowska (1992) was also adopted. The diagonal lines depicted in Figure 5.4 show the theoretical limits extrapolated for prograde sec toral modes and adjusted for the effects of rotation. The thick line shows the region of instability for models which have normal Helium abundance. The thin lines illustrate the region of instability for models which are overabundant in Helium. The two largest peaks in the Fourier map of 92 Tau fail within these limits. Therefore, if the high-degree mode is prograde and sectoral then both the low- and high-degree modes oscillate with about the same frequency in the corotating frame of the star. The amplitudes of both the photometric and radial velocity variations of  92  Tau at  13.7 cycles day’ are large. It is therefore unlikely that the variations at this frequency could arise from a mode with £ > 2. The analysis of the line-proffle variations has  Chapter 5. The Line-Profile Variations of 92 Tauri  113  also revealed a large peak with apparent azimuthal order  ii =  3.6 at nearly the same  frequency. If the variations detected as line-profile distortions are produced by the same mode that gives rise to the radial velocity and photometric variations then the most likely mode of oscillation is €  =  2. However, 02 Tau is known to oscillate with several modes  and the frequency resolution of the CFH observations is not sufficient to distinguish between them. In order to fully resolve all the frequencies, observations lasting several days are required. In Chapter 7, the results from a 4-day multisite campaign on 62 Tau are presented. The high-degree mode with th a prograde sectoral mode (say £  =  =  9.2 and £‘  =  16.1 cycles day’ can be reproduced with  8 or 9) which has a frequency in the corotating frame  of the star that is not very different from the frequencies observed for the low-degree modes. P-modes (trapped in the envelope) are expected to produce these variations. However for this to be the case, the radial order of the mode must be equal to or smaller than the radial order of the low-degree modes (i.e., n  =  2 or 3). Since frequency increases  with £ this requirement is not easily satisfied. Similar difficulties were encountered with the interpretation of the ‘r Peg spectra presented in Chapter 4. The observations of both stars await theoretical calculations in order to resolve the ambiguity.  Chapter 6  Four Line-Profile Variables Re-Visited  6.1  Introduction  The observations by Walker et al. (1987) of the first four line-proffle variables: 21 Mon,  v  UMa, and  2 ic  o1  En,  Boo were obtained with high resolution and high signal-to-noise  but the analysis offered by Walker et al. was very preliminary. In this chapter, the a new representation of line-profile variations of these stars is provided using the new two-dimensional Fourier technique. 6.1.1  HR 1298:  o1  En  1 Eridani is a rapidly rotating S Scuti star Omicron F211-III and apparent magnitude V  =  (v  sin i  =  98 km s) of spectral type  4.04. With photometric observations, Jorgensen  et al. (1971), Jorgensen & Norgaard-Nielsen (1975), and Poretti (1989) have detected multiple frequencies of oscillation in this star ranging from 6 to 14 cycles day . High1 resolution spectroscopic observations made by Yang & Walker (1986) established  o1  En  as the first high-degree line-proffle variable among the S Scuti stars. During these obser vations, Yang & Walker had attempted to measure the precise radial-velocities using a technique wherein a reference spectrum was superimposed on the stellar spectrum. They found, however, that the accuracy of the measurements was limited by the presence of ripples which traveled through the line profiles. These ripples were identified as high degree modes of oscillation.  o1  En was re-visited in the study by Walker et al. (1987)  114  Chapter 6. Four Line-Profile Variables Re-Visited  115  when the star was observed on two successive nights. No other spectroscopic observations have been reported for this star. 6.1.2  HR 2707: 21 Mon  21 Mon is classified as a F2Vn S Scuti star with apparent magnitude V projected rotational velocity v sin i  =  =  5.45 and  . It was discovered to be variable with a 1 130 km s  period of 0.11 days by Eggen (1968) who also noted variations in the amplitude. Gupta (1973) identified two frequencies of oscillation for this star at 10.0 and 13.3 cycles day 1 with a beat period at 0.3 days. The multiperiodicity of 21 Mon was confirmed by Stobie et al.  (1977) who identified five frequencies of variability which were interpreted as  nonraclial pulsations. Walker et al. (1987) observed 21 Mon and found large amplitude line proffle variations. 6.1.3  HR 3888: v UMa  Upsilon UMa is a bright (V  =  3.80), rapidly rotating (v sin i  =  110 km s ) S Scuti 1  star classified as F2IV. Very few photometric studies of v UMa have been reported. Danziger & Dickens (1967) established a timescale for variations at about 7.6 cycles 1 and noted that beating between multiple modes seemed likely. day  Walker et al.  (1987) established the presence of line-proffle variations in this star. Recently Korzennik et al. (1994) obtained fiber-optic echelle spectra of v UMa during four consecutive nights. Using a variation of the Fourier method already published by Kennelly et al. (1992), they identified modes of oscillation with effective azimuthal order ranging from 2 up to 12, with frequencies between 10.5 and 15.0 cycles day that gradually increased with azimuthal order. They also reported substantial night-to-night changes in the amplitudes of the modes of oscillation.  Chapter 6. Four Line-Profile Variables Re-Visited  8.1.4  HR 5329:  2  Boo  2 Bootis is a rapidly rotating S Scuti star (v sin i Kappa A8 IV with an apparent magnitude of V period as  =  116  =  =  115 km s_ ) of spectral type 1  4.54. Breger (1979) lists its photometric  0.066 days with an amplitude of L\m  =  0.03 magnitudes, although a  16-day beat period has also been reported (e.g., Desikachary et al. 1971). More recently, Jones et al. (1993) obtained several nights of CCD photometry of  ,2  Boo from a single  site and identified six frequencies of oscillation. However, an independent analysis of this data (Kurtz 1994) identified only two frequencies with certainty.  Boo is a spectroscopic  binary but its orbital parameters are not well determined. The period of the system is listed as P  =  1791.23 days. In fact, few spectroscopic studies have been made of this  object. However,  Boo was one of the first line-proffle variables discovered by Walker  et al. (1987). These data were analyzed previously by Kennelly et al. (1991) by fitting synthetic profiles to the data by a method of trial and error. The same data is analyzed in this chapter using the new Fourier technique. 6.2  Fourier Analysis of the 1987 CFHT Observations  The 1987 CFHT observations (HJD 2446838  -  39) were made in the same fashion as  those obtained for this thesis. (Some details may be found in Chapter 3 but for a more complete account see Walker et al. 1987.) Since the 1987 CFHT observations were not obtained specifically for this thesis, only the final results of the Fourier analysis will be presented in this chapter. The Fourier maps for the four stars are shown in Figures 6.1 through 6.5. Observations of  1  En were obtained on two successive nights. The variations of this  star were the most complex of all the variable stars observed by Walker et al. and this is reflected in the Fourier maps presented in Figures 6.1 and 6.2. In these plots, the  Chapter 6. Four Line-Profile Variables Re-Visited  117  amplitudes are expressed as a fraction of the continuum level. The 3 sigma confidence levels were calculated (using Bootstrap Resampling) to be 0.0014 and 0.0017 relative to a continuum of unity for the two transforms. Peaks with amplitudes above these levels are drawn with thick contours and the contours are drawn at intervals of 0.0003. Walker et al. attributed the observed variations to a mode with £  =  16 and  ii =  21.3 cycles day 1 but  the representations presented in this chapter demonstrate that the observations cannot be interpreted in terms of a single mode. Many peaks at high azimuthal order are apparent in both sets of data but otherwise the patterns of variation during the two nights are not very similar. It is not possible to identify the same modes from both sets of data and an attempt to combine the data from the two nights has not proved illuminating. An analysis of the radial velocity variations of this star suggest frequencies of oscillation 1 (both days) and 15.4 cycles day near 8.9 cycles day 1 (first day). In Figures 6.1 and 6.2, the peaks which might correspond to these variations can not be identified. Clearly, the oscillations of  o1  En will only be solved with a series of very extensive observations.  The Fourier representation of 21 Mon is shown in Figure 6.3. Contours with ampli tudes above the 3 sigma confidence level (0.0021) are drawn with thick lines. One very large peak with amplitude 0.0106 relative to the continuum can be clearly identified at apparent azimuthal order th  =  10 and frequency E  cantly from those of Walker et al. who finds £  =  =  13. These results differs signifi  8 and v  =  9.6 cycles day. No peaks  were identified which might correspond to the radial velocity variations of this star (at 1 and 16.7 cycles day 10.0 cycles day ). 1 The results for v UMa are presented in Figure 6.4 where two peaks with amplitudes greater than the 99% confidence level are seen to dominate its spectrum. One peak at low azimuthal order  (i$ =  3.5) corresponds well to the frequency of radial velocity  variations at 11.1 cycles day. The second peak has apparent azimuthal order  i =  7.2  . The interpretation by Walker et al. in terms of a single 1 and frequency 12.0 cycles day  Chapter 6. Four Line-Profile Variables Re-Visited  50  I  I  I  I  118  I  I  I  I  I  oc -  L  Th  I10  0  o  5  1  25  Apparent Azimuthal Order Figure 6.1: Fourier Representation of the Line-Profile Variations in 1 En (I). The line-profile variations of o1 Eri (HJD 2446838) were transformed in both space and time to produce this contour map of the apparent azimuthal orders and frequencies of oscillation. The contours are drawn at intervals of 0.0003 relative to a continuum of unity. Those contours with amplitudes above the 99% confidence level (0.0014) are plotted with thick lines. A theoretical upper limit to the pulsation frequencies of prograde sectoral modes is plotted for a model with standard He abundance (thick solid line).  Chapter 6. Four Line-Profile Variables Re-Visited  119  c 20  Apparent Azimuthal Order Figure 6.2: Fourier Representation of the Line-Profile Variations in o1 En (II). See Figure 6.1 (HJD 2446839). Contours are drawn at intervals of 0.0003 relative to a continuum of unity. Contours with amplitudes above the 99% confidence level (0.0017) are plotted with thick lines.  •-  C.  r-  o  •  o  c,  ..—‘  CD  oq  C..  E  0 $2  )*  0  0  .  -  CD  C.  .-  0  CD  CD  CD  •1 CD  ‘1  •ç  0  CT) C.1  • . , i CT) -4 0  CD  c4-  C.4  CT)  CT) C4  ‘•-  o  (  -‘(b 0 Q q  b’I  (T)  ‘CD  •  —‘  CD  0 ‘-1  c,I  0  cQ  CD  ci’  0  0 0  0  CZ 0  0  Apparent Frequency (cyc day ) 1 C)’ 0  0  I.  CT)  CT)  I.-.  Chapter 6. Four Line-Profile Variables Re-Visited  5J  —.  I  40  I  I  I  121  I  I  I  I  I  I  I  I  I  I  9  20  I 0  0  5  I  I  I  I  I  10 15 20 Apparent Azimuthal Order  25  Figure 6.4: Fourier Representation of the Line-Profile Variations in v UMa. See Figure 6.1. The contours are drawn at intervals of 0.0005 relative to a continuum of unity. Contours with amplitudes above the 99% confidence level (0.0014) are plotted with thick lines.  Chapter 6. Four Line-Profile Variables Re-Visited  122  50  40  C)  30 C) 0  z 0  20 ci U.’ U.’  10  0  0  5  10 15 20 Apparent Azimuthal Order  25  Figure 6.5: Fourier Representation of the Line-Profile Variations in Boo. See Figure 6.3. The contours are drawn at intervals of 0.0002 relative to a continuum of unity. Contours with amplitudes above the 99% confidence level (0.0017) are plotted with thick lines.  Chapter 6. Four Line-Profile Variables Re-Visited  mode with £  =  16 and v  =  123  15.2 cycles day 1 does not fit the observations. Radial velocity  variations were detected at 5.1 cycles day 1 but there is no corresponding peak in the Fourier map. Variation at this frequency might result from beating between two closely spaced (unresolved) frequencies near 11 cycles day . 1 The Fourier spectrum of  Boo is presented in Figure 6.5. Only one peak has an  amplitude greater than the 99% confidence level. This peak (at  112.  =  11.5 with  Li =  23  cycles day) agrees very with the mode identified by Kennelly et al. (1992) using a method of line-proffle fitting (t  =  12, v  Walker et al. find £  =  22.2 cycles day .) In addition, there appears also to 1  =  14 and z’  =  22.5 cycles day9. (In the original analysis,  be some evidence for a line-proffle variations resulting from the a low degree mode and perhaps even a mode at an intermediate value of £. However, the amplitudes of these modes lie below the 99% confidence level. 6.3  Discussion  It was not the purpose of this chapter to a present detailed analyses of the variations observed in the four stars:  o1  En, 21 Mon, v UMa, and  2 i  Boo, but rather to further  demonstrate the power of the two-dimensional Fourier method. The original analysis of the data presented in this chapter by Walker et al. (1987) attempted to identify oscillation modes of the four stars by tracing the motion of bumps within the profiles. At best only a single mode of could be identified with this method. The later analysis by Kennelly (1991) successfully demonstrated that the observed variations could in fact be modeled with line-profile variations resulting from nonradial pulsations. However, the method of proffle fitting which was used in that analysis was practical only for those stars which were dominated by a single periodicity (Kennelly 1990). In this Chapter, the re-analysis of the proffle variations using the new two-dimensional Fourier technique has  Chapter 6. Four Line-Profile Variables Re-Visited  124  Table 6.1: Characteristics of the Observed Stars  Star 1  En 21 Mon v Uma 2 Boo ç  0 M  Ti,,  log g  0 L/L  0 R/R  0 M/M  1.63 0.99 1.35 0.85  7250 7300 7200 8000  3.8 3.6 3.6 3.5  19 35 25 39  2.8 3.7 3.2 3.3  1.8 2.0 1.9 1.2  proved to be very effective, especially for the identification of multiple modes. The observed power spectra of the four stars were compared with those expected from theoretical models of Dziembowski (1990b). The luminosity of each star was determined from the absolute magnitudes which were derived from ubvy/3 photometry using the Crawford caiibrations in Chapter 3. (In lieu of a photometric magnitude, the absolute magnitude of  Boo was calculated as the average magnitude derived from period-  luminosity colour relation and from parailax measurements.) The effective temperatures and gravities of each star were determined from the models of Philip and Relyea (1979) assuming standard solar abundances. From these values, the radii and masses of the four stars were estimated. Table 6.1 summarizes the results. Dziembowski’s models are not specific enough to describe well all four stars. For the cooler stars (o 1 Eri and v Uma) lower limits to the oscillation frequencies could not be determined from Dziembowski’s illustrations. Nevertheless, useful comparisons can be made. The regions of unstable frequencies (for prograde sectoral modes) were extrapolated to high degree (for a model with inclination i = 90°) and are illustrated in Figures 6.1 to 6.5. The oscillation modes identified in  Boo, 21 Mon and v UMa fail  within bands of power which are consistent with Dziembowski’s models. The oscillations  Chapter 6. Four Line-Profile Variables Re-Visited  of  o1  125  En are not as easily described. Modes of oscillation covering quite a broad range in  frequencies are apparent for this star. These could possibly be explained as nonsectoral and retrograde high-degree modes. In afl cases, the frequencies of the modes observed with high th are lower than expected for p-mode oscillations.  Chapter 7  The MUSICOS Observations of  7.1  92  Tauri  Introduction  Intensive photometric observations (Breger et al. (1987), Breger et al. (1989), and Kovacs & Paparo (1989)) have shown that  92  Tau oscillates in as many as five low-degree modes  separated by less than 1.5 cycles day . Consistent frequencies and amplitudes were 1 derived from each of these investigations, suggesting that the modes present in  92  Tau  are stable over a. time scale of years. In Chapter 5, observations obtained during one night at CFHT were presented which demonstrated the presence of line-proffle variations in the spectra of  92  Tauri owing to low- (t  2  —  3) and high-degree oscillations. These  observations lasted only five hours and were therefore insufficient to resolve the individual frequencies of this interesting star. This could only be achieved with observations taken over several days. It is also necessary that the observations be free of gaps in order to eliminate one-day aliases which would otherwise confuse the period analysis. Such observations are only possible through multi-site campaigns. 7.2  The Goals of MUSICOS  MUSICOS (MUlti-SIte COntinuous Spectroscopy) is an international collaboration in terested in areas of astronomical research requiring continuous, high-resolution spec troscopy. The MUSICOS strategy has been 1) to organize international campaigns, 2) to build a prototype fiber-fed, high-resolution, echelle spectrograph (Baudrand & Böhm  126  Chapter 7. The MUSICOS Observations of 02 Tauri  127  1992), and 3) to install duplicates of the MUSICOS spectrograph on 2m class telescopes around the world. Stages 1 and 2 have been completed. An overview of the first MUSI COS campaign has been prepared by Catala et al. (1993). ‘7.3  Organization of the MUSICOS 92 Campaign  The second MUSICOS campaign was planned for December 1992 and coordinated by Claude Catala and Bernard Foing. The format of the second campaign resembled that of the first and consisted of three separate projects having similar observational require ments. Four days were to be devoted to each of the following projects during 12 successive days at multiple telescopes: 1) Rotational Modulation of Chromospheric Lines in the Pre-Main Sequence Star AB Aur, 2) Mode Identification in 62 Tau from Line-Profile Variations, 3) Surface Imaging of the RS CVn system HR1099. The main MUSICOS sites were: Observatoire de Haute Provence (OHP) in France, the William Herschel Telescope (WilT) in the Canary Islands, Penn State Observatory, Kitt Peak, the University of Hawaii telescope (UH), and the Xinglong observatory in China. The principle investigators (PIs) of each project were responsible for the preparation of individual observing proposals which were combined and submitted to time allocation committees at each observatory. Two transportable, fiber-fed spectrographs were avail able for the campaign. The new MUSICOS, echelle spectrograph was transported to the Hawaian site, while the ISIS spectrograph was sent to China. The observations of HR 1099 were planned in conjunction with JUE observations. The proposals were successful at all the core sites but were not necessarily granted the full 12 nights of observing time. Seven nights were awarded at the UH telescope and  Chapter 7. The MUSICOS Observations of 92 Tauri  128  it was decided to devote these nights to the observations of AB Aur and HR 1099 which would benefit more from the multi-wavelength character of the echelle data. Four nights were awarded at WilT of which one night was to be devoted to AB Aur and the rest to 92  Tau. Observing strategies were provided by the PIs and distributed to the observers at  all sites to insure the quality and consistency of the observations. Requirements for the observations were described in terms of three priorities governing the exposure times, signal-to-noise ratio, and resolution. If necessary these requirements could be relaxed in reverse order to maintain high quality data. The requirements for the  92  Tau spectra  were: Table 7.1: MUSICOS 92: Requirements  1. Resolution:  R> 30, 000  (mm 20,000)  2. Signal-to-Noise  S/N> 500  (mm 100)  3. Exposure Times  t, < 600 sec  (max 1200 sec)  In this case, the exposure times could be lengthened in order to provide higher signal-tonoise, and lower values of signal-to-noise were permitted as long as high resolution was maintained. The wavelength coverage of Ti II 4501.3  A,  Fe II 4508.3  92  A,  Tau was restricted to include the spectral features:  and Fe II 4515.3  A  which suffer little blending with  neighbouring lines and are therefore useful for studying line-proffle variations. How ever, large-wavelength coverage was encouraged in order to facilitate the measurement of  Chapter 7. The MUSICOS Observations of 62 Tauri  129  Table 7.2: MUSICOS 92: Sites and Instruments Site  Telescope  Instrument  Detector  Xinglong  2.16m  OHP  1.52m  WHT  4.2m  Tektronics 512x512 CCD One-D 2048 CCD 1280x1180 EEV CCD  KittP  1.5m McMath  ISIS spectrograph Aurélie spectrograph Utrecht Echelle spectrograph Stellar spectrograph  Nights  4 4 3 4  1000x3000 CCD  radial-velocity variations using a cross-correlation technique. Instructions regarding the sequence for observing stellar, bias, fiat-field and calibration spectra were specified prior to the campaign. ‘7.4  The MUSICOS Observations  Observations of 62 Tau were collected during December 07  -  11 UT (Heliocentric Julian  Dates: 2448964.0—2448968.0) from four of the six core MUSICOS sites in China, France, the Canary Islands, and Kitt Peak. As already stated, the 2.2m UH telescope did not participate in this part of the campaign and the 1.6m BMO telescope at Penn State ob tained no observations owing to poor weather. A summary of the instruments is provided in Table 7.1. The observations are summarized in Table 7.2 and the spectral coverage from each site is illustrated in Figure 7.1. Additional details about the observations and instruments is provided in the next section which describes the reduction of the data. The observers at the main sites were: Jiang Shiyang, Zhao Fuyuan, Bernard Foing  Chapter 7. The MUSICOS Observations of 82 Tauri  130  Table 7.3: MUSICOS 92: Observations of Site  Xinglong OHP WHT Kitt Peak  Reciprocal Dispersion (A/pixel)  Spectral Coverage  0.0654 0.0332 0.0419 0.132  33 68 83 106  (A)  Exposure Times (sec)  600 600 100 100  —  —  —  —  900 1200 600 300  82  Tau  S/N  Number of spectra  100-200 100-300 100-400 300-500  74 65 146 334  (Xinglong), Claude Catala (OHP), Eric Houdebine (WHT), Jim Neff (Kitt Peak). Ad ditional observations obtained at Vainu Bappu Observatory, India by K.K. Ghosh were also successful but were of much lower resolution. The reciprocal dispersion of these data was 0.511  A pixel . 1  In addition, some photometric observations were obtained in China  but are not presented here. The phase coverage from the combined sites was very good with a duty cycle of almost 80 percent. In many instances, overlapping observations were obtained. In Figure 7.2, the window functions for each of the MUSICOS sites and for the combined sites are illustrated. The one-day aliases which are strong in the observations from individual sites, are significantly reduced in the window function of the combined sites. The aliases could not be completely eliminated because a small gap in the coverage occurs each day across the Pacific. Also, no observations were made from WilT during the first day of the campaign.  Chapter 7. The MUSICOS Observations of 62 Tauri  131  1  0.4 4300  4400  4500  4600  Wavelength Figure 7.1: Spectral Coverage for the Individual MUSICOS Sites. The spectral region observed at each of the MUSIC OS sites is indicated relative to a spectrum obtained at the Dominion Astrophysical Observatory. Two orders of the echelle data from WHT are indicated as overlapping regions. .5  Reduction of the MUSICOS Spectra  The data were reduced using routines within IRAF. In many instances, the reduction routines had to be adjusted to accommodate differences in the spectrographs, image slicers and detectors at different sites. In general, however, the pre-processing was quite similar for each site. Bias levels (and darks whenever available) were subtracted from the every exposure. Stellar exposures were divided by the spectrum of a continuum source to remove pixel-to-pixel variations. Cosmic rays were removed. Wavelength calibrations were assigned based on the dispersion relations of arcs obtained preceding and succeeding the stellar exposure. Finally, the spectra were normalized to unit continuum. A detailed description of the observations, instruments, and reduction procedure follows. 7.5.1  Reduction of the Xinglong Data  The fiber-fed ISIS spectrograph was transported to China for use on the 2.16 m Xinglong telescope. The image slicer was a Bowen-Wairaven type producing 6 non-adjacent slices. The detector was 512 x 512 Tektronics CCD. A tungsten filament lamp was used for  Chapter 7. The MUSICOS Observations of 62 Tauri  132  1 0.5  0 1 0.5 ci)  0 1  .-l  0.5  —  S  0 1  C.) 0 ci)  0.5  0 1 0.5 0 1 0.5 0 —10  —5 0 5 Frequency (cycles day’)  10  Figure 7.2: The Observational Window Functions. The window functions for the individual and for the combined sites of the MUSICOS campaign are illustrated. Strong one-day aliases can be expected from the individual data sets but these are significantly reduced in the window function of the combined data set. The bottom figure shows the window function for data after correcting for differences in time resolution between sites.  Chapter 7. The MUSICOS Observations of 62 Tauri  133  flat-field corrections and a Th-Ar lamp for wavelength calibration. Both the light from the star and from the calibration lamp are fed through the optical fiber. A drift of one pixel per night due to flexure can be expected from the ISIS spectrograph. The bias level, obtained at the beginning and end of the night was subtracted from each image. The dark current was measured from the unexposed portions of the CCD frames and subtracted from the data. Two-dimensional flat-fields were normalized to one dimension before being used to remove pixel-to-pixel variations in the stellar spectra. Since the spectra were slightly inclined to the columns of the CCD, to maximize the signal the spectra were extracted by tracing the six apertures produced by the image slicer. The intensity within the central four apertures was summed to form a single one-dimensional spectrum. (The signal in the outer apertures was too low to include.) The calibration spectra were extracted by summing individual columns. The dispersion relation for each spectrum was calculated from a weighted average of the preceding and succeeding calibration spectra. The stability of the spectrograph measured from the drift in the arcs over the whole night was typically 2 km s 1 (Figure 7.3). 7.5.2  Reduction of the OHP Data  Spectra from the 1.52m OHP telescope were obtained with the Aurelie coud spectro graph with a 3000 gr/mm holographic grating. A Bowen-Wairaven image slicer which produces 5 adjacent slices was used during the observations. The detector was a one dimensional photodiode array with CCD read-out (2048 pixels of size 13 x 750 microns). The read-out noise of the detector is between 250 to 280 electrons. The gain is 40 elec trons per ADU. The detector is linear to better than 0.005 for the whole dynamic range (32000 ADU). The dark current is reported to be negligible. The bias level is known to be extremely sensitive to temperature but many biases were obtained during the night to compensate. The spectrum of a tungsten lamp was obtained for flat-field corrections.  Chapter 7. The MUSICOS Observations of 62 Tauri  134  A thorium-argon lamp was used for wavelength calibrations. Because the optical path of the stellar light and that of the calibration lamp are different, a slight shift in wavelength can be expected between the calibration and stellar spectra. The weather at OHP during the campaign was not ideal and the observations were sometimes interrupted by clouds. During times of extreme cloudiness dark exposures were obtained. The dark current was typically 4 ADU for a 20 minute exposure but changed by about 10 ADU during the night with no apparent trend. The dark current was larger than expected and amounted to approximately 0.5% of the stellar continuum but it was not possible to determine and subtract the appropriate level from each stellar exposure. Therefore, only the bias frames were subtracted from the data. Master flatfield frames were constructed from the individually recorded exposures and normalized to àne before being used to remove pixel-to-pixel variations from the stellar data. In this way, the relative signal levels of the stellar exposures were maintained when differing flat fields were used. The dispersion relation was determined for each spectrum using the Th-Ar arc spectra. The spectrograph was measured to drift by up to 2 km s during the course of the night (Figure 7.3), but the trend was systematic and could be corrected by assigning a dispersion relation to the stellar spectra interpolated from the preceding and succeeding arc exposures. ‘7.5.3  Reduction of the WHT Data  The Utrecht echelle spectrograph was used at the William Herschel 4.2 m telescope during the MUSICOS campaign. The observations were recorded on a 1280 x 1180 EEV CCD detector. The central wavelength was about 4909  A.  The lines of interest appeared  in order number 126 with free spectral range 4502 to 4538  A.  Flat-field spectra were  obtained at the beginning and end of the night and during times of poor weather. Bias exposures and calibration spectra were obtained throughout each night. Observations  Chapter 7. The MUSICOS Observations of 62 Tauri  135  were scheduled only during the last three nights of the campaign on 62 Tau. The weather at WilT during the first night of observation was poor. A few spectra obtained on this night were summed in order to improve their S/N while maintaining a combined integration time less than 1200 seconds. The WilT data were reduced using the echelle routines within IRAF. As a primary step, the spectra were trimmed and the zero-point level and bias were subtracted. The orders were inclined to the columns of the CCD and cross sections of the two-dimensional spectra were found not to be evenly distributed across a given order. In many cases, much of the signal from a stellar exposure extends outside the region of the CCD exposed during the flat-field exposures. Because of this, the flat field was used to define the locations of the apertures for all stellar and calibration spectra. The amount of scattered light, measured from the regions between the orders was found to be quite large, amounting to 10-20 percent of the true signal in the case of the lamps. The pattern of the scattered light was extremely lumpy and would have introduced many artificial features into the stellar spectra if not properly removed. The most successful method of removing the scattered light was to extract the stellar and flat-field exposures separately while measuring the ‘sky’ background from the region in between orders. Only the four orders centered on the spectral region of interest were extracted from the two-dimensional data. Echelle data are prone to fringing in the red spectral region which cannot be removed with flat fields. This was true of the WHT data. However, fringes also appeared in the extracted spectra of the lamps even in the region of interest near 4500  A.  The same  pattern could not be seen in the stellar data. Dividing the stellar spectra by these lamps introduced the fringe pattern into the stellar data. The pattern of fringes was not regular along the length of the spectrum but only appeared strong at the blue end of the data. Nothing could be done to salvage the lamps and, in the end, the flat fields had to be discarded. To remove the pixel-to-pixel variations in the stellar data, the  Chapter 7. The MUSICOS Observations of 62 Tauri  136  following procedure was adopted. A mean spectrum was constructed for each order from  au spectra obtained during the night. The noise pattern in each of these spectra was isolated by dividing the data by a smoothed version of itself. The resulting pattern was multiplied by a function which had been fit to the orders of the extracted lamp. The result was adopted as the new flat field and was divided into each spectrum obtained during the night. It should be noted that this procedure is potentially very dangerous because it could introduce artifacts around the positions of the lines. However, with minimal smoothing, no such artifacts were detected but a significant improvement in the signal-to-noise of the individual spectra was achieved. It was later discovered that fringing has been reported in other WHT runs. The problem seems to originate in the neutral density ifiter used during flat-field exposures. Stellar exposures should not have been affected. The stability of the WilT spectrograph was measured from the calibration spectra and is illustrated in Figure 7.3. A drift of 4.5 km s 1 was typical during one night’s observation. 7.5.4  Reduction of the Kitt Peak Data  The observations at Kitt Peak were obtained with the 1.5 m McMath-Pierce telescope. The detector was a 800 x 800 CCD binned by 4 x 1 and windowed to 47 x 800. A CR 600 dichroic ifiter and a BC 38 order sorting filter were used during the observations. Bias frames and flat fields were obtained at the beginning and end of the night. A thoriumargon lamp was used as a calibration source and exposures were obtained throughout the night. The reductions proceeded as follows. A master bias frame was constructed and sub tracted from the data. The dark level for each exposure was determined from the Un exposed parts of the CCD on either side of the data. A smooth function was fit to the  Chapter 7. The MUSICOS Observations of 62 Tauri  : I  I  I  I  I  137  I  I  I  I  I  I  I  I  2 0  —2  E :  I  8964  : I  I  8964.2 I  I  I  I  I  8964.4  I  I  I  I  I  8964.6 I  I  I  I  I  8964.8 I  I  I  I  8965 I  I  2 U)  S  0 —2  .. —  : 8965  0  0  —  2  L’  I  I 8965.2  I 8965.4  I  I  8966.2  8966.4  I  — —  — — —  %,.,:._:.  : .—  ........  .—.———  I  I  8965.6  I  I  I  8965.8  8966  8966.8  8967  I  0 —2 8966  : I  I  I  I  I  8966.6 I  I  2  0 —2 8967  8967.4 8967.6 8967.2 8967.8 Heliocentric Julian Date (—2440000)  8968  Figure 7.3: Stability at the MUSICOS Sites. The stability of the spectrograph used for the MUSICOS observations is illustrated in this figure. Straight lines connect the measure shift in the calibration spectra relative to some mean value for each site: Xinglong (solid), OHP (dotted), WHT (long dash), Kitt Peak (short dash). In most cases, there appears to be a smooth trend which can be removed from the observations by interpolating and assigning a weighted dispersion relation. Sudden jumps in the OHP curve are caused by refilling the detector with liquid nitrogen. A sharp shift in the Xinglong spectrograph at the beginning of the night goes off scale and is indicated by a circle. In this case, only one observation was obtained during the interim of the first two calibration spectra.  Chapter 7. The MUSICOS Observations of 92 Tauri  138  dark-signal and subtracted from the data. The two-dimensional flat fields were normal ized and divided into the stellar frames before extraction to insure that the pixel-to-pixel variations were correctly removed and to maintain the relative intensities of the stellar spectra in each column. The stellar data were collapsed into one-dimension by summing the exposed columns. The calibration data were extracted in a similar way. The spectra were calibrated using the dispersion relations of the arcs obtained prior to and following the stellar exposure. The stability of the spectrograph was measured and found to drift in a systematic way by up to 4 km s (Figure 7.3). ‘7.5.5  Corrections to the Line-of-Sight Velocity  Corrections for the heliocentric velocity and for the stellar and orbital motion of the bi nary system were applied to the observed spectra. When a dispersion relation is assigned to a spectrum using the packages within IRAF, the data is interpolated onto a wave length scale with a constant increment between the sampled points causing the data to become smoothed. By using a fifth order polynomial to interpolate the data the amount of smoothing is minimized, however every effort was made to avoid repeated smoothing of the data during successive steps of the reduction. Heliocentric corrections were made to remove contributions to the radial-velocity for the motion of the Earth. The velocity , and the velocity of the primary component in the 1 of the stellar system, 37.74 km s binary system were obtained from the fit to the orbital radial velocity curve (see Chapter 1 during the course of the 5). The velocity of the primary changed by about 1 km s observations. 7.5.6  Continuum Normalization  The task of normalizing the stellar spectra was complicated by the fact that very little true continuum exists for the spectrum of  92  Tau. Absorption lines are numerous and  Chapter 7. The MUSICOS Observations of 82 Tauri  139  broadened by the rotation of the star such that much of the spectrum is a blend of many absorption lines. The region around 4508  A is one of the few regions for which relatively  little blending occurs. It is important that the continuum be assigned to each spectrum in a consistent manner in order to properly measure the variations which occur within the line proffles. The choice of continuum can also affect the mode identification if as a result, the magnitude of the projected rotational broadening is incorrectly measured. The usual method of fitting the continuum to a series of spectra is to fit the continuum of all spectra to a limited number of specially chosen points. However, if the continuum is in fact the product of blended absorption lines which undergo variations then it is likely that the chosen continuum points are also variable. To avoid this problem an alternative method has been used to normalize the MUSICOS spectra. Spectra from each site of the campaign were treated separately. Data obtained on different nights were rebinned to the same wavelength dispersion to correct for shifts occurring between nights. All spectra from a given site were averaged together, weighted by the signal level in order to ensure the highest possible signal-to-noise. The mean spectrum was normalized to a somewhat arbitrary continuum. In this way, a single mean spectrum was generated for the data from Xinglong, OHP, WilT, and Kitt Peak. Data obtained at the Vainu Bappu observatory had much lower resolution than any of the data obtained from the core MUSICOS sites but it also had a much larger spectral coverage. The mean spectrum obtained from this site was used to define the continuum  for  92  Tau. The continuum of the mean spectrum from Kitt Peak was corrected to  this level by convolving it with a Gaussian to degrade the resolution then resampling both spectra in their common wavelength region at the dispersion of the Vainu Bappu data.  From the ratio of the two mean spectra, a low-order correction function can  be determined, resampled at the original dispersion, and applied to the original mean spectrum from Kitt Peak. The same procedure was applied to the spectra from the  Chapter 7. The MUSICOS Observations of 92 Tauri  140  other sites to bring them into reasonable agreement with Kitt Peak. Only large scale differences in the continuum of the spectra were removed in this way. In all cases, small scale differences which likely result from residual scattered light remained. The individual spectra from a given site were normalized as follows. Each spectrum was divided by the rectified mean for that site. Line-profile variations appeared as smallscale ripples in the resulting frames. Large scale variations due, for example, to differences in the illumination of the spectrograph during the course of the night were removed by fitting a low-order function to the residuals. This function was resampled at the dispersion of the original data and used to rectify each spectrum. A slightly different approach had to be adopted to normalize the echelle data from WHT. Only the two orders with spectral regions covered by the Kitt Peak data were considered. The mean spectra for both orders were normalized as described above to bring them into agreement with Kitt Peak. The individual spectra in both orders were also normalized as described above. Finally, the additional step of combining the two orders together was achieved by using a weighting function in the overlapping region. The consistency of the spectra, normalized in this way is generally found to be quite good. The absolute mean deviation is defined as the sum of the absolute value of each spectrum minus the mean, normalized by the number of spectra and is a good indicator for the presence of line-profile variations. When applied to the MUSICOS data, variations are clearly seen to be confined to the positions of the line profiles. The continuum is constant. Some difficulties were encountered with data from OHP for which non-stellar features, possibly due to scattered light, were seen to vary in intensity throughout the night. The impact of these features was minimized by excluding the affected region from the fit to the continuum. It is also worthy of note that the normalization procedure is achieved without further interpolation of the spectra.  Chapter 7. The MUSICOS Observations of 02 Tauri  7.5.7  141  Spectral and Temporal Resolution  The dispersion, signal-to-noise, and the integration times of the MUSICOS spectra cover a wide range of values. To analysis the data as a whole, requires that the spectra from each site be brought to a common spectral resolution. The data with the lowest spectral resolution were obtained at Kitt Peak. The spectra from the other sites can reduced to this dispersion by convolving with an appropriate Gaussian profile and resampling the data. The instrumental resolution from each site was measured from the width of the emission features in the calibration spectra. The appropriate width of the Gaussian smoothing functions required to bring all data to the same resolution with that obtained at Kitt Peak were calculated as follows:  =  V  —  The measured resolution  (c/FWHM) from each site and the width of the correction function are listed in Table 7.4. (Alternatively, the spectra could be brought to a common dispersion without any loss of resolution by interpolating the low-resolution data.) Table 7.4: MUSICOS 92: Spectral Resolution  Site  Xinglong OHP WHT Kitt Peak  Reciprocal Dispersion (A/pixel)  Observed Resolution  Instrumental FWHM (pixels)  Correction FWHM (pixels)  0.0654 0.0332 0.0419 0.132  32600 46900 46200 22600  2.1 2.9 2.3 1.5  2.2 5.3 4.0 0.0  The mean period of oscillation of  82  Tau is approximately 1.7 hours. Ideally, proper  sampling of the variations from this star requires exposure times no greater than 600  Chapter 7. The MUSICOS Observations of 62 Tauri  142  seconds in order to insure about 10 observations per cycle. It was perhaps an oversight that the restriction on the exposure times was given third priority during the campaign, after requirements on the resolution and S/N. Exposure times of up to 1200 seconds were permitted and some sites obtained much longer exposure times than other sites. Taking longer exposures causes phase smearing and reduces the amplitude of the observed variations. To ensure uniformity, a subset of the data was constructed by adding spectra such that the effective integration times of all spectra was between 600 and 1200 seconds. With the data binned in this way, the number of spectra obtained from each site becomes: Xinglong, 75, OHP, 62, WilT, 55, and Kitt Peak, 49. Thus, the number of observations obtained at each site are approximately equal and therefore the resulting spectrum will not be heavily weighted by the observations obtained at any one site (e.g., Kitt Peak). 7.6  Radial-Velocity Variations  Measurements of the radial velocity variations of  92  Tau were obtained using the cross-  correlation routine within IRAF with the mean spectrum from each site as the template. The accuracy of the measurements depends on the signal-to-noise of the observations, the dispersion, and the spectral coverage. The radial velocity curve derived from the convolved but unbinned data set is illustrated in Figure 7.4. Each panel in this figure shows the observations obtained during each day of the campaign. Observations from the four main sites are represented by different symbols. The amplitude of the variations is  1 km s with clear evidence for beating. In a few instances, observations obtained  at multiple sites were coincident. At these times, generally good agreement is found between the measurements from the different sites although some systematic effects can be identified. The Fourier amplitude spectrum was computed from the radial-velocity data derived  Chapter 7. The MUSICOS Observations of 82 Tauri  143  from the unbinned data using a routine for unequally spaced data (Matthews & Wehlau 1985) and is illustrated in the top panel of Figure 7.5. Also shown is the window function for the data set. A number of peaks are present, some of which occur at similar frequencies to those identified in previous photometric investigations, namely at 13.2 and 13.7 cycles . However, power at lower frequencies not identified in the photometric studies is 1 day also apparent in the radial velocity data. In particular, the peak near 12.3 cycles day’ is especially strong. Confidence levels for the observations were calculated using the method of Bootstrap Resampling. The 3 sigma confidence level is illustrated as a dashed line in the figure and specifies the amplitude above which there is a 99% chance that the observed peaks could not result from random fluctuations. The data were analyzed in the following way. The highest peak was identified in the Fourier transform. Then a least squares fit was made to the radial velocity measure ments using the approximate frequency of this peak as a starting value and allowing the frequency, amplitude, and phase to vary. Once the best fit had been obtained, the data were pre-whitened by subtracting the variations from the data. The Fourier transform of the new data set was examined to identify the frequency of the next most significant peak (relative to a recalculated 99% confidence level). Then a fit was made to the origi nal data using two sinusoids. The procedure continued until no obvious peak remained above the three-sigma level within the frequency band from 0 to 25 cycles day . The 1 procedure is illustrated step-by-step in Figure 7.5 for the convolved/unbinned data set. Five frequencies were derived with certainty and are listed in Table 7.5. As confirmation of these results and in order to investigate the possibility of system atic effects introduced by individual sites, subsets of the data were analyzed using the procedure described above. The results from the unbinned data set are heavily weighted towards the Kitt Peak observations because so many more observations were obtained at that site. The results of an analysis using the binned data are presented in Table 7.5.  Chapter 7. The MUSICOS Observations of 02 Tauri  144  I:  2 0 —2 I 8964  I 8964.2  I  I  I  I  8964.4  8964.6  8964.8  8965  8965  8965.2  8965.4  8965.6  8965.8  8966  I  I  8966  8966.2  2 Cl)  S  0 —2  •1-  .—  C) 0  —  2 0 —2  I  I  I  I  I  8966.4  I  I  I  I  I  I  I  I  I  8966.6 I  I  I  I  8966.8  I  I  I  I  I  8967  I  2 0 —2 I  8967  I  I  I  8967.2 8967.4 8967.6 8967.8 Heliocentric Julian Date (—2440000)  I  I  I  8968  Figure 7.4: The Radial Velocity Variations of 02 Tau. Radial velocity shifts of the spectra relative to a mean spectrum for each site were measured using a cross-correlation routine. In this figure, the symbols have the following meaning: circle = Xinglong, square = OHP, cross = WilT, triangle = Kitt Peak. Good agreement between data sets is obtained during times when overlapping observations were obtained. The solid curve is the result of a least-squares fit to the data with five sinusoidal functions.  Chapter 7. The MUSICOS Observations of 92 Tauri  145  1 0.5 0 —10  DATA  —  —5  0  5  10  1 Period  ci) DATA  —  2 Periods  .4W)  DATA  —  3 Periods  .—  C)  0  a) —  ct5  DATA  0.5  —  4 Periods  -  o DATA  —  5 Periods  0.5  o  0  5  10 15 Frequency (cycles day ) 1  20  25  Figure 7.5: The Spectral Window and Amplitude Spectra of 92 Tau The ampli tude spectra of 92 Tau are shown before and after the removal of each detected frequency. The spectral window function for the observations is shown at the top. The dashed line indicates the 3 sigma confidence level for the observations. Five frequencies were identi fied from the data.  Chapter 7. The MUSICOS Observations of 92 Tauri  :  2 0 —2  I  I  :  0  :  S  I  I  8964.2 I  I  2 : 0  I  I  I  <  •.  I  x  I  I  I  I  I  I  0.  U  I  I  I  I  x  I  I  8964.8 I  I  8965  I  I  s  x  0 o  I  8964.6  I  bo  Oo  ø’o  0  -  I  I  I  8964.4  I  I  I  U  U  : :  0  I  0 0  cpoQDOo  —  8964 -I  I  146  —2 :  4) .—  I  8965  0  0  :  0  I  8965.2 I  I  2 : 0  I  I  8965.4  I  I  I  I  I  8965.6 I  I  I  I  I  I  I  I  8965.8 I  I  I  I  I  8966 I  ‘Sc  -  0  °  0 0  .  —2 :  I  I  8966 :  I  I  8966.4  8966.2 I  I  I  I  I  I  I  I  I  8966.6 I  I  I  I  8967  8966.8 I  I  I  I  I  I  I  2 0  0  L  q%% j•  X)Sc  —2 :  8967  I  I  I  8967.4 8967.2 8967.6 8967.8 Heliocentric Julian Date (—2440000)  %: I  -  I  8968  Figure 7.6: Residual Velocities after Removing Five Frequencies. The residuals after removing five frequencies from the observed radial velocity variations of 92 Tau is shown. The symbols are the same as Figure 7.4. Systematic differences between sites can account for much of the scatter.  Chapter 7. The MUSICOS Observations of 62 Tauri  147  Table 7.5: MUSICOS 92: Analysis of the Radial Velocity Variations  Unbinned in Time Frequency Amplitude Phase (c/d) (km/s) (cycles)  11.75 12.32 13.25 13.71 14.33  0.50 0.62 0.48 0.50 0.21  0.90 0.29 0.75 0.30 0.75  Binned in Time Frequency Amplitude (km/s) (c/d)  11.71 12.31 13.23 13.67 14.32  Phase (cycles)  0.45 0.57 0.45 0.41 0.30  0.67 0.15 0.59 0.22 0.27  The same five frequencies are present. The unbinned data was also analyzed after ex cluding in turn each one of the sites. Again the same frequencies are identified although in some instances aliasing has become a problem. These results are listed in Table 7.6. Finally, the data from each site were analyzed independently but in this exercise only one frequency could be derived with any certainty. In this case, the frequency at 13.2 1 was the one that usually appeared with the largest amplitude. cycles day In Figure 7.4, the curve of best fit has been drawn overtop the observations. The residuals after subtracting five frequencies are shown in Figure 7.6. The scatter in the residuals is still quite large but much of this is due to systematic differences between the sites. There is no clear indication of additional frequencies although systematic trends remain. The Fourier transform of the residuals is compared to the data in Figure 7.7. 1 are observed in This illustration demonstrates that no variations above 15 cycles day the data. In Figure 7.8, the frequencies derived from the radial velocity variations are compared with those values obtained during the photometric campaign of Breger et al. (1989).  Chapter 7. The MUSICOS Observations of 92 Tauri  11111  I aj  0.5  lIt  hlhhhhlt  0  5  10  15  50  55  60  65  148  lillIlill  1(11  20  25  30  35  40  45  50  70  75  80  85  90  95  100  lilp  I  I  I  I  IllillillIl  II.  -  11111111111  11111111111111  jIll  0.5  100 lilt  105  110  111111  115 III  120  125  130  135  140  145  150  111111 liii 11111*11111111111 IIj  o.:’ 150  155  160  165  170  175  180  185  190  195  200  ) 1 Frequency (cycles day  Figure 7.7: Amplitude Spectrum Before and After Prewhitening. The amplitude spectrum before (thin line ) and after prewbitening (thick line ) are ifiustrated. No additional frequencies are apparent.  Chapter 7. The MUSICOS Observations of 82 Tauri  149  Table 7.6: MUSICOS 92: Further Analysis of the Radial Velocity Variations  Xinglong Frequency (c/d)  -  OHP Kitt Peak Amplitude (km/s) -  11.76 12.34 13.28 13.71  Xinglong Frequency (c/d)  11.70 12.33 13.15 13.66  0.53 0.65 0.50 0.58  -  OHP WHT Amplitude (km/s) -  0.45 0.70 0.43 0.44  Xinglong Frequency (c/d)  -  WHT Kitt Peak Amplitude (km/s) -  11.77 12.32 13.23 13.75  OHP Frequency (c/d)  11.74 12.26 13.30 13.69  0.54 0.61 0.56 0.49  -  WHT  Kitt Peak Amplitude (km/s) -  0.49 0.53 0.62 0.58  Three frequencies are found to be consistent at 13.2, 13.7, and 14.3 cycles day’. Al though velocity-to-light ratios may not be meaningful when comparing data taken years apart, the values for these three frequencies are 73, 111, and 78 km s mag . The vari 1 1 in the photometry was not recovered in the ation which appeared at 13.48 cycles day MUSICOS observations but this may reflect the inferior frequency resolution. Likewise, the frequency with the smallest photometric amplitude is not recovered. Nominally the velocity-to-light ratios for these frequencies are less than 38 and 83 km s mag . Two 1 new frequencies are found in the MUSICOS data. The reality of these peaks is confirmed by their appearance in subsets of the data. Apparently the assumption made in the  Chapter 7. The MUSICOS Observations of 62 Tauri  introduction, that the frequencies of  92  150  Tau are stable over long periods of time is not  necessarily true. Support for the conclusion of two new modes in  92  Tau is provided by the further  analysis of the 1989 photometric data. From the same data, Breger (1993) identifies two additional peaks near 12.2 and 12.8 cycles day 1 but with amplitudes below that which he considered reliable. In an independent analysis of the photometric data, Jinxin Hao . The amplitude 1 (1994) found additional peaks at 12.175, 12.836, and 12.390 cycles day 1 was even larger than the smallest peak identified of the variation at 12.175 cycles day 1 was removed by Breger (1989). (In addition, before the variation at 12.836 cycles day 1 could be observed.) In light of these from the data, an alias at 11.739 cycles day results, it seems likely that the frequencies observed in the MUSICOS data are real and have grown significantly in amplitude since the time of the photometric observations. 7.1  Line-Profile Variations  The analysis of line-profile variations in  92  Tau can provide information about both the  frequencies and modes of oscillation. In addition, modes of degree higher than 3 can be studied which would otherwise go undetected in photometric and radial velocity mea surements. In this section, a new representation of the line-profile variations is provided through the application of the Two-Dimensional Fourier technique. The spectroscopic data obtained from each of the sites are shown in Figures 7.9 through 7.23. In these figures, the information within three lines (M501, M508, and A4515) has been averaged in order to improve the signal-to-noise. Each plot shows the data from one site on a given day. By presenting the data in this way, comparisons between the different sites can be made. The variations of 92 Tau are obviously complex. Sometimes the variations are not very apparent. At other times, a clear progression of  Chapter 7. The MUSICOS Observations of 82 Tauri  I  I  I  I  I  I  I  I  —3  I  I  I  I  I  I  I  I  10  I  I  I  I  I I I 1i I  —2  —1  I  I  11  12  MUSICOS: I  I  I  I  I  0  1  I  I  I  13  I  I  I  151  I  I  I  I  I  I  I  I  I  I  I  I  1  I  2  1 1 1 I  I  I  14  I  I  3  I  15  16  Frequency (cycles daf’) Figure 7.8: Comparison of the MUSICOS Frequencies with Breger (1989). The frequencies derived from the MUSICOS observations are compared to the results from the Breger et al. (1989) photometric campaign. Also shown are the frequencies discovered by further analysis of the campaign data (Breger 1993). Three of the five MUSICOS frequencies agree very well with the photometry. At the top, a model for the rotational splitting of an £ = 2 mode is shown. traveling features can be observed. The variations are also shown as residuals, calculated by subtracting the mean proffle for each site. The spectra have already been corrected for the systemic and orbital velocity of the binary system. In Chapter 5, the projected rotational broadening was considered to be bounded between 65 and 75 km s based on measurements of the width of the line profiles and the width of the line-profile variations. Measurements of v sin i based on the MUSICOS observations suggest similar values. Here a value of v sin i  =  1 has 75 km s  been adopted. Residual variations within the line profile (—v sin i to +v sin i) relative to the mean proffle were determined for the )45O1, X4508, and )4515 line proffles and averaged to  Chapter 7. The MUSICOS Observations of 92 Tauri  1’11’1’1•  I  152  I  I  I  I  I.  1  1  0.9  0.9  0.8  0.8  00.7  0.7  0)  0.8  0.6  0.5  0.4  0.4  0.3  0.3  0.2  0.2  0.1  0.1  0  —0.1  0  I  0.05  0.02  —0.1 I  •  I  •  •  I  •  I  •  4506450745084509 45104511  Wavelength  I  •  I  •  I  •  I  •  I  •  450645074508450945104511  Wavelength  Figure 7.9: Time Series Observations from Xinglong (HJD 2448964). The line-proffle variations and corresponding residuals from the spectra obtained during the first day of the MUSICOS campaign with Xinglong telescope are shown. The line pro files at M501 A, )45O8 A, and )4515 A were averaged to produce this representation of the line-profile variations of 92 Tau. In this figure and those that follow, the times of observation are indicated on the y-axis in days. One full day is shown so as to allow comparisons between the different sites. The error bars give a measure of intensity in units of the continuum.  Chapter 7. The MUSICOS Observations of 82 Tauri  1’1•1’1•1  ,  153  •  I  11111,  1  1  0.9  0.9  0.8  0.8  cO.7  0.7  0.6  0.6  0.5  0.4  0.4  z  0.3  0.2  0.2  0.1  0.1  0-  0  J  —0.1 ,  0.02  0.05  —0.1 I  •  I  I  I  I  I  4506 4507 4508 4509 4510 4511  Wavelength  •  •  I  •  I  •  4506 4507 4508 4509 4510 4511 Wavelength  Figure 7.10: Time Series Observations from OHP (HJD 2448984). The line-proffle variations and corresponding residuals from the spectra obtained during the first day of the MUSICOS campaign from Observatoire d’Haute Provence are shown. (See Figure 7.9)  Chapter 7. The MUSICOS Observations of 82 Tauri  I’ll’.  I  154  I  I  •1  •  I  •  1  1  0.9  0.9  0.8  0.8  oO.7  0.7  CD  CD  0.8 V  0.5  0  0.4  0.4  -  C)  0.3  0.2  0.2  0.1  0.1  0  0 -  JO.05  10.02  —0.1  —0.1 I  4506 4507 4508 4509 4510 4511  Wavelength  .  I  .  .  I  .  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.11: Time Series Observations from Kitt Peak (HJD 2448964). The line-profile variations and corresponding residuals from the spectra obtained during the first day of the MUSICOS campaign from the McMath Telescope at Kitt Peak are shown. (See Figure 7.9)  CO  ‘—  I.-’  ‘  CD  0’  ,  0’  :-  t  (DCDCD  I-  O0Q  -ao  CD  I— I—  C)’  0  I—  Ci’  ..  C.1  0  .I-  I-•  0)  0  2.o’  CD  CD  .  j  0  C,’ 0 0)  (D4OQ  oIicn  0  0mi  ôO  clo  cCf  p  o  O  I  l3  CO  l3  C3  01  C)’  0)  0)  -1  -2  Heliocentric Julian Date (—2448964.0)  0  CO  P 0)  CO  CO  C7( 0  Chapter 7. The MUSICOS Observations of 62 Tauri  I  ‘  •  I  I  I  I  2-  1.9  1.8  ol.7  I  I  156  I  •  I  •  I  I  I,  -2  -  -  -  -  -  -  -  -  -  -  -  -  -  -  1.9  1.8  1.7  1.6  cs  1.5  1.4  1.4  .2 13’ C) 0  1.2  1.2  1.1  1.1  1  0.9  —1  -  J  0.05  0.02 -  I  I  I  I  I  4506 4507 4508 4509 4510 4511  Wavelength  I  I  0.9  I  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.13: Time Series Observations from OHP (HJD 2448965). The line-profile variations and corresponding residuals from the spectra obtained during the second day of the MUSICOS campaign from Observatoire d’Haute Provence are shown. (See Figure 7.9)  Chapter 7. The MUSICOS Observations of 62 Tauri  1•11•1’1  1’11•11’  2  157  -: 1.9  1.9 -  1.8  I.  -2  -  -  1.8  -  1.7  -  o1.7  -  (0  0:’  (0  1.6  V  1.5  1.4  z  -  C)  1.2  1.1  1 0.02  0.05  0.9  0.9  -  I  .  I  4506 4507 4508 4509 4510 4511  Wavelength  I  I  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.14: Time Series Observations from WHT (HJD 2448965). The line-proffle variations and corresponding residuals from the spectra obtained during the second day of the MUSICOS campaign from the William Herschel Telescope are shown. (See Figure 7.9)  Chapter 7. The MUSICOS Observations of 62 Tauri  ol.7  158  -  -  1.7  C) C’2  1.5  1.5  1.4  -  -  -  -  1.4  0 -  -  1.2  1.1  1  -  -  -  -  1.2  1.1  -1  -  10.05  1.3  10.02  0.9  I_0.9 11111111  4506 4507 4508 4509 4510 4511  Wavelength  1111111  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.15: Time Series Observations from Kitt Peak (HJD 2448965). The line-proffle variations and corresponding residuals from the spectra obtained during the second day of the MUSICOS campaign from the McMath Telescope at Kitt Peak are shown. (See Figure 7.9)  Chapter 7. The MUSICOS Observations of 02 Tauri  1’11•1’1•  I  I  159  I  •  I  •  I  •  I  3  3  2.9  2.9  2.8  2.8  o2.7  2.7  0)  2.6 q)  4-2.5  2.5  2.4  z  2.4  2.3  2.3  2.2  2.2  2.1  2.1  -  2  1.9  2 0.02  0.05  1.9 I  I  I  I  I  4506 4507 4508 4509 4510 4511  Wavelength  I  I  I  I  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.16: Time Series Observations from Xinglong (HJD 2448966). The line-profile variations and corresponding residuals from the spectra obtained during the third day of the MUSICOS campaign from Xinglong telescope in China are shown. (See Figure 7.9)  Chapter 7. The MUSICOS Observations of 82 Tauri  ‘  I  160  I  •  I  I  I  •  I.  3  3  2.9  2.9  2.8  2.8  2.7  -  -  -  -  -  -  2.7  CD CD  2.6  2.6  2.5  2.4  2.4 C)  2.3 q) C)  2.2  -  2.2  2.1  2.1  2  20.05  10.02 1.9  1.9 I  I  I  •  I  •  I  •  I  •  4506 4507 4508 4509 4510 4511  Wavelength  I  I  •  I  •  I  •  I  •  I  •  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.17: Time Series Observations from OHP (HJD 2448966). The line-proffle variations and corresponding residuals from the spectra obtained during the third day of the MUSICOS campaign from Observatoire d’Haute Provence are shown. (See Figure 7.9)  ‘-4.. +  —  +  ‘—  Co  P  ::i.  .c  CD  0  t-  Cl) CD  o  i  I_-I  0  ‘. .  CD  c_  0  CD  -  CDCD  -  00 CDp  + ‘—p0  CD  p  (D  O ,. b(D  .  CD  Cl) ( O(n  I—  (02  CD  -  (DO  ‘-  I  Cl)  —‘  CD  .  p  CD  0  I ‘-  C)’  0  4-  C,’  0’  0  C)’  0)  0’  I 4-  0’  0  I  C)’  C)’  0  C,’  0  C,’  cc  cc  C’)  C’)  ‘-  C’)  C’)  3  l3 C’)  CO  ‘3  CO  C’)  1’)  C)’  C’)  C3 0’  0)  C’)  0  -.2  C’)  Heliocentric Julian Date (—2448964.0)  C’)  C’) CD  cc  C’) Co  I.  I-’  0)  0  Cl)  Q 0  Chapter 7. The MUSICOS Observations of 62 Tauri  162  3  3  2.9  2.9  2.8  2.8  o2.7  2.7  CO 0) CO  2.6  2.6  a)  2.5  2.4  2.4  -  C)  2.3  2.3  2.2  2.2  2.1  2.1  2  2  1.9  1.9 4506 4507 4508 4509 4510 4511  Wavelength  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.19: Time Series Observations from Kitt Peak (HJD 2448966). The line-proffle variations and corresponding residuals from the spectra obtained during the third day of the MUSICOS campaign from the McMath Telescope at Kitt Peak are shown. (See Figure 7.9)  CD CDCD  p  CD  •g.  Eo  CD  CD  +•OQ OCDO  oq-o  oq  Cl28CV  fLJj  .  I— I.  (71  0  I-  (7’ ° CD  ,  g  0  I-.  (7’  0  C,’  CD  0  0  (7’  CO  C3  I-  0 F  F°  CO  0 F  C.,  0 F  C  o’  C.  CO  C.7  Li  CO  Heliocentric Julian Date (—2448964.0)  CO  CO  CO  CO CD  I-.  0  0 0  CD  ÷  CD  .  0  I.  -  •  CDCD  • oo  CD  0  CD  CD  0.0  E  °CD  oq  pCD  CiQ  b..  CD Oq  CD  Ci)  01  CD  J.  01  0  Cii  Ci’  0 -.2  01  0 0)  I  0  01  CD  .  0 (D  •  Cii  0)  (0  CD  CO  CO  CO  CO  CO 1’)  CO  CO CO  CO CO  CO  CO  CO C)’  CO  CO 0)  CO 0)  CO -2  CO -.2  Heliocentric Julian Date (—2448964.0)  CO  0)  CO CD  CD  0  Ci)  0 0  oq  CD  q.  CD  •  o  Cl)  CD  (DCD  00 CD  OIh  CD  o  o  o  QCD  :fc1)  CD  ‘  0  WI  c,  I  CD  0  CD  Co  ..  0  CD 0  CD  0  CD  CD  0’  Pj  0’  CD 0  CO  I-  0 l’3  l’3  CO CO  C.  CO  CO CD  CD  CO  -2  CO  0)  -.2  Heliocentric Julian Date (—2448964.0)  CO C  CO CD C;’  0  0  0 0  CD  Chapter 7. The MUSICOS Observations of 82 Tauri  166  4  4  3.9  3.9  3.8  3.8  03.7  3.7  0)  3.8 V .I-3.5  3.5  3.4 -  3.3  3.2  3.2  3.1  3.1  3  3  2.9  2.9 4506 4507 4508 4509 4510 4511  Wavelength  4506 4507 4508 4509 4510 4511  Wavelength  Figure 7.23: Time Series Observations from Kitt Peak (HJD 2448987). The line-profile variations and corresponding residuals from the spectra obtained during the fourth day of the MUSICOS campaign from the McMath Telescope at Kitt Peak are shown. (See Figure 7.9)  Chapter 7. The MUSICOS Observations of 92 Tauri  167  improve the signal-to-noise. The residuals were tabulated as a function of the surface angular coordinate and the resulting data were transformed in both time and space to produce a two dimensional Fourier representation of the data. Figures 7.24 and 7.25 show grey-scale representations of the results. Since the resolution in apparent azimuthal order is limited to th  =  4, the peaks appear as long strips. Most of the power is associated  with several peaks lying between 11 and 17 cycles day . These peaks are more clearly 1 seen in Figure 7.26 where slices at constant apparent azimuthal order are shown in each panel. The Fourier window function for each slice is equivalent to a one-dimensional temporal window function (see Figure 7.5). The 99% confidence level was calculated for the transform using Bootstrap Resampling and is indicated by the dashed line in each panel. The frequencies derived from the radial velocity variations are indicated in the third panel of Figure 7.26 where th  =  2. Recall that the simulations of Chapter 2  suggested that radial modes of oscillation will be represented by peaks near  1.76 due  to the effects of projection. Most of the power seems to be associated with low-degree modes, however some modes possibly as high as  =  8 are also revealed.  The window function for the two-dimensional Fourier transform should be free of strong aliases and therefore many of the peaks seen in Figure 7.27 likely result from real independent oscillations. However, because the modes are very closely spaced in both frequency and degree it is difficult to identify all the modes which are present. 7.8  Discussion  In this section, an interpretation of the observed spectrum of modes in  92  Tau is offered.  Conclusions are drawn from the information provided from both the MUSICOS obser vations and from the results of previous photometric campaigns. All of the observed  Chapter 7. The MUSICOS Observations of 82 Tauri  168  Apparent Azimuthal Order Figure 7.24: The Two-Dimensional Fourier Representation of 82 Tau. The line-profile variations of 82 Tau were transformed in both space and time to produce this grey-scale map of the apparent azimuthal orders and frequencies of oscillation. The largest peak occurs at a frequency of 14.61 cycles day’ and has a maximum value of 0.0045 relative to a continuum of unity.  Chapter 7. The MUSICOS Observations of 92 Tauri  169  Apparent Azimuthal Order Figure 7.25: A Detailed Two-Dimensional Fourier Map of 92 Tau. The spectrum of modes is shown in more detail in this illustration.  Chapter 7. The MUSICOS Observations of 62 Tauri  170  0.004 0.003 0.002 0.001 0  0.004 0.003 0.002 0.001 0  0.004 0.003 0.002 0.001 0 0.004 0.003 0.002 0.001 0  10 20 Frequency (cycles day)  30  0.004 0.003 0.002 0.001 0  12  0.004 0.003 0.002 0.001 0  13  0.004 0.003 0.002 0.001 0  14  0.004 0.003 0.002 0.001 0  15  I  0  10 20 Frequency (cycles day)  30  Figure 7.26: Slices of the Two-Dimensional Fourier Map of 92 Tau. The Fourier amplitude spectrum is shown as a function of frequency for values of the apparent az imuthal order ranging from iii. = 0 to 15. The frequencies of the radial velocity variations = 2 panel where the amplitude of the radial modes is expected to are indicated in the be near maximum. Some of these frequencies can be identified with peaks in the Fourier spectrum. Most of the power appears at low azimuthal order but at least two high-degree peaks are apparent at higher frequencies.  Chapter 7. The MUSICOS Observations of 2 Tauri  171  frequencies are assumed to originate with the primary of the binary system as demon strated by the CFHT observations presented in Chapter 5. The conclusions presented here must be considered tentative. To fully explain the oscillation properties of  62  Tau  would require the observed frequency spectrum to be matched to theoretical models which were not available for this thesis. Confirmation, especially of the new frequencies discovered in this thesis, from additional multisite observations is also especiaily desired. The low-degree modes, detected as either radial velocity or light variations, span a range of frequencies from 11.75 to 14.61 cycles day . From a comparison with the models 1 of Fitch (1981), Breger (1989) concluded that the observed (photometric) frequencies were compatible with radial modes of order, n  =  2 or 3. The ratio of the lower to upper  frequencies is 0.80 which is about equal to the expected ratio between the second and third overtones of radial oscillation. However, the mode of oscillation which gives rise to the frequency detected at 14.61 cycles day 1 can not be radial because it does not give rise to detectable radial velocity variations. Therefore, it is likely that there is at most only one radial mode present among the observed frequencies. The remainder of the modes must correspond to rotationally split, low-degree, nonradial modes. Although photometric and radial velocity measurements are sensitive, in principle, to modes with £ higher than £  3, many authors (e.g., Goupil et al. 1993) choose not to consider modes =  2 or even 1 when attempting to match the observed frequencies with  theoretical models. To justify this simplification, it is argued that (for constant amplitude on the stellar surface) the amplitude of the variation detected in integrated light decreases with increasing nonradial degree. However, this assumption can be misleading and in the analysis of the  92  Tan data it is recognized that even £  =  3 modes might give rise to  observable light and velocity variations. The magnitude of the rotational splitting in for the analysis of  92  92  Tau could potentially be a problem  Tau. The projected rotational velocity is v sin i  . The 1 75 km s  Chapter 7. The MUSICOS Observations of 92 Tauri  radius of the star is R  172  3.751?. Assuming an inclination of  of the first order splitting between modes with Am  =  900,  then the magnitude  1 is about 0.40 cycles day . The 1  frequency resolution of the MUSICOS data is not very different from this value which is close enough to half a day to introduce confusion with aliases. Fortunately, the resolution of the photometric data is extremely good and those frequencies identified both as radial velocity and light variations may be assumed to be reliable. It is curious to note that the variation at v 1 by At’ day At’  =  =  =  13.69 cycles day 1 differs from the variation at v  0.46 cycles day 1 and from the variation at z’  2 * 0.46 cycles day’. Also, the variations at v  1 are separated by At’ cycles day  =  2  *  =  =  =  13.23 cycles  14.61 cycles day 1 by  14.32 cycles day 1 and v  =  13.48  0.42. If these separations are due to rotational  splitting then the second order effects would have to be small. To identify the modes detected in the MUSICOS data, the NRP-code described in Chapter 2 was used to model the observed two-dimensional amplitude spectrum. In the Fourier representation of  92  Tau, the largest peak is that which appears at 14.61  . This peak has an apparent azimuthal order of 1 cycles day  it =  3.8. It is not detected  as a radial velocity variation but the frequency does match one of the smallest peaks found by Breger in his photometric analysis. This frequency is interpreted as an £  =  3 mode of  oscillation. It would be very difficult to otherwise reproduce the amplitude of the line proffle variations without introducing measurable radial velocity variations. The observed peak was reproduced with synthetic line-proffle variations resulting from a prograde, sectora.l mode with £  =  3 and frequency v  =  14.614 cycles day’. The amplitude of the  velocity variations on the stellar surface was determined to be 1.37 km s . 1 The signature of the sectoral £  =  3 mode was removed from the observed spectrum  by subtracting the synthetic Fourier amplitude spectrum. Additional modes could be identified from the residuals. The peaks which remain at low values of apparent azimuthal order are very crowded and are therefore difficult to interpret. However, two peaks at  Chapter 7. The MUSICOS Observations of 82 Tauri  high apparent azimuthal order, (th, z))  =  173  (6.98, 15.01) and (8.62, 16.16) are well separated  from the others. A new model was found that could reproduce both the £  =  3 mode and  the peaks at high degree. The high-degree peaks were modeled with prograde, sectoral modes with £  =  6 and £  =  8. The amplitude of the surface velocity variations was  estimated to be 0.67 and 0.44 km s 1 for the two modes. After subtracting the threemode model from the observed Fourier spectrum, significant power remained only at frequencies below 14 cycles day . 1 Assuming that the identification of the £  =  3, m  =  —3 mode is correct and neglecting  the second order effects of rotational splitting, the frequencies of the modes corresponding to m  =  —2, —1, 0, +1, +2, and +3 would approximately be 14.21, 13.81, 13.41, 13.01,  . According to this picture, the frequencies detected at 1 12.61, and 12.21 cycles day 1 could be interpreted as the m 13.69 and 13.23 cycles day  =  —1 and 0 modes. The  1 shows up as the second highest peak in the two-D Fourier frequency at 13.69 cycles day map. This peak has apparent azimuthal order th  =  3.4. It is also detected quite strongly  as a radial velocity variation. On the other hand, the variation at 13.23 cycles day 1 does not produce strong line-proffle variations but does show up strongly in both radial velocity and photometry. A model was developed based on these mode identifications, however the results were not able to produce a very satisfactory fit to the observations. The oscillation properties of this star are summarized in Figure 7.28 where the varia tions detected as photometric or radial velocity variations are indicated as line segments extending from £  =  0 to 3 and those detected as line-proffle variations are illustrated  with rectangular boxes. The modes identified from the simulations are shown as solid dots. Although the MUSICOS data have demonstrated evidence for two new frequencies arising from modes of low degree, identified two high-degree modes of oscillation, and demonstrated the existence of £  =  3 variations, it is disappointing that a more complete  description of the oscillations of this star could not be found. Further progress could be  Chapter 7. The MUSICOS Observations of 92 Tanri  18  174  ‘  17  16 -4  15— a)  ——--—-I-—-.  I  C)  C)  14 ‘..:  I  •...  C) 0)  z 13 ci)  12  11  10  0  I  I  I  1  2  3  I  I  I  •  4 5 6 7 Nonradial Degree  8  9  10  Figure 7.27: The Frequency Spectrum of 92 Tau. The variability of 92 Tau is summarized in this illustration. Those frequencies identified both as radial velocity vari ations and photometric variations are identified by the thick solid lines extending from azimuthal orders 0 to 3. Variations only seen in photometry are indicated by the thin dashed lines. Variations only seen in radial velocity are indicated by the thin solid lines. The peaics identified in the two-dimensional Fourier spectrum are illustrated as boxes. The size of the box generously allows for the uncertainties in the positions of these peaks. The solid dots show the modes determined from model fitting.  Chapter 7. The MUSICOS Observations of 92 Tauri  175  made by matching the theoretical spectrum to the observations. A third photometric campaign on  92  Tau is scheduled for the near future. In conjunction with these observa  tions, simultaneous multisite spectroscopic observations are also being planned. It will be interesting to compare the observed frequency spectrum obtained from the upcoming observations with that obtained during the MUSICOS campaign.  Chapter 8  Summary and Conclusions  The S Scuti stars are 1.5 to 2.5M® stars which lie within the boundaries of the Cepheid instability strip near the main sequence. They are unstable to low-order p-mode and to g-mode oscillations. The theoretical frequency spectra for S Scuti stars are extremely rich and the spectra become increasingly complex as the stars evolve off the main sequence. For evolved stars, most low-degree modes are of mixed nature and propagate like p-modes in the envelope of the star but change character to resemble g-modes in the core of the star. On the other hand, high-degree modes are predicted to be trapped as either pure g-modes in the core of the star or as pure p-modes in the stellar envelope. The number of modes detected observationally always falls short of the number which is predicted to be unstable. The nonlinear effects of coupling between modes at the surface with g-modes in the core has been suggested as a possible explanation for the process of mode selection. This mechanism may also explain why only a third of the stars in the S Scuti instability strip are observed to be variable. The relative sparsity of observed oscillations compared to that which is predicted theoretically poses a serious problem for S Scuti seismology. Although, the information provided by multisite photometric campaigns about the frequencies of low-degree oscilla tions can potentially be very precise, it is often insufficient to unambiguously identify the oscillation modes which are characterized by the three quantum numbers n, 1, and m. It is therefore vitally important that the modes of oscillation be determined observationafly.  176  Chapter 8. Summary and Conclusions  177  The identification of modes in S Scuti stars has been central to this thesis. In par ticular, high- and low-degree modes were studied by exploiting the spatial information available in the Doppler-broadened proffles of rapidly rotating stars. In these stars, the velocity variations on the stellar surface introduce bumps in the profiles which are ob served to travel from blue to red wavelengths in a time series of spectra. Often the variations appear very complex. However, by transforming the observations in both time and ‘Doppler space’ to the Fourier domain, a representation of the variations is revealed which can be easily interpreted. In Fourier space, each mode of oscillation is represented by a peak which is specified by its ‘apparent frequency’: i and its ‘apparent azimuthal order’:  i.  The properties of the two-dimensional Fourier representation were explored  using numerical simulations. Because of projection effects, the mapping between the ac tual modes and those which are observed is not linear. However, the 2-D representation can still provide a good indication of the nonradial degree of the modes provided the pro jected rotational broadening can be accurately determined. Meanwhile, the azimuthal orders can be distinguished if rotational splitting can be detected. In principle, modes as high as £  v sin i  =  10 or 20 can be detected in stars with projected rotational velocities,  . 1 60 to 200 km s  The seismology of S Scuti stars is particularly appeaiing because variations occur with observable amplitudes and because the stars within the instability strip exist at very interesting stages of evolution, as the stars evolve off the main sequence and be gin to burn hydrogen in a shell. In a survey of all rapidly rotating stars lying in and around the boundaries of the S Scuti instability strip, the frequency of high-degree oscil lations among stars in the strip was examined. In particular, the possibility that many of the photometrically constant stars could in fact be variable with high-degree modes was considered. However, the only stars which showed clear indications of line-profile  Chapter 8. Summary and Conclusions  178  variations were those which were already known to be variable from photometric investi gations. Those stars which seemed to be in a more advanced stage of evolution displayed variations with larger amplitudes than did those stars which were located on the main sequence. The discovery of main sequence line-proffle variables is important because the theoretical frequency spectra for these stars should be less complex and therefore easier to match with the observations. The line-profile variations of several S Scuti stars (r Peg, 62 Tau, UMa, and  2  o1  Eri, 21 Mon, v  Boo) were investigated in this thesis based on observations from CFHT.  Using the two-dimensional Fourier technique, modes of oscillation possibly as high as £  =  15 or 20 were identified. However, the interpretation of the variations remains  unclear. For example, the high-degree modes observed in r Peg were found to lie within a narrow band of frequencies. These observations were shown to be consistent with prograde, sectoral modes (1  =  —m) if the modes all have the same frequency in the  corotating frame of the star. However, even before the correction is made for rotation, the frequencies of the observed modes seem to be too low to be compatible with those expected for high-f, p-mode oscillations. Similar results were obtained for the other five stars. Theoretical investigations into this apparent discrepancy are greatly desired. The MUSICOS observations of 62 Tau have brought efforts in S Scuti seismology to a new level, proving that multisite spectroscopic campaigns are possible, practical, and capable of providing vital information necessary for the identification of oscillation modes. With the addition of the MUSICOS results, 02 Tau is possibly the most thor oughly observed S Scuti star to date. In spite of this fact, calculations of the theoretical oscillation spectrum specific to this star have never been made. Now more than ever these calculations are especially needed. In the future, even greater progress could be made by combining photometric and spectroscopic observations. Very extensive photo metric observations (lasting several weeks) could be used to obtain precise information  Chapter 8. Summary and Conclusions  179  about the frequencies of low-degree oscillations. 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