. [2.3] The degree of the pulsation mode is defined as the index, \u00a3, of the associated Legendre polynomial and m = \u2014\u00a3, \u2014 \u00a3 + 1, ...\u00a3. 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\u2122(cos 9) cos(m<\/> + at) [2.4] xe - kAr\u2014PP(cos 9) cos(m(j) + at) [2.5] Ou , \u201e mPf m(cos6) . . , . xs = -kAr \u2014 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 \u2014 P\u2122(cos 9) s in (m^ + at) [2.8] Ou , A mP\" l(cos8) . , VJ, = -akAr cos(m 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\/\u00a3l >> 1), corresponding to slow rotators or high order p\u2014modes, then the Coriolis force can be adequately included wi th a single correction term such that 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\u2014modes prevail. The more slowly rotating 8 Scuti stars, which appear to pulsate wi th p\u2014modes, 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\u00b0 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-\u00a3(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 \u00a3 = 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 (.\u00a3 < 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 \u00a3 = 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 \u00a3. 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 \u00b1 ) ) [3.2] where 6\u00b1 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 = \u2014 \u2014 (sin 8 sin i cos d> + cos 8 cos 1) \u2014 r \u2014(cos 9 sin i cos 6 \u2014 sin 8 cos i) dtK ' dV 1 20 +r s in#[\u2014] i n ( s inz sin<\/>) \u2022 [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 \u00a3 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: \u00a3 = \\m\\ = 2, \u00a3 = \\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 '\u00a9I 6 W ' So \u00ab \u00a9 \u00b0 o o o o o \u00b0\u00ae \u00a9 O o ffiffl ? I \u00a9 , \u00a9Oft, i - 10 \u2022 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: \u00a3 = |m| = 2, b) High-degree: \u00a3 = \\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\u2014rr - i \u2014 i \u2014 1 _ 4495 4500 Waue l ength -[ 1 1 1 r -i\u2014r'l i \u2022 1111 I \u2022 ' \u2022 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 \u00a3 and m. For each of these figures a l l the other parameters were held constant: vsini \u2014 100 km\/s , i = 90\u00b0 , 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 \u2022 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 (\u00a3 \u2014 \u2014m). 29 N R P P r o f i l e s : 1 = 1 2 \u2122i 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 ' \u2022 i \u2022 \u00bb 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 (\u00a3 \u2014 \u2014m). The second diagram demonstrates the effect of m on the profiles for a fixed value of \u00a3. The thi rd diagram shows the effect of \u00a3 for a fixed value of m. Notice i n Figure 3.3 c) that the tesseral mode {\u00a3 = |m | + 1) produces twice as many bumps, but wi th lower amplitude. These arise from contributions above and below the equator 180\u00b0 out of phase. A s \u00a3 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 \u00a3 > 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 --\u2014 -1. J l_ Ua mp =0 . 0-U a m p - 1 . 0 Ua mp = 2 . \u2022 Ua mp =3 . fj-Ua mp = 4 . 0 Ua mp = 5 . \u2022 \u2022 ' ' 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 = \u2022 . \u2022 k - 0 . 2 k =0 . 4 k = 0 . 6 k = \u2022 . 8 k = 1 . \u2022 _ J i i i _ _ i i i i _ - i i \u2022 \u2022 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 \u2022 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 = \u2014m = 14 and k = 0. Other relevant parameters are: u s i n i = 115 km\/s , i \u2014 90\u00b0. 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 '> Q C . \u2014 I 0 CO C| .008 CD .+_> .006 a \u2022 i-H .004 3 .002 0 \"J o i i 1 i 11 1 \u2022 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=\u201410 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 _ \u2014 ! ! I i i ! i 1 1 1 \u2014 ; 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\u20141 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 \u2014 i i \u2022 . 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=\u201440 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=\u201450 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\u2014ri ; v=+50 k m \/ s ; .008 \u2022 i\u2014l ^\u2022006 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 \u00b10 .001 v U M a 0.089 \u00b10 .002 0.0332 \u00b1 0.0009 21 Mon 0.077 \u00b10 .006 o1 Er i (1 s t ) 0.094 \u00b10 .004 o 1 Er i (2 n d ) 0.11 \u00b1 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\u2014I -50 \u2014 \u2022100 \u2014 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 \u00b1 0.04 0.041 \u00b1 0.006 13 \u00b1 2 v U M a 0.63 \u00b1 0 . 0 5 0.033 \u00b1 0.004 19 \u00b1 3 0.30 \u00b1 0.06 0.090 \u00b1 0 . 0 0 6 3 \u00b1 1 21 Mon 0.79 \u00b1 0.06 0.074 \u00b1 0.018 11 \u00b1 3 o1 Eri \u2014 \u2014 \u2014 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(\u00aboi + 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 \u00b10 .002 A i = 0.83 \u00b1 0.07 v U M a Pi = 0.090 \u00b1 0.001 A i = 2.22 \u00b1 0.06 P2 = 0.198 \u00b1 0.007 A 2 = 1.88 \u00b1 0 . 0 5 21 Mon Pi = 0.10 \u00b1 0 . 0 1 A, = 2.2 \u00b1 0 . 6 P2 = 0.06 \u00b1 0 . 0 1 A2 = 1.0 \u00b1 0 . 6 o 1 Er i Pi = 0.112 \u00b10 .004 A i = 1.07 \u00b1 0 . 0 8 ( 1 s t night) P2 = 0.065 \u00b1 0.002 A 2 = 0.94 \u00b1 0 . 0 7 o 1 E r i Pi = 0.113 \u00b10 .006 A i = 0.69 \u00b1 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\u00b0. The limb darkening is chosen to be (3 \u2014 0.6. The waves are assumed to be prograde and sectorial (\u00a3 = \u2014m). 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 \u2022 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 \u2022 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 . \u2022 , , i i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 ' \u2014 i \u2014 i \u2014 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) \u2022 16 day beat period 0.0681 0.039 \u2022 alias ? 0.0668 0.028 Elliot (1974) \u2022 beats or changing period - Moreno (1980) 0.0762 0.02 Ibanoglu et.al. (1983) \u2022 small colour variations v UMa: Period Amplitude Reference Notes (days) (mag.) 0.132 0.05 Danziger and Dickens (1967) \u2022 beats 21 Mon: Period Amplitude Reference Notes (days) (mag.) 0.11 0.035 Eggen (1968) \u2022 variable amplitude 0.0999 0.026 Gupta (1973) \u2022 0.3 day beat period 0.0750 0.008 ?! 0.0895 0.019 Stobie et.al. (1977) \u2022 variable ampl.\/freq. 0.1599 0.013 V \u2022 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) \u2022 very uncertain 0.100 0.01 55 0.0815 0.05 Jorgensen et.al. (1975) \u2022 non-unique 0.1291 - 55 0.0720 0.007 Poretti (1989) \u2022 nrp suspected 0.144 0.006 55 \u2022 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 \u00a3 = 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 \u00a3 = 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\u00b0 was again assumed. Table XII lists the light amplitude corresponding to the geometrical variations for cases with \u00a3 = 0 and 2 for all the stars. No net changes in stellar disk area result from the \u00a3 = 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) \u00a3 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) \u00a3 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) \u00a3 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) \u00a3 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, 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 \u00a3 \u2014 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 \u2014 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? Continued investigations of rapidly rotating stars in the 8 Scuti strip promises to greatly improve our understanding of the oscillations of nrp in general. 96 REFERENCES Baade, D. 1984, Asiron. Astrophys., 135, 101. Balona, L . A . 1981, M.N.R.A.S., 196, 159. Balona, L . A . 1986a, M.N.R.A.S., 219, 111. Balona, L . A . 1986b, M.N.R.A.S., 220, 647. Balona, L . A . 1987, M.N.R.A.S., 224, 41. Balona, L . A . and Engelbrecht, C A . 1985, M.N.R.A.S., 214, 559. Balona, L . A . and Stobie, R.S. 1979, M.N.R.A.S., 187, 217. Breger, M . 1979, Pub. A.S.P., 91, 5. Breger, M . 1990, Delta Scuti Star Newsletter, 2, 13. Christy, R.F. 1967, A. J., 72, 293. Cowling, T . G . 1941, M.N.R.A.S., 101, 367. Cox, J.P. 1976, Ann. Rev. Astron. Astrophys., 14, 247. Danziger, I.J. and Dickens, R.J. 1967, Ap. J., 149, 743. Desikachary, K. , Parthasarathy, M . and Kameswara Rao, N. 1971, P.A.S.P., 83, 832. Dziembowski, W. 1977, Acta Astronomica, 27, 203. 97 Eggen, O.J. 1968 I.B.V.S., 250. Elliot, J .E . 1974, A.J., 79, 1082. Fahlman,G.G. and Glaspy, J.W. 1973, Astronomical Observations with Television-Type Sensors, J .W. Glaspy and G . A . H . Walker eds. Gies, D.R. and Kullavanijaya, A. 1988, Ap. J . , 326, 813. The Global Oscillation Network Group 1985, Report, National Solar Observatory, National Optical Astronomy Observatories. Griffin, R. and Griffin, R. 1979, A Photometric Atlas of the Spectrum of Procyon, (Yorkshire: The Scolar Press). Gupta, S.K. 1973, Obs., 93, 192. Hubeny, I. 1988, Computer Physics Communications, 52, 103. Ibanoglu C. , Ertan, A . V . , Tunca, Z., Turner, O., and Evren, S. 1983, Rev. Mexi-cana Astron. Astof., 5, 261. Jorgensen, H.E . , Johansen, K . T . , and Olsen, E . H . 1971, Acta Astron., 12, 223. Jorgensen, H.E. and Norgaard-Nielsen, H.U. 1975, Astron. Astrophys, 19, 235. Kambe, E . and Osaki, Y . 1988, Pub. Astr. Soc. Japan, 40, 313. Kubiak, M . 1978, Acta Astron., 28, 153. Ledoux, P. 1951, Ap. J., 114, 373. 98 Lee, U. and Saio, H. 1986, M.N.R.A.S., 221, 365. Lee, U. and Saio, H. 1987a, M.N.R.A.S., 224, 513. Lee, U. and Saio, H. 1987b, M.N.R.A.S., 225, 643. Lee, U. and Saio, H. 1989, M.N.R.A.S., 237, 875. Lee, U. and Saio, H. 1990, (preprint). Matthews, J . M . and Wehlau, W . H . 1985, Pub. A.S.P., 97, 841. Millis, R .L. 1966, I.B.V.S., 137. Moreno, M . A . 1980, Rev. Mexicana Astron. Astro}., 5, 19. Osaki,Y. 1971, Pub. Astr. Soc. Japan, 23, 485. Osaki, Y . 1974, Ap. J., 189, 469. Pekeris, C L . 1983, Ap. J., 88, 189. Poretti, E . 1989, Astr. and Astrophys., 220, 144. Pritchet, C.J . , Mochnacki, S. and Yang S. 1982, Pub. A.S.P., 94, 733. Smeyers, P. 1984, Theoretical Problems in Stellar Stability and Oscillations, ed. A Noels and M . Gabriel (Liege: Institut d'Astrophysique), p68. Smith, M.A. 1977, Ap. J., 215, 574. Smith, M.A. 1985, Ap. J., 297, 206. 99 Stamford, P.A. and Watson, R.D. 1976, Pub. Astr. Soc. Australia, 3, 75. 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 \u00ab0 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\u2014i\u2014i\u2014r 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\u2014i\u2014i i\u2014 i\u2014 i\u2014 i\u2014i 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 \u2014 3.84. b) Synthetic spectum with vsins' = 110. 105 .008 .008 .004 .008 0 .008 .008 .004 -\u00ab-> .\u2014i 0 CO \u00a3 -008 CD _l_> .008 .!=! .004 CD 0 0 8 o .+J .008 .1\u2014 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 \u2022 ' \u2022 \u2022 i i 1 T^I i i 1 i n i 1 I P T \u2014 \\ Z~l~i\u20141 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\u2014i 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 \u2022 i \u2022 \u2022 i v i I i * * t \u2014 r l i i i i 1 i i\u2014i\u2014 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 -\u2014: ! i i \u2022\u2022 i ! i I i \u2014 ; v = - 2 0 k m \/ s ; i i i i 1 i i i r l i i V i n i i i \u2014 i \u2014 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 \u2014 n \u2014 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 \u2014 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=\u201440 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 \u2022 N - T i ' . 1 I I I | \u2022 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 \u2022100 \"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\u2014r ~ i\u2014i\u2014 r\u2014i\u2014i\u2014i\u2014i\u2014|\u2014r 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-> \u2022 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 -+-> \u2022f -H 0 \u2022 r - 1 '.005 T) o 3 cd .01 .005 .01 .005 .01 .005 T \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i i i \u2014 i \u2014 i i i | i \u2014 i \u2014 i i | i i i \u2014 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 \u2014 I I I I \u2014 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-> \u2022 i\u2014I 0 V) \u2022 I\u2014( .002 73 o S0 0 8 6 CTj .002 0 .006 .004 .002 0 .008 .004 .002 7 1 \u2014 i \u2014 i \u2014 i \u2014 \u2014 i | i \u2014 i i i j \u2014 I \u2014 I \u2014 I i | i I I\u2014rz 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\u2014i\u20141 1 | . i \u2014 i \u2014 i \u2014 | \u2014 i \u2014 i \u2014 i \u2014 | \u2014 | \u2014 I \u2014 i \u2014 I \u2014 I \u2014 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 \u2014 r \u2014 i \u2014 i \u2014 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 -\u2014 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 \u2022 | 1 1 1 1 | 1 1 1 1 | 1 1 1 1 1 1 1 1 1. v=\u201450 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 \u2014 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 \u00a3 008 CD _t_> .006 .r-H .004 C D 0 0 8 T3 o .+_\u00bb .008 ^ . 0 0 6 \u00a3 0 0 4 ^ . 0 0 2 0 .008 .006 .004 .002 0 .008 .006 .004 .002 0 n\u2014i\u2014r\u2014i |\u2014i i i i | i\u2014i i i\u2014| i i\u2014i i | i i i\u2014rz : 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 _ \u2022 i \u2022 i i \u2022 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 \u2022 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. \u2014 i >. X ~ i V ''111 \u2014 ; \/ \\ 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\u2014f\" . 1 1 1 1 | 1 1 1 1 | - 1 1 1 11 \u20141\u20141\u2014T 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 \u2022 i _ ; v = \u00bb + 3 0 k m \/ s ; ' \\ \\ i i * i i i 1 i i i \u2022 ' 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=\u201450 k m \/ s ; r i i i i 1 i i i i 1 i i i V 1 \u2022 i i i 1 i i i\u2014r-= 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 \u2014 i \u2014 i \u2014 i \u2014 i \u2014 r \u2014 i \u2014 i \u2014 i \u2014 | \u2014 i \u2014 i \u2014 i \u2014 r \u2014 | \u2014 i \u2014 i \u2014 i \u2014 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\u2014i\u2014i\u2014i\u2014|\u2014r\u2014i\u2014i\u2014i\u2014|\u2014i\u2014i i i | i i i r-q -.0005 t \u2014 I \u2014 I \u2014 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 \u2014 i \u2014 i \u2014 i\u2014 i\u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i \u2014 i i i i r \" i \u2014 i \u2014 r 100 50 \u2022rH O o I \u2014 I CD > 0 -50 \u2014 X X x X X -100 X X X \u2022150 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 \u2022150 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. ","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0098314","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Astronomy","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https:\/\/open.library.ubc.ca\/terms_of_use.","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Nonradial pulsations of rapidly rotating [delta] Scuti stars","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/29140","type":"literal","lang":"en"}]}}