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Nonradial pulsations of rapidly rotating [delta] Scuti stars Kennelly, Edward James 1990

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NONRADIAL PULSATIONS OF RAPIDLY ROTATING 6 SCUTI STARS By EDWARD JAMES KENNELLY B.Sc, University of Western Ontario, 1988 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Geophysics of Astronomy We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1990 ©Edward James Kennelly, 1990 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. Department of Geophysics and Astronomy The University of British Columbia Vancouver, Canada Date Sept. 21, 1990 DE-6 (2/88) Abstract Time series of high resolution C F H T spectra of four 8 Scuti stars are examined for consistency wi th the presence of nonradial pulsations (nrp). Each series ex-hibits a progression of subfeatures moving from blue to red through the absorption lines. We have reproduced the profile variations using a geometrical model which imposes sectorial modes on the surface of the star. Model ing of the low-degree modes is guided by radial-velocity variations and the known photometric varia-tions. Synthetic spectra generated wi th the appropriate Teff and log g for each star are used as input for the model. In this way, the entire wavelength region covered by the observations can be reproduced and the effects of blending on the nrp profiles are included explicitly. The extension from a single-line model to one generated over a wide spectral region provides a much more sensitive comparison wi th the observations. In general, we find that the data can be reproduced by the combination of a high-degree mode (£ > 8) and a low-degree mode [i < 2). The intrinsic line widths and v sinz together set a l imit on the resolution of the stellar surface and by including this resolution i n our treatment we can estimate the velocity amplitude of the oscillations (~ 5 km/s ) . We find that low values of k (< 0.1) expected for p-mode oscillations are consistent wi th the observations. A possible relationship between the periods of the high- and low-degree modes is noted, the question of uniqueness is addressed and comparisons are made wi th models invoking starspots. For at least one of the stars (K2 Boo) , it is impossible to fit the observations by a starspot model without assuming unrealistic values of radius or equatorial velocity. n Table of Contents ABSTRACT . . ii LIST OF TABLES v LIST OF FIGURES vi ACKNOWLEDGEMENTS x 1. Line-Profile Variable Stars 1 1.1 Introduction 1 . 1 . 2 6 Scuti Stars 4 2. NRP Theory 6 2.1 General Theory . . . 6 2.2 Rotating Stars 13 2.3 Light Variations 15 2.4 Previous Investigations 17 3. Profile Modeling 19 3.1 Description of Program 19 3.2 Testing the Program 22 3.3 Reduction Techiques 24 3.4 Synthetic Spectra 27 3.5 Parameters and Profiles 28 3.6 Resolution of Surface Features 37 4. Observations 42 4.1 Data Aquisition 42 4.2 Data Reduction 43 i n 5. Analysis 52 5.1 Stellar Models 52 5.2 Profile Variations 59 5.3 Radial-Veloci ty Variations ." 66 5.4 NRP Models .' 69 5.5 Light Variations 77 6. Discussion 83 6.1 Uniqueness 83 6.2 Super Period? 88 6.3 Alternative Models 89 7. Conclusion 92 References 97 Appendix A . v U M a 101 Appendix B . 21 M o n I l l Appendix C . o 1 E r i 120 i v L I S T O F T A B L E S I. Observations 43 II. B u m p Ampl i tude in Observed Line Profiles 51 III. Rota t ional Broadening of Observed Line Profiles 53 I V . Stellar Parameters from ubvy Photometry 56 V . Support from Parallax Measurements 57 V I . Results from Ampl i tude Spectra 62 V I I . Line-Profile Variations 65 V I I I . Radial-Velocity. Variations 68 I X . NRP Models: Summary 76 X . Publ ished Photometric Periods 78 X I . Radial-Veloci ty and Photometric Periods 80 X I I . Geometrical Light Variations 82 X I I I . Effect of Inclination on Oscil lat ion Parameters 85 X I V . Star Spot Models 91 v L I S T O F F I G U R E S 1.1 Pulsat ion in the H - R Diagram 3 2.1 Veloci ty M a p of Surface Oscillations 10 2.2 Identification of NRP Modes -. 12 3.1 Format ion of NRP Bumps by Velocity Perturbations 23 3.2 A T i m e Series of Mode l Profiles a) profiles, b) residuals " 25 c) mean, d) mean deviation 26 3.3 NRP Line Profiles: Mode Dependence a) sectorial modes (£ = m) 29 b) effect of m 30 c) effect of £ . 31 3.4 NRP Line Profiles: Dependence on Input parameters a) velocity amplitude 33 b) incl inat ion 34 c) fc-value 35 3.5 The Effect of k on the Residuals and M e a n Deviat ion 36 3.6 Resolution of the Stellar Surface a) bump amplitude 38 b) mean deviation amplitude 39 3.7 Temporal Resolution 41 4.1 Da ta Reduct ion a) raw data, b) normalized data, c) rectified data 45 4.2 K2 Boo Observations a) spectra, b) residuals 47 v i c) mean, d) mean deviation 48 4.3 K2 B o o Observations of the A4508 line a) residuals, b) mean deviation 50 5.1 K2 Boo Synthetic Spectrum a) intrinsic, b) rotationally broadened 58 5.2 Fourier Ampl i tude Spectrum of K2 Boo a) at 10 k m / s spacing 60 b) mean spectrum 61 5.3 Line-Profile Variations of K2 Boo 64 5.4 Radial-Veloci ty Variations of K2 Boo 67 5.5 K2 B o o M o d e l of the A4508 line a) residuals, b) mean deviation 71 5.6 Comparison of Mode l wi th Observations of K2 Boo (A4508) a) residuals 72 b) power spectrum 73 5.7 NRP M o d e l for K2 Boo a) spectra, b) residuals 74 c) mean, d) mean deviation 75 A . l v U M a Observations a) spectra, b) residuals 102 c) mean, d) mean deviation ' 103 A.2 v U M a Observations (A4508) a) residuals, b) mean deviation 104 A.3 v U M a synthetic spectrum a) intrinsic, b) rotationally broadened 105 vi i A.4 Fourier Amplitude Spectrum for v U M a a) at 10 km/s spacing 106 b) mean spectrum 107 A.5 Line-Profile Variations of v U M a a) high degree 108 b) low degree 109 A. 6 Radial-Velocity Variations of v U M a 110 B. l 21 Mon Observations a) spectra, b) residuals 112 c) mean, d) mean deviation 113 B.2 21 Mon Observations (A4508) a) residuals, b) mean deviation 114 B.3 21 Mon synthetic spectrum a) intrinsic, b) rotationally broadened 115 B.4 Fourier Amplitude Spectrum for 21 Mon a) at 10 km/s spacing 116 b) mean spectrum 117 B.5 Line-Profile Variations of 21 Mon 118 B. 6 Radial-Velocity Variations of 21 Mon 119 C. l o 1 E r i (1st night) Observations a) spectra, b) residuals 121 c) mean, d) mean deviation 122 C.2 o 1 E r i (2nd night) Observations a) spectra, b) residuals 123 c) mean, d) mean deviation . 124 vm C.3 o 1 Er i (1st night) Observations (A450S) a) residuals, b) mean deviation 125 C.4 o 1 Er i (2nd night) Observations (A4508) a) residuals, b) mean deviation 126 C.5. o 1 Er i synthetic spectrum a) intrinsic, b) rotationally broadened 127 C.6 Fourier Amplitude Spectrum of o 1 Er i (1st night) a) at 10 km/s spacing 128 b) mean spectrum 129 C.7 Fourier Amplitude Spectrum, of o 1 Eri.(2nd night) a) at 10 km/s spacing 130 b) mean spectrum 131 C.8 Fourier Amplitude Spectrum of o 1 Er i (both nights) a) at 10 km/s spacing 132 b) mean spectrum 133 c) window function, d) residual 134 C.9 Line-Profile Variations of o 1 Eri (1st night) 135 C I O Line-Profile Variations of o 1 Eri (2nd night) 136 C . l l Radial-Velocity Variations of o 1 Eri (1st night) 137 C.12 Radial-Velocity Variations of o 1 Er i (2nd night) 138 ix A C K N O W L E D G E M E N T S Yay! It's done... Thanks to my supervisor, Gordon Walker, for suggesting the topic and for his insight throughout its completion. Thanks to Jaymie "The Poob" Matthews for keeping me sane in times of crisis, for his good (and bad) suggestions and for his supurb ski l l as an editor. Thanks to " R E T I C E N T - M a s t e r " Stephenson Yang who, wi th my supervisor, obtained the data. Thanks to Ivan Hubeny for providing the model atmosphere and synthetic spectra code. Thanks to Gerry Grieve for his help wi th the nrp program and other computer-related stuff. I 'd also like to thank all the astronomy-types, especially my fellow stooges B r a d " C u r l y " Gibson and P h i l "Moe" Hodder for being nuttier than I am, Nadine Dinshaw for her friendship, and Claudia Mendes de Oliveira for caring, as well as Dave Woods, Gordon Drukier, James Brewer, Dan Hurley, K a l p a n a (and J im) Gi l roy and Derek Richardson. Thanks also to my friends Carola Schultheiss, Chris Langdon, and Dave Butler for giving me reasons to not work on my thesis and support when I was. Special thanks goes to my family; M o m , Dad , Stacey and R u t h (and my evil twin Jerome) for their support. x Chapter 1 LINE-PROFILE VARIABLE STARS 1.1 I n t r o d u c t i o n Developments in instrumentation over the past decade have made possible new dis-coveries in high resolution spectroscopy. Observations of rapidly rotating pulsating stars with high spectral and time resolution has revealed a new class of variables stars, the "line-profile variables". The shapes of their absorption lines change cycli-cally with time. Low-resolution studies have detected changes in line asymmetry. High-resolution studies have unveiled stars with subfeatures (or bumps) traveling through their rotationally broadened line profiles. The bumps in the star's profiles appear to be related to features on or near the surface of the star. The rotation introduces wavelength shifts of the light from different positions on the stellar sur-face to produce a one-dimensional mapping of the surface in each line profile. This effect was dubbed "Doppler Imaging" by Vogt and Penrod (1983). 1 The traveling bump phenomenon has been observed among rapid rotators in at least two instabili ty regions of the Hertzsprung-Russell diagram. B o t h the early-type O B variables and the S Scuti variables exhibit similar line profile variations (see Figure 1.1). The first detection of the moving bump phenomenon was made for the 0 9 . 5 V star C O p h by Walker, Yang and Fahlman (1979). Soon after, Walker et al. (1982) discovered similar variations in the second star, a V i r (B1.5IV) . Almost al l studies have been devoted to the detection and understanding of such variations i n the early-type stars. A number of theories have been proposed to account for the presence of trav-eling features in the profiles, such as temperature variations on the stellar surface (spots), occultations of the surface by circumstellar "blobs" of material , and trav-eling surface oscillations. Vogt and Penrod (1982) have discussed and applied these models to their observations of £ Oph. The possibility that surface spots could cause the variations was first proposed by Walker, Yang and Fahlman (1979) but was retracted (Walker, Yang and Fahlman 1981) because the motion of the bumps through the line profiles was too rapid to be caused by rotation alone. They suggested instead that the variations could be due to obscurations of the star by blobs of material in a circumstellar disk, although they also mentioned the possibility that surface oscillations might be responsible. Vogt and Penrod (1982) obtained good agreement wi th observed line profiles using an occultation model only if the blobs were "spokes" of enhanced density. However, simultaneous pho-tometry failed to show the light variations predicted by this scenario, while models invoking surface oscillations were successful in reproducing the observations. The latter mechanism - nonradial pulsations (nrp) - is widely accepted as the correct explanation of the line-profile variations. Of course, no one mechanism 2 - 6 - 4 1 1 1 1 1 1 f \ 0 , B variable c h e i d s t r i p , y?Cephei, ^ n » ^ s t a r s ( ) / / / _ - 2 \ \\ i // Early-type / / - V supergiants / / / — 0 "Variable DO \ / / / ~ / \ \ v £ < J Set stars 1 I / c s ^ ^ A p stars J 2 ' Giant _ branch 4 ^\'' ( \ / / N & S u n — 6 _ \Var iab le Jl \^ V N DQ jl M a i n sequence^ -8 N ~ Variable DA CS<v whito dwarf \soquence 1 1 1 1 1 r O BO AO FO GO KO MO ~ 1 0 5 27000 10400 7200 6000 5120 3750 Spectral Type (T.rc) F I G U R E 1.1. Pulsation in the H-R Diagram. Both the early-type OB stars and the 8 Scuti stars have been found to display trav-eling subfeatures in the absorption line profiles of their spectra and are associated with known regions of stellar pulsation. Unno et al. (1989). 3 need be responsible for the variations nor must it be the same for al l types of stars, but nrp theory is usually consistent wi th the observations. 1.2 8 S c u t i S tars The 8 Scuti variables are A - F type stars ly ing just above the main sequence in the H - R diagram (Figure 1.1). Thei r photometric periods range from about 30 minutes to 5 hours wi th a mean period of about 2 hours. Typ ica l broadband light amplitudes are around 0.02 mag. Radial-velocity variations are on the order of 1 km/s . B o t h radial and nonradial modes of oscillation have been identified and very often the stars are multiperiodic. Because of this property, 8 Scuti stars would be prime candidates for stellar seismology whereby the internal structure of the star is inferred from the oscillations on the surface. Approximately one-third of the stars i n the 8 Scuti instabil i ty strip are variables. Most of these identifications have been for large-amplitude low degree modes. Perhaps a l l stars are unstable to nrp but the amplitudes may be too low and the degree too high to detect i n many cases. Very litt le is understood about what would drive the high-degree pulsations and the factors that select only certain modes. A good review of the properties of 8 Scuti stars is given by Breger (1979). Dur ing a project to measure precise radial velocities of 8 Scuti stars, Yang and Walker (1986) found that the star o 1 E r i (F2II-III) displayed features moving through its line profiles. Walker et al. (1987) followed up this discovery wi th observations of four rapidly rotating 8 Scuti stars (including o 1 E r i ) , a l l of which showed regular progressions of traveling subfeatures. Because of the extremely 4 high quality of the spectra, these four stars are ideal candidates for testing the theories which have been invoked to explain the variations. Th i s study is an analysis and interpretation of the variations observed i n the 8 Scuti stars: K2 BOO, v U M a , 21 M o n , and o 1 E r i . 5 C h a p t e r 2 N R P T H E O R Y 2.1 G e n e r a l T h e o r y Consider a star, constructed such that some sort of driving mechanism causes it to pulsate. (The specific driving mechanism for the pulsations is not an issue here.) The pulsations manifest themselves as variations of stellar radius, temperature and brightness. If the pulsation amplitude depends on the angular position at the surface of the star, then the pulsation is termed "nonradial". In theory, perturba-tions from the stationary equilibrium state can be specified by the constraints of conservation of mass, momentum, and energy, and by the expressions for radiative transfer and gravitation. However, in general the problem is very complex and solutions have been obtained only for the simplest cases. The idea that stars could pulsate dates back to 1878 when Ritter proposed it as an explanation of observed stellar light variations. In 1918, Eddington laid the foundations of the theory of conservative (adiabatic) free radial oscillations of gaseous spheres. However, until the work of Pekeris (1938) and Cowling (1941), the importance of nonradial oscillations had been overlooked. Cowling's paper 6 introduced the now fundamental concepts of the gravity and pressure (or acoustic)' modes: two types of oscillations for which gravity and pressure act as the restoring force. A good review of nonradial oscillations can be found i n Smeyers (1984); more detail is available in Unno et al. (1989). Under the assumptions of spherical symmetry and small oscillations, the time, angular and radial components of the pulsations are separable. The solution of the radial motions can be very complicated but the time and angular dependencies are more simply described. A position wi th in the star at time, t, is specified by the spherical coordinates (r, #,</>). The time dependence of the pulsations takes the form exp( ia i ) , where a is the frequency of oscillation; the angular dependence is described by spherical harmonics. Two solutions are possible: toroidal or spheroidal modes. Toroidal modes have only angular displacements and no temperature variations (and therefore no light variations). Spheroidal modes possess both radial and angular displacements and can produce light variations. For toroidal modes, the displacement, £, due to pulsation is given by where £ r and £/j specify the functional dependence on radius, r , of the pulsations in the radial and horizontal (i.e. tangential) directions, and Y™(0, <f>) are spherical harmonics expressed in terms of the associated Legendre polynomials P ™ ( c o s # ) as 1 d d [2-1] and for spheroidal modes by [2.2] 7 Yem(d, <j>) = P € m ( cos 9)eim<t>. [2.3] The degree of the pulsation mode is defined as the index, £, of the associated Legendre polynomial and m = —£, — £ + 1, ...£. Al though both types of modes are possible and may be equally important (especially for O B stars) we wi l l treat only spheroidal modes i n the following analysis. The observed photometric variations indicate that such modes must be present (at least for low degree modes). If we assume that the light from the star emerges from a thin atmosphere, then we can restrict our analysis to the surface of the star. The displacement of the surface due to the pulsations is described by xr = ArP™(cos 9) cos(m</> + at) [2.4] xe - kAr—PP(cos 9) cos(m(j) + at) [2.5] Ou , „ mPf m(cos6) . . , . xs = -kAr — sm(md> + at) 2.6 s i n 6 and the corresponding velocity fields are vr = -aArPP(cos 6) s'm(rri(j) + at) [2.7] ve = -ahAr — P™(cos 9) s in (m^ + at) [2.8] Ou , A mP" l(cos8) . , VJ, = -akAr cos(m<p + at). [2.9] sm8 8 where Ar is the surface displacement in the radial direction. The phase velocity of the oscillations is given by [2.10] We have introduced the parameter k i n the above equations as the ratio of the horizontal to radial velocity amplitudes. Theoretically its value is given by (Ledoux 1951) which is a direct result of the boundary condition on the pressure at the surface of the star. The value of k is related to the pulsation constant Q — U(P//5Q) where II is the pulsation period i n days by (Unno et al. 1989) There has been a long-standing problem associated w i th k for early-type line-profile variables. The value necessary to match the observations is always very much smaller than the theoretical value. Lee and Saio (1990) c la im to have alleviated this problem somewhat by introducing temperature variations as well as geometrical variations. Examples i l lustrat ing the appearance of the stellar surface for different modes of oscillation are depicted in Figure 2.1. The maps show the surface velocity projected along the line of sight. The value of \m\ specifies the number of nodes encircling the star, while £ — \m\ is the number of nodes intersecting the poles. Cancellat ion effects and l imitations of spectral resolution prohibit some of these k = GM/R3 a2 [2.11] 9 a. 1 = Iml = 1 b. 1 = 5, Iml = 3 c. 1 = Iml = 12 d. 1 = 20, Iml = 10 F I G U R E 2.1. Velocity Map of Surface Oscillations. Four nonradially pulsating stars with a) £ = \m\ = 1; b) £ — 5, \m\ = 3; c) £ = \m\ = 12; and d) £ = 20, |m| = 10 are illustrated by a map of the velocity amplitude along the line of sight as a function of position on the surface of the stars. Dark areas represent motion away from the observer, light areas represent motion towards the observer. 10 modes from being observed. In practice, we assume that the source of traveling bumps is a "sectorial" mode (£ — \m\) for which the amplitude is maximized. Nrp theory predicts two classes of possible modes, wi th a special mode sep-arating the two groups. High frequency p—mode oscillations are generated by prorogat ing acoustic waves which are preferentially concentrated towards the sur-face of the star where they are free to propagate. Deeper into the star they become attenuated. The radial modes are a particular case of p—modes. A t low frequen-cies, g—modes can propagate deep wi th in the star and are attenuated as they near the surface. There are two types of g—modes which are either stable or unstable depending on whether or not the star is everywhere stable or unstable against con-vection. The stable modes are propagating gravity waves labeled as g+— modes; the unstable modes are manifestations of the convective motions and are called g~ — modes. Each mode is given a subscript, the radial order, which is related to the number of nodal surfaces between the centre and the surface of the star. The fundamental or /—mode (not to be confused wi th the fundamental (zeroth-radial-order) harmonic of the other modes) oscillates at a frequency intermediate between the p— and g—modes, and is caused by boundary waves at the stellar surface. Figure 2.2 illustrates the frequency dependence on the degree and radial order of the three types of modes. Osaki (1974) and Lee and Saio (1986, 1987a, 1987b, 1989) have shown that overstable convective modes generated in the convective core can penetrate out to the envelope of a rotating massive star. (Rapid rotation in the convective core tends to stabilize g~ — modes. Resonance coupling between these modes and surface oscillations might then occur.) The modes could then be detected as nonradial pulsations. Variations of O B stars are interpreted as g—mode pulsations. However, for the less massive 8 Scuti stars, p—modes are probably the origin of 11 F I G U R E 2.2. Identification of NRP Modes. The basic modes of oscillation: p (pressure), g (gravity) and f (fundamental), are illustrated as a function of frequency (o^), radial order (k), and degree (£). Cox (1976). 12 the variability. Because of the inverse relationship of Q wi th frequency, large Q (and k) values are expected for O B stars, while for the 6 Scuti stars one would expect small values. 2.2 R o t a t i n g S tars The rotation of the star not only reveals its nrp nature as traveling bumps in the Doppler broadened line profiles, but also introduces complications in the oscillations which exist on the surface of the star. For rotating stars, the apparent phase velocity derived from the observations can be related to the true phase velocity of the wave by = + [2.13] where Q, = (vsini/Rsini) is the rotational velocity of the star and the indices rot and in refer to the rotating and inertial frames of rest. According to Equat ion 2.10, this can be rewritten as ain = a - mVt [2.14] The convention used here is that (for a > 0 and Q, > 0), m < 0 corresponds to prograde waves (in the same direction as rotation) and m > 0 corresponds to retrograde motion. The derivation of the above relation has neglected the effects of the Coriolis force; however for rapidly rotating stars this effect can be important. A rigorous 13 theoretical treatment of nrp in a rotating star has not yet been performed because of the difficulty involved. Lee and Saio (1990) have performed the most complete analysis to date, which includes correction terms for the Coriolis force only up to second order. Even wi th this approximation, a general treatment of the problem is complicated. However, i f wave speed of the pulsations is large compared to the rotat ion of the star (i.e. a/£l >> 1), corresponding to slow rotators or high order p—modes, then the Coriolis force can be adequately included wi th a single correction term such that <Tin = a - mQ(l - C) [2.15] where C is given by For high-degree modes wi th a large value of k (i.e. g—modes) then C is given by C = 1/(1 + £{£ + 1))- Let t ing = kt\r (see equations 2.4 - 2.6) where k is a constant throughout the star gives ' 2k + k2 r l+£(£ + l)k2 1 J The above assumption is arbitrary since k is determined by the surface boundary conditions. However, it is adequate to determine the order-of-magnitude size of the Coriolis correction term. Ideally the correction term is sufficiently small that uncertainties in other parameters govern the accuracy of the result. If a/Q. is small, then higher-order effects must be taken into account. Lee and Saio (1990) show that i n this case, it becomes impossible to specify a single £ 14 value for any given pulsation. A l l modes wi th £ > m must be summed together. The effect of adding modes is to concentrate the pulsation even more towards the equator of the star. In addit ion, the star's rapid rotation is expected to generate surface toroidal waves. This problem almost certainly applies when investigating nrp in O B stars for which g—modes prevail. The more slowly rotating 8 Scuti stars, which appear to pulsate wi th p—modes, can be described by the first-order approximation. The surface displacement and velocity equations for a rotating star are es-sentially the same as equations 2.4 to 2.9 but the oscillation frequency a in these equations must be replaced by the apparent frequency, ain i n the inertial frame. The bumps i n the line profiles result from the redistribution in wavelength of absorption features from the rotational Doppler shifts due to the velocity pertur-bations of the waves. 2.3 Light Variations Large-amplitude temperature variations at the surface of the star associated wi th the pulsations could also produce traveling bumps in a rotationally broadened line profile as well as photometric variations. In this case, the bumps would result from the changes i n absorption line strength wi th temperature over the surface of the star. For 8 Scuti stars the temperature variations associated wi th high-degree modes are expected to be small but w i l l be significant for low-degree modes. Bo th temperature and geometrical effects contribute to the photometric variations of the star. 15 Balona and Engelbrecht (1985) have summarized the theoretical analysis of light valuations arising from nonradial pulsations. Add i t iona l references include Dziembowski (1977), Balona and Stobie (1979) and Ba lona (1981). Assuming a 180° phase lag between the surface brightness and the radius variations, then the light variation arising from nonradial oscillations can be expressed as A m = 1.084(/6< - gt) AVosc 2iraR Here A m is the peak-to-peak light variation i n magnitudes and AV0SC is the full velocity amplitude on the surface of the star. The parameter / is the ratio of the surface brightness to radius variation amplitudes. Let t ing f = 0 gives the min imum light variation arising only from geometrical effects i n the size of the stellar disk. The functions be and ge are given by (Balona 1981) be = (2 - B)IX + 1.5/?I2 [2.19] gt = [2-£(e+l)]bi [2.20] where B is the l imb darkening coefficient and In= f finPe(vW. [2-21] Jo Balona and Engelbrecht (1985) applied this formula to three O B stars to estimate the light variation caused by geometrical variations of low-degree modes wi th a given velocity amplitude. Simultaneous photometric and spectroscopic observations of 8 Scuti stars could be used to estimate the value of / for a given 16 velocity amplitude on the surface of the star and allow for both temperature and velocity variations in the analysis of low degree modes. 2.4 Previous Investigations Although photometric and spectroscopic studies of 8 Scuti stars have found them to be r ich in nonradial pulsations, most of the analytical techniques used to identify modes were first applied to the O B stars. Ledoux (1951) first used the idea of nrp to describe the variations in 3 C M a (a 3 Cephei star) by an £ = 2, \m\ = 2 mode plus rotation. Chris ty (1967) invoked nrp to explain line doubling in 3 Cephei stars. The first theoretical line profiles were calculated by Osaki (1971) for comparison to 3 Cephei stars. This inspired a series of theoretical calculations by Stamford and Watson (1976, 1977), Smi th (1977), Kub iak (1978), Balona (1986a,b, 1987), Kambe and Osaki (1988) and finally Lee and Saio (1990) for which nrp line profiles were generated. W i t h improvements in spectroscopic resolution, modes of increasingly high degree have been detected and the models extended to handle the new data. Direct comparisons between theoretical profiles and observed profiles have been attempted by Smith (1977), Vogt and Penrod (1983) and Baade (1984) to identify nrp modes for specific stars. These investigations adopt a trial-and-error procedure to match theoretical profiles to the observations although the uniqueness of the results from this method is often questioned. Smi th (1977) tried to detect low-degree modes (.£ < 4) as variations in line asymmetry in the slowly rotating B-type variable 53 Per. More recently/ Smith (1985) claimed to have identified high degree modes (up to £ = 16) in the rapidly rotating star a V i r (Spica). 17 The four stars observed by Walker et al. (1987) are the only 8 Scuti stars known to possess high degree variations. A preliminary analysis was performed by measuring the travel rates of subfeatures to infer the modes of oscillation. The next logical step is to reconstruct the variations using theoretical models. However, before proceeding, two alternative techniques of mode identification are worth noting. Balona (1986a, 1986b, 1987) proposed a method whereby the mode is derived from a Fourier analysis of the first two or three moments of the observed line profile. Unfortunately, this technique becomes less effective for high values of £. A different approach was taken by Gies and Kullavanijaya (1988) who measured up to four modes associated wi th the line-profile variations of the B 0 . 5 V star e Per. They perform a Fourier analysis of the intensity variations at each point in the line profile (relative to the mean profile) for a long time series of observations extending over several days. W i t h such long time series, the technique can be sensitive to multiple mode frequencies. 18 Chapter 3 PROFILE MODELING 3.1 D e s c r i p t i o n o f P r o g r a m A F O R T R A N computer program has been developed at U B C over several years to simulate absorption line profiles of a rotating, nonradially pulsating star. The model has existed i n many versions and a number of people have been involved^ wi th i t , including D . Thibaul t , G . Fraser, D . Francis, G . Grieve, S. Yang , and E . Kennelly. The model at present is'" strictly geometrical (although, it is a simple matter to include temperature variations) and the pulsations of the star are de-scribed by spherical harmonics. This is not the first program of its k ind . The previous chapter recounted the history of nrp line profile programs. A t present the Tokyo group (e.g. Lee and Saio 1990) are leading the way in nrp theory and have been working to produce more and more sophisticated models. We have used the published model profiles of K a m b e and Osaki (1988) as a comparison to test our nrp program. The model procedure itself is relatively simple. The unperturbed star is repre-sented by a rotating (oblate) spheroid. The surface is divided into many segments 19 specified by lines of latitude and longitude. Spherical harmonics are imposed as a perturbation of this geometry and the positions of the segments are monitored as the pulsation proceeds with time. The velocity along the line of sight of each segment can be translated into a Doppler shift by the formula The light from each segment is assumed to be described by the same intrinsic line profile (for a star without temperature variations). The intensity of each profile is then weighted by the effective area of the segment, by limb darkening and by gravity darkening. A standard lirnb darkening law is used, where /(costfj.) cx (1 - 3(1 - cos0 ± ) ) [3.2] where 6± is the angle between the line of sight and the normal to the surface of the star and f3 is the limb darkening coefficient. Gravity darkening can be described by I oc {g/gpoiarV [3.3] where the exponent 7 is usually set to 1. Profiles from all segments of the visible disk are Doppler-shifted in wavelength and summed together to yield the line profile for the entire star. The line-of-sight velocity of a point on the surface of the rotating, pulsating star is dr dd z = — — (sin 8 sin i cos d> + cos 8 cos 1) — r —(cos 9 sin i cos 6 — sin 8 cos i) dtK ' dV  1 20 +r s in#[—] i n ( s inz sin</>) • [3.4] This quantity (minus the rotation term) is mapped i n Figure 2.1. The tradit ional convention that z < 0 for motions away from the observer is adopted. The intrinsic (input) line profile is in reality determined by the conditions wi th in the atmosphere of the star. For simple models, the intrinsic line profile can be represented by a Gaussian of the form Joe e x p [ - ( A A / A A 0 ) 2 ] [3.5] where A A 0 = A 0 V & / c [3.6], and Vf, is the (thermal) broadening velocity (~ 5 km/s ) . The ratio, V ^ / v sins', determines the ultimate resolution of the surface features in the line profile. The true velocity amplitude of the oscillations can be recovered from the observations only after the resolution has been taken into account. This aspect w i l l be discussed further in section 3.6. More realistic nrp models require synthetic line profiles (or entire spectral regions) generated from model atmospheres as input. In this way, the effects of intrinsic line wid th on the resolution (and line blending) are directly included in the analysis. Matching the synthetic spectra to observed spectra also provides a check on the parameters of mass and radius used in the model. The model star is specified by its mass, radius, v sinz, incl inat ion, l imb dark-ening coefficient and gravity darkening coefficient. Each mode of pulsation is 21 specified by its £ and m values, wave speed, velocity amplitude, amplitude ratio between the horizontal and radial displacements (i.e., k) and phase. Thus 12 pa-rameters are required to specify a single geometrical pulsation mode. The inclusion of temperature variations would introduce two more parameters: amplitude and phase of the temperature curve. 3.2 Testing the Program A l l aspects of the program were tested for agreement wi th the expected be-haviour in simple cases. A n extremely efficient technique in early tests was to examine the profile resulting only from contributions at the equator of the star. It was also useful to compare profiles generated by our program wi th the results of K a m b e and Osaki (1988). The construction of the line profiles can be illustrated by considering only the equatorial cross section of the model star and the resultant profile. Figure 3.1 shows such a cross section and the radial component of the pulsation for three cases: £ = \m\ = 2, £ = \m\ = 14, and combination of the two modes. Each point in the graphs represent a segment on the surface of the star. The top diagrams indicate the surface velocity variations due to pulsation alone as a function of azimuthal angle. The middle diagrams show the velocity along the line of sight for each segment, including rotation, and pulsation. The bot tom diagram shows the line profile resulting from the perturbations introduced by the pulsations on the rotational velocities. The redistribution of light in velocity (wavelength) space due to the pulsations produces bumps in the profile of the star where the segments 'pile 22 b) O '©I 6 W ' So « © ° o o o o o °® © O o ffiffl ? I © , ©Oft, i - 10 • 10 v e 1 o c i t y v e 1 o c i t y -200-100 0 100 200 ; -200-100 0 100 200 -200-100 0 100 200 v e l o c i t y u e l o c l t y v e l o c i t y -200-100 0 100 200 -200-100 0 100 200 -200-100 0 100 200 u e 1o c i t y v e l o c i t y v e l o c i t y F I G U R E 3.1. Formation of NRP Bumps by Velocity Perturbations. Three situations are illustrated: a) Low-degree: £ = |m| = 2, b) High-degree: £ = \m\ = 14, and c) Combined high- and low-degree modes. The velocity per-turbations on the surface of the star (top) are combined with the rotation of the star to produce the net velocity projected along the line of sight (middle). The associated Doppler shifts redistribute the light and produces "bumps" in the line profiles (bottom). 23 up ' at nearly the same velocity. For a schematical representation of this process, refer to Vogt and Penrod (1983). When two modes are present, the sum of the perturbations of the individual mode determines the positions of the bumps in the line profile. Separate tests were performed for the purely tangential components of pulsations. The nrp program is designed such that time series of model profiles can be generated for comparison wi th observations. The phase can be incremented at constant intervals or can be updated from times read as input to the program. A n example is shown in Figure 3.2 a). The progression of bumps in the line profile moving from blue to red is readily apparent. The profiles in the figure have the same input parameters as those published in Figure 1 of Kambe and Osaki (1988); both the models agree very well. 3.3 Reduction Techniques In the analysis of line-profile variations two quantities are calculated which are even more sensitive to the presence and nature of the bumps traveling through line profiles: spectral residuals and the absolute mean deviation. The mean profile of a t ime series wi l l not show nrp bumps if averaged over enough profiles at different phases. The mean profile then resembles the rotationally broadened profile of a star without pulsation. A time series of residuals can be constructed by subtracting the mean profile from each profile. The absolute mean deviation is defined as: D e v = Ezli \(profile)j - (mean)] 24 F I G U R E 3.2. Time Series of Model Profiles. a) A time series of model profiles demonstrating the motion of bumps through the line profiles. b) A time series of residuals, corresponding to the profiles of (a), constructed by subtracting each profile from the mean profile of the series. fi 1 I s u a -} u J 25 i n c OJ ZD c CD T r 77—rr - i — i — 1 _ 4495 4500 Waue l ength -[ 1 1 1 r -i—r'l i • 1111 I • ' • I - i L 4505 i 1 1 r _ i i i i _ 4495 4500 4505 Waue l ength F I G U R E 3.2. Time Series of Model Profiles. c) The mean profile generated from the time series of model profiles illustrated in figure (a). d) The mean deviation, denned by Equation 3.7, for the same time series of model profiles. 26 (Unlike the variance, this parameter weights the variations evenly throughout the line profile.) The shape and amplitude of this parameter can be used to gather information about the pulsation parameters. Figure 3.2 b) illustrates the time series of residuals which correspond to the profiles of Figure 3.2 a). The mean and mean deviation for the series are presented in Figure 3.2 c) and d). 3.4 S y n t h e t i c S p e c t r a The nrp program accepts synthetic spectra encompassing many lines to be used as the intrinsic profile input describing the light coming from each part of the stellar surface. The spectra are generated from model atmospheres tailored to the known or estimated parameters for each star. The spectra provide both a test of the stellar parameters and a treatment of resolution and blending effects in the observed spectrum. B o t h the model atmosphere program, T L U S T Y , and the program to generate synthetic spectra, S Y N S P E C , were provided by Dr . Ivan Hubeny. A s well , a program which produces rotationally broadened profiles from the intrinsic spectra, R O T I N S , was supplied. Hubeny (1987) gives a detailed description of the program T L U S T Y . The model assumes a plane-parallel, horizontally homogeneous atmosphere in radia-tive and hydrostatic equil ibrium. Al though it was not necessary for this work, the program allows for departures from L T E for selected atomic and ionic energy levels. A complete-linearization technique is used to solve the system of coupled, nonlinear equations specified by radiative transfer, hydrostatic equil ibrium, sta-tistical equil ibr ium and radiative equil ibrium. Initially, an LTE-grey model is 27 generated and the equations are solved iteratively unt i l the program converges to the final L T E model. (This model can then be used as the starting point for gener-ating n o n - L T E models.) The program allows great flexibility in choosing chemical species, ionization degree, and the energy levels and transitions to be considered. A l l of this information was kindly supplied by Hubeny for stars in the region of the S Scuti strip of the H - R diagram. A l l that remained was to provide the effec-tive temperature, Teff, surface gravity, logg, and microturbulent velocity for each star. The program S Y N S P E C uses computed model atmospheres (e.g., from the program T L U S T Y ) to calculate synthetic spectra i n a chosen wavelength region. It uses the same input as T L U S T Y , specifying the basic model parameters and the choice of explicit atoms, ions, energy levels, transitions and degree of sophistica-tion. Again , al l but T e / / , log g and microturbulent velocity were already provided. Addi t iona l files supply information concerning the desired wavelength region and the absorption lines present therein. The line list used is essentially a subsection of the Kurucz-Peytremann line list for the wavelength region 4460-4520A. 3.5 P a r a m e t e r s a n d Prof i les The effects of varying the input parameters on the artificial line profiles were examined first. Figures 3.3 a), b) and c), illustrate profiles generated for different combinations of £ and m. For each of these figures a l l the other parameters were held constant: vsini — 100 km/s , i = 90° , Vamp = 5.0 k m / s , and k = 0.0 and the intrinsic line profile was represented by a Gaussian. The first diagram 28 N R P P r o f i l e s : 1 = - m -i 1 1 1 1 1 r - i i I i i i i _ 1 = - m = 2 0 1 = - m = 1 6 1 = - m = 1 2 1 = - m = 8 1 = - m = 4 1 = - m = 0 • i i i 4 4 9 5 4 5 0 0 4 5 0 5 N a u e l e n g t h 4 5 1 0 F I G U R E 3.3. NRP Line Profiles: Mode Dependence, a) Sectorial modes (£ — —m). 29 N R P P r o f i l e s : 1 = 1 2 ™i 1 1 r T 1 1 1 1 r i ! : ' ' l i I I m = - 1 2 m = - 1 0 m -m -m m = - 8 - 6 - 4 - 2 JL J i i i ' • i • » 4 4 9 5 4 5 0 0 4 5 0 5 W a u e l e n g t h F I G U R E 3.3. NRP Line Profiles: Mode Dependence, b) The effect of m on the profiles. 4 5 1 0 30 N R P P r o f i l e s : m = - 1 2 " i 1 1 1 1 1 1 1 1 1 1 r I ! ' 1 i l l ! 1 = 1 2 1 = 1 3 1 = 1 4 1 = 1 5 1 = 1 6 1 = 1 7 J i i i _ _ i i i i i i _ 4 4 9 5 4 5 0 D 4 5 0 5 U a u G l e n g t h 4 5 1 0 F I G U R E 3.3. NRP Line Profiles: Mode Dependence, c) The effect of I on the profiles. 31 depicts profiles generated for sectorial modes (£ — —m). The second diagram demonstrates the effect of m on the profiles for a fixed value of £. The thi rd diagram shows the effect of £ for a fixed value of m. Notice i n Figure 3.3 c) that the tesseral mode {£ = |m | + 1) produces twice as many bumps, but wi th lower amplitude. These arise from contributions above and below the equator 180° out of phase. A s £ increases, cancellation from different latitudes decreases the amplitude of the bumps. According to Lee and Saio (1990), second-order Coriolis corrections require that a l l modes wi th £ > m be considered for a rapidly rotating star. The effect is to concentrate the pulsation towards the equator and to decrease the amplitude of the bumps. Figures 3.4 a) and b) illustrate profiles for which velocity amplitude and in-clination have been varied. As expected the amplitude of the bumps increases wi th the velocity amplitude but decreases as the inclination is increased. This remains one of the fundamental difficulties wi th nrp modeling. It is not possible to separate the effects of the inclination and pulsational amplitude on the bump amplitude based on profile data alone. Some other information is necessary to estimate the inclination. The effect of k on the profiles is demonstrated in Figure 3.4 c) and in Figure 3.5. The amplitude of the pulsations i n the azimuthal direction begins to dominate as k increases. The line-of-sight contribution of this component becomes significant towards the l imb of the star. Thus greater variation can be expected from the wings of the line as k increases. It is apparent in Figure 3.4 c) that as k is increased the bumps towards the wings of the profile grow stronger, but this effect is even more obvious in the residuals and the mean deviation plot. The mean deviation in Figure 3.5 develops a pronounced dip at line center as k increases. One might expect be able to use this signature to estimate k from the observations. However, 32 N R P P r o f i l e s : U a m p -I 1 r T r T r \ -: ! 1 i ' I M t --— -1. J l_ Ua mp =0 . 0-U a m p - 1 . 0 Ua mp = 2 . • Ua mp =3 . fj-Ua mp = 4 . 0 Ua mp = 5 . • • ' ' 4 4 9 5 4 5 0 0 4 5 0 5 Ula U G l e n g t h 4 5 1 0 F I G U R E 3.4. NRP Line Profiles: Dependence on Input Parameters, a) Velocity amplitude. 33 N R P P r o f i l e s : i T 1 1 1 1 r r -| 1 1 1 1 1 r ZD i n c OJ ! : > ! I i I I i -I l_ i = 9 0 i = 8 0 i = 7 0 i = 6 0 i = 5 0 i = 4 0 _ i i i i i _ 4 4 9 5 4 5 0 0 4 5 0 5 Wa ve l e n g t h 4 5 1 0 F I G U R E 3.4. NRP Line Profiles: Dependence on Input Parameters, b) Inclination. 34 N R P P r o f i l e s : k -i 1 1 r 1 1 1 1 1 1 1 1 1 1 r ! I I 1 I I l \ I k = • . • k - 0 . 2 k =0 . 4 k = 0 . 6 k = • . 8 k = 1 . • _ J i i i _ _ i i i i _ - i i • • 4 4 9 5 4 5 0 0 4 5 0 5 Waue l e n g t h 4 5 1 0 F I G U R E 3.4. NRP Line Profiles: Dependence on Input Parameters, c) The ratio of horizontal to radial pulsations, k. 35 0 Ye 1o c i t y • UG 1O C i t y F I G U R E 3.5. The Effect of k on the Residuals and Mean deviation. a) Time series of residuals and corresponding mean deviation plot for k = 0.0. b) Time series of residuals and corresponding mean deviation plot for k = 0.3. c) Time series of residuals and corresponding mean deviation plot for k = 0.6. in practise the existence of multiple modes complicates the appearance of this parameter. 3.6 R e s o l u t i o n o f Surface Fea tures Recall ing from Figure 3.1 how light (i.e., absorption lines) from each part of a rotating star add together to produce a 'Doppler image' of the surface, it is clear that both the wid th of the intrinsic line, Vj and the amount of the rotational broadening, v sin i largely determine the resolution on the surface of the star. The resolution improves wi th greater values of v sinz and narrower intrinsic lines. Thus, the ratio: Vb/vs'mi is a measure of the resolution of the surface features. The velocity amplitude of the waves on the surface of the star is reflected i n the amplitude of the bumps in the line profiles as il lustrated i n Figure 3.4 a). However, the bump amplitude is also influenced by the resolution. To determine the velocity amplitude from the features in the line profiles it is therefore crucial to take into account the resolution. This is accomplished by using synthetic spectra to specify the intrinsic line wid th , V j . Figure 3.6 a) illustrates the dependence of the amplitude of the blimps on the velocity amplitude on the stellar surface and the intrinsic line width. The bump amplitude is defined as the mean of the differences between maximum and min imum intensities i n a time series of residuals. The example illustrated in these and subsequent diagrams is for a single mode wi th I = —m = 14 and k = 0. Other relevant parameters are: u s i n i = 115 km/s , i — 90°. The equivalent width of the line used in the example is 26.OA. ( A linear relationship exists between 37 Intrinsic Line Width: V b (km/s) F I G U R E 3.6. Resolution of the Stellar Surface. a) The dependence of the bump amplitude in the residuals on the velocity ampli-tude and intrinsic line width. 38 .25 r-' o 2 .15 '> <o a c .05 Q I I I I I I I I I I I I I I I I I I I I I I I 1 I I I L 0 2 4 6 8 10 12 14 Velocity Amplitude: V a m p (km/s) o -4-> <a > Q C <S (U i i i \ — i — i — i — i — r 5 10 15 Intrinsic Line Width: V b (km/s) 20 F I G U R E 3.6. Resolution of the Stellar Surface. b) The dependence of the maximum amplitude of the mean deviation on the velocity amplitude and intrinsic line width. 39 amplitude of the variations and the equivalent width (i.e., strength) of the line.) Figure 3.6 b) illustrates the same dependence for the amplitude of the absolute mean deviation in the line. For a given model, if the intrinsic line width is known and the bump amplitude is measured, then the velocity amplitude can be readily determined from the plots. In addition to the fundamental limit on resolution at the stellar surface, in-strumental resolution will also influence the amplitude of the variations. However, this effect is easily approximated by convolving the model profile generated by the nrp program with a Gaussian profile to represent smoothing by the instrumental resolution. Finally, time averaging introduces an additional smoothing of the observed bump amplitudes of the observations. The model profiles are instantaneous but the observations represent integrations over 10-15 minutes (necessary to obtain sufficiently high signal to noise in the spectra). The averaging process will de-crease the bump amplitude but the amount is small. A number of model profiles were generated at very short time intervals, then averaged to approximate this ef-fect. The results are shown in Figure 3.7. The average residual does not show any significant difference from the median instantaneous model apart from a slight de-crease in amplitude. We choose to neglect this minor effect in this study, although it may be important in other data sets with different timescales. 40 -A/V\A~ - -AA/Vv--A/VVv. - A A A A --AAAA-^\f\f\f^ — - A A / v ^ - A A A ^ ^ A A A ^ -—^r\f\r^^-: — 100 2 0 0 3 0 0 p i x e l 2 0 0 p i x e l F I G U R E 3.9. Temporal Resolution. A series of residuals generated at short time intervals (top) are averaged together to simulate the effect that a finite exposure time has on the amplitude of the bumps. A comparison between the mean and the median residuals (bottom) indicates that no significant effect results from the finite exposure times. 41 Chapter 4 OBSERVATIONS 4.1 D a t a A c q u i s i t i o n Time series of spectra were obtained at C F H T by Walker et al. (1987) for four 8 Scuti stars with the f/8.2 camera of the coude spectrograph using an RL1872F/30 E G & G Reticon detector. The spectra have a dispersion of 0.035A per pixel. The exposure times were ~ 15 minutes achieving a S/N ~ 500. The observations are listed in Table I. Observations were made on one and a half nights. The star, o1 Eri, for which traveling subfeatures had been discovered previously (Yang and Walker 1986) was observed on both nights. Line profile variations were discovered in three other stars: 21 Mon, v UMa, and K2 BOO. Observations of a fifth star, 57 Tau, during the same run did not show obvious line profile variability, although these data suffer from poorer S/N. Using improved reduction techniques Yang (private communication) has been able to detect variations just above the noise level. 42 T A B L E I. Observations o1 Eri 21 Mon v UMa K2 Boo magnitude 4.04 5.45 3.80 4.54 Spectral Type F2II-III F2Vn F2IV A8IV Series Length (days) 0.181,0.087 0.115 0.160 0.099 No. of Observations 28,14 12 17 16 Exposure Time (s) 500 525 800 850 S/N per diode 544 227 820 450 43 4.2 Data Reduction The data reduction program RETICENT (Pritchet, Mochnacki, and Yang 1982) was used to process the data. This package was designed to reduce and analyze spectroscopic data recorded specifically with a Reticon detector. The quality of the CFHT data is immediately obvious from Figure 4.1 a) which shows one of the raw spectra obtained for K2 BOO, with only the baseline subtracted to reduce the detector readout noise. The presence of subfeatures in the absorption lines is unmistakable. Differences in the responses of individual diodes introduce another source of noise into the observations. These variations are removed by dividing the data by a continuum source (lamp). Figure 4.1 b) shows the spectrum after that correction. The data is read out from four separate circuits, or video lines. The first circuit reads out diodes one, five, nine, the second circuit reads diodes two, six, ten, and so on. Small systematic differences between the four lines introduce a four-point noise pattern into the data. An additive four-line normalization removes this source of variation. Cosmic ray spikes were removed from the data, by locating each affected pixel and replacing its intensity with the mean value of the neighbouring pixels. A two-step process was performed to rectify the data. First, all the spectra in the time series for a given star were averaged to produce a mean spectrum. A third-degree polynomial was fitted to points chosen to be on the continuum. Dividing the mean spectrum b}^  the polynomial fit yields a rectified mean. The second step was to rectify all the individual spectra in the time series. Each spectrum was divided by the mean for the series, then a polynomial was fitted 44 O r" • . . . i . . . . t . . . . i 5 0 0 1 0 0 0 1 5 0 0 5 0 0 1 0 0 0 1 S 0 0 5 0 0 1 0 0 0 1 5 0 0 FIGURE 4.1. Data Reduction. a) Raw spectrum (data - baseline) for K2 BOO. b) Spectrum of K2 Boo after normalization c) Rectified Spectrum of K2 BOO. 45 to the residuals. A single polynomial was not sufficient to describe the shape of the residuals, especially at high pixel values. Instead, spline fits were obtained by breaking the residual into two parts and fitting polynomials to each section. The curves were combined and smoothed to yield a continuous function which adequately described the form of the residual. Dividing the each spectrum by its polynomial fit yielded a rectified version of the data as illustrated in Figure 4.1 c). Calibration of the data from pixel-space into wavelength-space was performed using the Thorium-Neon arcs taken throughout the night. Normally it is important to obtain arc spectra often throughout the night because drifts in wavelength over the detector will occur. Relatively little drift occurred during these two nights of observations. At most the positions varied by 0.3 km/s during any given time series. It is possible to correct for this by fitting polynomials to the drift curves; however, the level of drift was negligible compared to the stellar variations. For the reader's convenience, only the data for the star n2 Boo is presented in the following chapters. Data for the other three stars can be found in Appendices A, B, and C. The results obtained for all four stars will be discussed in Chapters 6 and 7. Figure 4.2 a) is the time series of observations obtained for K2 BOO. Each spectrum has been convolved with a Gaussian to smooth the data. The ability to 'stack-plot' spectra using R E T I C E N T helps to reveal the presence of traveling bumps in the line profiles. Each spectrum in the figure is positioned vertically according to the median time of observation. The numbers listed down the right-hand-side are measurements of time in fractions of a Julian day. It is often helpful to view the diagram with the eye just above the plane of the page, in so called 'Walker-vision'. 46 F I G U R E 4.2. K2 Boo Observations. a) The motion of the bumps through the line profiles is illustrated in this time series of spectra. The numbers down the right-hand-side represent time in Julian days with respect to JD 2,446,838. The error bar indicates the intensity scale relative to the continuum. b) A time series of residuals was generated by sutracting each spectrum in the series by the mean spectrum for the series. Traveling bumps are visible at ~ 1 % of the continuum. 47 J , L 4 4 8 0 4 5 0 0 4 5 2 0 4 4 8 0 4 5 0 0 4 5 2 0 W a u s l e n g t h F I G U R E 4.2. K2 Boo Observations. c) The Mean spectrum for the time series. d) The mean deviation, obtained by calculating the mean absolute deviations the residuals. 48 For a sufficiently long time series, the presence of traveling bumps would be completely removed from the mean spectrum. Figure 4.2 c) shows the mean spec-trum for the time series of K2 Boo. A time series of residual spectra was generated by subtracting this mean from each individual spectrum. The resulting plot, pre-sented in Figure 4.2 b) dramatically illustrates the presence of traveling bumps within the line profiles, even in the weaker lines. The absolute mean deviation was also generated for the time series (see Figure 4.2 d). This provides a useful measure of the amplitudes of the variations in the lines. The chosen spectral region is rich with absorption features. Line identifica-tions were performed from comparisons with the spectrum of Procyon (Griffin (1979)). A Mgll-Fel blend around A4482 dominates the spectrum. Although many of the lines suffer from blending, there are four absorption features which are relatively free from blends in which the behaviour of the nrp bumps is clear-est. These are: A4476.061 (Fel doublet), A4501.278 (Till), A4508.289 (Fell), and A4515.342 (Fell). Each was investigated separately, and the residuals, mean, and mean deviation were obtained, as well as the line depth, equivalent width and bump amplitude. The data for Fell A4508 is presented in Figure 4.3 for K2 Boo. The line depths, equivalent widths and bump amplitudes for all the stars are tabulated in Table II. 49 Ue 1 Ue 1 o c i t y F I G U R E 4.3. K2 Boo Observations of the A4508 Line, (see Figure 4.2) a) Time series of residuals. b) Mean deviation. T A B L E II. B u m p Ampl i tude i n Observed Line Profiles K2 Boo: Line Depth Equivalent Ampl i tude (% of cont.) W i d t h (mA) (% of cont.) A4476 2.83 68 1.36 A4501 8.64 226 2.00 A4508 6.15 155 1.56 A4515 5.10 169 1.53 v U M a : Line Depth Equivalent Ampl i tude (% of cont.) W i d t h (mA) (% of cont.) A4476 4.65 86 1.24 A4501 10.62 293 1.83 A4508 7.46 232 1.47 A4515 6.00 252 1.32 21 M o m Line Depth Equivalent Ampl i tude (% of cont.) W i d t h (mA) (% of cont.) A4476 - - -A4501 10.16 319 2.62 A4508 7.37 203 2.28 A4515 - - -o1 E r i ( 1 s t night): Line Depth Equivalent Ampl i tude (% of cont.) W i d t h (mA) (% of cont.) A4476 5.85 124 1.69 A4501 13.00 311 2.10 A4508 8.71 205 1.73 A4515 6.84 209 1.60 o1 E r i (2nd night): Line Depth Equivalent Ampl i tude (% of cont.) W i d t h (mA) (% of cont.) A4476 5.64 128 1.42 A4501 12.52 328 1.93 A4508 8.77 202 1.53 A4515 6.47 180 1.47 51 Chapter 5 ANALYSIS The analysis procedure was relatively simple: (i) generate a model for each star; (ii) estimate the nrp modes from the observed line profile variations, and (ii) fit theoretical nrp models to the observations. The goal was to identify the nrp modes and determine their amplitudes and periods. 5.1 S te l l ar M o d e l s The projected rotational velocity (v sinz) of each star was estimated by match-ing the width of the observed mean line profiles wi th theoretical profiles of known v sin i. The nrp program was used to generate a grid of rotationally broadened line profiles (but without any nrp modes). The uncertainty in determining u s i n i by this technique is about ± 5 k m / s (5 % for the rotation rates of the four programme stars). The values are listed in Table III. In some cases, the values differ from the published values (e.g., Bright Star Catalog) but are probably more reliable because of the high dispersion of the observations. 52 T A B L E III. Rotational Broadening: v sin i Star v sins' (km/s) K2 Boo 115 ± 5 v U M a 110 ± 5 21 Mon 130 ± 5 o1 Er i 110 ± 5 The inclinations of the four stars are not known. None of the four stars is in a close binary system so the inclination can not be derived from orbital characteristics. However, the v sin i values fall close to the mean for 8 Scuti stars (~ 100 km/s) and well below the observed maximum value of ~ 200 km/s (Wolff 1983), so we can use statistical arguments to place limits on the inclination. Good estimates of the mass and radius of the stars are necessary for the analysis, since they affect the derived values of the periods and amplitudes. We have used empirical calibrations of ubvy photometry to place estimates on these parameters. Breger (1990) relates the Teff, logg, and absolute magnitude Mv to the Stromgren indices (3 and C\ by the following equations: log Teff = 0.5242/9 - 0.0027cx + 2.4347 [5.1] logg = 4.91876 - 2.6298Cl - 7.3982 [5.2] Mv = -9c i + 153 - 32.22. [5.3] The values of 3 and C\ were obtained from Breger (1979). Uncertainties in T e / j , log g and Mv were estimated by assigning reasonable photometric errors to 3 and Ci ( ± 0 . 0 2 ) . From these results, the luminosity L , radius R, and mass M are derived from the basic formulae: l o g ( L / L 0 ) = O.4(M 0 - Mv) [5.4] 54 R/RQ = {TITQY L/LQ [5.5] M/MQ = g/gQ L/LQ [5.6] {T/TQY where M© = 4.75, T© — 5800 and logg© = 4.44. Here we have chosen to neglect the small bolometric correction for A - F stars. Table IV lists these quantities and their associated uncertainties for the four stars. Parallax measurements are a check of the calculated values of My. The absolute magnitude of the star is related to its apparent magnitude by where 7r is the parallax in seconds of arc. We have ignored effects of interstellar absorption. Although measured parallaxes can be highly uncertain at distances greater than ~ 50 pc, comparisons between values of My determined from equa-tions 5.3 and 5.7 are a useful consistency check of the stellar models. Table V shows that the agreement between the observed and model parallaxes is reasonably good. The calculated values of T e / / and logg for each star can also be substanti-ated by comparing rotationally broadened synthetic spectra with the observations. Model atmospheres were generated for each star using the calculated values of Tef f and log g, from which synthetic spectra covering the observed spectral region were produced. (The programs were described in Section 3.3. These model spectra were rotationally broadened by the measured values of v sinz. Intrinsic and broad-ened synthetic spectra for K2 BOO are presented in Figure 5.1. The broadened spectrum reproduces the observed spectrum (Figure 4.2) but differ significantly My = m + 5 + 5 log 7T [5.7] 55 T A B L E I V . Stellar Parameters from ubvy Photometry o1 E r i 21 M o n v U M a K2 B o o (b-y) 0.197 0.184 0.196 0.116 P 2.730 2.743 2.730 (2.818) C l 1.022 0.878 0.830 0.789 Teff ( K ) 7200 ± 80 7400 ± 80 7200 ± 80 8100 ± 90 loggr ( c m / s 2 ) 3.95 ± 0 . 1 1 3.78 ± 0 . 1 1 3.84 ± 0 . 1 1 3.77 ± 0 . 1 1 Mv (mag.) 1.69 ± 0.35 1.02 ± 0.35 1.26 ± 0 . 3 5 0.85 ± 0.35 L / L Q 17 ± 3 31 ± 4 25 ± 4 36 ± 5 R/RQ 2.7 ± 0 . 2 3.4 ± 0.3 3.2 ± 0 . 3 3.1 ± 0 . 3 M/MQ 2.3 ± 0 . 5 2.6 ± 0.5 2.6 ± 0 . 5 2.0 ± 0 . 4 56 T A B L E V . Parallaxes Star Parallax ( T^exp ( ) arcsec) <h, (") o1 Eri 0.034 0.033 21 Mon 0.013 0.021 v U M a 0.031 0.041 K2 Boo 0.018 0.010 1. Bright Star Catalog F I G U R E 5.1. K 2 B o o Synthetic Spectrum. a) Intrinsic synthetic spectrum wi th Teff = 8100 and log g = b) Synthetic spectrum wi th v sinz = 115. 58 at the short-wavelength end. More importantly the synthetic spectra appear to adequately represent the four isolated lines in all the program stars, although the relative intensities do not always match (especially for A4476). This may reflect the difficulty in determining the true continuum of the observed spectra. 5.2 Profile Variations If the subfeatures within the line profiles are separated in time by At, then the intensity of the residuals at any given position (e.g., at line center) should vary with a period At. A Fourier transform of the variation should display a peak at the frequency corresponding to the time spacing At at all positions within the profile. If a sufficiently long series is obtained, one will also be sensative to multiple periods and their beating. This approach was adopted by Gies and Kullavanijaya (1987). In the work here, the profiles of A4508 were sampled at 10 km/s intervals between -50 to +50 km/s and Fourier amplitude spectra were obtained with a periodogram routine for equally spaced time series (Matthews and Wehlau 1985). The results for K BOO are displayed in Figure 5.2 a) and the mean amplitude spectrum is shown in Figure 5.2 b). The dashed line is the spectral window assuming a single period of 0.0455 days. The limited number of spectra in each series, the poor frequency resolution and the possible complication of additional modes are probably responsible for the slight differences in the peak at different positions in the line profiles. However, for all stars except o1 Eri , a single frequency dominates at most positions. At values determined by this method are summarized in Table VI. 59 .008 .006 .004 .002 0 .008 .006 .004 -+-> . — I 0 CO C| .008 CD .+_> .006 a • i-H .004 3 .002 0 "J o i i 1 i 11 1 • i i 1 11 i 1 i i i _ i i i i | i i i i | i i i i | i i i i | i i i i . ; v=0 k m / s ; l l l l l l l l l l l l l l l i l l l l . i i i i | i i i I ]-r I I I | I I l l II l l l . ; v=—10 k m / s \ ' \ I I I I i i i i 1 i i i i 1 i i i i 1 r V f i J : l 1 1 1 | 1 1 1 1 | 1 1 1 1 1 1 1 1 1 1 I I 1 1. ; v=+10 k m / s ; 7\ 1 1 1 1 t 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I _ . i i i i | i i i i i i i i i i i i i i i i i i I _ — ! ! I i i ! i 1 1 1 — ; v=-20 k m / s \ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~ . 1 1 1 1 I'i \"t -1—1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. ; v=+20 k m / s ; 7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I I I i " . 1 1 1 1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _ ; v=-30 k m / s \ ' \ i i i I i i i i 1 i i i i 1 i i i T - 4 f - T — i i • . 1 1 1 1 J ' ! 1 ! l~r | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 . ; v=+30 k m / s ; i i i i i i i i i i i i i r ^ i i i i i i i i i . i i i i | i i i i | i i i i | i i i i r r i i i _ ; v=—40 k m / s jj - ' / ' 1 1 \ . ; i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i " : I I I t | - \ l I l | l l I l | l l l l l M 1. - v =+40 k m / s ; V r - * ! i i 1 i i i i 1 i i i i 1 V i i i 1 i i i i -_ i i i i 1 i i I i 1 i i i i i i i i i 1 i i i i _ - v=—50 k m / s -J \ i i i 1 i i i i 1 i i i i T i i i i T i i i i -: I I 1 I | I 1 1 1 | 1 1 1 1 | 1 i i 1 | 1 1-I—ri ; v=+50 k m / s ; .008 • i—l ^•006 B 004 ^.002 0 .008 .006 .004 .002 0 .008 .006 .004 .002 0 10 20 30 40 10 20 30 40 frequency (c/d) F I G U R E 5.2. Fourier Amplitude Spectrum for K2 BOO. a) The variation of the residuals at a given position in the line profile (A4508) were measured and transformed into the Fourier domain. The series of plots illustrates the amplitude spectrum at 10 km/s intervals. 60 .008 At = 0.0455 days f r e q u e n c y ( c / d ) F I G U R E 5.2. Fourier Ampli tude Spectrum for K2 BOO. b) The mean amplitude spectrum (solid line) from those il lustrated i n Figure (a). The dashed line is the window function for the period, At. T A B L E V I . Results from Amplitude Spectra Star At (days) K Boo 0.045 ±0 .001 v U M a 0.089 ±0 .002 0.0332 ± 0.0009 21 Mon 0.077 ±0 .006 o1 Er i (1 s t ) 0.094 ±0 .004 o 1 Er i (2 n d ) 0.11 ± 0 . 0 1 If we assume that the velocity perturbations are small, then the positions of the subfeatures as they move through the profiles can be described by a sine curve with amplitude usini. The period of the curve, P , will be the time it takes for a wave crest to encircle the star relative to the observer. This is related to the apparent (inertial) frequency by a{n = m/P, where m is the degree of the mode. We obtain an estimate of P by measuring the positions of the subfeatures in a given line (usually A4508 is chosen but sometimes it is necessary to combine many lines) and then fitting the positions of each subfeature to V = (v sin i) sm(2Tr(t0+t)/P). [5.8] The subfeature positions were read from the R E T I C E N T screen displays of the residuals. The values were tabulated and sine curves fitted to the observed positions using a least-squares fitting package, O P D A T A , kindly made available by P. Bennett. The unknowns, t0 and P, were determined for each feature. The best value of P is taken to be the mean for all the curves. The mean separation in time, At, between the subfeatures was also determined from the curves of best fit. The corresponding sectorial mode is then given by \m\ = P/At. [5.9] Theoretical curves calculated from the measured values of P and At are su-perimposed on the observed positions for K2 BOO in Figure 5.3. The initial phase of the wave can be estimated from the spacing of the bumps at the beginning of the time series. We note that the time series of v UMa indicates both high-and low-degree traveling bumps (Figure A.2). In order to specify both possible modes in this star, the fitting procedure was performed twice; once using highly 63 150 100 "i i r -4-> o o r—I -50 — •100 — l i I | I 1 I 1 1 1 vsini = 115. + / - 5. P = 0.53 + / - 0.04 .. dt = 0.041 + / - 0.006 : m = 13.0 + / - 2.0 .1.12 Julian Date 1.14 1.16 1.18 F I G U R E 5.3. Line-Profile Variations of K2 BOO. The period and mode are estimated by fitting a sine curve to the subfeature positions in the time series of residuals of the A4508 l ine. T A B L E VII Line-Profile Variations Star P At m (days) (days) K Boo 0.53 ± 0.04 0.041 ± 0.006 13 ± 2 v U M a 0.63 ± 0 . 0 5 0.033 ± 0.004 19 ± 3 0.30 ± 0.06 0.090 ± 0 . 0 0 6 3 ± 1 21 Mon 0.79 ± 0.06 0.074 ± 0.018 11 ± 3 o1 Eri — — — 65 smoothed residuals (Figure A.5). In this case, the positions of both the minima and maxima are mapped. Although, the technique works well if there is a single mode dominating the line profile variations, if the star pulsates in multiple modes of nearly equal amplitude, the pattern becomes very complicated. Such may be the case for o1 Eri (Figures C.9 and C.10) for which no satisfactory sine curve could be obtained to describe the subfeature motions. The results from this section are summarized in Table VII. 5.3 R a d i a l - V e l o c i t y V a r i a t i o n s Low-degree modes are expected to produce radial-velocity variations of the profiles. The strongest lines in our spectra allow very accurate radial-velocity curves to be determined. Radial-velocity shifts relative to the mean spectrum were obtained for each time series using the Fahlman-Glaspey (1973) technique within R E T I C E N T . This technique determines the shift between two spectral profiles by minimizing the difference function between one profile and the other shifted at small intervals relative to the first. O P D A T A was used to fit the radial-velocity curves with sinusoids of the form JV VRV = Yl^Ai sin(27r(«oi + t)/Pl)). [5.10] The radial-velocity curve for K2 BOO is shown in Figure 5.4 and the results for all the stars are summarized in Table VIII. 66 F I G U R E 5.4. Radial-Velocity Variations of K2 BOO. The velocity shift for the time series of spectra measured with respect to the mean spectrum. T A B L E VIII. Radial-Velocity Variations Period (days) Amplitude (km/s) K Boo P i = 0.067 ±0 .002 A i = 0.83 ± 0.07 v U M a Pi = 0.090 ± 0.001 A i = 2.22 ± 0.06 P2 = 0.198 ± 0.007 A 2 = 1.88 ± 0 . 0 5 21 Mon Pi = 0.10 ± 0 . 0 1 A, = 2.2 ± 0 . 6 P2 = 0.06 ± 0 . 0 1 A2 = 1.0 ± 0 . 6 o 1 Er i Pi = 0.112 ±0 .004 A i = 1.07 ± 0 . 0 8 ( 1 s t night) P2 = 0.065 ± 0.002 A 2 = 0.94 ± 0 . 0 7 o 1 E r i Pi = 0.113 ±0 .006 A i = 0.69 ± 0.05 ( 2 n d night) 68 Multiple periods may be present for v UMa, 21 Mon and o1 Eri. More likely is that the radial-velocity curves are nonsinusoidal. By allowing for harmonics of the fundamental period and fitting curves by the data can be described well. Note that the low-degree line-profile variations detected in v UMa are also visible as radial-velocity variations. Also, the observations of o 1 Eri raise the question of mode stability. Two sets of observations are available for this star on successive nights. The radial-velocity curves indicate that one of the variations has disappeared while the other has persisted at a lower amplitude. Longer time series will be necessary to investigate this interesting star. 5.4 N R P M o d e l s It is instructive to rewrite equations 2.15 and 2.11 to describe the oscillations in terms of the observed quantities: N VRV = J2(A> s i n ( 2 7 r ( * o i + t)/P)) 1 A t mv sin i (1 - C)[0.01976] [5.12] P< + Rsini osc k = 74.44 MPl osc [5.13] 69 In the above formulations, P0Sc is the oscillation period of the wave in days, At is the bump spacing in days, m is the mode, vsins' is in km/s, and R and M are the radius and mass in solar units. The values of R, M, vsmi, m and P are read by the nrp program. The inclination is assumed to be 90°. The limb darkening is chosen to be (3 — 0.6. The waves are assumed to be prograde and sectorial (£ = —m). The value of k is determined from the theoretical relations 5.12 and 5.13. An initial guess is provided for the velocity amplitudes of all the the modes. A single line (A4508) is selected from the synthetic spectrum for each star to serve as the input profile. Times for the model are selected to match the median times of observation. The program then generates a series of output profiles which can be compared with the observations. The profiles are converted into RETICENT-readable files and analyzed in the same way as the observations. Residual, variance, and mean profile plots are produced. The equivalent width, line depth and bump amplitude are measured. All of these are compared with the observations. The goodness of fit is judged by visual inspection with the model residuals superimposed on the observations. Parameters are adjusted and new profiles are generated until an adequate match is achieved after several iterations. The program is run once more using the entire synthetic spectrum as input and the time series compared to the observed series. Figure 5.5 shows the model time series of residuals and variance in the line A4508, while Figure 5.6 compares the modeled residuals and the mean Fourier spectrum of the variations with the observations. Figure 5.7 illustrates the time series of profiles and residuals and the mean and variance generated for the en-tire spectral region. The apparent existence of underlying multiple modes in the time series of the other three stars made comparison of models and observations 70 b) a o c o :> Q c QJ 5Z OJ • 2 0 0 Ue 1 o c i t y F I G U R E 5.5. NRP Model of the A4508 Line for K2 BOO a) Time series of residuals. b) Mean deviation. (see Figure 4 • 2 0 0 V E L O C I T Y ( k m / s ) F I G U R E 5.6 Comparison of Model with Observations of K2 BOO (A4508). a) Time series of residuals. Both the observed and the modeled variations have been normaized to the depth of the line, (see Figure 4.2) 72 F I G U R E 5.6 Comparison of Model with Observations of K2 BOO (A4508). b) Amplitude spectrum. a) ^4 4 4 6 0 4 4 8 0 : 4 5 0 0 U a u e l e n g t h 4 5 2 0 F I G U R E 5.7. NRP Model for K2 BOO. (see Figure 4.2) a) Time series of spectra. b) Time series of residuals. h . . . i . • , , i i — i — i — i — i — i — i — i — i — i — i — ' — i — i — a 4460 4470 4480 4490 _ 4500 - 4510 4520 4460 4470 4480 4490 4500 4510 4520 W a u e l e n g t h F I G U R E 5.7. NRP Model for K2 BOO. (see Figure 4.2) c) Mean spectrum. d) Mean deviation. 75 T A B L E IX. NRP Models: Summary Star R M (RQ) ( M 0 ) v sini (km/s) m At (days) p x osc (days) Vamp (km/s) jfc <5 K2 Boo 3.1 2.0 115 -12 0.0445 0.071 3.4 0.025 19 0.02 low 0.067 ? ? ? ? ? v UMa 3.2 2.6 110 -19 0.033 0.057 3.0 0.02 26 0.02 -3 0.090 0.107 15.0 0.07 14 0.03 low 0.198 ? ? ? ? ? 21 Mon 3.4 2.6 130 -10 0.077 0.162 ... 0.13 8 0.04 low 0.10 ? ? ? ? ? low 0.06 ? ? ? ? ? o1 Eri 2.7 2.3 110 high ... ... low 0.112 ? ? ? ? ? low 0.065 ? ? ? ? ? impractical. Extended observations are required to determine the nature of the additional modes before reasonable models can be calculated. The results of the nrp analysis of all the stars are summarized in Table IX. 5.5 L i g h t V a r i a t i o n s Periods and amplitudes determined from photometric studies of these four 6 Scuti stars are, generally, either ill-determined or inconsistent. A literature search for data on each star was performed with the aid of the SIMBAD database. Table X lists the published photometric results. Periods, amplitudes and references are provided. Good agreement exists among the various studies of K2 Boo, but the periods and amplitudes are not well defined among the other stars. The periods derived from radial-velocity variations measured in this study (section 5.3) are compared with the published photometric periods (Breger 1979) in Table XI. When more than one radial-velocity variation is evident, the one with largest amplitude is chosen for comparison. The period-luminosity relation for 6 Scuti stars predicts that the absolute magnitude is related to the (photometric) period of oscillation by (Breger 1979) Mv = -3.052 log P + 8.456(6 - y) - 3.121. [5.14] Included in Table XI are the expected periods based on their derived absolute magnitudes. Good agreement between the photometric and radial-velocity periods is found for K2 BOO and 21 Mon. 77 T A B L E X. Published Photometric Variations K2 Boo: Period Amplitude Reference Notes (days) (mag.) 0.069 0.04 Millis (1966) 0.0731 0.040 Desikachary etal. (1971) • 16 day beat period 0.0681 0.039 • alias ? 0.0668 0.028 Elliot (1974) • beats or changing period - Moreno (1980) 0.0762 0.02 Ibanoglu et.al. (1983) • small colour variations v UMa: Period Amplitude Reference Notes (days) (mag.) 0.132 0.05 Danziger and Dickens (1967) • beats 21 Mon: Period Amplitude Reference Notes (days) (mag.) 0.11 0.035 Eggen (1968) • variable amplitude 0.0999 0.026 Gupta (1973) • 0.3 day beat period 0.0750 0.008 ?! 0.0895 0.019 Stobie et.al. (1977) • variable ampl./freq. 0.1599 0.013 V • suspected nrp 0.1065 0.007 5) 0.2700 0.006 J ? 0.1507 0.006 )! 78 T A B L E X (cont.). Published Photometric Variations o 1 Eri:. Period (days) Amplitude (mag.) Reference Notes 0.0756 0.02 Jorgensen et.al. (1971) • very uncertain 0.100 0.01 55 0.0815 0.05 Jorgensen et.al. (1975) • non-unique 0.1291 - 55 0.0720 0.007 Poretti (1989) • nrp suspected 0.144 0.006 55 • addditional modes 0.166 0.006 55 79 T A B L E XI. Radial-Velocity and Photometric Periods Period (days) Star PRV Pphot p 1 exp K2 Boo 0.067 0.066 0.105 v U M a 0.090 0.132 0.128 21 Mon 0.100 0.100 0.142 o 1 Eri 0.112 0.081 0.093 80 Equation 2.18 can be employed to calculate the light variation arising from geometrical distortions of the projected disk due to the oscillations. The light vari-ation arising from high-degree modes is naturally quite small due to cancellation effects. For example, geometrical distortions of the £ = 12 mode identified in K2 Boo would result in a peak-to-peak light variation of only 0.00005 magnitudes, far below current detection limits. We shall therefore restrict our discussion to light variations due to low-degree modes. The low-degree modes cannot be identified uniquely from radial-velocity vari-ations alone. We assume that radial and nonradial modes of degree as high as £ = 2 could be the source of the variations. Velocity amplitudes on the surface of the star necessary to produce the observed radial velocity variations were deter-mined using the nrp program. A time series of profiles with a low degree mode of oscillation was generated and the radial velocity variations in the line profiles were measured using the Fahlman-Glaspey (1973) technique. The process was repeated to produce a grid of models for each of the low-degree modes at different velocity amplitudes and values of vsini. An inclination of 90° was again assumed. Table XII lists the light amplitude corresponding to the geometrical variations for cases with £ = 0 and 2 for all the stars. No net changes in stellar disk area result from the £ = 1 mode so that no (geometrical) light variations are associated with this mode. 81 T A B L E XII. Geometrical Light Variations K2 Boo: A m = 0.03 R.V. (km/s) a p -1 osc (days) Vamp (km/s) A m AV A m (mag.) 0.83 0 0.000 0.067 1.2 0.00093 0.002 0.83 2 0.026 0.074 2.8 0.00060 0.003 v UMa: A m = 0.05 R.V. (km/s) £ k p 1 osc (days) Vamp (km/s) A m AV A m (mag.) 2.22 0 0.00 0.090 1.2 0.00120 0.007 2.22 2 0.06 0.101 7.4 0.00080 0.006 1.88 0 0.00 0.198 2.7 0.00265 0.014 1.88 2 0.3 0.231 6.2 0.00182 0.011 21 Mon: A m = 0.04 R.V. (km/s) £ k p x osc (days) Vamp (km/s) A m AV A m (mag.) 2.2 0 0.00 0.10 3.1 0.00126 0.008 2.2 2 0.06 0.11 7.3 0.00082 . 0.006 1.0 0 0.00 0.06 1.4 0.00076 0.001 1.0 2 0.02 0.07 3.4 0.00049 0.002 o1 Eri (1st ni ght): A m = 0.03 R.V. (km/s) £ k p -* osc (days) Vamp (km/s) A m A V A m (mag.) 1.07 0 0.00 0.112 1.5 0.00178 0.005 1.07 2 0.15 0.129 3.6 0.00121 0.004 0.94 0 0.00 0.065 1.4 0.00103 0.003 0.94 2 0.05 0.072 3.2 0.00067 0.002 o1 Eri (2nd night): A m = 0.03 R.V. (km/s) £ k p x osc (days) Vamp (km/s) A m AV A m (mag.) 0.69 0 0.00 0.113 1.0 0.00179 0.004 1.07 2 0.15 0.130 2.3 0.00122 0.003 82 Chapter 6 DISCUSSION 6.1 U n i q u e n e s s Nrp theory has been applied to the four 8 Scuti stars to explain the line profile and radial velocity variations in the spectra. The technique of comparing model line profiles with the observations to infer the mode, period, and amplitude of oscillation has been demonstrated for the star K2 BOO. We now consider the uniqueness of the models we have derived and discuss the implications of the assumptions we have adopted. Our treatment of nrp in the presence of rapid rotation neglected all effects of the Coriolis force greater than first order (Equation 2.15). This approximation is justified if the ratio of the pulsation frequency, aosc to the apparent oscillation frequency, 0. is large, as would be the case for high-degree p-modes. In Table IX, we list the ratios of these two quantities, <TOSC/Q. For example, crosc/fl = 19 at i = 90° for K2 Boo. We conclude that p-modes are responsible for the high-degree variations and that our assumption is justified. 83 We have used the theoretical relationship of Equation 2.12 to estimate the value of k. For rc2 Boo, a very low value of k = 0.02 ± 0.01 is implied for the high-degree mode. This value is consistent with the shape of the observed residuals and the variance plot. The other stars appear to be complicated by additional modes of oscillation, making it difficult to confirm the theoretical values of k but it appears as if the so-called k—problem associated with OB type line profile variables does not arise for the 8 Scuti stars. We have applied the nrp theory with the goal of determining the pulsation frequencies of the waves on the surface of these stars. The relationship between the apparent period of variability and the true oscillation period is given by Equation 5.12. The best model was obtained for the star K2 BOO. In this case the oscillation period is estimated to be Posc = 0.071 ± 0.009, assuming i = 90° . The apparent period of oscillation can be determined accurately from the power spectra of a long time series. The accuracy of the solution is then limited by the uncertainties in m, R, vsini and i. Estimates of the errors in Poac and k were calculated under the assumption that the inclination, with an adopted value of 90°, introduces no uncertainty. Be-cause the inclination is in fact highly uncertain, we choose to address its effects separately. If we assume that the observed upper limit of 200 km/s on the distribu-tion of v sin i in 8 Scuti stars represents a maximum in their rotational velocities, then the minimum inclination can be derived from the ratio (usini)/200. For K2 Boo, this limit is im%n = 35° . Conceivably the inclination can take on any value between this and 90° . Table XIII illustrates how inclination affects the derived quantities of k, P o s c , and Vamp for K2 BOO. The effect does not cast any doubts on the assumptions of the theory or the p-mode nature of the oscillations, but there is unavoidable uncertainty in Posc due to the unknown inclination in these models. 84 T A B L E XIII . Effect of Inclination on the Oscillation Parameters i p A osc Vamp Jb Vosc/£l Q (deg.) (days) (km/s) 90 0.071 3.4 0.025 19 0.018 80 0.071 3.5 0.025 19 0.018 70 0.074 3.9 0.027 17 0.019 60 0.078 4.7 0.030 15 0.020 50 0.085 6.5 0.036 12 0.022 40 0.101 10.0 0.050 9 0.026 85 The nrp modes were assumed to be sectorial. The basis for this assumption is that such modes yield the largest variation in the line profiles for a given velocity amplitude (see Figure 3.3). It is possible that another mode with £ ^ \m\ is responsible for the variations if the velocity amplitude on the surface of the star is permitted be larger than derived here. For example, a tesseral mode (£ — \rn\ + 1) with \m\ = 6 and £ = 7 would produce the same pattern in the line profiles as an £ = \m\ = 12 mode. In this case the alternate features in the profiles arise from contributions above and below the stellar equator which are 180° out of phase. Such a mode could describe the variations in K2 Boo. Furthermore, the bumps in the observed residuals seem to decrease in amplitude as they move to the line wings more quickly than expected from the model profiles for sectorial modes. This is especially the case for the star 21 Mon, as demonstrated by its peculiar variance function. Such an effect may result from nonsectorial modes of oscillation, such as an £ = 12, |m| = 10 mode in this case. Another assumption was that temperature variations are also negligible. For high degree modes, temperature effects are not expected to be very significant. For low-degree modes, with generally larger velocity and light amplitudes, they are more important. It was found that geometrical distortions alone could not account for the published light variations. However, the inclusion of temperature variations into the analysis of the stars would only serve to introduce two additional \ parameters into a problem already wrought with complications. The published photometric periods and amplitudes are useful but simultaneous photometry and spectroscopy is required to treat the problem of temperature variations with any degree of confidence. The nrp model as it has been presented here may be an oversimplification of reality in line profile variables. Observed line profiles often show bumps which 86 vary in amplitude as they progress through the profile. Often a single bump may die out completely. Sometimes features can be observed to separate or even move backwards against the flow of the other bumps. Naturally, if one is allowed to impose enough modes on the surface then any irregular behaviour can be accounted for by simple nrp theory. Alternatively, it is conceivable that the theory is not sufficient to describe oscillations which are not stable for long periods of time. Shocks may develop or oscillations may die out quickly, to be replaced by other modes. An example of this type of behaviour can be seen in the residuals of v UMa (Figure A.2). Twice during the series a given feature appears to separate into two while the preceding feature seems to disappear. Is this evidence of a shock occurring on the star creating an additional wave crest and then re-establishing its former structure by wiping out one of the pre-existing waves? More likely the behaviour arises from the interaction between the high- and low-degree modes. The addition of the velocity perturbations of these modes may mask the num-ber of features present in the high-degree modes on the stellar surface. Features would seem to appear and disappear as both modes traveled around the star at their respective rates. This would result in an error in the measured value of m. Unfortunately difficulties in modeling the low-degree mode prohibits drawing a firm conclusion. (The bumps associated with the low-degree mode seem to die out as they move through the profile.) More observations and the inclusion of temperature variations are required to resolve the situation. 87 6.2 Super Pe r iod? The concept of a "superperiod" was introduced by Smith (1985) for line-profile variables. It is defined as mPosc = constant for all £ = —m modes of oscillation and represents the time required for a wave to encircle the star once. This concept has been applied to the observations of multiperiodic OB stars to explain the apparent relationships between observed frequencies. If such a restriction exists, then it must be linked to the excitation mechanism driving the pulsations. The problem with the rapidly rotating OB stars is that the rotation frequency dominates the apparent oscillation frequencies. Under these circumstances, it is difficult to measure the wave speeds. The oscillation frequencies of the 8 Scuti stars are very much greater than the rotation frequencies, making them much easier to distinguish. The periods of oscillation in each of the four stars analyzed here appear to be related to each other but not according to the superperiod as defined for the OB stars. A 1:1, 2:1, ... relationship between the periods is consistent with the derived values although the uncertainty in many of the quoted periods is high. The most reliable results were obtained for K2 BOO. Our results indicated that both the high {£ = 12) and the low {£ = 0, 1, or 2) degree modes pulsate with a period of ~ 0.071 days. This implies that the high-degree mode is of low radial order and the low-degree mode of high order (see Figure 2.2). Perhaps the existence of one mode might excite the other. The periods of the other three stars can be incorporated with this idea if the periods of oscillation obey a 1:2:3:4 ratio for the radial, £ = 2, and higher-degree modes. 88 6.3 A l t e r n a t i v e M o d e l s In Chapter 1 we listed some of the theories which have been proposed to explain line profile variations. Nrp is by far the most widely accepted explanation for such variations. An unfortunate aspect of nrp theory is that it is very difficult to rule out, with certainty in any case, due to the flexibility introduced by the wave speed. The "spoke" model, in which elongated blobs of material occult the surface of the star to produce line profile variations suffers from the same sort of flexibility as the nrp model but requires more stringent limits on the inclination of the star. In this case, the rate of variability can be adjusted by changing the distance from the star at which the obscuring material orbits. (The greater the distance the larger the inclination limit.) On the other hand, the "spot" models which invoke temperature and brightness differences require that the features responsible for the variations be fixed to the surface of the star and travel at exactly the rotation rate, so that the observed period of the bumps in the fine profiles is the rotation period of the star. This restriction makes the spot model easiest to test. If we assume that the spots are distributed about the equatorial region and that the period of the traveling bumps is equal to the rotation period, then the equatorial velocity, Ve (in km/s), and inclination, i, are given by V e ~ (0.01976) P  [6"1] . . v sin i r , sim = ——— [6.2J where RE is the equatorial radius (in RQ), P is the measured period (in days) and usinz is the rotational broadening (in km/s). The critical velocity, V c r t < , (in km/s) 89 of the star, for which the centrifugal and gravitational acceleration at the surface are equal and oppositely directed, is given by where M is the mass (in MQ). Applying this model to the four 8 Scuti stars, we find that small inclinations and unrealistically large equatorial velocities are required to fit the observations. For example, K2 BOO would need an inclination of only 23° and an equatorial velocity of about 300 km/s. The results for all the stars are presented in Table XIV. The velocities are well above the maximum value (~ 200 km/s) expected for 8 Scuti stars. As a further test, the nrp program has been adapted such that spots can be imposed on the surface of the star. Preliminary models indicate that in order to produce the large amplitudes of the bumps in the line profiles, very elongated spots stretching nearly from pole to pole with large brightness contrasts must be invoked. Such a, fixed pattern so extensive in latitude is unlikely to persist on the surface of a rotating star in the presence differential rotation. These arguments safely eliminate the starspot model as the explanation of rapid line-profile variations in 8 Scuti stars. 90 T A B L E X I V . Star Spot Models Star M R Prot v sin i Ve Vcrit i (Mo) (Ro) (days) (km/s) (km/s) (km/s) (deg.) K2 Boo 2.0 3.1 0.53 115 296 351 23 v U M a 2.6 3.2 0.63 110 257 394 25 21 Mon 2.6 3.4 0.77 130 223 382 30 o 1 Er i 2.3 2.7 ... 110 ... 403 ... 91 Chapter 7 CONCLUSIONS We have successfully applied Nrp theory to explain the line-profile variations in four 8 Scuti stars: K2 BOO, V UMa, 21 Mon and o1 Eri. Spectral residuals and the absolute mean deviation of the variations in the profiles were used to characterize the nature of the high-degree variations. By calculating Fourier spectra of these variations and by fitting the motion of the subfeatures through the profiles, the mode and apparent period of the oscillation were found. Invoking nrp theory for a rotating star then leads to the true periods of oscillation. A geometrical nrp model was developed which reconstructs the line profile variations of a nonradially pulsating star from synthetic spectra. By using syn-thetic spectra as input, the dependence of surface resolution on the intrinsic widths of the lines is automatically taken into account so that the velocity amplitude on the stellar surface can be better estimated from the amplitudes of the profile vari-ations. We have shown in detail how the model was applied to K2 BOO and the mode and period of oscillation were refined by this technique. High frequencies and low k-values were found for all stars, in agreement with expectations for p-mode oscillations (in contrast to the OB line-profile variables for which g-mode oscillations prevail). The nrp theory for a rotating star becomes 92 simplified in the case of p-modes since the high-order terms of the Coriolis force can be neglected. Low-degree modes were detected as radial-velocity variations in the four stars. It was determined that the geometrical distortions associated with the required surface velocity amplitudes are not sufficient to reproduce the observed light vari-ations. Temperature variations must be a significant effect in these modes. A correlation between the periods of oscillation of the high- and low- degree modes has been noted. Although the uncertainty in the derived periods may be large, a significant harmonic relationship between the oscillation periods of the various modes is evident. This differs from the concept of the superperiod invoked for O B stars where the (degree of the mode) x (the period of oscillation) is argued to be a constant. If the relationship for 8 Scuti stars is real, then it undoubtably is relevant to the excitation mechanism for the pulsations. Accurate models of the other three stars and additional 8 Scuti stars are called for to establish the validity of this result. Alternative models to explain the variations have been discussed. Star spots on the stellar surface are found to be inconsistent with the observations. The observed periods would imply a rotational velocity which is much too high for the 8 Scuti stars and also require a very low inclination. A summary of the findings for each star is provided below. K2 B o o This star displayed the simplest variations of all the stars. The profile model-ing technique was demonstrated for K2 BOO and the mode, period, and amplitude of oscillation were derived. A single high-degree mode with I = — m = 12 is 93 sufficient to explain the line profile variations. The period of oscillation was de-termined to be Posc = 0.071 ± 0.009 days and the k-value is 0.02 ± 0.01. The velocity amplitude on the stellar surface was estimated to be Vamp = 3.5 ± 1.0 km/s. A single low-degree mode was identified from the radial velocity variations of the spectrum with an apparent period of 0.067 ± 0.002 days. These variations could result from a radial or low-degree nrp mode of oscillation (£ = 0, 1, 2 with periods 0.067, 0.070, 0.074 respectively). The oscillation periods of the high- and low-degree modes are suspected to be equal. v U M a The significant feature of this star is that both a low-degree and a high-degree mode are visible as line-profile variations. The low-degree mode, which also man-ifests itself in radial velocity, is consistent with an 1 — —m = 2 mode. The period of oscillation of this mode is P03C = 0.101 days. Attempts to model this mode with the nrp program were not very successful, and we conclude that temperature vari-ations are likely to be significant. Due to the difficulty in properly removing the low-degree mode, the high-degree variations could not be easily specified. These variations arise from a mode with degree I > 16 with a period between of ~ 0.055 days but the appearance and disappearance of features in the profiles makes this mode difficult to identify. The possibility of nonlinear effects was discussed but it seems more likely that the combination of this mode with a low-degree mode of comparable amplitude masks the number of features in the profile. A second low-degree mode with period 0.198 days is suggested by the radial velocity variations. The periods found are consistent with a 1:2:4 relationship between the low- and high-degree modes. 94 21 M o n Since this is the shortest time series, with the lowest signal-to-noise, it is diffi-cult to draw solid conclusions about this star. Three apparent modes of oscillation were detected from the line-profile and radial-velocity variations. The high-degree mode appears to be an £ — 10 mode with a period of 0.162 days. The two low-degree modes have periods of 0.11 and 0.06 days. 1:2:3 relationship can not be ruled out. The low-amplitude variations in the wings may be a result of the low signal-to-noise or possibly nonsectorial modes of oscillation. o 1 E r i Observations of this star were obtained on two consecutive nights. However, the complexity of the profile variations in this star is such that the the identity of the high-degree modes of oscillation could not be identified with certainty. There appear to be several large-amplitude modes which interact with each other. This is indicated by the large amount of power at many frequencies in the Fourier transform of the variations. Two low-degree modes were detected on the first night as radial-velocity variations with periods of 0.112 and 0.065 days. However, by the second night, the amplitude of one had decreased and the other disappeared. Due to the shortness of the time series (1-2 oscillation cycles) it is difficult to discuss the stability of the modes. T h e Future Time series observations of ~ 1 — 2 hours have revealed much about the nature of high- and low-degree nrp in rapidly rotating 8 Scuti stars. Stars with a single high-degree mode of oscillations can be effectively modeled based on even short time series. However, if the star pulsates in more than one high-degree mode (e.g. o1 Eri), then it becomes more difficult to identify the periods of the oscillations 95 from the line profile variations. A long time series (~ 10 hr) obtained over one or more nights would significantly improve the resolution of the frequencies in the Fourier spectrum of the variations. With observations of this length, the apparent relationship between the pulsation periods could be confirmed. At any rate, a complete spectrum of oscillation frequencies would be an important step towards understanding the excitation mechanism and the internal structure of 8 Scuti stars through the application of stellar seismology. In order to properly model the low-degree modes, both temperature and geo-metrical effects must be taken into account. Modifications to the nrp program to include temperature variations would be trivial, but the extra degree of complexity introduced by the new parameters would make a unique solution more difficult. Simultaneous photometry could resolve some of the ambiguities in the low-degree variations. Finally, since the four stars discussed here are the only 8 Scuti stars investi-gated for high-degree variability so far, a more complete survey of the phenomenon is called for. Many questions remain to be answered. Is there a connection between the high- and low-degree oscillations? Could many of the apparently 'constant' stars in the 8 Scuti strip be variable with high-degree oscillations? Does rotation play a role in exciting or determining the modes of oscillation? Are the modes stable over long time scales? 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Stamford, P.A. and Watson, R.D. 1977, M.N.R.A.S., 180, 551. Stobie, R.S., Pickup, D.A. , and Shobbrook, R.R. 1977, M.N.R.A.S., 170, 389. Unno, W. , Osaki,Y., Ando, H . , Saio, H. and Shibahashi, H. 1989, Nonradial Os-cillations of Stars, second edition, University of Tokyo Press, pl9. Vogt, S.S. and Penrod, G.D. 1983, Ap. J., 275, 661. Walker, G . A . H . , Moyles, K. , Yang, S., and Fahlman, G . G . 1982, Pub. A.S.P, 94, 143. Walker, G . A . H . , Yang, S., and Fahlman, G .G. 1979, Ap. J., 233, 199. Walker, G . A . H . , Yang, S. and Fahlman, G .G. 1981, Proc. Workshop on Pulsating B Stars, Nice Obs., ed. M . Auvergne et al. Walker, G . A . H . , Yang, S., and Fahlman, G . G . 1987, Ap. J., 320, L139. Yang, S. and Walker, G . A . H . 1986, Pub. A.S.P., 98, 1156. 100 Appendix A vUMa The observations and analysis of v UMa are presented as a series of diagrams in this section. Time series data for the entire spectral region and the selected Fell A4508 line are illustrated. Power spectra and subfeature position curves of line profile variations and radial velocity curves are shown. 101 4480 4500 4520 U a u e l e n g t h F I G U R E A . l v UMa Observations, (see Figure 4.2) a) Time series observations. b) Time series of residuals. 0. 9DB 0.915 0. 921 0. 929 0. 939 0. 948 0.9G0 0. 970 0.979 0. 9B9 1 . DOB 1.017 1. 027 1. 037 1. 048 1.058 1 . 0GB 4480 4500 U l a u e l e n g t h 4520 «0 b) 4 4 8 0 4 5 0 0 U l a u e l e n g t h 4 5 2 0 F I G U R E A . l v U M a Observations, (see Figure 4.2) c) Mean spectrum. d) Mean deviation. 103 a) o Ue loc i ty 2 0 0 Ue l o c i t y F I G U R E A.2 v U M a Observations of the Line A4508. (see Figure 4.2) a) Time series of residuals. b) Mean deviation. .2 -l—i—i—r j L_l L 0 4 4 6 0 4 4 7 0 I fvmr J I I I I I I 1 I I I I I 1 I I 1 I u 4 4 8 0 4 4 9 0 4 5 0 0 wavelength [A] 4 5 1 0 4 5 2 0 1.05 "l i i i I i i i i I i i i i I i i i i i i—i—i i— i— i— i—i r .75 l_j I i i I i i i i I i i I i I i i I i I i L_I i I ' ' i i_ ' 4 4 6 0 4 4 7 0 4 4 8 0 4 4 9 0 4 5 0 0 4 5 1 0 4 5 2 0 wavelength [A] F I G U R E A.3 v UMa Synthetic Spectrum. a) Intrinsic synthetic spectum with Teff = 7200 and log g — 3.84. b) Synthetic spectum with vsins' = 110. 105 .008 .008 .004 .008 0 .008 .008 .004 -«-> .—i 0 CO £ -008 CD _l_> .008 .!=! .004 CD 0 0 8 o .+J .008 .1— i ^.008 S 004 ^.008 0 .008 .008 .004 .008 0 .008 .006 .004 .008 0 . 1 1 1 1 | 1 1 1 1 | 1 1 1 1 ] l " T 1 1 ) 1 1 1 I T I v=0 k m / s ; -"*r"i i • ' • • i i 1 T^I i i 1 i n i 1 I P T — \ Z~l~i—1 1 |~ 1 1 1 1 | T ' l"1 1 | 1 1 1 1 | 1 1 1 1 . ; v=0 k m / s ; ' " f i i i 1 i i i i 1 1—i i i 1 i n i 1 I I r " i . i i i i I I i i i | i i i i | i i i i | i i i i . ; v=-10 k m / s ; ' ' I I I • i • • i v i I i * * t — r l i i i i 1 i i—i— i . 1 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 I I 1 1 1 . ; v=+10 k m / s [ :-T~+^I i 1 i i i i 1 n i i 1 i r i i 1 i r~T~-r" L 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 -—: ! i i •• i ! i I i — ; v = - 2 0 k m / s ; i i i i 1 i i i r l i i V i n i i i — i — t - ' f i ~ _ 1 r I I 1 1 1 I 1 I I I 1 1 1 1 1 1 1 I 1 1 1. 2 v=+20 k m / s ; > T T Y I 1 i i n 1 i i i i i K i I T — n — r " . i i i i i i i i i i i i i i i i i i i r i i . \ / \ v=-30 k m / s ; i i M 1 I 1 1 i | 1 1 1 — r - ^ - ' T i " T T T * r i i i i i i i i i i i i i i i i i i i i . 1 v=+30 k m / s ; r i i V i 1 i i i i 1 ^ / i i i 1 i p i^ i 1 i i Pr -. I I I I I I I I I I I I I T I T I I I I . ; / \ v=—40 k m / s ; - / ' '' ' V -i i y i i i i IVI i i s\ i i i i i i i i i -. i i r i i i i i i i i i i i i i i i i i i i i i j '- v=+40 k m / s ; i i i i 1 ' i i l 1 l i l l 1 l l l l i l l 1 i . 1 1 1 1 1 1 1 1 1 1 1 1 I I 1 1 1 1 1 1 1 1 1 1 . \ ss. v=-50 k m / s \ / < i i i 1 i i i i 1 i V i i 1 i i i i 1 • N - T i ' . 1 I I I | • 1 1 I | I I I 1 | 1 1 1 1 | I 1 1 1. ; v=+50 k m / s ; r i i i V I.J. i i i 1 i Y i . i r N - H - n ~ T - 4 ^ T ~ T ^ : 10 80 30 40 0 10 80 30 40 frequency (c/d) F I G U R E A.4. Fourier Amplitude Spectrum for v UMa. a) The variation of the residuals at a given position in the line profile were mea-sured and transformed into the Fourier domain. The series of plots illustrates the amplitude spectrum at 10 km/s intervals. 106 FIGURE A.4. Fourier Amplitude Spectrum for v UMa. b) The mean amplitude spectrum (solid line) from those illustrated in Figure (a). The dashed line is the window function for the period, At. i i i i r i I r 150 100 50 -50 •100 "l i i | i i i | i i r~j i i i | i r v s i n i = 110. + / - 5. P = 0.63 + / " 0.05 .. dt = 0.033 + / - 0.004 : m = 19.0 + / - 2.8 J I I i i i I i i i I i i i J I I I I L I I I I I .92 .94 .96 .98 1 Ju l i an Date 1.02 1.04 1.06 FIGURE A.5. Line-Profile Variations of v UMa. a) The period an mode are estimated by fitting a sine curve to the subfeature positions in the time series of residuals. i—r ~ i—i— r—i—i—i—i—|—r vsi'ni = 110. + / - 5. P = 0.30 + / - 0.06 d t = 0.090 + / - 0.006 m = 3.3 + / - 0- 7 I i i i i i i i i i i i r -i-> • rH o > 150 100 50 - 5 0 - 1 0 0 .9 .92 .94 .96 .98 1 Ju l i an Date 1.02 1.04 1.06 F I G U R E A . 5 . Line-Profile Variations of v U M a . b) The line profile was highly smoothed to remove the high degree variations. A sine curve was fit to both the minimum and max imum points i n the resulting residuals in order to estimate the low degree mode. F I G U R E A . 6 . Radial-Velocity Variations of v U M a . The velocity shift for the time series of spectra measured w i t h respect to the mean spectrum. Appendix B 2 1 Mon The observations and analysis of 21 Mon are presented as a series of diagrams in this section. Time series data for the entire spectral region and the selected Fell A4508 line are illustrated. Power spectra and subfeature position curves of line profile variations and radial velocity curves are shown. I l l to 4 4 8 0 4 5 0 0 4 5 2 0 U l a u e l e n g t h F I G U R E B . l . 21 Mon Observations, (see Figure 4.2) a) Time series observations. b) Time series of residuals. J , I i L 4480 4500 4520 4 4 8 0 4 5 0 0 4 5 2 0 W a v e l e n g t h F I G U R E B . l . 21 Mon Observations, (see Figure 4.2) c) Mean spectrum. d) Mean deviation. 113 a) 200 Ue 1o c i t y b) F I G U R E B.2 . 21 M o n Observations of the Line A4508. a) T ime series of residuals. b) Mean deviation. (see Figure 4.2) 4460 4470 4480 4490 4500 4510 4520 wavelength [A] F I G U R E B.3. 21 Mon Synthetic Spectrum. a) Intrinsic synthetic spectum with Teff = 7400 and log g = 3.78. b) Synthetic spectum with vsini = 130. 115 .01 .005 -v=0 k m / s .01 .005 -+-> •f -H 0 <D . o t -t-> • r - 1 '.005 T) o 3 cd .01 .005 .01 .005 .01 .005 T — i — i — i — i — i — i — i i i — i — i i i | i — i — i i | i i i — r fT \ I I | I I I I | M l I | I 1 I I | I I 1 I I I | 1 I I I | I II I | I I I 1 | I I I I ~ i — I I I I — r I 1 1 1 1 I 1 1 1 1 I 1 ' 1 1 I v=+10 k m / s v=+30 k m / s -v=+40 k m / s -10 20 30 40 10 20 30 40 frequency (c/d) F I G U R E B.4. Fourier Amplitude Spectrum for 21 Mon. a) The variation of the residuals at a given position in the line profile were mea-sured and transformed into the Fourier domain. The series of plots illustrates the amplitude spectrum at 10 km/s intervals. 116 .01 f r e q u e n c y ( c / d ) F I G U R E B.4. Fourier Ampli tude Spectrum for 21 M o n . b) The mean amplitude spectrum (solid line) from those il lustrated in Figure (a). The dashed line is the window function for the period, At. F I G U R E B.5 . Line-Profile Variations of 21 M o n . The period an mode are estimated by fitting a sine curve to the subfeature positions in the time series of residuals. Julian Date F I G U R E B.6 . Radial-Velocity Variations of 21 M o n . The velocity shift for the time series of spectra measured w i th respect to the mean spectrum. Appendix C o'Eri The observations and analysis of two nights of data for o1 Eri are presented as a series of diagrams in this section. Time series data for the entire spectral region and the selected Fell A4508 line are illustrated. Power spectra and subfeature position curves of line profile variations and radial velocity curves are shown. 120 b) 4480 4500 W a u e l e n g t h 4520 44B0 4500 W a u e l e n g t h 4520 F I G U R E C . l . o 1 E r i (1st night) Observations, (see Figure 4.2) a) T ime series observations. b) T ime series of residuals. 4 4 8 0 4 5 0 0 U i g e l e n g t h 4 5 2 0 F I G U R E C . l . o1 Eri (Is* night) Observations, (see Figure 4.2) c) Mean spectrum. d) Mean deviation. 122 4 4 8 0 4 5 D 0 4 5 2 0 U a u e l e n g t h F I G U R E C.2. o 1 Eri ( 2 n d night) Observations, (see Figure 4.2) a) Time series observations. b) Time series of residuals. 4 4 8 0 4 5 0 0 4 5 2 0 U a u e l e n g t h c) 4480 4500 4520 d) Mean deviation. 124 F I G U R E C.3. o1 Eri (1st night) Observations of the Line A4508. (see Figure 4.2) a) Time series of residuals. b) Mean deviation. 0 200 Ue 1 o c i t y F I G U R E C.4. o 1 Er i ( 2 n d night) Observations of the Line A4508. (see Figure 4.2) a) Time series of residuals. b) Mean deviation. F I G U R E C.5. o1 Eri Synthetic Spectrum. a) Intrinsic synthetic spectum with Teff = 7200 and log g = 3.95. b) Synthetic spectum with usinz = 110. 127 .009 .004 .002 0 .000 .004 -t-> • i—I 0 V) • I—( .002 73 o S0 0 8 6 CTj .002 0 .006 .004 .002 0 .008 .004 .002 7 1 — i — i — i — — i | i — i i i j — I — I — I i | i I I—rz v=0 k m / s * I I Y I I I I Y I I I I I I M I I V ^ i T i I I I " 7T—i—1 1 | . i — i — i — | — i — i — i — | — | — I — i — I — I — p i I I I . v=0 k m / s ' i i Y i l i i r i l i i i i M i i V T T i i i r _ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _ v=-10 k m / s -i i V i 1 i i i i 1 i i i \ l * n i i i S — r — i — i — T " i i i i i i i i i i i i i I i i i i i i i i i . v=+10 k m / s -' i i i i f i i i i 1 I I I I r I I I I P T I 1 .1 1 t"t - | 1 1 1 1 j 1 1 1 1 ^ 1 1 1 I I 1 I I 1. v=-20 k m / s -- : ! I ' ' ; s i I i _2 i i i i 1 i i i i 1 I i i i 1 I I | I 1 I I I y [ . I I I I I I I I I I I I I I I I I I I I I . v=+20 k m / s -' i i i i 1 1 i i V 1 i i i i 1 i i i i 1 i i i i .1 I 1 1 I I I I I I I I | I I I I 1 I I I I . v=-30 k m / s -' i i i i 1 i i i i 1 i i i i 1 i i t i [ i i i i 1 1 1 1 1 1 1 1 I I 1 1 1 I I 1 1 1 1 1 1 1 . v = + 3 6 k m / s -i i i i 1 i i i i 1 V i i i 1 i i i i 1 i i M i _ 1 1 1 1 l 1 1 1 1 1 1 1 l l 1 l l l l 1 l l l l _ v=-40 k m / s -— J ' 1 1 ' \ x \ _I f t i i i 1 i i i i I i i i i 1 t i i i 1 \ 1 _ l l l l | i l l l L r l l l | L l l l l l l l I _ v=+40 k m / s -l 1 " l 1 1 l 1 l l 1 1 1 1 1 1 1 1 1 1 1 11 . 1 1 1 1 • | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 1 1 1 1 1. v=—50 k m / s -f i i i i 1 i i i i 1 i i i i 1 i i i i n i V r _ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 _ v=+50 k m / s -\004 10 20 30 40 10 20 30 40 frequency (c/d) F I G U R E C.6. Fourier Ampl i tude Spectrum for o 1 E r i (1st night)., a) The variation of the residuals at a given position i n the line profile were mea-sured and transformed into the Fourier domain. The series of plots illustrates the amplitude spectrum at 10 k m / s intervals. 128 .005 .004 Q> .003 .002 "i i r i i i i i i i r " i i r At = 0.0945 days .001 series length: 0.181 days — resolution: 5.52 c/day 20 30 f r e q u e n c y ( c / d ) 50 F I G U R E C.6. Fourier Ampli tude Spectrum for o 1 E r i (Ist night). b) The mean amplitude spectrum (solid line) from those il lustrated i n Figure (a). The dashed line is the window function for the period, At. .008 .006 .004 .002 0 .008 .006 .004 ^ . 0 0 2 . ^ H 0 w £ 008 CD _t_> .006 .r-H .004 C D 0 0 8 T3 o .+_» .008 ^ . 0 0 6 £ 0 0 4 ^ . 0 0 2 0 .008 .006 .004 .002 0 .008 .006 .004 .002 0 n—i—r—i |—i i i i | i—i i i—| i i—i i | i i i—rz : v = 0 k m / s ; n i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i* -1 1 1 1 | 1 1 1 1 1 1 | 1 1 1 1 | 1 1 1 1" ; v=0 k m / s ; I i i 1 i i i 1 i i i i 1 i i i i 1 i i i i _ • i • i i • I i i 1 i i i i i I i i i i i i i i _ ; / \ v = - 1 0 k m / s ; 1 I 1 1 1 I I I 1 1 I I 1 1 1 1 1 1 1 1 i i 1 1" 1 1 1 1 1 | 1 1 1 1 | 1 1 1 1 II 1 1 1 II 1 "1-1 . ; v = + 1 0 k m / s ; r\ i i i 1 i • i i i 1 i i i i 1 i i i i 1 i i i i " .1 1 1 1 | 1 I I 1 | 1 1 1 1 | 1 1 1 1 M 1 1 1. — i >. X ~ i V ''111 — ; / \ v = - 2 0 k m / s \ i i i i 1 i i i i 1 i i i i 1 i i i i 1 i ii—f" . 1 1 1 1 | 1 1 1 1 | - 1 1 1 11 —1—1—T t 1-f 1 1 1 \ . '- v = + 2 0 k m / s \ > f i i i 1 * i i i 1 i i i t l i i i i 1 N^I i i -. I I I I | I I I I | I I I I I I I I I I I I I I . ; v = - 3 0 k m / s \ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r l 1 1 l l l Y " n i i i | i i i i | i i i i | i i i i 1 i i i • i _ ; v = » + 3 0 k m / s ; ' \ \ i i * i i i 1 i i i • ' 1 1 1 1 ' 1 1 < _ i i i i i i i i i i i i i i i i i i i i l l i i _ ; v = - 4 0 k m / s ; i i i 1 i i i i 1 i i i i 1 i i i i 1 \ i" _ l i l l | i l l l 1 l 1 1 1 1 1 1 1 1 1 1 1 1 1. ; v = + 4 0 k m / s ; X i i T l i i i i 1 i i i t 1 i i i i 1 i i i i -. i i i i i i i i i | i i i i i i i i i i f n i i _ ; v=—50 k m / s ; r i i i i 1 i i i i 1 i i i V 1 • i i i 1 i i i—r-= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . - v = + 5 0 k m / s ; 10 20 30 40 10 20 30 40 frequency (c/d) F I G U R E C.7. Fourier Ampl i tude Spectrum for o 1 E r i ( 2 n d night), a) The variation of the residuals at a given position in the line profile were mea-sured and transformed into the Fourier domain. The series of plots illustrates the amplitude spectrum at 10 k m / s intervals. 130 F I G U R E C .7. Fourier Ampl i tude Spectrum for o1 E r i ( 2 n d night). b) The mean amplitude spectrum (solid line) from those i l lustrated i n Figure (a). The dashed line is the window function for the period, At. frequency (c/d) F I G U R E C.8. Fourier Amplitude Spectrum for o1 Eri (both nights), a) The variation of the residuals at a given position in the line profile were mea-sured and transformed into the Fourier domain. The series of plots illustrates the amplitude spectrum at 10 km/s intervals. 132 i — i — i — i — i — r — i — i — i — | — i — i — i — r — | — i — i — i — i | i i i r ' I i I i I l l i i I i i i L_ J i i i i I i i i I 0 10 20 30 40 50 f r e q u e n c y ( c / d ) F I G U R E C.8. Fourier Ampli tude Spectrum for o 1 E r i (both nights), b) The mean amplitude spectrum from those illustrated in Figure (a). .006 c) d) frequency ( c /d ) .003 .0025 i i i r - i—i—i—i—|—r—i—i—i—|—i—i i i | i i i r-q -.0005 t — I — I — L l l I I I I I I I 1 I I i i I i i 1 L 10 20 30 40 50 frequency (c /d ) F I G U R E C .8. Fourier Amplitude Spectrum for o1 Eri (both nights). c) The window function at the period At. d) The difference between the amplitude spectrum and the window function. 134 150 i i i I i i i I i r i — i — i — i— i— i — i — i — i — i — i — i i i i r " i — i — r 100 50 •rH O o I — I CD > 0 -50 — X X x X X -100 X X X •150 i i i 1 i i i I i i i I i i i 1 i i i I i i i I i i i 1 i i i i .7 .72 .74 .76 .78 .8 .82 Julian Date .84 .86 .88 F I G U R E C.9. Line-Profile Variations of o 1 Eri (1 s t night). The period an mode are estimated by fitting a sine curve to the subfeature positions in the time series of residuals. 150 i l i i i i l i i i i I I I I I I I I I I I I I I I I I i i i i i I I I I I ~i r 100 50 o o CD > x x X X -50 - 1 0 0 x x •150 I i i i i I i i i i 1 i i i i I i i i i I i i i i I i i i .7 .71 .72 .73 .74 .75 Julian Date I i i i i 1 ' i i i I i i i i .76 .77 .78 .79 F I G U R E C.10. Line-Profile Variations of o1 Eri (2 n d night). The period an mode are estimated by fitting a sine curve to the subfeature positions in the time series of residuals. F I G U R E G i l . Radial-Velocity Variations of o1 Eri (Is* night). The velocity shift for the time series of spectra measured with respect to the mean spectrum. 1 P I = 0.113 + / - 0-006 i i I i i i i I i i i i I i i i i I i i i i I i i i i I i i i i 1 i i i i I i i i I I I l .7 .71 .72 .73 .74 .75 .76 .77 .78 Julian Date F I G U R E C.12. Radial-Velocity Variations of o 1 Er i (2nd night). The velocity shift for the time series of spectra measured with respect to the mean spectrum. 

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