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Asteroseismic tuning of the magnetic star HR 1217 : understanding magnetism and stellar structure through.. 2010
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Title | Asteroseismic tuning of the magnetic star HR 1217 : understanding magnetism and stellar structure through MOST spacebased photometry |
Creator |
Cameron, Christopher J. |
Publisher | University of British Columbia |
Date Created | 2010-08-23 |
Date Issued | 2010-08-23 |
Date | 2010 |
Description | The chemically peculiar A (Ap) stars show extreme examples of astrophysical processes that have only recently been studied in detail in one other star—the Sun. These stars exhibit spectral anomalies caused by diffusion of some ionic species in a stellar atmosphere threaded by a strong (~ kG), organized magnetic field. A subset of the Ap stars rapidly oscillate (roAp) with periods ranging from 5 to 25 minutes. One of these roAp stars, HR 1217, is well studied with data from two global (ground-based) photometric campaigns that led to asteroseismic evidence of magnetically perturbed oscillation modes. This was the motivation to make HR 1217 a MOST space mission target. Our analysis of the almost 30 days of near-continuous MOST photometry on HR 1217 reveals a number of new periodicities that show spacings of ~ 15, 2.5, and 1.5 μHz. These new frequencies can be interpreted as magnetically perturbed oscillations and potentially second order spacings that could constrain the age and the magnetic interior of the star for the first time. These data are collected with a 95% duty cycle and reach a precision of 6 μmag, making this by far the best photometric data set on HR 1217. In addition, we present a grid of almost 52,000 stellar pulsation models including a large range of magnetic dipole field strengths (1-10 kG). This is the largest grid of stellar pulsation models of any Ap star to date and is critical to the interpretation of the MOST photometry. Our models can match the MOST observations to a fractional accuracy of about 0.05% with a mean deviation between theory and observation of a few μHz. A unique model match to the MOST observations could not be found. The results highlight the sensitivity to physics that has not been usually incorporated in Ap interior models, and the complex nature of the interaction of globally organized magnetic fields with stellar pulsation eigenmodes. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Electronic Theses and Dissertations (ETDs) 2008+ |
Date Available | 2010-08-23 |
DOI | 10.14288/1.0071166 |
Degree |
Doctor of Philosophy - PhD |
Program |
Astronomy |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2010-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/27647 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/24/items/1.0071166/source |
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Asteroseismic Tuning of the Magnetic Star HR 1217 Understanding magnetism and stellar structure through MOST spacebased photometry by Christopher J. Cameron B. Sc., Saint Mary’s University, 2001 M. Sc., University of British Columbia, 2004 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Astronomy) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August, 2010 c© Christopher J. Cameron 2010 Abstract The chemically peculiar A (Ap) stars show extreme examples of astrophys- ical processes that have only recently been studied in detail in one other star — the Sun. These stars exhibit spectral anomalies caused by diffusion of some ionic species in a stellar atmosphere threaded by a strong (∼ kG), organized magnetic field. A subset of the Ap stars rapidly oscillate (roAp) with periods ranging from 5 to 25 minutes. One of these roAp stars, HR 1217, is well studied with data from two global (ground-based) photometric campaigns that led to asteroseismic evidence of magnetically perturbed os- cillation modes. This was the motivation to make HR 1217 a MOST space mission target. Our analysis of the almost 30 days of near-continuous MOST photometry on HR 1217 reveals a number of new periodicities that show spacings of ∼ 15, 2.5, and 1.5 µHz. These new frequencies can be interpreted as magnetically perturbed oscillations and potentially second order spacings that could con- strain the age and the magnetic interior of the star for the first time. These data are collected with a 95% duty cycle and reach a precision of 6 µmag, making this by far the best photometric data set on HR 1217. In addition, we present a grid of almost 52,000 stellar pulsation models including a large range of magnetic dipole field strengths (1–10 kG). This ii Abstract is the largest grid of stellar pulsation models of any Ap star to date and is critical to the interpretation of the MOST photometry. Our models can match the MOST observations to a fractional accuracy of about 0.05% with a mean deviation between theory and observation of a few µHz. A unique model match to the MOST observations could not be found. The results highlight the sensitivity to physics that has not been usually incorporated in Ap interior models, and the complex nature of the interaction of globally organized magnetic fields with stellar pulsation eigenmodes. iii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi Statement of Collaboration . . . . . . . . . . . . . . . . . . . . . . xvii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The sounding of stellar interiors . . . . . . . . . . . . . . . . 2 1.1.1 Nonradial stellar oscillations . . . . . . . . . . . . . . 4 1.2 The chemically peculiar A stars . . . . . . . . . . . . . . . . . 16 1.2.1 Magnetic fields, diffusion, & Ap stars . . . . . . . . . 17 1.2.2 Stellar seismology & Ap stars . . . . . . . . . . . . . . 21 1.3 The roAp star HR 1217 . . . . . . . . . . . . . . . . . . . . . 29 1.4 Thesis goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 iv Table of Contents 2 The MOST mission: Data reduction techniques . . . . . . . 38 2.1 An overview of the MOST mission . . . . . . . . . . . . . . . 38 2.2 Photometry & time-series analysis . . . . . . . . . . . . . . . 40 2.3 Applications of CAPER to other MOST science . . . . . . . 46 2.3.1 The roAp star HD 134214 . . . . . . . . . . . . . . . . 47 2.3.2 The B supergiant HD 163899 . . . . . . . . . . . . . . 48 2.3.3 The SPBe stars HD 127756 & HD 217543 . . . . . . . 49 3 MOST & HR 1217 . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.1 The light curve reduction . . . . . . . . . . . . . . . . . . . . 62 3.2 On the rotation period of HR 1217 . . . . . . . . . . . . . . . 66 3.3 The frequency analysis . . . . . . . . . . . . . . . . . . . . . . 75 4 Modelling stellar oscillations . . . . . . . . . . . . . . . . . . . 98 4.1 Stellar evolution models . . . . . . . . . . . . . . . . . . . . . 98 4.1.1 Defining the parameter space . . . . . . . . . . . . . . 100 4.2 Pulsation models . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.2.1 A new search method for eigenmodes . . . . . . . . . . 113 4.3 The magnetic effects . . . . . . . . . . . . . . . . . . . . . . . 119 5 Best matched models . . . . . . . . . . . . . . . . . . . . . . . . 138 5.1 The matching procedure . . . . . . . . . . . . . . . . . . . . . 138 5.1.1 Seismic observations & models of the Sun . . . . . . . 140 5.1.2 Seismic observations & models of HR 1217 . . . . . . 144 5.2 Matching the true modes of HR 1217 . . . . . . . . . . . . . . 171 5.2.1 Matches to a sub-set of the MOST frequencies . . . . 187 5.3 On discriminating between models . . . . . . . . . . . . . . . 201 v Table of Contents 6 Summary, conclusions, & future work . . . . . . . . . . . . . 215 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Appendices A HD 127756 and HD 217543 parameter tables . . . . . . . . 245 B Frequency lists in cycles day−1 . . . . . . . . . . . . . . . . . . 253 C Bootstrap distributions . . . . . . . . . . . . . . . . . . . . . . 263 D Magnetic perturbation (Saio) plots . . . . . . . . . . . . . . . 279 vi List of Tables 3.1 Frequency model parameters identified using the MOST data. 80 3.2 The frequency separations δ† identified in the MOST data. . . 86 3.3 Unresolved frequencies identified in the MOST data. . . . . . 93 4.1 Some properties of selected models. . . . . . . . . . . . . . . . 107 5.1 Properties of the most probable models of HR 1217 — M series.157 5.2 Properties of model MM1. . . . . . . . . . . . . . . . . . . . . 164 5.3 Selected frequencies from the MOST data — A. . . . . . . . . 173 5.4 Properties of the most probable models of HR 1217 — U series.179 5.5 Properties of models UU1 and UU2. . . . . . . . . . . . . . . . 185 5.6 Properties of models GS and MS. . . . . . . . . . . . . . . . . 190 5.7 Selected frequencies from the MOST data — B . . . . . . . . 194 5.8 Properties of model PGS. . . . . . . . . . . . . . . . . . . . . 198 5.9 Properties of model LPC. . . . . . . . . . . . . . . . . . . . . 207 A.1 Frequencies identified in the star HD 127756 . . . . . . . . . . 246 A.2 Frequencies identified in the star HD 217543 . . . . . . . . . . 249 B.1 Table 3.1 with frequencies in units of cycles day−1 . . . . . . . 254 B.2 Table 3.2 with frequencies in units of cycles day−1 . . . . . . . 256 vii List of Tables B.3 Table 3.3 with frequencies in units of cycles day−1 . . . . . . . 258 B.4 Table 5.3 with frequencies in units of cycles day−1 . . . . . . . 262 viii List of Figures 1.1 A schematic representation of surface spherical harmonics. . . 8 1.2 Eigenfunctions of a low-order g-mode and a high-order p-mode from a solar model. . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 A schematic showing frequency spacings. . . . . . . . . . . . . 14 1.4 Schematic diagram of the oblique rotator model. . . . . . . . . 19 1.5 A theoretical HR diagram showing the approximate location of roAp stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.6 Variation in the pulsation amplitude and the magnetic field strength of HR 1217. . . . . . . . . . . . . . . . . . . . . . . . 25 1.7 Schematic amplitude spectra of HR 1217 from past data. . . . 32 2.1 MOST light curve of HD 127756 . . . . . . . . . . . . . . . . 53 2.2 The Fourier amplitude spectrum of the HD 127756 light curve 54 2.3 Zoomed region around the largest peak in the DFT of the HD 127756 light curve. . . . . . . . . . . . . . . . . . . . . . . . . 55 2.4 A comparison of bootstrap distributions — SPBe star HD 127756 56 2.5 MOST light curve of HD 217543 . . . . . . . . . . . . . . . . 58 2.6 Fourier amplitude spectrum of the HD 217543 light curve. . . 59 2.7 A comparison of bootstrap distributions — SPBe star HD 217543 60 ix List of Figures 3.1 Photometric reduction steps using the MOST data. . . . . . . 65 3.2 The reduced HR 1217 light curve. . . . . . . . . . . . . . . . . 67 3.3 Likelihood ratio for the rotation period of HR 1217. . . . . . . 71 3.4 Bootstrap distributions of the rotation frequency of HR 1217. 74 3.5 The Fourier amplitude spectrum of the HR 1217 light curve. . 79 3.6 Normalized pulsation frequency amplitudes as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 3.7 Time-frequency plot of the HR 1217 data. . . . . . . . . . . . 90 3.8 Comparison between resolved and unresolved frequencies. . . . 92 4.1 HR diagram showing the extremes of the model parameters. . 105 4.2 Histograms of model properties. . . . . . . . . . . . . . . . . . 106 4.3 Small spacings for the models listed in Tab. 4.1. . . . . . . . . 114 4.4 Eigenfrequencies calculated using two different methods. . . . 120 4.5 The cyclic behaviour of the magnetic perturbations. . . . . . . 126 4.6 The magnetic perturbations for a 5 kG magnetic field. . . . . 127 4.7 The magnetic perturbations to ∆ν for B = 0 kG and 5 kG. . . 131 4.8 The magnetic perturbations to ∆ν for all B values. . . . . . . 132 4.9 The magnetic perturbations to δν for B = 5 kG. . . . . . . . . 135 4.10 The magnetic perturbations to δν for all B values. . . . . . . . 136 5.1 Standard Solar model (SSM) frequencies compared to the Birm- ingham Solar Oscillations Network (BiSON) observations. . . . 143 5.2 Probability plots with σ scaled by factors of 1 and 1000. . . . 149 5.3 Probability plots with σ scaled by a factor of 50. . . . . . . . . 151 5.4 Probability plots using the Laplace distribution. . . . . . . . . 153 x List of Figures 5.5 The median frequency deviation vs model probability . . . . . 156 5.6 νM1 - νMOST vs. νMOST . . . . . . . . . . . . . . . . . . . . . . 161 5.7 νM3 - νMOST and νMM1 - νMOST vs. νMOST . . . . . . . . . . . 165 5.8 Saio plots for models M1 and M2. . . . . . . . . . . . . . . . . 168 5.9 Saio plots for models M9 and MM1. . . . . . . . . . . . . . . . 169 5.10 Probability plots for models matched to νMOST (Tab. 5.3) — A.177 5.11 Probability plots for models matched to νMOST (Tab. 5.3) — B.178 5.12 νU1 - νMOST vs νMOST and a Saio plot for model U1. . . . . . 183 5.13 νUU1 - νMOST vs νMOST . . . . . . . . . . . . . . . . . . . . . . 186 5.14 Saio plots for models UU1 and UU2. . . . . . . . . . . . . . . 187 5.15 νGS - νMOST vs. νMOST . . . . . . . . . . . . . . . . . . . . . . 191 5.16 Saio plots for models GS and MS. . . . . . . . . . . . . . . . . 192 5.17 Probability plot for models matched to νMOST (Tab. 5.7). . . . 196 5.18 νPGS - νMOST vs. νMOST . . . . . . . . . . . . . . . . . . . . . . 199 5.19 νM3 - νMOST and νM6 - νMOST vs. νMOST . . . . . . . . . . . . 201 5.20 Minimum frequency deviations required to match 50 and 75% of the MOST observations. . . . . . . . . . . . . . . . . . . . . 204 5.21 Minimum frequency deviations required to match 100% of the MOST observations. . . . . . . . . . . . . . . . . . . . . . . . 205 5.22 νLPC - νMOST vs. νMOST . . . . . . . . . . . . . . . . . . . . . . 208 5.23 Histograms of matched frequencies — all B values and B = 0. 210 5.24 Histograms of matched frequencies — B = 2 kG and B = 4 kG.211 5.25 Histograms of matched frequencies — B = 6 kG and B = 8 kG.212 C.1 Bootstrap distributions for fit parameters 1 and 2. . . . . . . . 264 C.2 Bootstrap distributions for fit parameters 3 and 4. . . . . . . . 265 xi List of Figures C.3 Bootstrap distributions for fit parameters 5 and 6. . . . . . . . 266 C.4 Bootstrap distributions for fit parameters 7 and 8. . . . . . . . 267 C.5 Bootstrap distributions for fit parameters 9 and 10. . . . . . . 268 C.6 Bootstrap distributions for fit parameters 11 and 12. . . . . . 269 C.7 Bootstrap distributions for fit parameters 13 and 14. . . . . . 270 C.8 Bootstrap distributions for fit parameters 15 and 16. . . . . . 271 C.9 Bootstrap distributions for fit parameters 17 and 18. . . . . . 272 C.10 Bootstrap distributions for fit parameters 19 and 20. . . . . . 273 C.11 Bootstrap distributions for fit parameters 21 and 22. . . . . . 274 C.12 Bootstrap distributions for fit parameters 23 and 24. . . . . . 275 C.13 Bootstrap distributions for fit parameters 25 and 26. . . . . . 276 C.14 Bootstrap distributions for fit parameters 27 and 28. . . . . . 277 C.15 Bootstrap distributions for fit parameters 29. . . . . . . . . . . 278 D.1 Saio plots showing the best fit models M1 and M2. . . . . . . 280 D.2 Saio plots showing the best fit models M3 and M4. . . . . . . 281 D.3 Saio plots showing the best fit models M5 and M6. . . . . . . 282 D.4 Saio plots showing the best fit models M7 and M8. . . . . . . 283 D.5 Saio plots showing the best fit models M9 and M10. . . . . . . 284 D.6 Saio plots showing the best fit models M9 and MM1. . . . . . 285 D.7 Saio plots showing the best fit models U1 and U2. . . . . . . . 286 D.8 Saio plots showing the best fit models U3 and U4. . . . . . . . 287 D.9 Saio plots showing the best fit models U5 and U6. . . . . . . . 288 D.10 Saio plots showing the best fit models U7 and U8. . . . . . . . 289 D.11 Saio plots showing the best fit models U9 and U10. . . . . . . 290 D.12 Saio plots showing the best fit models UU1 and UU2. . . . . . 291 xii List of Figures D.13 Saio plots showing the best fit models GS and MS. . . . . . . 292 D.14 Saio plots showing the best fit models PGS and LPC. . . . . . 293 xiii Acknowledgements This thesis would not have been possible without the support of my PhD supervisor Jaymie Matthews. He has gone beyond the call of duty as a supervisor and a mentor to make sure that I was exposed to cutting edge research, and that my work was made known to other researchers in our field. I learned a lot from him. There are times in a person’s life when they find out who their real friends are — Jaymie Matthews is a real friend. My sincerest thanks. Thank you to the MOST science team for giving me the opportunity to work with such high quality data. I can only hope that you will be as pleased with the results as I am. Margarida Cunha, thank you for spending a significant amount of your time to teach me about your research on the roAp stars. I have benefited greatly from scientific discussions with Hideyuki Saio. He has read many versions of this thesis and has encouraged my research at every phase. Rainer Kuschnig wasn’t know as the ‘fun boy’ of the MOST lab for noth- ing. He is a great friend that has guided my research and patiently listened to my many rants over the years. To my other cohort in the MOST lab, Jason Rowe: Our science discussions have spurred many great ideas. You are one of the few people that can interpret my ramblings as I try to work xiv Acknowledgements out the details of my new projects. Your friendship and honest advice are always welcome and appreciated. I was fortunate to meet many interesting people while studying at UBC. The science discussions with other graduate students were an important part of my growth as a researcher; but, I also managed to have some fun times outside of the academic environment . . . I would especially like to thank Kelsey Hoffman, Mark Huber, Jason Kalirai, Thomas Pfrommer, and Jason Rowe for all of those fun times. My parents, Joey and Marie Cameron, and my sister, Cheryl Cameron, have continually given me support and have encouraged me in everything I have ever done. Without them I would not have been able to finish this project. I hope they are as proud of me as I am of them. Jennifer McGillivray, my wife, has been by my side since before my grad- uate school days. She has moved with me from one side of Canada to the other, to Europe, and back again. Her devotion and patience mean the world to me. There would be no thesis without your loving motivation. Thank you for taking this journey with me. xv Dedication For my son, Owen. xvi Statement of Collaboration The following is intended to notify the reader of publications in which work from this thesis appears. It also serves to identify people who have collabo- rated on various aspects of this work. In chapter 2 (§ 2.2 and 2.3) the time-series analysis techniques and model interpretations were published in the following works: Walker et al. (2005), Saio et al. (2006), Cameron et al. (2006), King et al. (2006), Saio et al. (2007) and Cameron et al. (2008). The coauthors of those publications (see the bibliography) should be credited as collaborators on the work presented in chapter 2. My contribution to the above publications was primarily the frequency analysis, light curve analysis, some model interpretation and writ- ing of the manuscripts. Data and models from this thesis have been presented and discussed at a number of conferences since 2004. Dr. D. B. Guenther pro- vided the stellar pulsation and evolution software packages (JIG and YREC) used to calculate models in chapters 4 and 5. Dr. M. S. Cunha provided the software used to calculate the magnetic perturbations to the oscillation frequencies in chapters 4 and 5. I performed the model calculations and in- terpreted the results presented in those chapters. Dr. J. F. Rowe should be credited with the preliminary photometric reduction of the HR 1217 data. This thesis has benefited from the revisions suggested by my PhD ex- xvii Statement of Collaboration amining committee members: Drs. Hiromoto Shibahashi, Michael Bostock, and Jeremy Heyl. My supervisory committee, Drs. Jaymie Matthews, Neil Balmforth, Brett Gladman, and Scott Oser have helped guide and refine the final format and content of this thesis. xviii Chapter 1 Introduction This thesis covers both observational (chapters 2 and 3) and theoretical (chapters 4 and 5) studies of the seismology of the pulsating, magnetic, chemically-peculiar, A-type star HR 1217. An attempt at modelling and interpreting the many physical processes that must be explored to fully de- scribe the observed properties of HR 1217 is a daunting task. With this in mind, we compare the unique observations of the pulsations in HR 1217 obtained by the MOST satellite (chapter 2) to pulsation models that focus on the magnetic effects that influence the properties of the calculated, and observed, pulsation modes. There have been many comprehensive reviews on the subject of stellar seismology, and in the past few years there has been a growing number of reviews focusing on the variable, chemically peculiar A stars as well. These will be referenced extensively as necessary sources of background information on both the theory and measured properties that detail the rapidly oscillating Ap (roAp) star phenomenon. In this introductory chapter we briefly outline stellar seismology, the pecu- liar A stars, and the history of the observations of HR 1217 that are relevant to this work. Further emphasis on the details of data acquisition, reduction methods and on the modelling of stellar oscillations (in the context of HR 1 1.1. The sounding of stellar interiors 1217) are found in the later chapters of this thesis. 1.1 The sounding of stellar interiors Observations of stars are typically only skin deep. Spectroscopic measure- ments of star light give a glimpse of a razor-thin layer of the stellar surface from which astronomers infer the chemical constitution of the star, its aver- age surface temperature and some of its mean properties, e.g., surface gravity. Integrated light measurements (photometry) can give astronomers informa- tion about the radius of the star, assuming its distance is known and an estimate of its temperature is available. In the early 1960s Leighton et al. (1962) noticed there were spatially incoherent, wave-like disturbances on the surface of the Sun with periods near 5 minutes. This oscillation period should be compared to the longer dynamical timescale1 of approximately one hour that can be estimated for the Sun. Ulrich (1970) and Leibacher & Stein (1971) were the first to interpret these oscillations as sound waves produced in the solar interior that resonate in acoustic cavities defined by changes in the physical state, e.g., pressure and density, of the solar material with increasing depth. These sound waves, also known as p-modes because pressure is the restoring force, produce standing wave patterns as they propagate and provide the possibility of seismically 1The dynamical period of a self gravitating sphere is inversely proportional to the square-root of the mean density of the material ( √ R3/GM) and represents the time it would take for a particle on the surface of the sphere to free-fall to the centre. The expression for this ‘free-fall’ time contains the mass M and the radius R of the sphere and the physical dimensions of the problem are scaled by the gravitational constant G. 2 1.1. The sounding of stellar interiors measuring the variation of sound speed throughout the recesses of the Sun. Seismology of the Sun (helioseismology) was then possible and astronomers were at last able to lift the optically thick veil that hid the interior of a star. Since these early beginnings, millions of p-modes have been identified in the Sun and measurements of its internal rotation rate, the depth of its convective zone, constraints on element diffusion, and the run of sound speed have all been inferred. Recent reviews on helioseismology are provided by Christensen-Dalsgaard (2002) and Basu & Antia (2008). The stellar (non-solar) equivalent of helioseismology is asteroseismology. Many stars show both photometric and spectroscopic variations that are consistent with p-modes and longer period g-modes (buoyancy waves with gravity taking the role of the restoring force). Some examples include: δ Scuti stars (oscillation periods of P ∼ 0.5–2 h), RR Lyrae stars (P ∼ 1.5–24 h), β Cephei stars (P ∼ 3–6 h), variable white dwarfs (P ∼ 100–1600 s) and supergiants (P ranges from hours to days). Cox (1980) gives a detailed de- scription of primarily radial (spherically symmetric) stellar oscillations while the general, nonradial, case of oscillations with both horizontal and radial displacements is discussed in detail by Unno et al. (1989). An introduction to many types of variable stars and the theory of stellar oscillations can be found in Christensen-Dalsgaard (2003), and review articles by Gautschy & Saio (1995, 1996), Cunha et al. (2007), and Aerts et al. (2008) also provide general asteroseismic discussions and some recent results. Below I summarize the properties of stellar oscillations that are important for the interpretation of the rapidly oscillating Ap (roAp) stars (P ∼ 5–30 min) — A particular type of variable star that is described in more detail in § 1.2.2. One roAp 3 1.1. The sounding of stellar interiors star, HR 1217, is the focus of this research project. 1.1.1 Nonradial stellar oscillations The properties of stellar oscillations are introduced in this section along with the asymptotic relations that connect the properties of individual oscillation modes, frequencies, and spacings to the mean properties of a star. The notation we use follows closely to that of Unno et al. (1989). Both Unno et al. (1989) and Christensen-Dalsgaard (2003) should be consulted as detailed, recent references outlining the physics of stellar oscillations. More concise reviews are provided by Gautschy & Saio (1995), Shibahashi (2005), and Basu & Antia (2008). Stellar pulsations are described by the usual fluid equations. They are the continuity equation, the momentum equation (a.k.a. the equation of mo- tion or the Euler equation), an energy equation that details the flow and transfer of energy in the fluid, and Poisson’s equation relating the gravita- tional potential, Φ, of the fluid to its internal density, ρ. These equations are represented, in the order presented above, by ∂ρ ∂t = −∇ · (ρv) (1.1) ρ ( ∂ ∂t + v · ∇ ) v = −∇p− ρ∇Φ ∂p ∂t + v · ∇p = c2s ( ∂ρ ∂t + v · ∇ρ ) ∇2Φ = 4piGρ where the gas pressure and fluid velocity at a given time t are given by p and v, respectively. All variables are functions of depth within the model, i.e., 4 1.1. The sounding of stellar interiors radius r, or mass shell m(r). The energy equation (the third equation in the set of Eqs. 1.1) is given in the adiabatic approximation so that no heat gains or losses occur within the fluid over an oscillation cycle. In this approximation the adiabatic sound speed cs is related to the gas pressure and density by√ Γ1p/ρ, with the first adiabatic exponent defined as Γ1 = (∂ ln p/∂ ln ρ)S. The adiabatic approximation is commonly used to model stellar oscillations. In this thesis we focus on the magnetic perturbation to the acoustic oscillation spectrum of variable stars and the matching of those perturbed modes to high quality data (see chapters 4 and 5). Because we are not exploring the excitation mechanism of these magnetic oscillation modes, any non-adiabatic effects will, for simplicity, be ignored. A non-rotating, non-magnetic star has spherical symmetry. This is also approximately true for slow rotation and if a small magnetic field is present. Assuming that oscillations about this static, symmetric, reference state have small amplitudes compared to the radius of the star, we can perform a nor- mal mode analysis of Eqs. 1.1 by linearizing the variables, substituting them back into Eqs. 1.1, and keeping only first order terms (see, for example, Unno et al. 1989). We do this so we can describe a complex oscillation pattern on the surface of a star as the superposition of a number of small, independent, oscillations. Supposing the linearized perturbations vary with a time depen- dence proportional to exp(iσt), with (angular) oscillation frequency σ, and that the radial and angular components are separable, the pulsation variables (for example, pressure) can be expressed as f(r, θ, φ, t) = f0(r) + f ′(r)Y ml (θ, φ) exp(iσt) (1.2) The perturbation f ′(r) to the spherically symmetric background state f0(r) 5 1.1. The sounding of stellar interiors in Eq. 1.2 is evaluated at a fixed radius r. This is called an Eulerian pertur- bation. We can also define the Lagrangian perturbation as a perturbation calculated in a reference frame moving along with the fluid. If a fluid element moves a small distance ξ = δr from the reference position r0, the Lagrangian perturbation can be written as δf(r) = f(r0 + ξ)− f0(r0) (1.3) By expanding f(r0 + ξ) to first order about the background state, Eq. 1.3 becomes δf(r) = f(r0) + ξ · ∇f0 − f0(r0) (1.4) The difference f(r0)− f0(r0) is just the Eulerian perturbation (f ′) evaluated at the radius r0. Placing f ′(r0) into Eq. 1.4, we arrive at δf(r) = f ′(r0) + ξ · ∇f0 (1.5) This expression relates the Lagrangian and Eulerian perturbations. Each of the pulsation variables can be expressed in either of the Eulerian or La- grangian reference frames. The choice is usually one of mathematical or numerical convenience. The linearized version of Eqs. 1.1 (and a discussion on the boundary conditions needed to complete the solution set) are found in § 4.2 (see Eqs. 4.2). The angular coordinates in Eq. 1.2 are described by spherical harmonics Y ml (θ, φ). The index ` is commonly called the degree of the mode and divides the surface of the star into regions oscillating in opposite phase. If ` = 0, the oscillation mode is radial, or more precisely, spherically symmetric. Photometric observations of pulsating stars generally show the contributions 6 1.1. The sounding of stellar interiors from low-` modes because the contribution from modes with ` & 3 averages out when the light from the stellar disc is summed. The index m is the azimuthal order of the mode and represents the number of longitudinal nodal lines on the stellar surface. Because the spherical harmonics are proportional to exp(imφ), the phase velocity is( ∂φ ∂t ) phase = − σ m (1.6) The sign of m indicates the direction in which the wave travels. A standing wave pattern can be formed on the surface of the star by superimposing two waves travelling in opposite directions because the background state is, to first order, spherically symmetric. These indices are quantized with m taking integer values from −` to +`, where ` is a whole number. If m = 0 all nodal lines on a sphere are lines of latitude (zonal modes) and if m = ` they are all lines of longitude (sectoral modes). If m takes on values between these extremes, the modes are known as tesseral modes and of the ` nodal lines, ` − |m| are lines of latitude. Examples are drawn schematically in Fig. 1.1 for the cases ` = 3 and m = 0, 1, 2, and 3. Because the modes are degenerate with respect to m, an agent such as rotation is needed to define an axis that breaks the spherical symmetry and lifts the 2`+ 1 degeneracy. 7 1.1. The sounding of stellar interiors Figure 1.1: A schematic representation of different surface spherical harmonics. The up- per left sphere shows harmonics having an angular degree of ` = 3 and an azimuthal order of m = 0 while the upper right sphere illustrates ` = 3 and m = 1. The lower left and right portions of the diagram show ` = 3, m = 2 and ` = 3, m = 3, respectively. The (-) and (+) represent regions (separated by nodal lines) that oscil- late in opposite phase. Note there are ` − |m| lines of latitude on each sphere. (This image was taken from the wikipedia website http://en.wikipedia.org/wiki/File: Harmoniques_spheriques_positif_negatif.png and is available under the GNU Free Documentation License.) Along with ` and m, there is another index that describes the character of an individual component of the linear analysis outlined above. The equations describing the nonradial stellar oscillations can be reduced to a fourth order 8 1.1. The sounding of stellar interiors differential equation with the oscillation frequency (squared) as an eigenvalue of the system. The eigenvalues and eigenfunctions of the system are ordered into two groups (p and g-modes) that are indexed by n. Each frequency2 has associated with it a displacement vector (or eigenfunction) with components ξn,`,m = ( ξr(r)Y m l , ξh(r) ∂Y ml ∂θ , ξh(r) sin θ ∂Y ml ∂φ ) exp(iσt) (1.7) The index n is the radial order of the oscillations and corresponds to the number of radial nodes (zeros) in the eigenfunction ξn,`,m. In general, p- modes (or acoustic waves) have frequencies that increase with increasing n. Their frequencies are greater than the dynamical frequency √ GM/R3. The p-modes have their largest radial displacement near the surface of a star. The g-modes (or buoyancy waves), on the other hand, have periods that increase with decreasing n and are longer than the dynamical period √ R3/(GM). The amplitude of the radial eigenfunction is largest near the centre of a star for g-modes. This behaviour is illustrated in Fig. 1.2 for a p and g-mode calculated using a model of the Sun. The total displacement on the surface of a variable star can, in principle be described by summing Eq. 1.7 over all values of n, `, and m associated with the eigenfrequencies that most closely match the observed oscillation periods of that star. 2A modelled frequency, or eigenvalue, may also be referred to as an eigenfrequency, eigenmode, or simply a mode. 9 1.1. The sounding of stellar interiors Figure 1.2: Eigenfunctions (presented here as the radial displacement ξ normalized by radius r) of a low-order g-mode (red) and a high order p-mode (black) calculated using a model of the Sun. Each mode has an angular degree of ` = 1 and is axisymmetric (m = 0). The radial order (n) gives the number of zero crossings of the eigenfunctions. (Negative n values are used for g-modes by convention.) The g-mode has its largest amplitude near the centre of the model while the high-order p-mode has its largest amplitude near the surface. Using the terminology defined above we will now consider the properties 10 1.1. The sounding of stellar interiors of p-modes in more detail because they are important to the interpretation of the data and models presented in chapters 4 and 5. It can be shown (see Christensen-Dalsgaard 2003) that the radial wave number, kr, of a p-mode is approximately k2r = σ2 c2s − `(`+ 1) r2 (1.8) The second term in the above is identified as the horizontal wave number, kh. As a wave with an oscillation frequency σ travels inward from the surface of a star toward its centre, the local sound speed increases with increasing temperature. The horizontal wavenumber begins to dominate, causing the wave front to slowly bend until the wave refracts upward again. At this turning radius, rt, the radial wave number kr goes to zero and the wave has only a horizontal displacement. Put another way r2t = c2s(rt)`(`+ 1) σ2 (1.9) Thus, for a given frequency and angular degree, a wave will bend differently from star to star (or from model to model) depending on how the sound speed changes as a function of radius within the star (or model). This is the essence of stellar seismology — A method by which the interior structure of a star can be estimated by observing oscillations having different σ and `. When studying the properties of stellar oscillations it is useful to focus on subsets of the calculated oscillation spectra that have special properties. For example, modes with a high angular degree behave differently than modes with a low angular degree. Of particular importance to this work are high- order, low-degree modes. Modes are described as being high-order (or high- overtone) when n is large. In the case of p-modes, high-order modes are 11 1.1. The sounding of stellar interiors essentially radial and have vertical wave numbers of kr ≈ σ/cs near the stellar surface. These modes tend to have oscillation frequencies of a few mHz (a period of 5 min corresponds to a frequency of 3.3 mHz) and are consistent with oscillations observed in the Sun. The number of nodes a given eigenfunction has between the centre and surface of a star can be estimated by adding up the number of nodes per wavelength there are in that radius range (see Hansen & Kawaler 1994). That is, n ∝ ∫ R 0 2dr/λ ≈ ∫ R 0 krdr (1.10) This relation gives a hint that the frequency of a given mode is related to the number of nodes the eigenfunction has because kr ∝ σ. A careful asymptotic analysis of high-overtone, low-degree, p-modes by Tassoul (1980, 1990) gives νn,` ∼= ( n+ ` 2 + β ) ∆ν − [A(`+ 1/2)2 − ] ∆ν2 νn,` (1.11) with ∆ν = ( 2 ∫ R 0 dr cs(r) )−1 (1.12) and A = 1 4pi2∆ν ( κ− ∫ R 0 dcs dr d ln r ) (1.13) where we now make the transition between angular frequency σ and the observed oscillation frequencies ν = σ/2pi. In this asymptotic form (see also Christensen-Dalsgaard 2003) constants β, , and κ are small and depend on the surface structure of the star. Equation 1.11 is valid for adiabatic oscillations about a spherically symmetric star and for p-modes with n `. This asymptotic relation holds when n/` ∼ 10 or greater. 12 1.1. The sounding of stellar interiors Equation 1.11 tells us that frequencies of the same degree and with orders that differ by one are approximately equally spaced by ∆ν (= νn+1,` − νn,`). The quantity ∆ν is known as the large spacing and it will be used extensively in this thesis. If the modes have orders that differ from each other by one and ` values that alternate between even and odd values, Eq. 1.11 provides a frequency spacing of ∆ν/2. If, however, consecutive ` values are either all even or odd, the frequencies are spaced by about ∆ν. Another spacing that has diagnostic importance is the second-order, or small spacing, δν. Modes with ` values differing by two and n values differing by one (νn,` − νn−1,`+2) have nearly the same frequency. Both the large and small spacings are illustrated schematically in Fig. 1.3, where a picket fence pattern of frequencies similar to that observed, for example, in the Sun is shown. (For completeness, a solar model yields values of ∆ν ∼ 135 µHz and δν ∼ 10 µHz.) Frequencies generally increase in value as n and ` increase. Figure 1.3 presents both large and small spacings, shown between consecutive overtones, depending on the degree difference between adjacent modes. It is important to note that both the large and small spacings are functions of both frequency and degree and are not strictly constant over a large frequency range for a given star or model. 13 1.1. The sounding of stellar interiors ν ν(n, ") ν(n, "+ 1) ν(n, "+ 2) ν(n, "+ 3) ν(n+ 1, "− 1) ν(n+ 1, ") ν(n+ 1, "+ 1) ∆ν δν Figure 1.3: A schematic showing frequency spacings. Frequencies increase to the right in this diagram. Modes with common radial overtone n are shown as either solid or dashed vertical lines. Both the large and small spacings (∆ν = νn+1,` − νn,` and δν = νn,` − νn−1,`+2, respectively) are shown. In general, increasing either n or ` causes an increase in frequency and the larger the degree of the mode, the lower its observed amplitude. The spacing parameters can easily be isolated in a calculated oscillation spectra by noting the radial overtones and the degrees of the modes. More important is the physical interpretation of these parameters and what the spacing between observed oscillation periods can tell us about the star. Physically the large spacing ∆ν is the inverse of the time it takes sound to cross the diameter of the star. The sound speed can be approximated as√ p/ρ and is dimensionally the same as √ GM/R. Placing this into Eq. 1.12 shows the large spacing is on the same order as the inverse of the free-fall (or dynamical) timescale and is related to the mean density (M/R3) of the star. Gabriel et al. (1985) showed from a grid of stellar evolution models that the 14 1.1. The sounding of stellar interiors large spacing is approximately ∆ν = (0.205± 0.011) ( GM R3 )1/2 Hz (1.14) with G, M , and R given in SI units. Matthews et al. (1999) rewrote the large spacing in terms of a star’s effective temperature Teff and luminosity L to give ∆ν = (6.64± 0.36)× 10−16M1/2T 3effL−3/4 Hz (1.15) In this form L and M are in solar units and Teff in degrees Kelvin. If a regular frequency spacing is observed in an oscillating star it can be used to determine the star’s luminosity independent of other distance determination methods like trigonometric parallax. This method of asteroseismically determining a star’s luminosity assumes there is some constraint on the star’s mass and a measurement its effective temperature is available. The wavenumber (Eq. 1.8) of a high-overtone, low-degree mode has es- sentially no contributions from ` near the stellar surface. The potential to probe the deep stellar interior, while at the same time cancelling out the small contributions of ` near the stellar surface, motivates the isolation of the small spacing. Computing the frequency difference νn,` − νn−1,`+2 in Eq. 1.11 shows δν ∝ ∫ R 0 1 r dcs dr dr (1.16) Close to the centre of a star, the leading contribution to the small spacing comes from the 1/r factor in the integrand of Eq. 1.16. As a star evolves, its interior composition changes through nuclear burning (increasing the amount of He in the core) and the sound speed gradient, weighted by this 1/r term, is sensitive to this evolutionary change. As a star ages, the mean molecular 15 1.2. The chemically peculiar A stars weight in its core increases causing a decrease in both the local sound speed and in the small spacing. Therefore, the small spacing can be exploited as a diagnostic of stellar age (e.g., Provost 1984, Gough 1987, Christensen- Dalsgaard 1988, and Guenther & Demarque 1997). Observations of the large and small spacings allow the unique determina- tion of a star’s evolutionary status and position on the HR diagram. 1.2 The chemically peculiar A stars We next discuss a subset of A-type stars that have a number of distinguishing characteristics that strongly influence their pulsation spectra. The Ap stars are spectroscopically (chemically) peculiar A-type stars. Also known as CP2 stars (Preston, 1974), the class actually ranges from late B to early F spectral type with a temperature range of about 7, 000 K . Teff . 14, 000 K. The peculiarities in these stars are observed as spectral line strength anomalies that are significantly different from the majority of other stars and are interpreted as abundance enhancements or depletions on the stellar sur- face. The peculiar line strengths correspond to photospheric overabundances up to 105 times the corresponding solar value and under abundances down to 10−2 times the solar value. Near the main sequence there are a number of chemically peculiar stars, with the Ap stars making up less than about 10 percent of the bright stars on the main sequence (North et al., 2008). Each group is defined by a spe- cific temperature range, magnetic field strength, spectral line anomalies, and their pulsational properties. As an example, the hotter Ap stars, with tem- 16 1.2. The chemically peculiar A stars peratures ranging from about 10,000–14,000 K, can exhibit either Si or Hg and Mn peculiarities in their spectra. The Ap Si stars have a detectable magnetic field while the Ap HgMn stars generally do not. The cooler stars with Teff . 10, 000 K are classified as Ap SrCrEu stars and show very strong line strengths for the rare earth elements (La to Er) Sr, Cr, and Eu. Both hot and cool Ap stars seem to have weak He lines, if they are present at all. An overview of the Ap class of stars is provided by Wolff (1983) and Kurtz & Martinez (2000). 1.2.1 Magnetic fields, diffusion, & Ap stars The Ap stars show magnetic fields that appear to be global, predominately dipolar, and have strengths ranging from about 0.3 kG to 30 kG (Landstreet, 1992a). These magnetic fields seem to be connected to the chemical peculiar- ities and the variability of these stars, both of which will be discussed below. A general discussion of magnetic fields in stars is given by Mathys (1989), Landstreet (1992a; 1992b and 1993), and more recently by Wade (2006). In the presence of a magnetic field, spectral lines having a wavelength, λ, can show splitting caused by the Zeeman effect. For magnetic field strengths up to a few tens of kG, the separation between the split spec- tral line components is ∝ λ2B (Landstreet, 1992b). Competing effects like thermal and Doppler broadening of spectral lines means that only a small subset of stars with B & 10 kG and small projected (rotational) velocities v sin(i) . 10 km s−1 show significant broadening from the Zeeman effect (Landstreet, 1992b). Examples of this average magnetic field modulus deter- mination are given by Mathys et al. (1997). 17 1.2. The chemically peculiar A stars When Zeeman components are not resolved it is possible to exploit the polarization properties of the magnetically perturbed lines. If a spectral line is observed in both left and right-circularly polarized (LCP and RCP respectively) light, the contribution to the spectral line from each of the unresolved, split components is different. The difference in the position of the mean wavelength of the line in RCP and LCP light provides a measurement of the line-of-sight component of the magnetic field averaged over the stellar disc. This measurement is known as the mean longitudinal field strength and is represented by Bl. For most Ap stars, Bl, and its variation with rotation of the star, are the only magnetic observations available because they are relatively easy to obtain (Landstreet, 1993)—The measurement is most sensitive to modest fields with simple structures and no a priori information about the line profile is needed. Bl varies periodically through rotation phase φ as Bl(φ) ∝ Bp (cos β cos i+ sin β sin i cosφ) (1.17) Here Bp is the polar field strength, assumed to be a centred dipole, i is the angle between the rotation axis and the line of sight, and β is the angle between the rotation and magnetic axis. The zero point of the rotation phase (φ = 0) is defined at the time of magnetic maximum. The geometry of this model, known as the oblique rotator model (Stibbs, 1950), is shown in Fig. 1.4. As the star rotates, the angle from the magnetic pole to the line-of-sight changes, leading to a modulation of Bl with the rotation of the star. 18 1.2. The chemically peculiar A stars β φ i Figure 1.4: A schematic diagram showing the oblique rotator model geometry. The rota- tion axis and dipole magnetic axis are labelled R and B, respectively. As the star rotates through a phase angle φ (the zero point of the phase is at the time of magnetic maximum) an observer (located to the right in the diagram) sees the magnetic field vary in strength. The aspect of the magnetic axis varies because it is inclined to the star’s rotation axis by an angle β and to the observer’s line-of-sight by an angle i. See the discussion in § 1.2.1 for more details. If there is a strong field (& 1 kG) we may also obtain information about its geometry from the transverse field component. In this case, the central, unsplit, spectral line component will saturate before the split components and the integrated line profile will have a net linear polarization (e.g., Leroy et al., 1993). From this net linear polarization one can, in some cases, obtain 19 1.2. The chemically peculiar A stars unique values for i, β and Bp defined in Eq. 1.17. The technique that exploits this characteristic of the Zeeman components is known as broadband linear polarization. The separation of elements through gravitational settling and radiative lifting of some ionic species was used to explain the observed abundance patches of Ap stars as early as the late 1960s and early 1970s (e.g., Michaud, 1970). In principle, any element that is heavier than the surrounding (mainly Hydrogen) mixture will sink under the influence of gravity. Exceptions occur for elements that have absorption lines at the wavelengths near the local flux maximum. In these cases, the elements may be levitated upward toward the stellar surface if the radiative forces are greater than the gravitational force. Once the elements reach an equilibrium position between the gravitational and radiative forces they may accumulate in sufficient amounts, resulting in abundance anomalies at these locations. General reviews of diffusion theory may be found in Vauclair & Vauclair (1982), Michaud & Proffitt (1993) and more recently in Turcotte (2003; 2005). The element segregation described above depends on a fragile equilibrium. If there are turbulent or convective velocities in the upper atmospheres of these stars that exceed the diffusion velocities of a few cm s−1 the abundance anomalies will simply be mixed away (Michaud, 1976). While the main se- quence A stars do not have large convective envelopes, the A giants do. This explains why the A-type giants lose their abundance peculiarities. For the diffusion mechanism in these stars to be efficient there must be a stabilizing mechanism against turbulence. It is believed that the magnetic field provides this stabilizing mechanism (e.g., Gough & Tayler 1966 and Balmforth et al. 20 1.2. The chemically peculiar A stars 2001). A magnetic field also influences the distribution of elements, caus- ing horizontally distributed abundances on the stellar surface near regions of horizontal magnetic field lines (Michaud et al., 1981). The effect of a mag- netic field on element diffusion is described in Babel & Michaud (1991) and Michaud (1996). For a recent discussion on the spectroscopic determination of abundance variations with depth in Ap star atmospheres see Ryabchikova (2008). 1.2.2 Stellar seismology & Ap stars Before the late 1970s observations of variable stars did not show p-modes with periodicities and frequency spacings like those observed in the Sun. This changed when Kurtz (1978, 1982) discovered the rapidly oscillating Ap (roAp) stars. These stars are variable both photometrically and spectro- scopically, have long term variability (with periods on the order of the stellar rotation period; ranging from days to years), and also show rapid oscillations with periods ranging from about 5 to 30 minutes. The amplitudes of the oscillations are generally small, with semi-amplitudes being less than about 10 mmag when seen through a Johnson B filter (Johnson & Morgan, 1953). The roAp stars tend to be the coolest members of the Ap class (usually showing enhancements in one or all of Sr, Cr, and Eu) and have masses of less than about 3.0 M. The approximate position of the roAp stars on the Hertzsprung-Russell (HR) diagram is shown in Fig. 1.5. Some of these stars are multi-periodic and exhibit asymptotic (see § 1.1.1) spacings indica- tive of the large spacing. While some oscillate with periods that seem to be stable over many years, other members of the class may pulsate with only 21 1.2. The chemically peculiar A stars one periodicity or have frequencies that vary over a timescale as short as a few days. To date there are about 40 known roAp stars. Some detailed reviews of roAp stars are provided by Kurtz (1990), Matthews (1991), Kurtz & Martinez (2000) and Kochukhov (2008). As Ap stars rotate they show both photometric and magnetic field strength variations. The magnetic fields of the Ap stars seem to play a critical role in both the micro- and macroscopic physics that influences these unique stel- lar structures. In the outer layers of the Ap stars the strong magnetic field produces pressures that are greater than the local gas pressure and it is ex- pected that this results in an observable effect on the rapid oscillations of the roAp stars (see, e.g., Cunha 2007). In fact, the magnetic field influences the observed oscillation spectrum of these stars in a number of ways. One of which is a ‘point-of-view’ effect where the observed frequencies are split into multiple components (akin to the Zeeman splitting of spectral lines) spaced by the rotation period of the star. Another is a physical perturbation to the displacements of the rapidly oscillating medium, producing frequency shifts (and spacings) that cannot be predicted by models that do not include mag- netic fields. These two examples are discussed below, with more details on the latter of these effects being provided in chapter 4. It was noted in § 1.2 and 1.2.1 that the magnetic field and the accumulated patches of certain elements on the surface of an Ap star were linked. As an Ap star rotates, the magnetic field strength varies as the magnetic pole comes in-and-out of an observer’s field-of-view. The spotty abundance patches on the star’s surface tend to lower the stellar flux, causing a decrease in the light an observer sees, while the spot (or a portion of it) is in view. Interestingly, 22 1.2. The chemically peculiar A stars Figure 1.5: A theoretical HR diagram showing the approximate location of the roAp stars. Solid lines are stellar evolution tracks calculated with the Yale Rotating Evolution Code (YREC) described in § 4.1. Vertical dashed lines show the upper and lower bounds of effective temperature (6, 000 K . Teff . 9, 000 K) where the roAp phenomenon occurs. The roAp stars are main-sequence (stars that burn Hydrogen in their cores) variables and have a mass range of 1.3 M .M . 3.0 M. 23 1.2. The chemically peculiar A stars the amplitude of the rapid stellar oscillations of the roAp stars is generally at maximum during this minimum of the mean light of the star. Each of the long-term variability (or mean light variations), the rapid variability, and the magnetic field strength are related to each other, and are functions of rotation phase. Figure 1.6 compares the variation in the amplitude of the rapid pulsation (∼ 6 minute period) to that of the magnetic field strength as a function of rotation phase (referenced to a rotation period of about 12.5 days) for the roAp HR 1217. The pulsation amplitude of HR 1217 is in phase with the magnetic field. (Later, in Fig. 3.6 of § 3.3, we show that the mean light amplitude; associated with the spotty abundance patches, and the pulsation amplitudes vary in anti-phase as HR 1217 rotates.) The modulation of the magnetic field strength is described well by the oblique rotator model of Stibbs (1950, see § 1.2.1). Following this model, Kurtz (1982) suggested that the amplitudes of the rapid variations may be modulated in a similar way, through what is now called the oblique pulsator model. 24 1.2. The chemically peculiar A stars Figure 1.6: (Upper panel) The variation in the pulsation amplitude of HR 1217 (through a B filter) as a function of the rotation phase (referenced to a rotation period of about 12.5 days) of the star. The magnetic field variation defined by Bl (shown as H e in the figure) in § 1.2.1 as a function of rotation phase is presented in the lower panel. The pulsations (∼ 6 minute period) have amplitudes that vary synchronously with the magnetic field strength. Pulsation data and magnetic field data were taken from Kurtz (1982) and Preston (1972) respectively. This figure was obtained with permission from Matthews (1991). In this model, the pulsation axis of the star is aligned with the magnetic axis and both are inclined to the rotation axis. As the star rotates the aspect of the pulsation and magnetic axis varies, modulating the pulsation amplitudes. The geometry is the same as that shown in Fig. 1.4 with the magnetic and pulsation axis being the same. Consider an axisymmetric (m 25 1.2. The chemically peculiar A stars = 0) pulsation mode oscillating with frequency ν. Kurtz (1982) showed the luminosity varies with rotation like ∆L L ∝ Pm` (cosα) cos [νt+ ϕp] (1.18) where Pm` is the associated Legendre polynomial, ϕp is an arbitrary oscillation phase, and α is the rotation phase angle zeroed at the time of pulsation or magnetic maximum. For a dipole pulsation mode ` = 1 and the Legendre polynomial is equal to cosα. Equation 1.18 may be expanded as ∆L L ≈ A0 cos (νt+ ϕp) + A1 [cos ({ν + Ω}t+ ϕp) + cos ({ν − Ω}t+ ϕp)] (1.19) where α was replaced by the product of the rotation rate and time, Ωt, and the two amplitude functions are defined as A0 = cos i cos β and A1 = (sin i sin β)/2. This model predicts that a single dipole mode is split into a triplet exactly spaced by the rotation period of the star. Generally, a mode with degree ` is split into 2` + 1 frequencies. The roAp stars do show this fine structure in their oscillation spectra (see, e.g., chapter 3). The shortcoming of this simple model put forth by Kurtz (1982) is that it does not explain the amplitude asymmetry between the split frequency components observed in some roAp stars. Equation 1.19 shows us that the split frequencies should have the same amplitude A1. The equal amplitudes of the rotationally split components is not observed in the Ap stars’ oscillation spectra. This is resolved if the effects of both rotation and the magnetic field on the eigenfunctions (through the Coriolis and Lorentz forces) are taken into account. The most recent contribution to the oblique pulsator model comes from Bigot & Dziembowski (2002). They used a non-perturbative approach 26 1.2. The chemically peculiar A stars to show that the centrifugal force is important in determining the frequency splitting while the Coriolis force is dominant in determining the amplitude asymmetries. Those authors also show that the pulsation, rotation, and magnetic axes are all inclined to each other, i.e., the magnetic and pulsation axis are not necessarily aligned. The geometry of roAp oscillations is reviewed by Cunha (2005). The effects of the strong magnetic field on the rapidly varying normal modes (i.e., those eigenfrequencies calculated in the case of no magnetic field or rotation) will be discussed in chapter 4. Recent theoretical reviews discussing the interaction between the magnetic field and the pulsation fre- quencies are given by Cunha (2007), Saio (2008) and Shibahashi (2008). In the interior of Ap stars, the gas pressure dominates the magnetic pressures. It is only in a small surface layer (& 0.95 r/R) where perturbations become significant enough to strongly modify the motions of the local plasma. With this in mind, a number of authors have attempted to model the effect of a strong magnetic field on high overtone stellar pulsations that have their largest amplitudes near the stellar surface. Roberts & Soward (1983) de- scribed an analytic matching procedure used to link the surface effects to those of pulsations in the stellar interior where the magnetic field can be ne- glected. Numerical experiments by Campbell & Papaloizou (1986) expanded upon the work of Roberts & Soward (1983) and paved the way for the mod- ern numerical simulations of Dziembowski & Goode (1996), Cunha & Gough (2000), Saio & Gautschy (2004), and Saio (2005). In general, the magnetic field modifies the surface of an Ap star to such a high degree that a single spherical harmonic cannot describe the displacement, which has a significant 27 1.2. The chemically peculiar A stars horizontal component. Each of the investigations above recognize that in the deep interior of the star, the magnetic and acoustic components of the dis- placement decouple and the magnetic component dissipates as it propagates inward, draining energy from the pulsation. The above studies conclude that there is a significant observable effect on the oscillation spectrum of an Ap star; however, there are some differences in the details. Cunha & Gough (2000) and Saio & Gautschy (2004), for example, notice a cyclic variation (or shift) to some frequencies in the calculated eigenspectrum. They differ mainly in the amplitude of that shift, with the method of Cunha & Gough (2000) giving larger shifts in frequency over the method used by Saio & Gautschy (2004) by about 30%. In this study, we use the method of Cunha & Gough (2000) to calculate the perturbations to the acoustic modes caused by the presence of a magnetic field. The method is described in § 4.3 and has been successfully tested against observations of HR 1217 (discussed in the following section). Finally, it should be noted that a number of authors have attempted to modify stellar evolution models in such a way as to excite the frequencies that are observed in roAp stars. Recent efforts include modifications to the atmospheric temperature stratification (Gautschy et al., 1998) to reflect pulsation modes, the inclusion of helium gradients and convective suppression of the stellar envelope (Balmforth et al., 2001), a magnetic and radiative damping (non-adiabatic) study of oscillations Saio (2005), and the inclusion of the effects of stellar winds and metal gradients in the stellar envelope (Théado et al. 2005,2009). While there have been some successes in exciting the high overtone modes in Ap stars, none of the studies is able to provide 28 1.3. The roAp star HR 1217 a globally applicable excitation mechanism for these stars. In this study we focus solely on adiabatic results; i.e, we ignore excitation, so that we can, for the first time, attempt to constrain the physical parameters of a roAp star using a large grid of magnetically perturbed pulsation models. 1.3 The roAp star HR 1217 This thesis focuses on observations and models of the roAp star HR 1217 (a.k.a HD 24712; DO Eri; B = 6.3 mag; δ = -12◦ 5′ 56.′′78; α = 03 hr 55 min 16.s128 ). This was one of the first Ap stars to be identified as a roAp star by Kurtz (1982) and has since become one of the most studied. Recently, high- quality data has provided information about the abundance, magnetic and photometric characteristics of HR 1217. The most recent photometric data is analyzed in this thesis and is, in many ways, an unrivalled photometric data set on HR 1217 (chapter 3). A review of the other recent observations of HR 1217 is presented below. The magnetic field of HR 1217 was measured by Bagnulo et al. (1995). They were able to model the magnetic field geometry for HR 1217 and give values of 137◦, 150◦, and 3.9 kG for i, β and Bp respectively. Estimated uncertainties on the above angular measurements are ≈ 2–3◦ and the uncer- tainly of the polar field strength is about 5%. The mean longitudinal field, Bl, for HR 1217 varies between ∼ 0.5 and 1.5 kG (Preston, 1972), and is illustrated in the lower panel of Fig. 1.6. Kochukhov & Wade (2007) discuss current magnetic and rotation properties of HR 1217. The most recent mag- netic field measurements are given by Lüftinger et al. (2008). They measure 29 1.3. The roAp star HR 1217 a magnetic field variation of 2.2 to 4.4 kG (no uncertainty given) based on a Doppler imaging analysis. The first thorough abundance analysis of HR 1217 was performed by Ryabchikova et al. (1997). Their results are consistent with the idea that the abundance enhancements on the surfaces of these stars are patchy. In particular they find the mean chemical abundances vary with the magnetic and rotational phase of the star. When compared to solar values, the rare earth elements are the most overabundant and show the largest variation over the rotation of the star. The iron peak elements, on the other hand, tend to be under abundant. An estimate of the metal content of HR 1217 based on the work of Ryabchikova et al. (1997) is presented in § 4.1.1. The first global photometric campaign to observe HR 1217 was headed by Don Kurtz and Jaymie Matthews (Kurtz et al., 1989). The goal of those observations was to achieve as much continuous coverage of the star as pos- sible so that gaps in their data would not affect their frequency analysis. They achieved a 29% duty cycle (compare this to the 95% duty cycle ob- tained in this work; see chapter 3) with 325 hrs of data spanning a 46-day period. The results of the campaign are shown schematically in the top panel of Fig. 1.7. There exists an ambiguity in identifying ∆ν from the observed frequency spacing (§ 1.1.1). If the modes are alternating between even and odd ` values we would expect to see a ∆ν ∼ 68 µHz. In fact, the alternat- ing spacing of 33 and 34 µHz is consistent with models having alternating even and odd modes (e.g., chapter 4). If the modes were all even or odd, we would expect that the spacing between adjacent modes would remain the same. Each of the frequencies in Fig. 1.7 are actually multiplets having fre- 30 1.3. The roAp star HR 1217 quencies spaced by approximately 0.9 µHz (the rotational frequency of the star). The unexpected frequency spacing (∼ 50 µHz) between the last two frequencies (between ∼ 2750–2800 µHz) is not consistent with asymptotic theory and the observed large spacings. The gap in the frequency spectrum of HR 1217 observed by Kurtz et al. (1989) is usually said to be a “missing mode”. Almost a decade after Kurtz et al. (1989) released their results the true large spacing value of ∼ 68 µHz was unambiguously determined. Using Eq. 1.15, Matthews et al. (1999) were able to calculate a parallax for HR 1217 based on an inferred large spacing of 68 µHz. Their predicted parallax of pi = 19.23±0.54 mas was shown to be consistent with the recent Hipparcos parallax (Perryman et al., 1997) of pi = 20.41± 0.84 mas (§ 4.1.1). 31 1.3. The roAp star HR 1217 Figure 1.7: Schematic amplitude spectra of HR 1217 oscillations from previous obser- vations. The upper panel shows the frequencies identified in the 1986 global observation campaign of HR 1217 (Kurtz et al., 1989). Frequencies are regularly spaced (see discussion on asymptotic spacing in § 1.1), alternating between ∼ 33.5 (green) and 34.5 (blue) µHz. The exception to the regular spacing is a spacing of ∼ 50 µHz represented by a red line in the top panel. The middle panel gives the frequencies found in the more recent 2000 global campaign (Kurtz et al., 2005). This campaign uncovered a “missing” frequency near 2800 µHz that fit with the previously identified asymptotic spacing of ∼ 33.5 µHz (represented by the red line in the middle panel). In addition to this discovery, Kurtz et al. (2005) discovered a frequency spacing of ∼ 2.5 µHz (the difference between the red and green lines) that is consistent with a second order asymptotic spacing for a main sequence A star. The furthest frequency spacing (blue line) of ∼ 14 µHz is inconsistent with the observed asymptotic spacings and may be explained by magnetic perturbations (see Cunha 2001 and § 4.3). The lower panel shows the results from recent spectroscopic observations of Mkrtichian & Hatzes (2005). Those authors recovered the previously observed periodici- ties of HR 1217 and suggested new frequencies with spacings of ∼ 34 (blue line) and 32 (green line) µHz may exist at the low frequency end of the oscillation spectrum. 32 1.3. The roAp star HR 1217 A success in the interpretation of the frequency spacings of HR 1217 was provided by Cunha (2001). She predicted that magnetic damping could be the cause of the missing frequency in the 1986 data set. She also showed that some frequencies may be shifted by approximately 10–20 µHz because of magnetic field effects (Cunha & Gough 2000, and Cunha 2001; further discussion is provided in § 4.3). In 2000, HR 1217 was selected to be observed in another global campaign (organized by the Whole Earth Telescope, WET; see Kleinman et al. 1996 and Kurtz & Martinez 2000) and a preliminary data reduction for this data set did find the missing frequency at ∼ 2795 µHz (Kurtz et al., 2002). Although it was long suspected, this was the first time it was shown that the influence of the magnetic field along with stellar rotation and surface chemical gradients plays a critical role in the pulsation characteristics of the roAp stars. These unique properties of the roAp stars could allow us the opportunity to model the magneto-acoustic pulsation modes and study in detail an environment that is significantly different to that found in other stars, including the Sun. The full analysis of the global WET campaign data (collected in 2000) is given in Kurtz et al. (2005). Their results are drawn schematically in the middle panel of Fig. 1.7. They collected at total of 35 days of data with a duty cycle of 36% and reached an amplitude precision of 14 µmag (one of the most precise ground based photometric studies to date). Along with the discovery of the missing mode, they also discovered a closely spaced frequency grouping of about 2.5 µHz that could not be explained by the oblique pulsator model. This “small” spacing is on the order of magnitude that one would expect for a second order asymptotic spacing discussed in § 1.1.1 for a main 33 1.4. Thesis goals sequence A star. This could be the first time such a small spacing was observed in a star other than the Sun. That being said, the diagnostic power of the small separation may be limited because the magnetic perturbations to the frequencies may be of the same order, or larger than, the small spacing (Dziembowski & Goode, 1996). This is further illustrated in § 4.3. A recent radial velocity study by Mkrtichian & Hatzes (2005) has also uncovered some new frequencies (shown in the lower panel of Fig. 1.7). Those observations covered about a month of time (from 22 Dec, 1997 to 30 Jan 1998 with two additional days in early Feb 1998) giving about 5 hours of observations per night for 9 nights over the month. They find new frequencies at ∼ 2553 and 2585 µHz. These frequencies were only identified in a subset of the authors’ data and are, admittedly, of low significance and of low frequency resolution. 1.4 Thesis goals Progress in the study of the Ap and roAp stars has been steadily increas- ing over the past few years with the development of new observational and theoretical tools. In this thesis we attempt to tie together the most recent photometric observations (chapter 3) on the roAp star HR 1217 (obtained by the MOST satellite, which is described in chapter 2) to theoretical stellar evolution and pulsation models that include the effects of a magnetic field (chapter 4). The data collected by the MOST mission has obtained both ultra-high precision and time sampling that has never been achieved before, on this 34 1.4. Thesis goals target (HR 1217), and for a vast majority of asteroseimic targets. With this data in hand, we have the following observational goals and questions: 1. It has been known since the early 1980s that HR 1217 is a multiperiodic variable with near equally spaced modes, alternating between ∼ 33.5 and 34.5 µHz. The most recent confirmation of this was the analysis of Whole Earth Telescope data by Kurtz et al. (2005), who achieved a duty cycle of 36% and reached an unprecedented precision of 14 µmag for a ground-based photometric study. We will be able to unambiguously identify the spacings in the MOST data because of its near continuous coverage and ultra-high precision. Are the already identified periodici- ties constant over time for all observations on HR 1217, or do they vary over time, pointing to a selective mode damping mechanism or some nonlinear interaction? 2. The strange, apparently non-asymptotic, spacing that was observed be- tween the highest two frequencies in HR 1217 remained a mystery until the recent results of Kurtz et al. (2005, 2002). Those authors identified a previously unidentified frequency near 2790 µHz that could only be explained as a magnetic perturbation Cunha (2001). Explaining that frequency (or lack of that frequency) has driven the theory of roAp star oscillations for more than two decades. Is this frequency identified in the MOST data on HR 1217 and is it stable since the last observations of Kurtz et al. (2005)? 3. We expect to observe a number of new periodicities in the data because of the high level of photometric precision and the near contiguous data 35 1.4. Thesis goals sampling of the MOST satellite. Can new periodicities be identified and, if so, can they be used to constrain the physics of this magnetic pulsator? 4. A recent spectroscopic study on HR 1217 by Mkrtichian & Hatzes (2005) identified new frequencies at ∼ 2553 and 2585 µHz. These fre- quencies are interesting because they approximately match the alter- nating ∼ 33.5 and 34.5 µHz pattern previously observed in HR 1217, but were not identified in the photometry of Kurtz et al. (2005). Does the MOST data uncover those potentially new periodicities? 5. In order to determine the reliability and uniqueness of the periodicities in a multiperiodic variable, we need to develop tools that can test the precision and resolution of our data set. Techniques we have developed for other MOST targets will be applied to the HR 1217 data. The res- olution of our data set and the assertion that closely spaced frequencies are spaced by the rotation rate of the star will be tested. A massive grid of A star evolution and pulsation models (chapter 4) has been constructed for this research project. The magnetically perturbed pulsation modes are calculated and are matched to the observed frequencies of HR 1217 (chapter 5) for the first time. The scope of this model grid eclipses any previous attempts to constrain the properties of HR 1217, or any other roAp star. The following theoretical questions will be answered in this thesis: 1. For the first time, a realistic stellar model grid, covering a vast swath of parameter space appropriate for matching the observable character- 36 1.4. Thesis goals istics of HR 1217, is constructed that includes the magnetic perturba- tions to the calculated normal modes of the models. Is the magnetic perturbation a necessary ingredient for the matching of a model to the observations? 2. Are the spacings observed by the MOST mission reproducible with the physics used to construct our model grid? 3. How are models matched to the observations and what are the sensi- tivities of those matching procedures? 4. Is there a preference as to what angular degree ` is matched to an observed frequency? 5. We test the calculations of the pulsation modes in our grid by using a new method developed by Kobayashi (2007) for the calculation of normal modes in a geophysical context. The methodology and the results of the observational and theoretical analysis of the HR 1217 observations from the MOST mission are outlined and discussed throughout the following chapters. The results are summarized in chapter 6 of this thesis. 37 Chapter 2 The MOST mission: Data reduction techniques The HR 1217 data presented in chapter 3 of this thesis was collected by the MOST satellite and analyzed using techniques developed over a period of about 5 years by the MOST team and its collaborators; including the author of this thesis. In this chapter we give an overview of the MOST mission, the photometric and time-series methods that are relevant to the HR 1217 data reduction, and finish with some examples of other MOST results that used the described data analysis methods. 2.1 An overview of the MOST mission The Microvariability & Oscillations of STars (MOST ) mission is Canada’s first space telescope3. Launched into a sun-synchronous orbit at an altitude of ∼ 820 km (orbital period = 101.413 min) on 30 June 2003, the micro- satellite (weighing about 54 kg with dimensions similar to that of a typical suitcase) functions as an ultra-precise light meter, that routinely obtains near 3Being a Canadian space mission it is also appropriate that the acronym MOST works equally well in French . . . Microvariabilité et Oscillations STellaire 38 2.1. An overview of the MOST mission continuous4, micro-magnitude, photometric measurements of oscillating stars and extrasolar planets. The instrument is a 15/17.3 cm Rumak-Maksutov telescope that directs light through a custom broadband filter, covering a wavelength range of 350– 700 nm,5 to two CCDs: One used for tracking and the other for scientific measurements. The experiment is outlined in Matthews et al. (2000), with more technical details provided by Walker et al. (2003). Originally, MOST produced scientific data in Fabry and direct imaging modes. In the Fabry mode, an image of the telescope entrance pupil (illumi- nated by starlight) is projected onto the science CCD, spreading light over ∼ 1500 pixels so as to limit tracking jitter and the effect of individual pixel variations (caused, for example, by radiation). The direct imaging mode provides in-focus photometry of fainter (& 6 mag) stars on the science CCD. The satellite, through software updates, was eventually (after about a year of operation) able to achieve sub-arcsecond pointing, allowing scientific data to be gathered on guide stars from the tracking CCD, greatly increasing the scientific output of the mission. MOST functioned in this way until early 2006, when the tracking CCD suddenly failed. It is suspected that a par- ticle hit damaged the CCD. Since that time, both science and tracking are performed on the science CCD. The first science results from MOST were published by Matthews et al. 4MOST has a Continuous Viewing Zone that ranges in declination from +36◦ ≥ δ ≥ −18◦. Stars in this declination range can, at most, be viewed continuously for up to 60 days. 5A Johnson B filter (Johnson & Morgan, 1953) is centred at about 445 nm with a full width at half maximum of ∼ 94 nm (see Binney & Merrifield 1998) 39 2.2. Photometry & time-series analysis (2004) and recent reviews of the mission results are given by Matthews (2007) and Walker (2008). They clearly illustrate the breadth of the scientific accom- plishments of the MOST instrument to date. Data on binary stars, rapidly rotating stars, stars with spots, magnetic stars, exoplanets, and even stable stars are providing new insights into exotic and, in many cases, otherwise untestable physical scenarios. With more than 1000 stars observed—about a quarter of which show detectable light variations — the scientific discoveries are on-going. 2.2 Photometry & time-series analysis Reducing MOST photometry is a challenging task because of stray-light contamination caused by Earth shine illuminating the CCDs. This parasitic addition to the stellar signal depends on the orbital phase and orientation of the satellite and varies with season. From one orbital period (∼ 101 min) to another, this contaminant changes its character, making it very difficult to model. The result is a superposition of orbital components (∼ 14 cycles day−1), along with their harmonics and aliases, and the stellar signal in the Fourier amplitude spectra (discussed below) of a target’s light curve. Methods have evolved since the launch of MOST to combat the stray-light effect. These are discussed in detail in Rowe et al. (2006b, 2006c), Reegen et al. (2006), and Rowe (2008). Typical photometric data techniques such as sky background removal (estimated from the dimmest ∼ 200 pixels), flat-fielding, and PSF modelling (using ∼ 150 of the brightest pixels) are also discussed in those 40 2.2. Photometry & time-series analysis papers. Photometric errors are estimated from Poisson statistics and are described, for example, in Everett & Howell (2001). The asteroseismic results are determined from relative amplitude fluctu- ations in the star’s light curve. Absolute magnitude determinations are not necessary, so limiting the stray-light contribution to the data is more impor- tant than applying mean corrections to the counts. In the work presented in this thesis the stray-light was suppressed by subtracting a smoothed pro- file created by phasing the data over a number of satellite orbits (see, e.g., Rucinski et al. 2004 and Rowe et al. 2006b). The larger the number of orbital bins used to find the correction, the less aggressive is the reduction of the orbital signal and its harmonics. This is because the mean stray-light con- tribution may not well represent the contribution from one orbit to another. By finding the correction to data phased over a small number of satellite orbital periods, we remove orbital signal averaged over shorter timescales, essentially over-smoothing the orbital component. The main advantage of this method is that no prior information about the structure of the orbital contamination is assumed. Any remaining long term trends can be removed by either subtracting a low-order polynomial or a spline fit to binned data. The application of this method is further discussed in § 3.1 (see Fig. 3.1 for an example of the stray-light contamination to the HR 1217 light curve). The frequency analysis of the reduced time-series is carried out using CAPER6(Cameron et al., 2006). CAPER is a collection of Fortran driver routines that use a Discrete Fourier Transform (DFT) as a frequency and amplitude estimation tool and nonlinear least squares fitting to refine the 6Controlled or Automated Period Estimation and Refinement 41 2.2. Photometry & time-series analysis identified parameters. The software package has been used in a number of recent publications (Walker et al. 2005, Saio et al. 2006, Cameron et al. 2006, King et al. 2006, Saio et al. 2007 and Cameron et al. 2008) and follows the same reduction philosophy as the popular, and time-tested, time-series anal- ysis packages Period98 (Sperl, 1998) and Period04 (Lenz & Breger, 2005). For a time-series of length T, the amplitude spectrum (calculated from the DFT) is defined by An(ν) = 2 T √ a2n + b 2 n (2.1) where an and bn are the imaginary and real parts of the Fourier transform and are calculated from an = T∑ j=1 f(tj) sin(2piνtj) and bn = T∑ j=1 f(tj) cos(2piνtj) (2.2) In these expressions f(tj) is the magnitude (or flux value) at time tj and ν is the current frequency of interest. The amplitude spectrum is calculated efficiently over a range of ν values using the algorithm proposed by Kurtz (1985). This algorithm is a modified version of the Deeming algorithm for computing the DFT of data with sampling gaps (Deeming 1975; also see Matthews & Wehlau 1985). Once the largest peak in the amplitude spectrum is identified, its phase can be calculated as φ = arctan(an/bn). The set of parameters (ν, A, φ) obtained from the DFT are next refined by nonlinear least squares fitting using the Levenberg-Marquardt method (Press et al., 1992)7 which minimizes χ2 between a fitting function and the data. A sinusoidal fitting function of the form Ffit(t) = A sin(2piνt+ φ) (2.3) 7The name of the Numerical Recipes routine used by CAPER is mrqmin 42 2.2. Photometry & time-series analysis is used by CAPER. Once Ffit is optimized, it is subtracted from the original data set. A Fourier spectrum of the residuals is calculated and a new parameter set is identified. These new parameters, in conjunction with the previous set(s), are re-fitted to the original time-series and the process is iterated to a predefined stopping criterion, usually set as a signal-to-noise (S/N) threshold. The fit is improved at each step based on the original data until there are no longer any meaningful changes in the residuals from the fit. A S/N threshold of ∼ 3.5 is commonly used in the asteroseismic com- munity as a lower limit to the significance of a detectable peak in the DFT. Peaks in the DFT with a S/N greater than this represent a detection & 2.5σ (Breger et al. 1993 and Kuschnig et al. 1997). This is adopted as a lower limit to the significance of the extracted periodicities in this work. The S/N for each recovered signal component is calculated (before prewhitening8 that component) as the mean amplitude in a box around the identified peak, sigma clipped until the mean converges. The clipping is done so that addi- tional high amplitude peaks near the identified frequency do not skew the local mean amplitude (noise) value. This method of calculating the S/N has two potential sources of uncertainty: 1. The width of the box used to av- erage the amplitude spectrum (the noise) and 2. Uncertainties in the fitted amplitude. The uncertainty in the noise calculation is estimated by varying the width of the averaging box in small steps and then calculating an average of each noise value along with a standard error on that average noise. Once 8Prewhitening is a term used to describe the subtraction of a sinusoidal function from the data. 43 2.2. Photometry & time-series analysis amplitude uncertainties are assessed from a bootstrap analysis (described below), we combine both to arrive at the final uncertainty in S/N. The uncertainties we calculate for time-series parameters derived from nonlinear least-squares methods depend on the noise of the data (which may be a combination of instrumental and random processes) and on the time sampling. It is also known that fitted phase and frequency parameters are correlated, leading to underestimated uncertainties in these parameters when calculated from a covariance matrix (see, e.g., Montgomery & O’Donoghue 1999). A bootstrap analysis is an ideal way to assess the uncertainties in fitted parameters. The procedure has recently been used in a number of MOST applications (Walker et al. 2005, Saio et al. 2006, Cameron et al. 2006, Rowe et al. 2006a, Gruberbauer et al. 2007, Saio et al. 2007, Cameron et al. 2008, Rowe 2008, Rowe et al. 2008a, and Rowe et al. 2008b) and was used by Clement et al. (1992) to estimate the uncertainties in Fourier parameters they derived for RR Lyrae stars. The bootstrap (see Wall & Jenkins 2003) produces a distribution for each calculated parameter by constructing a large number of light curves (usually > 10, 000) from the original data. No assumptions need to be made about noise properties of the data and individual photometric errors are not needed. Each new light curve is assembled by randomly selecting N points from the original light curve (also containing N points) with the possibility of replacement. In this way, the new synthetic light curves preserve the noise properties of the original data. The fit procedure is repeated for each new light curve, eventually building distributions in each of the fit parameters. Recently, Breger (2007) discussed the difference between frequency resolu- 44 2.2. Photometry & time-series analysis tion and the precision of fit parameters to time-series data. Traditionally one assigns a frequency resolution as T−1, also known as the Rayleigh criterion, where T is the length of the observing run. Loumos & Deeming (1978) sug- gests an upper limit of 1.5 T−1 be used when extracting peaks directly from the amplitude spectrum. This roughly corresponds to the spacing between the main peak of the window function9 and the peak of its first sidelobe. Lower, and arguably more realistic, estimates are used by Kurtz (1981), who estimates frequency resolution as 0.5 T−1 (approximately the half-width at half-maximum of a peak in the amplitude spectrum), and by Kallinger et al. (2008b), who suggest ∼ 0.25 T−1 can be used based on a large number of simulated data sets. Ultimately, the resolution of frequencies in Fourier space is a function of S/N (or significance) of each individual peak; see, for example, Kallinger et al. (2008b), and the above criteria are only estimates used as an average frequency resolution for all frequencies of a given data set. The precision of fitted parameters, on the other hand, can be estimated by refitting identified parameters to a large number of data sets created by sampling the fitted function in the same way as the data and adding random, normally distributed noise. This Monte Carlo procedure is used, for example, in Period04 (Lenz & Breger, 2005). A bootstrap analysis differs from such a Monte Carlo procedure in that there are no assumptions made about the noise of our re-sampled data sets (we only use the data) and that the frequency resolution of our data sets can be estimated because the windowing (determined from the temporal sampling 9The window function of the data is estimated as the DFT of a noise free sinusoid, sampled in the same way as the data, and varying at a period of interest. 45 2.3. Applications of CAPER to other MOST science of the data) of each of the re-sampled data sets is randomly changed. Thus, the uncertainties on each of the fitted parameters take into account both the inherent noise of the data and the sampling of the data as well. Put another way, the robustness of our fit parameters to the variations in the data (due to any source) is estimated. 2.3 Applications of CAPER to other MOST science The following sections review a number of published examples of MOST ob- servations that use the reduction methods described above, and that were used for the reduction of the HR 1217 observations collected by MOST. (In particular, each subsection outlines the results of a Fourier analysis using the CAPER software and a combined bootstrapping error analysis.) The results for the roAp star HD 134214 and the B supergiant HD 163899 are sum- marized and a more detailed frequency analysis of the slowly pulsating Be (SPBe) stars HD 127756 and HD 217543 is presented. Background informa- tion and a more thorough physical interpretation of these results are found in the references provided. Although the following analyses are common to those undertaken in this thesis, the results are independent of this work and reader may wish to proceed to the analysis of the HR 1217 data in chapter 3. 46 2.3. Applications of CAPER to other MOST science 2.3.1 The roAp star HD 134214 The roAp star HD 134214 was observed by MOST as a trial Direct Imaging target on 1 May 2006 and the results were published by Cameron et al. (2006). The star was in one of two target fields observed during each MOST satellite orbit (period = 101.413 min), so data were collected ∼ 47% of the observing time. In total there were 505 measurements spanning a 10.27 hrs base-line. HD 134214 was shown to be monoperiodic and oscillating at a frequency of 2948.97 ± 0.55 µHz. This is consistent with earlier ground based photometric campaigns (e.g. Kreidl et al. 1994). Recent results by Kurtz et al. (2006) suggest that HD 134214 is oscillating spectroscopically in up to 6 modes. Our observations can rule out additional photometric oscillations at a 2σ detection level of 0.36 mmag, based on S/N estimations described in § 2.2. Our estimated DFT noise level of ∼ 0.12 mmag for 10 hours of observations with MOST is comparable to that ob- tained by Kreidl et al. (1994) based on about 56 hours of photometry from four observatories over about 4 months. The excitation and selection of pul- sation modes in roAp stars is an open question. In the case of HD 134214, there are several avenues to be explored with respect to the additional modes seen spectroscopically: 1. Is it a case of radial velocities of certain elements and ionization stages being more sensitive to degrees of higher `? 2. If the sensitivity of the photometry can be improved sufficiently, will these modes also appear? 3. Is the broadband photometry just averaging over too wide an extent of the pulsating atmosphere? or 4. Might there be new physics in the upper atmospheres of (some) roAp stars to account for the differences observed? 47 2.3. Applications of CAPER to other MOST science Kurtz et al. (2007) presented new photometric measurements on HD 134214 and confirm some of the frequencies seen spectroscopically by Kurtz et al. (2006). The authors suggest that the additional frequencies may be below the detection limit of the short MOST run. As a result, MOST has re- observed HD 134214, surpassing the 10.27 hrs of data presented in Cameron et al. (2006) with over a month of data. The results are currently being analyzed and will hopefully shed new light on the questions raised by these works. 2.3.2 The B supergiant HD 163899 The MOST satellite observed the B supergiant HD 163899 (B2 Ib/II) as a guide star and detected 48 frequencies . 2.8 cycles day−1 with amplitudes of a few mmag and less (Saio et al., 2006). HD 163899 was observed as one of 20 guide stars used during the photometry measurements of WR 103 (HD 164270). Observations were made from 2005 June 14 to July 21 for a total of 36.6 days. Because the WR 103 field is outside the MOST Continuous Viewing Zone, the duty cycle was limited to about 50% of the 101 min orbit. The frequency range suggests g- and p-mode pulsations. It was generally thought that no g-modes are excited in less luminous B supergiants because strong radiative damping is expected in the core. Theoretical models calcu- lated by Saio and Gautschy (Saio et al., 2006) show that such g-modes are excited in massive post-main-sequence stars. Excitation by the Fe-bump in opacity is possible because g-modes can be partially reflected at a convective zone associated with the hydrogen-burning shell, which significantly reduces radiative damping in the core. The MOST light curve on HD 163899 shows 48 2.3. Applications of CAPER to other MOST science that such a reflection of g-modes actually occurs, and reveals the existence of a previously unrecognized type of variable, slowly pulsating B supergiants (SPBsg). Such g-modes have great potential for asteroseismology. HD 163899 is the first member of the SPBsg family, showing both g-modes and hybrid β Cephei-type (p-mode) behaviour. 2.3.3 The SPBe stars HD 127756 & HD 217543 MOST discovered SPBe (Slowly Pulsating Be) oscillations in the stars HD 127756 (B1/B2 Vne) and HD 217543 (B3 Vpe). The results are published in Cameron et al. (2008). HD 127756 and HD 217543 join the stars HD 163868 (Walker et al., 2005) and β CMi (Saio et al., 2007) as SPBe stars discovered by MOST. High radial order g-modes with pulsation frequencies in the co-rotating frame that are much smaller than the rotation frequency appear in groups that depend on the azimuthal orderm in a DFT. Theoretical models (see, for example, models calculated by Saio in Cameron et al. 2008) indicate that, in rapidly rotating stars, high-order g-modes are excited near the Fe opacity bump as in SPB stars. One difference from slowly rotating SPB stars is the fact that among the high-order g-modes, prograde modes are predominantly excited. These modes have frequencies of ∼ |m|Ω in the observer’s frame with Ω being the rotation frequency of the star. For m = −1 and −2, expected frequencies are grouped at Ω and 2Ω consistent with observed frequencies previously discovered in HD 163868 (Walker et al., 2005). The analysis of HD 127756 and HD 217543 also shows this behaviour, opening up the possibility of determining the rotation rates of the stars by matching the observed frequency groups to models of rapidly rotating stars. 49 2.3. Applications of CAPER to other MOST science HD 127756 was observed for a total of 30.7 days by MOST . The target field was outside the MOST CVZ so there is a gap during part of each 101.4 min satellite orbit resulting in a duty cycle of 30.7% (Note that our data sampling (∼ 20 s), even with gaps every orbit, thoroughly covers the time scale of the intrinsic variations (∼ 1 day period) of HD 127756, giving us an effective duty cycle of near 100%). The light curve is presented in Fig. 2.1 and shows clear variations with periods near 1 day and 0.5 day. Table A.1 lists the frequencies, amplitudes, phases, the 1 and 3σ uncertainties from the bootstrap analysis (using 100,000 realizations), and the S/N of the 30 most significant periodicities. The 1 and 3σ uncertainties are estimated for each parameter as the width of the region, centred on the parameter in question, that contains 68 and 99% of the bootstrap realizations, respectively. The fit is shown superimposed over two zoomed sections of the light curve labelled A and B in the lower panels of Fig. 2.1. The amplitude spectrum of the data along with the fitted points and the residuals from the fit are shown in the top panel of Fig. 2.2. Most of the frequencies gather into three groups; ∼ 0 cycles day−1, ∼ 1 cycles day−1, and ∼ 2 cycles day−1. This property is similar to the frequency groupings of the SPBe star HD 163868 (Walker et al., 2005). The lower panel of Fig. 2.2 plots the S/N for each of the identified periodicities and the window function of the data. Among the frequencies listed in Tab. A.1, ν1 = 0.0335 cycles day −1 and ν2 = 0.0739 cycles day −1 have the fewest observed cycles (close to the length of the run at 1/30.7 = 0.0326 cycles day−1) and are included to reduce the scatter in the residuals from our fit. They may not be genuine stellar oscillation frequencies but it should be noted that observational artifacts 50 2.3. Applications of CAPER to other MOST science associated with the baselines of other MOST observations, especially with such a relatively large amplitude of 7 mmag as in the case of ν1 here, have not been observed. A comparison of the closely spaced frequencies near ∼ 1 cycles day−1 to the window function and to our fit is given in Fig. 2.3. Notice that the peak with the largest amplitude has an asymmetric component that is wider than the window function. When that frequency (ν14) is prewhitened, significant power remains near that asymmetry and is fitted as ν13 (shown as the data point with the smallest amplitude in Fig. 2.3). These frequencies are spaced by ∼ 0.04 cycles day−1 which is greater than the Rayleigh criterion for our data (∼ 0.03 cycles day−1). The points are clearly separated in frequency within their respective 3σ errorbars. The peak labelled as Ax (ν15 in Table A.1) is spaced from ν14 at nearly the resolution limit suggested by Loumos & Deeming (1978) (∼ 0.06 cycles day−1). This peak is clearly resolved from ν14 and has an amplitude that is ∼ 4 times that of the first side lobe of the window function (labelled as Ay). Although the amplitude of this peak may be influenced by windowing of the data, the evidence suggests the frequen- cies are resolved. Frequencies ν18 and ν19 (shown circled in Fig. 2.3) have the smallest frequency separation and are barely resolved within their 3σ error- bars with a separation of ∼ 0.004 cycles day−1. The bootstrap distributions for these frequencies are plotted in Fig. 2.4 and shows the parameters are normally distributed and the frequency distributions of ν18 and ν19 nearly overlap. We suggest, based on our bootstrap distributions, that frequencies spaced by less than 0.5 T−1 ∼ 0.0163 cycles day−1 (within their 3σ errorbars) are at 51 2.3. Applications of CAPER to other MOST science the resolution limit of our data set. Using this resolution criterion, frequency pairs ν6 and ν7; ν18 and ν19; and ν23 and ν24 may not be resolved frequencies. Cameron et al. (2008) show that the resolution of these frequencies doesn’t affect the physical interpretation of the data. MOST observed HD 217543 as a guide star for a total of 26.1 days with a duty cycle of ∼ 33% (like HD 127756 the effective duty cycle is closer to 100%). Figure 2.5 shows the light curve with clear periods of ∼ 0.5 and ∼ 0.25 days with modulations characteristic of more complex multi- periodicity. The fit to the 40 most significant frequencies (see Tab. A.2) is shown in zoomed regions labelled A and B in the lower panels of the plot. Note that in Tab. A.2 there are 6 frequencies with S/N ranging from 3.09 to 3.38. These are below the S/N ∼ 3.5 limit described above and represent & 2σ detections (Kuschnig et al., 1997). They are included to illustrate that within the S/N errors plotted in the lower panel of Fig. 2.6 all identified frequencies reach the S/N ∼ 3.5 limit. These periodicities do not adversely influence the fit and do not change the physical interpretations of Cameron et al. (2008). Figure 2.6 shows an amplitude spectrum of HD 217543 in the top panel and the S/N of the identified frequencies and the window function of the data in the lower panel. As with HD 127756, most frequencies are grouped around three ranges; ∼ 0 cycles day−1, ∼ 2 cycles day−1, ∼ 4 cycles day−1. The second and the third frequency range is higher by a factor of ∼ 2 than the corresponding ones of HD 127756. Frequencies ν1 = 0.0269 cycles day −1 52 2.3. Applications of CAPER to other MOST science Figure 2.1: MOST light curve of HD 127756. The top panel shows the entire light curve spanning a total of 30.7 days. The middle and the bottom panels are expanded light curves for the portions A and B, respectively, indicated in the top panel. Solid lines indicate the fit of the 30 significant frequencies (Tab. A.1) from the frequency analysis of the full light curve. The short-term variability seen in the middle panel is a consequence of stray Earth shine modulated with the MOST satellite orbital period of ∼ 101.4 min. 53 2.3. Applications of CAPER to other MOST science Figure 2.2: (Top panel) The Fourier amplitude spectrum of the light curve of HD 127756. Filled (blue) circles with 3σ error bars are the fitted parameters (see Tab. A.1). The inverted dash–dot line is the residual amplitude spectrum obtained after the fit was sub- tracted from the light curve. (Lower panel) The S/N of the identified periodicities with 3σ uncertainties estimated from both the fitted amplitudes and frequencies and the mean of the amplitude spectrum (see § 2.2 for details). The light grey line represents the window function of the data centred on the frequency with the largest amplitude and scaled to the maximum S/N for clarity. 54 2.3. Applications of CAPER to other MOST science Figure 2.3: The zoomed region around the largest peak in the amplitude spectrum of HD 127756. The window function is shown as the inverted, dotted line and the fit is shown as points with 3σ errorbars. The asymmetry of the largest peak in the amplitude spectrum (width ∼ 0.052 cycles day−1 [dashed line]) is compared to the width of the window function (∼ 0.048 cycles day−1 [dash dot line]). The amplitude of the first sidelobe of the window function (labelled Ay) is ∼ 4 times smaller than the second largest peak in the amplitude spectrum at Ax. The resolution of frequencies ν18 and ν19 (both circled) is discussed in section 2.2. 55 2.3. Applications of CAPER to other MOST science Figure 2.4: A comparison of bootstrap distributions for parameter sets (ν18, A18, φ18) and (ν19, A19, φ19) [see Tab. A.1] for 100,000 realizations of the HD 127756 light curve. The top panels are distributions for the fitted phase (φ) while the middle and lower panels show distributions for the amplitude (A) and frequency (ν) parameters, respectively. In each panel symbols are shown (from top to bottom) for the 1σ (F) and 2σ () error intervals containing 68 and 95 % of the realizations (note that Tab. A.1 lists the 3σ, or 99%, error interval) centred on the fitted parameter. Below those symbols in each panel are the 1σ () errorbars obtained from the formula definition of standard deviation and the mean (•) of the distribution with the standard error on the mean. These distributions are shown because the frequencies are the closest to each other. 56 2.3. Applications of CAPER to other MOST science and ν2 = 0.0806 cycles day −1 are close to the length of run (1/26.1 = 0.0383 cycles day−1) but were included to reduce the residuals in the light curve. They may not be intrinsic stellar pulsations. Frequencies ν6 and ν7 of Table A.2 overlap within their 3σ uncertainties. The bootstrap distributions are given in Fig. 2.7 and show all parameters are normally distributed like those in Fig. 2.4 for HD 127756. However, the long tails on the frequency distributions suggest that these frequencies are not fully resolved. If we adopt the same resolution criterion as for HD 127756, frequencies spaced less than 0.5 T−1 ∼ 0.0192 cycles day−1 (within their 3σ errorbars) are at (or below) the resolution limit of our data set. This means frequency sets ν6 and ν7; ν19 and ν20; and ν34 and ν35 may not be fully resolved. The SPBe stars provide an opportunity to determine the rotation fre- quency without referring to v sin i (Cameron et al., 2008). It now seems possible to determine the rotation rates of these stars by matching the ob- served frequency groups to models of rapidly rotating stars. This is only a possibility now because of the high quality of the MOST data. Cameron et al. (2008) provide a detailed modelling effort for the SPBe stars already published by MOST team. However, even without presenting detailed mod- els here, we can say that HD 217543 rotates about twice as fast as HD 127756 based solely on the spacings between the frequency groups observed in Figs. 2.2 and 2.6. Unfortunately, detailed g-mode asteroseismolgy is not possible at this time because of the complexities involved in modelling these rapidly rotating stars and the long observation runs needed to resolve the 57 2.3. Applications of CAPER to other MOST science Figure 2.5: MOST light curve of HD 217543. (Top panel) The full light curve for a total of 26.1 days. The middle and the bottom panels show expanded light curves for the portions A and B (respectively) indicated in the top panel. Solid lines indicate the fit of the 40 most significant frequencies from the frequency analysis of the full light curve (see Tab. A.2). 58 2.3. Applications of CAPER to other MOST science Figure 2.6: Fourier amplitude spectrum of the light curve of HD 217543 and the identified frequency parameters from Table A.2. The panels and the meaning of the symbols are described in Fig. 2.2. 59 2.3. Applications of CAPER to other MOST science Figure 2.7: A comparison of bootstrap distributions for parameter sets (ν6, A6, φ6) and (ν7, A7, φ7) [see Tab. A.2] for 100,000 realizations of the HD 217543 light curve. Symbols are the same as those in Fig. 2.4. These distributions are shown because the fitted frequencies are the closest to each other. In this case, the long tails on the frequency distributions suggest that these frequencies are not fully resolved. 60 2.3. Applications of CAPER to other MOST science many g-modes excited in these stars. 61 Chapter 3 MOST & HR 1217 The data analysis methods discussed in chapter 2 are applied to the HR 1217 photometry collected by the MOST satellite. This chapter begins with an outline of the photometric reduction, followed by a discussion on the determination of the rotation period of HR 1217, and then finishes with a detailed frequency analysis of the photometric time-series. The results from this chapter will be interpreted, for the first time, using a dense grid of stellar models outlined in the following chapters. 3.1 The light curve reduction MOST observed HR 1217 near continuously as a Fabry target (§ 2.1) from Nov. 5 to Dec. 4, 2004. Integration times for the target were 30 sec and more than 72,000 data points were collected. The preliminary light curve reduction was performed (by Dr. Jason Rowe) using methods described in § 2.2. Stray- light was removed by binning the data at 35 orbital periods (∼ 2.5 days), then folding the binned light curve at the orbital period and removing the running mean trend from data (e.g.,Rucinski et al. 2004). Residual trends in the data were removed by subtracting a mean value, determined by splining 0.005 day bins, while at the same time, making sure that no periodicities were introduced or subtracted near the periods of interest. The interesting 62 3.1. The light curve reduction frequency range is the range where oscillations were previously identified (§ 1.3) in HR 1217, and approximately covers 2500 to 2800 µHz. Extreme outliers were clipped from the data, leaving a total of 68,095 data points, spanning a time of 29.02 days. These reduction steps are further described below, and are presented in Fig. 3.1. There are two ∼ 0.2 day gaps in the data caused by a particle hit on the CCD and a satellite software upgrade during the observations. The resulting duty cycle of the data, based on those gaps, is 95%. If the duty cycle is calculated as the total number of potential data points that could have been taken every 30 sec over 29 days, the result would be closer to 82%. Because the oscillation periods HR 1217 are about 6 minutes, our time sampling and coverage warrant the use of the 95% effective duty cycle. Along with stray-light, the long-term, rotational modulation — with a period near 12.5 days (§ 1.2.2, 1.3) — is removed from the light curve. Fig- ure 3.1 (left panels) shows the reduction steps used to arrive at the final light curve for HR 1217 plotted in Fig. 3.2. There are three steps in the light curve reduction. The first step, shown in the top panel (labelled A), shows the light curve with only a first sigma clip performed to remove the outliers. The light curve clearly shows repeating spikes in brightness that are superimposed over the 12.5 day, near sinusoidal, stellar rotation trend in the data. These light spikes are the result of stray Earth shine illuminat- ing the CCD during satellite observations and occur every ∼ 101 minutes. The second reduction step is the removal of those light spikes (stray-light) from the data. The folding of the light curve in 35 orbital bins, and the subtraction of the mean trend at each orbital phase, results in a light curve 63 3.1. The light curve reduction shown in the middle left panel (labelled B). At this step, the light curve only shows the rotational modulation in the HR 1217 data. There are a total of three mean-light maximums and two minimums covered over the 29 days of observations. The third, and final, reduction step is the removal of this long- term, or rotational, trend (with peak-to-peak amplitude of about 4 mmag) from the data. As stated above, this was done be removing a running mean (determined by splining binned data) from the light curve. The amplitude of the light variations that remain in the data; i.e., the light variations caused by the rapid, ∼ 6 min, oscillations (§ 1.2.2, 1.3), are less than 1 mmag. The final light curve is shown in the lower, left panel (labelled C), and illustrates what looks like a beat pattern in the data. During mean light minimum, the rapid oscillations reach their maximum amplitude. This will be shown, and discussed, later in this chapter. We next check that only the stray-light and rotation trends are removed from the data. In the right panels of Fig. 3.1, the DFTs of the corresponding light curves in the left panels are plotted. In order to show the reduction of the stray-light amplitude over a large range in frequency space, we have phased the DFT to the orbital frequency (∼ 14 cycles day−1) for each of these reduction steps and showed only a window in phase space that is ± 6 cycles day−1 wide and centred on the Nth orbital harmonic. (The value of N is indicated by the colour bar below the right panels in the figure.) The fre- quency range where pulsations occur in HR 1217 is inverted in those panels. The stray-light components, and their associated aliases, are dramatically reduced in amplitude by the final reduction step. Because of the phase win- dow used in the right panels of Fig. 3.1, the frequency peak associated with 64 3.1. The light curve reduction Figure 3.1: Photometric reduction steps using the MOST data of HR 1217. The left panels show the light curves of HR 1217 in three separate reductions phases described in § 3.1. The first light curve (labelled A) shows only a preliminary sigma clipping of the data. The middle panel (labelled B) shows the removal of the stray-light component. The final, lower panel, (labelled C) shows the light curve with both stray-light and the long-term, rotation modulation removed. The rotation modulation occurs over a period of about 12.5 days. The right panels show the phased DFTs for each of the reduction steps, folded at the orbital frequency of the satellite (∼ 14 cycles day−1). Each panel shows a phase window ± 6 cycles day−1 wide centred on the Nth orbital harmonic. The Nth harmonics are colour coded, with the colour legend below the lowest panel. The amplitude of the top right panel is set to the amplitude of the peak in the DFT that is associated with the rotation of HR 1217. The DFTs are inverted for frequencies in the range of 220 to 250 cycles day−1, where the known oscillation frequencies of HR 1217 are observed. Note the reduction of the orbital components and the daily aliases (vertical dotted lines) from panels A to C. 65 3.2. On the rotation period of HR 1217 the rotation period of HR 1217 is not shown. (It is approximately 14 cycles day−1 away from the first stray-light peak.) The amplitude of the first, upper right, panel is scaled to the amplitude of the peak associated with rotation (at about 0.08 cycles day−1) in the DFT. Its amplitude is much larger than the stray-light and pulsation amplitudes. The final, zoomed, version of the time-series is presented in Fig. 3.2. The top panel shows the light curve with an apparent beat pattern having a pe- riod of about 12.5 days (§ 3.2) and the lower panels are zoomed portions (about 0.6 hr) of the data. Over-layed on those zoomed panels is the fitted sinusoidal function (parameters are found in Tab. 3.1) obtained using the CAPER software (§ 2.2). The errorbars on the photometric data are esti- mated from Poisson statistics and are about 0.1 mmag. The fit is discussed in more detail below. 3.2 On the rotation period of HR 1217 The oscillations observed in roAp stars are described well by the oblique pulsator model (§ 1.2.2). As a result, the DFTs of most roAp star light curves show a number of periodicities that are spaced from each other by the stars’ rotation frequencies. (This is illustrated later in this chapter, in Fig. 3.8, for HR 1217.) The rotation period of HR 1217 has been measured by a number of authors. Kurtz & Marang (1987) determine a rotation period of 12.4572 ± 0.0003 days from photometric data collected from a variety of sources. A slightly longer period of 12.4610 days (no uncertainty given) was determined by Mathys (1991) based on the variation of magnetic field 66 3.2. On the rotation period of HR 1217 Figure 3.2: The reduced HR 1217 light curve. The entire (∼ 29 day) time-series is shown in the top panel after being reduced using procedures outlined in § 3.1. The lower two panels are expanded views of sections labelled A and B in the top panel, respectively, and each cover approximately 0.6 hr of data. The nonlinear least squares fit using a multi-sinusoid function (with parameters listed in Tab. 3.1) is over-plotted on the data points for sections A and B. Those fit parameters are described in § 3.3. Errorbars represent Poisson errors and are about 0.1 mmag. The duty cycle for the entire light curve is about 95%. 67 3.2. On the rotation period of HR 1217 strength over time. More recent measurements by Leone & Catanzaro (2004) and Ryabchikova et al. (2005) arrive at values of 12.4582 ± 0.0006 days and 12.45877 ± 0.00016 days, respectively. Assuming that the magnetic field and photometric variations should yield the same rotation period; i.e., there are no zero-point shifts that have to be taken into account, these rotation period determinations seem to be statistically inconsistent. The MOST data on HR 1217 covers about 29 days, corresponding to ∼ 2.3 rotation periods. Over that baseline there are three photometric maximums and two minimums (see Fig 3.1). Although the coverage of the MOST light curve is near 100%; something that has never been achieved before, the small number of rotation periods observed does not allow us to determine the rotation period of HR 1217 to the level of precision quoted in the above works. As a comparison between the completeness of the coverage of the MOST observations to the coverage of the data used in the works cited above, we note that Ryabchikova et al. (2005) – having the most precise period determination – arrive at a rotation period using 39 data points collected over about 50 years of magnetic field measurements. A single sinusoid is the simplest model that gives information about am- plitude, frequency, and phase. This model was used to estimate the rotation period of HR 1217 in the studies described above. A fit to a single sinusoid function using the MOST data, with the unbinned light curve (∼ 70,000 data points), and without the rotation trend removed (middle panel of Fig. 3.1), yields the following fit parameters: 1. Amplitude, Am = 1.1634×10−2 mag, 2. Rotation frequency, νm = 0.0807 cycles day −1 (period = 12.3916 days), and 3. Phase, φm = 3.9481 rad. (The phase is referenced to the time of 68 3.2. On the rotation period of HR 1217 the first data point of the observations = HJD 2451545.00 + 1769.42 days.) These are the minimum χ2 parameters calculated by CAPER (§ 2.2). The fit of a single sinusoid can reproduce most of the variation in the light curve (see Fig. 3.3) and results in a reasonable estimate of the rotation period of the star given that there are only a few cycles observed. However, with a reduced χ2 ∼ 17.5 the quality of the single sinusoid fit is not satisfactory. The residuals of the fit have a standard deviation of 2.7 mmag compared to the individual point uncertainties of about 1 mmag. The sharpness of the minimums is illustrated through the unprecedented time coverage of the HR 1217 photometry. Introducing additional Fourier components to our fit will improve the overall quality of the fitted parameters but our single frequency fit is precise enough to determine which variations are associated with the rotation of the star, given that only a few cycles are observed. The rotation period determined from our fit is shorter than the values measured by the sources listed above. We illustrate the difference between the MOST rotation period and previously determined rotation periods by looking at the likelihood of the fits. The top panel of Fig. 3.3 shows the HR 1217 light curve with the single sinusoid fit plotted over the data (green line). (It should be noted again that the peaks at the light minimums are much sharper than those predicted from a single sinusoidal model but this simple model gives us a rotation period that can be used to identify the effects of rotation in our oscillation spectrum.) By fixing the amplitude of our fit at Am = 1.1634×10−2 mag and then varying frequency and phase parameters, we can see how changing those quantities (frequency and phase) affects the likelihood of the model fit. For each (Am, ν, φ) parameter set we construct 69 3.2. On the rotation period of HR 1217 a χ2 measure from χ2(d;Am, ν, φ, σ) = N∑ i=1 (Am sin[2piνti + φ]− d(ti))2 2σ2 (3.1) where each of the i = 1 . . . N photometric points, measured at time ti, are represented by d. A similar χ2 measure can be constructed for the single, best-fit, set of parameters (Am, νm, φm). Labelling this χ 2 m, we construct a likelihood ratio as L/L0 = exp (−∆χ2) (3.2) where ∆χ2 = χ2 − χ2m. The individual photometric errors (σ in Eq. 3.1) are taken to be unity so we can explore the variation of parameters per unit error. This is done for clarity of presentation. Individual point uncertainties (σ ∼ 1 mmag) can be used and result is a very sharply peaked likelihood function near our best fitted parameters. Figure 3.3 (lower panel) shows the results of this analysis. The MOST fit and the previously determine values are represented by white dots in the centre of the contour plot. They are consistent with each other for this ideal case of unit residual error. The larger the ∆χ2 value is, the further the fit solution is from the minimum χ2m value. As discussed in § 2.2, the uncertainty in the rotation frequency can range from about 0.25/∆T = 0.009 cycles day−1 to 1.5/∆T = 0.05 cycles day−1. Using our best fit value of νm = 0.0807 cycles day −1, and those estimates of frequency uncertainty, the MOST measurement of the rotation period is 12.4 ± 1.4 days, or, 12.4 ± 7.6 days. Clearly we do not reach the level of precision on the measured rotation period of HR 1217 (using the MOST data) that the previous studies achieve (four decimal places). 70 3.2. On the rotation period of HR 1217 Figure 3.3: Likelihood ratio for the rotation period of HR 1217. The top panel shows the time-series with the fit of a single sinusoid over-plotted in green. The parameters of the fitted sinusoid represent the minimum χ2 solution using CAPER. The lower panel presents the contours for the likelihood ratio in frequency-phase (ν-φ) space. The likelihood ratio is defined by Eqs. 3.1 and 3.2. The white dots near the centre of contour plot show the separation between the MOST determination of the rotation frequency (0.0807 cycles day−1) compared to the previous measurements discussed in this in this section (∼ 0.0803 cycles day−1). 71 3.2. On the rotation period of HR 1217 The simplest explanation for the difference in the measured precision from all sources is that each data set spans a different length of time. For example, the magnetic data used by Ryabchikova et al. (2005) covers a period of about 50 yrs while the MOST data covers about 30 days. But why is the fitted rotation period so inconsistent given that we all fit a similar model to our data sets? Each group used observations that were spread (unevenly) over time. As a result, the sharp (non-sinusoidal) amplitude minimums seen in the MOST data (Fig. 3.3) were essentially smoothed out in their observations. To explore what affect the time sampling has on the fitted rotation period from the MOST data, we bin the data in different bin sizes (0 bins, 25 bins, and 100 bins), refit the parameters, and perform a bootstrap analysis using 25,000 realizations. By binning we increase the time between individual ob- servations and essentially smooth the sharp minimums observed in the light curve. Figure 3.4 gives the results of this experiment. The lower panel of Fig. 3.4 shows the the rotation frequency for all of the binned data. The previously determined values of the rotation frequency are over-plotted (with errors smaller than the point sizes), as is the approximate resolution (§ 2.2) of 0.25/∆T = 0.009 cycles day−1 (shown as the horizontal dashed line). In the first, 0 data bin case, the fit is determined to be 0.08073 ± 0.00008 cycles day−1. The 100 data bin case yields a rotation frequency of 0.081 ± 0.002 cycles day−1 and the 25 data bin case results in a rotation period of 0.081 ± 0.007 cycles day−1. The more the data is binned, the more consistent our rotation period determination is with those that were previously determined. This experi- 72 3.2. On the rotation period of HR 1217 ment is not used to improve agreement between our observations and those previously determined values. The precisions quoted in each study are sim- ply a function of the length of the observations. The inconsistencies in the calculated values comes primarily from the differences in the completeness of the time sampling between the various data sets. Binning the data gives a quick illustration of the sensitivity of rotation period determination to the sampling of the data. The high precision obtained for the fit using all of the MOST data reflects the fact that the fit parameters are insensitive to small variations in the data. This can be thought of as a standard error on a mean determination — the more data you have, the smaller the error you measure for that mean. However, in our case, a single sinusoid does not properly model the sharp minimums of the light curve and overstating the precision of the measurement without acknowledging the pitfalls of the model could be misleading. 73 3.2. On the rotation period of HR 1217 Figure 3.4: Bootstrap distributions of the rotation frequency of HR 1217. The panels show (from top to bottom) the bootstrap (using 25,000 realizations) distributions for the phase φ, amplitude A, and frequency ν parameters of the single sinusoid fit (shown in the legend of the top panel). The filled, dashed distributions represent the light curve of HR 1217 (Fig. 3.2) binned into 25 segments. The distributions with no fill are for the light curve with 100 equal bins and the black filled distributions is the fit using all of the data. Symbols in the lower panel are for previously determined rotation frequency values (see discussion in § 3.2). Note that the frequency resolution estimated from the length of our data set (given by ∼ 0.25/∆T ) is shown as the dashed horizontal line in the lower panel and is much larger than the errors obtained for any of the rotation frequencies presented. 74 3.3. The frequency analysis We will show in the next section that determining the rotation frequency of HR 1217 to better than 10−3 µHz (9 × 10−5 cycles day−1) does not in- fluence our asteroseismic analysis because our pulsation frequencies are not determined to better than that level of precision. 3.3 The frequency analysis The identification and refinement of the rapid variations in the HR 1217 data is done using the CAPER software that was described in chapter 2. The largest peaks in the DFT are sequentially identified, are used to determine an appropriate set of amplitude, frequency and phase parameters. A set of sinusoids characterized by those parameters is then subtracted from the light curve. Parameters are refined as they are identified by nonlinear least squares fitting to the original time-series until the identified parameters have a signal- to-noise (S/N) of less than about 3.5. This corresponds to a detection limit of & 2.5σ (Breger et al. 1993, Kuschnig et al. 1997). An analysis of the MOST photometry of HR 1217 yields a series of 29 periodicities that are listed in Tab. 3.1 and are shown plotted over a DFT in Fig 3.5. (Note that appendix B has all tables presented in this section with frequencies listed in cycles day−1 rather than in µHz.) Uncertainties for the identified parameters are determined from 100,000 bootstrap samples (see § 2.2). Appendix C shows all of the bootstrap distributions for the fitted parameters. In all cases the distributions are normal (Gaussian) in nature and all of the fitted amplitudes have a S/N > 3.5 within the determined un- certainties. The uncertainties we calculate for time-series parameters using 75 3.3. The frequency analysis nonlinear least-squares methods depends on the noise of the data, correla- tions between parameters, and on the time sampling. For example, it is known that fitted phase and frequency parameters are correlated, leading to underestimated uncertainties in these parameters when calculated from a co- variance matrix (see, e.g., Montgomery & O’Donoghue 1999). Typically the uncertainties obtained by using only the diagonal elements of the covariance matrix are underestimated by a factor of 2 or more. The uncertainties de- termined for the parameters identified in the WET (Whole Earth Telescope) observations of HR 1217 (Kurtz et al., 2005) were determined from the di- agonal elements of the covariance matrix and have an average value of 0.017 µHz. The frequencies extracted from the MOST data have an average (1σ) uncertainty of 0.032 µHz. We have checked that the uncertainties of the pa- rameters calculated using the MOST data are about the same as those from the parameters identified from the WET data if the covariance matrix is used to calculate the uncertainties. Therefore, we are confident that our bootstrap analysis gives realistic error estimates that take into account correlations be- tween our frequency and phase parameters. The 1σ frequency uncertainties are listed in all data tables along with the 3σ values. As discussed in § 2.2, the uncertainty (commonly calculated as the frequency resolution of the data) on extracted periodicities from a time-series can range from 0.25/∆T to 1.5/∆T , depending on the preference of the investigators. The bootstrap analysis gives a measure of the precision of the fitted parameters and gives an indication of the frequency resolution of our data. To be conservative we will generally make reference to our 3σ bootstrap uncertainties when discussing the fitted frequency parameters found in this thesis. 76 3.3. The frequency analysis The MOST mission has uncovered the following new frequencies in the spectrum of HR 1217 (Tab. 3.1): ν1 = 2603.832 µHz, ν8 = 2684.832 µHz, ν9 = 2685.659 µHz, ν13 = 2718.984 µHz, ν17 = 2723.469 µHz, ν20 = 2756.035 µHz, and ν21 = 2756.925 µHz. The average difference between the frequencies observed by Kurtz et al. (2005) for HR 1217 and the MOST results is 0.08 µHz. The average 3σ error on that difference is ± 0.09 µHz. This indicates that all frequencies identified by both the ground-based and space- based campaigns are consistent with each other and shows that those periods common to each of the data sets are stable since the initial ground-based col- laboration of Kurtz et al. (1989). In addition, the frequencies near ∼ 2790 µHz (see Fig. 1.7) that were present in the WET data of Kurtz et al. (2005), but were not found in the Kurtz et al. (1989) data, are identified in the MOST data. (ν22 through ν27 in Tab. 3.1) This suggests that those frequen- cies are stable over a baseline of at least 4 years. A direct comparison between the photometric amplitudes observed by MOST and Kurtz et al. (2005) is not possible because the data sets were measured using different photometric filters that average over different parts of the visible spectrum. However, the MOST data shows the frequency (∼ 2720 µHz) having the largest corresponding amplitude (∼ 760 µmag) is still the dominant peak in the DFT of the MOST and the WET’s 2000 data. This is a change from the 1986 global campaign data (Kurtz et al., 1989) where the largest amplitude of 1.069 ± 0.018 mmag occurred at a frequency of 2687.60 ± 0.06 µHz. Although amplitude differences have been shown between the campaigns of Kurtz et al. (1989) and Kurtz et al. (2005), the net power (sum of the square amplitudes) is conserved. The precision of 77 3.3. The frequency analysis the amplitude determinations from the WET data was 14 µmag. The most precise amplitude determined from the MOST data is 6 µmag; a value that is less than half of the quoted precision of the WET data, even without taking into account the differences in the precision calculations. 78 3.3. The frequency analysis Figure 3.5: (Top Panel) The Fourier amplitude spectrum of the HR 1217 light curve. Filled circles with 3σ error bars are the fitted parameters (see Tab. 3.1). The inverted dash–dot line is the residual amplitude spectrum obtained after the fit was subtracted from the light curve. (Lower Panel) The signal–to–noise (S/N) of the identified periodicities with 3σ uncertainties estimated from both the fitted amplitudes and the mean of the amplitude spectrum. The light grey line represents the window function of the data centred on the frequency with the largest amplitude and scaled to the maximum S/N for clarity. 79 3.3. T h e freq u en cy an aly sis Table 3.1: Frequency model parameters identified using the MOST data. Both 1 and 3σ uncertainties on the fitted parameters are estimated from 100,000 bootstrap realizations. The phases are referenced to the time of the first observations of HR 1217 (= HJD 2451545.00 + 1769.42 days). See appendix B for frequencies in units of cycles day−1. # ν [µHz] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 2603.832 39 2.22 3.8 ± 0.028 0.080 6 15 0.34 0.93 2 2618.600 50 3.29 4.2 ± 0.024 0.066 6 16 0.27 0.72 3 2619.495 125 5.70 9.2 ± 0.015 0.039 6 16 0.11 0.29 4 2620.440 68 1.68 6.3 ± 0.028 0.074 6 16 0.28 0.74 5 2651.998 96 3.36 7.5 ± 0.017 0.045 6 16 0.14 0.35 6 2652.924 365 5.56 24.2 ± 0.004 0.011 6 16 0.04 0.10 7 2653.883 92 2.07 7.8 ± 0.012 0.033 6 16 0.10 0.27 8 2684.832 43 2.02 3.7 ± 0.054 0.145 6 16 0.38 1.10 9 2685.659 31 2.83 3.1 ± 0.091 0.241 9 26 0.55 2.18 10 2686.560 123 1.26 8.9 ± 0.011 0.029 7 17 0.10 0.26 11 2687.420 467 4.46 29.7 ± 0.004 0.009 6 16 0.03 0.08 12 2688.364 139 1.13 10.9 ± 0.009 0.023 6 16 0.07 0.18 13 2718.984 51 4.50 5.8 ± 0.051 0.137 6 16 0.25 0.68 14 2719.957 319 6.04 23.3 ± 0.006 0.014 6 16 0.05 0.13 continued on next page 80 3.3. T h e freq u en cy an aly sis Table 3.1: continued # ν [µHz] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 15 2720.914 764 1.83 41.6 ± 0.002 0.005 6 16 0.02 0.05 16 2721.822 177 4.59 12.4 ± 0.011 0.027 6 16 0.10 0.26 17 2723.469 41 2.08 3.1 ± 0.034 0.097 6 16 0.39 1.10 18 2754.470 110 1.81 8.5 ± 0.016 0.043 6 16 0.17 0.45 19 2755.393 118 3.45 8.9 ± 0.015 0.039 6 16 0.17 0.45 20 2756.035 67 2.75 6.2 ± 0.038 0.107 6 16 0.36 1.05 21 2756.925 39 5.14 3.1 ± 0.045 0.159 6 17 0.53 1.67 22 2787.799 24 4.72 2.7 ± 0.127 0.317 6 16 0.90 2.59 23 2788.872 37 0.24 3.6 ± 0.034 0.096 6 16 0.29 0.80 24 2789.673 33 5.56 3.3 ± 0.047 0.157 7 16 0.38 1.16 25 2790.406 37 2.85 4.4 ± 0.073 0.196 7 18 0.45 1.24 26 2791.384 82 5.00 6.8 ± 0.033 0.085 7 17 0.29 0.74 27 2792.249 57 1.47 5.6 + 0.054 0.206 9 24 0.35 1.06 . . . . . . 2792.249 57 1.47 5.6 − 0.054 0.178 9 25 0.35 1.06 28 2805.650 84 1.10 6.9 ± 0.015 0.038 6 15 0.15 0.37 continued on next page 81 3.3. T h e freq u en cy an aly sis Table 3.1: continued # ν [µHz] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 29 2806.592 81 2.09 7.1 ± 0.018 0.048 6 16 0.16 0.44 82 3.3. The frequency analysis Table 3.2 lists the calculated differences between the various frequencies identified in the MOST data. We adopt the symbol δ† as a general spacing between two frequencies. There are 19 potential frequency spacings having a value of δ† ∼ 0.9 µHz. In the previous section (§ 3.2) we identify the rotation period of HR 1217 to be ∼ 12.5 days, corresponding to a rotation frequency of 0.08 cycles day−1 or 0.93 µHz. The average of those 19 poten- tially rotationally split frequencies is 0.924 ± 0.047 µHz, and is consistent with the observed rotation rate of HR 1217. As stated at the end of § 3.2, the rotationally split frequencies measured in the MOST data do not reach the same level of precision (. 10−3 µHz) as that on the rotation rate quoted by other authors (see § 3.2). Using one author’s value for the rotation rate over another’s does not change the frequencies determined from the MOST data. Frequency spacings δ†1 through δ † 5, δ † 8 and δ † 26 of Tab. 3.2 show alternating values of ∼ 33.5 and ∼ 34.5 µHz. These spacings were also identified in previous studies of HR 1217 (e.g.,Kurtz et al. 2005) and are interpreted as half of the large separation, ∆ν. In chapter 1 (§ 1.1.1) we state that the large spacing is related to the mean density of the star by Eq. 1.14. Because of this, Matthews et al. (1999) was able estimate the asteroseismic parallax of HR 1217 and showed that it was consistent with the measured Hipparcos parallax (§ 1.3). These consistent measurements on the position of HR 1217 in the HR diagram will be used to define the relevant model parameter space in § 4.1.1. The alternating values of ∼ 33.5 and ∼ 34.5 µHz suggests that the observed modes are alternating between even and odd angular degree. Another set of recurring frequency spacings are ∼ 1.5 and ∼ 2.5 µHz (δ†19 through δ†23). Main sequence models of A-type stars (e.g., § 4.2) have small 83 3.3. The frequency analysis spacings (§ 1.1.1) that are about 3 µHz and higher. This small spacing can potentially constrain the age of the star because its value is most sensitive to changes in the composition of the stellar core as the star evolves. Are these observed spacings actually rotational splittings? Consider the average rotation frequency derived from all of the (near rotationally split) frequency spacings in Tab. 3.2. Its value is 0.924± 0.047 µHz and multiplying it by 2 and 3 yields 1.848 ± 0.094 µHz and 2.772 ± 0.141 µHz, respectively. This suggests that δ†19 and δ † 23 are potentially rotational components in the set of frequencies ν8 through ν12 (Tab. 3.1). However, the frequency spacing between ν10 and ν11 (δ † 15) is not consistent with a rotational spacing, even using 3σ uncertainties. Likewise, the frequency spacing between ν19 and ν20 (δ†29) is the most inconsistent with a rotational splitting in Tab. 3.2. So if these are related to rotation there are other anomalies between adjacent frequencies that need to be explained. That, coupled with observations of Kurtz et al. 2005, who also see δ†23 ∼ 2.5 µHz, leads us to believe that these additional spacings cannot be explained away as rotational components in the oscillation spectrum. Because of the low amplitudes associated with those potential small separations, including them in our nonlinear fitting routine does not influence the other parameters in the fit. It should also be noted that interpreting these spacings as small separations may not be appropriate in strongly magnetic stars (e.g., § 1.3 and 4.3). Kurtz et al. (2005) discovered a frequency spacing of ∼ 14 µHz that was explained by Cunha (2001) as a potentially magnetically perturbed mode. This spacing has also been identified in the MOST photometry along with another possible magnetic spacing of 15.663 µHz (δ†6) at the opposite end 84 3.3. The frequency analysis of the spectrum. These modes spacings do not conform to the alternating asymptotic pattern that is observed between the other periodicities in the MOST data and will be modelled in the following chapters of this thesis. 85 3.3. T h e freq u en cy an aly sis Table 3.2: The frequency separations δ† of the frequencies identified in the MOST data. Columns with headers νi - νj use i and j to denote the frequency number from Tab. 3.1. The average rotation separation (spacings 9 . . . 18 & 27 . . . 35) is 0.924 ± 0.047 µHz (3σ = 0.130 µHz). See appendix B for frequencies in units of cycles day−1. # νi - νj δ † [µHz] σδ† 3σδ† # νi - νj δ† [µHz] σδ† 3σδ† 1 6 - 3 33.429 0.015 0.041 19 11 - 8 2.588 0.054 0.145 2 11 - 6 34.495 0.006 0.014 20 11 - 9 1.761 0.091 0.242 3 15 - 11 33.494 0.004 0.010 21 17 - 15 2.555 0.034 0.097 4 19 - 15 34.479 0.015 0.040 22 21 - 19 1.532 0.047 0.164 5 23 - 19 33.479 0.037 0.104 23 26 - 23 2.512 0.047 0.128 6 3 - 1 15.663 0.032 0.089 24 28 - 26 14.266 0.036 0.093 7 17 - 8 38.637 0.064 0.174 25 17 - 9 37.811 0.097 0.260 8 21 -17 33.456 0.056 0.187 26 26 - 21 34.459 0.056 0.181 9 3 - 2 0.895 0.029 0.077 27 16 - 15 0.908 0.011 0.028 10 4 - 3 0.945 0.032 0.084 28 19 - 18 0.923 0.022 0.058 11 6 - 5 0.926 0.018 0.046 29 20 - 19 0.642 0.041 0.114 12 7 - 6 0.958 0.013 0.035 30 21 - 20 0.890 0.059 0.192 13 9 - 8 0.827 0.106 0.281 31 23 - 22 1.072 0.131 0.332 continued on next page86 3.3. T h e freq u en cy an aly sis Table 3.2: continued # νi - νj δ † [µHz] σδ† 3σδ† # νi - νj δ† [µHz] σδ† 3σδ† 14 10 - 9 0.901 0.091 0.243 32 24 - 23 0.802 0.058 0.184 15 11 - 10 0.860 0.012 0.030 33 26 - 25 0.978 0.080 0.213 16 12 - 11 0.945 0.010 0.025 34 27 - 26 0.866 0.063 0.197 17 14 - 13 0.973 0.051 0.138 35 29 - 28 0.942 0.023 0.061 18 15 - 14 0.957 0.006 0.015 87 3.3. The frequency analysis In § 1.2.2 it was noted that the roAp stars show amplitude modulation that is a direct result of modes being excited preferentially near a mag- netic pole that is inclined to both the rotation axis of the star and to the line-of-sight of the observer. The amplitude modulation of the sinusoidal components identified in the MOST data is shown in Fig. 3.6. The time- series is divided into 30 segments and the amplitudes and phases are refitted at a fixed frequency for the modes identified in Tab. 3.1. Rotation modu- lation is clearly seen for the periodicities with the three largest amplitudes but the other frequencies have corresponding amplitudes that are too small to properly see the variation. In general, all modes follow the apparent “beat” period of ∼ 12.5 days in the data and show maximum amplitudes at minimum brightness (maximum magnitude) of the light curve. This is also confirmed by a time-frequency analysis presented in Fig. 3.7. In that figure, the DFT is calculated for a 4 day segment of the data and then a moving average DFT is made by advancing the data segment by 0.1 days and then recalculating the DFT. There is clear substructure surrounding the highest amplitude frequency. Because the time coverage of the MOST data is near 100%, we can see the amplitude modulations in the light curve of HR 1217 in an unprecedented way. 88 3.3. The frequency analysis Figure 3.6: Normalized pulsation frequency amplitudes as a function of time. The HR 1217 time-series is broken into 30 segments and the amplitudes of the frequencies listed in the figure are then refitted for each data segment. Plotted are the fitted amplitudes per 3σ error. The light curves of the HR 1217 data with and without the rotational modulation removed are shown above the computed amplitudes using an arbitrary scaling. Note that the amplitude maxima correspond to the maximum magnitudes (minimum brightness) of the light curve with the rotation modulation and also to the maximum “beat” amplitude of the light curve with the rotation modulation removed. 89 3.3. The frequency analysis Figure 3.7: Time-frequency plot of the HR 1217 data. The top panel is the DFT of the HR 1217 data with the red line representing the median noise in the DFT over the data range presented above. The lower plot is a moving box average DFT of the data. The box width is 4 days and it is moved at 0.1 day intervals. The contours represent S/N of the moving box. 90 3.3. The frequency analysis Finally, we discuss the resolution of the observed periodicities in the MOST data and the potential for the identification of other, closely spaced frequencies. During the frequency analysis there were a number of frequen- cies identified that were spaced from their neighbour frequencies by about one half of the rotation frequency of HR 1217. To assess the resolution of these frequencies we perform a bootstrap analysis (using 100,000 realiza- tions) on the data but keep these frequencies as part of the fitted solution. In order to converge the fit with these additional frequencies we had to lower the convergence criteria on the χ2 fitting routine in CAPER by 2 orders of magnitude, allowing the fit to relax to its final solution using a lower tol- erance. The fit with all of the components is given in Tab. 3.3 (frequencies having numbers greater than 30 are the unresolved components). The results for the top 5 most prominent (largest amplitude) frequency groups is shown in Fig. 3.8 with the fits using both the resolved and unresolved frequencies over-plotted on the DFT. The addition of the extra, unresolved frequencies causes the fit to become unstable and the new frequencies clearly interact with the previously identified, resolved, parameters of Tab. 3.1. The identifications of the unresolved frequencies that are about half of the rotation period of the star are highly suspect given the resolution of the data set. While there may be independent frequencies still to be discovered in the data, at this time these are more likely a convolution of closely spaced modes (spaced by the rotation period of the star) that are amplitude modulated. This interpretation is in line with that of Kurtz et al. (2005). 91 3.3. The frequency analysis Figure 3.8: Comparison between resolved and unresolved frequencies. A bootstrap analysis using 100,000 realizations is shown for the five largest amplitude components in the HR 1217 data. Black points are for the final frequency solution (Tab. 3.1) and the red points are the unresolved frequency components found in the data (Tab. 3.3). Green dots in the top panel show the bootstrap points. Note that the frequencies common to both Tabs. 3.1 and 3.3 agree within the uncertainties while the additional frequencies tend to adversely affect the fit, and in some cases overlap each other. 92 3.3. T h e freq u en cy an aly sis Table 3.3: Unresolved frequency model parameters identified using the MOST data. Both 1 and 3σ uncertainties on the fitted parameters are estimated from 100,000 bootstrap realizations. Frequencies with numbers greater than 30 are the unresolved components. The phases are referenced to the time of the first observations of HR 1217 (= HJD 2451545.00 + 1769.42 days). See appendix B for frequencies in units of cycles day−1. # ν [µHz] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ 1 2603.831 38 2.22 ± 0.027 5 0.26 0.077 16 0.11 2 2618.611 45 3.25 ± 0.023 5 0.21 0.070 17 0.50 3 2619.440 106 5.96 ± 0.025 12 0.20 0.055 15 0.75 4 2620.384 64 2.07 ± 0.042 5 0.34 0.083 24 1.42 5 2652.012 95 3.20 ± 0.014 5 0.16 0.049 17 0.36 6 2652.923 364 5.59 ± 0.004 5 0.04 0.013 16 0.08 7 2653.863 81 2.13 ± 0.020 9 0.14 0.060 17 0.31 8 2684.876 43 1.67 ± 0.044 5 0.35 0.084 30 0.95 9 2685.682 40 3.02 ± 0.049 10 0.44 0.153 16 0.11 10 2686.536 111 1.37 ± 0.020 10 0.10 0.042 36 0.12 11 2687.416 475 4.50 ± 0.004 9 0.05 0.009 16 0.23 12 2688.340 131 1.30 ± 0.018 8 0.17 0.036 16 0.88 13 2718.943 54 5.18 ± 0.042 8 0.52 0.115 16 0.50 continued on next page 93 3.3. T h e freq u en cy an aly sis Table 3.3: continued # ν [µHz] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ 14 2719.956 320 6.11 ± 0.004 5 0.06 0.012 28 0.58 15 2720.920 761 1.81 + 0.005 8 0.02 0.014 36 0.12 . . . . . . 2720.920 761 1.81 − 0.005 8 0.02 0.013 36 0.12 16 2721.852 184 4.45 ± 0.014 6 0.13 0.030 16 0.23 17 2723.448 31 2.33 ± 0.028 9 0.27 0.085 16 0.88 18 2754.444 108 1.99 + 0.022 5 0.18 0.049 16 0.50 . . . . . . 2754.444 108 1.99 − 0.022 5 0.18 0.049 16 0.45 19 2755.361 112 4.02 ± 0.028 6 0.22 0.066 28 0.58 20 2756.149 53 1.75 + 0.044 12 0.40 0.412 25 1.29 . . . . . . 2756.149 53 1.75 − 0.044 12 0.40 0.114 25 1.29 21 2756.898 33 5.35 ± 0.031 6 0.29 0.115 16 1.24 22 2787.801 24 4.73 ± 0.047 5 0.44 0.143 15 1.37 23 2788.871 36 0.19 ± 0.030 5 0.28 0.089 15 0.84 24 2789.676 34 5.48 ± 0.036 5 0.31 0.107 15 0.95 25 2790.395 35 3.02 ± 0.034 5 0.30 0.100 16 0.87 26 2791.377 82 5.04 ± 0.015 5 0.14 0.045 16 0.41 continued on next page 94 3.3. T h e freq u en cy an aly sis Table 3.3: continued # ν [µHz] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ 27 2792.243 58 1.50 ± 0.025 5 0.22 0.078 16 0.69 28 2805.651 84 1.10 ± 0.013 5 0.12 0.038 15 0.36 29 2806.593 82 2.09 ± 0.013 5 0.13 0.039 15 0.37 30 2619.628 36 4.63 + 0.157 12 0.72 0.286 24 1.65 . . . . . . 2619.628 36 4.63 − 0.157 12 0.72 0.286 24 4.38 31 2620.823 31 5.19 ± 0.099 6 0.54 0.190 17 1.50 32 2652.948 10 2.08 + 1.073 8 2.00 1.968 15 4.13 . . . . . . 2652.948 10 2.08 − 1.073 8 2.00 1.968 10 2.08 33 2653.374 51 3.47 ± 0.034 5 0.31 0.113 17 0.99 34 2686.004 57 2.84 ± 0.033 8 0.27 0.139 24 1.20 35 2687.912 55 2.10 ± 0.042 7 0.47 0.100 18 0.92 36 2718.852 25 3.63 + 0.112 8 1.22 0.335 27 2.63 . . . . . . 2718.852 25 3.63 − 0.112 8 1.22 0.335 25 2.73 37 2720.730 47 2.06 + 0.065 16 0.51 0.195 47 1.68 . . . . . . 2720.730 47 2.06 − 0.065 16 0.51 0.195 44 1.68 38 2721.327 77 3.06 ± 0.035 10 0.20 0.084 21 0.76 continued on next page 95 3.3. T h e freq u en cy an aly sis Table 3.3: continued # ν [µHz] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ 39 2754.877 48 3.48 ± 0.059 6 0.49 0.138 18 1.28 40 2755.633 55 6.07 + 0.086 17 0.49 0.291 42 1.76 . . . . . . 2755.633 55 6.07 − 0.086 17 0.49 0.291 42 2.57 41 2792.612 28 0.19 ± 0.049 5 0.46 0.148 15 1.39 96 3.3. The frequency analysis The frequencies identified in the MOST data on HR 1217 are consis- tent with those previously discovered in two earlier ground-based campaigns. However, because of the unprecedented precision of the MOST data, we were able to identify a number of additional frequencies that need to be interpreted in the context of models of magnetic stars. The following two chapters describe the modelling procedure used in this work and the efforts put forth to theoretically match and interpret the MOST observations. Like the MOST data, the parameter space appropriate to the oscillations of HR 1217 is explored in an unequalled, and unique, way. 97 Chapter 4 Modelling stellar oscillations In this chapter a grid of evolution and pulsation models of A-type stars is calculated exploring a variety of parameters, constrained by the observed properties of HR 1217. The effect of a magnetic field on the resulting pul- sation frequencies is estimated, for the first time, using an extensive grid of stellar models. The matching of models to the observed periodicities of HR 1217 is described in chapter 5. 4.1 Stellar evolution models Stellar models are calculated using the Yale Rotating Evolution Code (YREC) in its non-rotating configuration. YREC solves the mechanical, conservation, and energy transport equations of stellar structure using the Henyey relax- ation scheme (Henyey et al. 1959; also see Larson & Demarque 1964 and Kippenhahn & Weigert 1994). These equations are standard and may found in a number of texts on stellar structure (e.g, Kippenhahn & Weigert 1994). The nuclear cross sections, element diffusion, the equation of state and opac- ities available to YREC were discussed recently by Demarque et al. (2008). The code is flexible enough to allow automatic computation of a number of stellar evolutionary tracks with the convective treatment or atmospheric pa- rameters of one’s choice. For example, Fig. 1.5 shows a number of evolution 98 4.1. Stellar evolution models tracks on an HR diagram that were computed using YREC. Tracks of stellar models with parameters suited to those observed in HR 1217 are shown in Fig. 4.1 and are discussed in § 4.1.1. The density of a stellar model is related to the other material functions (temperature and pressure) that are calculated from the stellar structure equations using a tabulated equation of state (EOS). YREC interpolates be- tween OPAL EOS tables (Rogers1986, and Rogers, Swenson, & Iglesias 1996) with scaled solar composition (Grevesse & Noels, 1993) in order to obtain the appropriate densities for a particular model. These EOS tables men- tioned above are commonly used, but other tables are also available. Guzik & Swenson (1997) discuss the effects that different EOS tables have on the calculated solar pulsation frequencies. They find the effect is less than about 1 µHz. This type of detailed comparison between oscillation frequencies and different input physics is most fruitful for helioseismic investigations because the global parameters of the Sun are well known. In asteroseismic inves- tigations we make the assumption that the physics included and tested in helioseismic investigations are valid for other stars. The energy transport equations of stellar structure require information about the opacity of the stellar plasma. These are interpolated from two separate tables depending on the local temperature in the model. If the temperature of a mass shell in a given model is greater than logT = 4.1, the OPAL opacities (Iglesias & Rogers, 1996) are used. The low-temperature (molecular) opacity tables of Alexander & Ferguson (1994) are used to obtain the opacity for temperatures less than logT = 4.1. Linear interpolation is used to smoothly transition between the two tables. 99 4.1. Stellar evolution models The models calculated in this work are evolved from the zero age main sequence (ZAMS) to a point near the end of the main sequence. Each model generated has ∼ 3000 shells evenly distributed over the interior, envelope and the atmosphere. The model interior represents approximately the inner 99% of the model by mass while the envelope makes up the other 1%. The over- lying atmosphere is calculated assuming the Eddington grey approximation, which is a frequency independent temperature-optical depth (T -τ) relation. YREC is used in its non-rotating configuration even though real stars do rotate. Because the roAp stars are generally moderate rotators, and because the observed pulsation modes are high-order, rotation has little affect on the frequencies. Shibahashi & Saio (1985) estimate the effect of rotation on the calculated pulsation modes to be < 10−3 (frequency units) so the additional complexity of adding rotation to the calculations would not yield useful information. 4.1.1 Defining the parameter space Stellar models are calculated for a given mass, luminosity, effective temper- ature and element composition (specified as a mass average). In most cases spectroscopic observations give information about the effective temperature and the heavy metal (elements with atomic numbers > 2) content of a star. The luminosity can be estimated for stars using parallax measurements and the mass is only tightly constrained if the star belongs to a binary from which an orbital solution can be derived. There is also the potential to use asteroseismic observations to further constrain the evolutionary status of a star through parameters like the large spacing ∆ν described in chapter 1 (see 100 4.1. Stellar evolution models Matthews et al. 1999). Even with information on all observable quantities, the standard treatment of convective energy transport (mixing length theory of Böhm-Vitense 1958) adds a free parameter known as the mixing length parameter α. This parameter sets the number of pressure scale-heights a convective element rises before releasing its heat to the surrounding plasma. Traditionally, α is calibrated by calculating a solar model, having the same input physics as the model grid that is being calculated, that reproduces the solar radius and luminosity. For example, a value of α = 1.6 is appropriate for a solar model without element diffusion or overshoot (Guenther, 2002). Cunha et al. (2003) examined a set of stellar models for HR 1217 in hopes of determining its evolutionary status. Their results are complementary to those of this thesis and there is an overlap in the choice of model parameters presented below. The model physics described above was also chosen to be similar to their choices for an easier comparison. This study explores the magnetic effects on the pulsation frequencies for models of HR 1217 for the first time and significantly extends the number of models and the breadth of the parameter space explored when compared to earlier asteroseismic studies on HR 1217. The mass of HR 1217 was estimated previously by a number of authors using a variety of assumptions. Shibahashi & Saio (1985) estimated a mass of 2.0 ± 0.5 M. (This mass was also adopted by Matthews et al. 1999.) Wade (1997) gives a tighter constraint of 1.8 ± 0.3 M assuming that HR 1217 is a rigid rotator and the recent results of Kochukhov & Bagnulo (2006) find 1.55 ± 0.03 M based on a statistical analysis of a sample of Ap stars. Each of these mass determinations are based on stellar evolution models that 101 4.1. Stellar evolution models come close to the observed luminosity and effective temperature of HR 1217. The models calculated in this study have masses of 1.3 to 1.8 M in 0.05 M steps. The effective temperature of HR 1217 is estimated from two sources. Ryabchikova et al. (1997) use their spectral synthesis code to estimate an effective temperature of Teff = 7250 K (with no uncertainty provided). The other estimate comes from the Strömgren photometry of Moon & Dworetsky (1985). Matthews et al. (1999) used those measurements to estimate an effective temperature of 7400 ± 100 K. We adopt a value of Teff = 7400+100−200 K to cover the span in temperature determined in those earlier works. This is identical to the effective temperature estimate used by Cunha et al. (2003) and is consistent with a more recent estimate of Teff = 7350 K (Lüftinger et al., 2008) using the same methods as Ryabchikova et al. (1997). The luminosity of HR 1217 is determined from its Hipparcos (Perryman et al., 1997) parallax. Matthews et al. (1999) derive a luminosity of 7.8 ± 0.7 L from a parallax of 20.41 ± 0.84 mas (milli-arc seconds). An asteroseismic parallax can also be estimated from Eq. 1.15. Using the above estimates on Teff and mass, and assuming a large spacing of ∼ 68 µHz we arrive at a luminosity of 8.2+1.6−1.5 L. If half the large spacing (∼ 34 µHz) is used, the asteroseismic luminosity is 20.7+4.0−3.8 L. The latter value is inconsistent with the more accurate Hipparcos value. All physical parameters used in our model grid are also consistent with those determined by Kochukhov & Bagnulo (2006). A difficulty arises when determining the metallicity for the Ap stars be- cause the metal content estimated from spectroscopic observations can dra- 102 4.1. Stellar evolution models matically change over the magnetic (rotation) phase of the star. There is also no reason to believe that the interior metal content of an Ap star is reflected by the observed, patchy, surface values. The spectroscopy of Ryabchikova et al. (1997) indicates [Fe/H] ≈ 0.32 ± 16% (an average of Fe I and II lines at both Bmax and Bmin). Assuming Fe is a tracer of the interior metal con- tent of a star, a heavy metal mass fraction of Z = 0.008 is calculated. If it is (naively) assumed that the average of all of the metal abundances from Ryabchikova et al. (1997) for each of the magnetic maximum and minimum phases represent the interior heavy metal content of HR 1217, we estimate a value of Z ∼ 0.022. Thus, the inferred interior Z for HR 1217 can vary over a large range depending on how the surface metallicity is used. For completeness we use Z = 0.008 to 0.022 in steps of 0.002 in our evolutionary models and adopt a hydrogen mass fraction of X = 0.700 to 0.740 in steps of 0.020. The large extent of our parameter space in composition encompasses the uncertainties from the observed [Fe/H] above. Finally, mixing length parameters of α = 1.4, 1.6, and 1.8 are used in the model grid. If a star possesses a convective envelope, α may be used to set the adiabatic temperature gradient at the base of the convective zone. Luminosity is a function of temperature and radius so varying α will slightly change the radius of the model because it changes the temperature at the base of the convective zone (Guenther et al., 1992). Because the envelopes of A stars are essentially radiative, any structural changes from the different values of α are small when compared to the effects introduced by varying the other model parameters. All models are plotted on the theoretical Hertzsprung-Russell (HR) di- 103 4.1. Stellar evolution models agram in Fig. 4.1. The Hipparcos luminosity and the effective temperature determination discussed above constrain the location of HR 1217 in the HR diagram. The error box outlining that area in the HR diagram is shown as a black box in Fig. 4.1. There were approximately 105 stellar models generated in the model grid. Detailed models (containing information necessary for the pulsation analysis; see § 4.2) were output in age steps of 0.05 Gyr from the ZAMS, decreasing the number of models by an order of magnitude. Of those models, a total of 569 fall within the HR 1217 error box. Those 569 models are used in the pulsation and magnetic perturbation (§ 4.3) calculations. Figure 4.2 shows histograms of the mass, radius, and composition (Z/X) of the models. The mass is nearly evenly distributed, with slightly more models at 1.65 M. Composition is also nearly uniform in its distribution. For comparison, in the Grevesse & Noels (1993) mixture (Z/X) = 0.0245 and the metallicity is Z = 0.0175. The lower panel of Fig. 4.2 plots the mean sound speed of the models, offset by the mixing length parameter α, as a function of Z/X. All models are split evenly for each of the mixing length values used in the calculations (α = 1.4, 1.6, and 1.8). The large spacing of the models (∆ν ∼ (M/R3)1/2) ranges from ∼ 57–81 µHz. Some physical characteristics of the models that are closest to the four corners of the errorbox, along with the model that is closest to the centre of the box, are listed in Tab. 4.1. Those models show a range of physical parameters and will be used in the following sections as examples illustrating the properties of the oscillation spectra and magnetic perturbations to those model spectra. 104 4.1. Stellar evolution models Figure 4.1: A theoretical HR diagram showing the extremes of the explored parameter space. The black error box represents the Hipparcos luminosity and the effective temper- ature appropriate for HR 1217 (see § 4.1.1 for details). Contours show lines of constant ∆ν and five evolutionary tracks are selected showing how position on the HR diagram is affected by changing composition and mass. The metallicity has the greatest effect on the evolutionary status of the stellar model. A track shifts toward the lower right of the HR diagram (making the stars cooler and less luminous) by increasing Z. 105 4.1. Stellar evolution models Figure 4.2: Histograms of model properties. The three smaller panels at the top of the figure show how mass (solar units), radius (solar units) and composition (Z/X) are dis- tributed within the Hipparcos error box (see Fig. 4.1). The lower panel plots mixing length α plus the ratio of the average dynamical speed squared to that of the sun as a function of composition. That quantity is plotted to illustrate that the models are nearly equally split in mixing length values (α = 1.4, 1.6 and 1.8). 106 4.1. S tellar evolu tion m o d els Table 4.1: Some properties of selected models. Models A and B have luminosities and effective temperatures that are the closest to the upper left and right corners (respectively) of the error box shown in Fig. 4.1. Model C is the closest to the centre of the error box while models D and E are nearest to the lower left and right corners, respectively. # M/M X Z Age (Gyr) α Mconv.core/M? Mconv.env./M? log10 (Teff) log10 (L/L) log10 (R/R) ∆ν (µHz) Acoustic cut-off (µHz) A 1.30 0.700 0.008 2.15 1.8 0.000 0.0 3.8577 0.9284 0.2727 57.60 1393.52 B 1.45 0.720 0.008 1.35 1.8 0.118 0.0 3.8740 0.9289 0.2403 66.96 1703.88 C 1.55 0.740 0.012 1.10 1.8 0.142 0.0 3.8691 0.8912 0.2312 71.53 1908.77 D 1.40 0.700 0.010 1.40 1.8 0.119 0.0 3.8576 0.8518 0.2346 67.80 1790.08 E 1.65 0.740 0.018 0.45 1.8 0.148 0.0 3.8730 0.8523 0.2039 80.76 2280.42 107 4.2. Pulsation models Convection can influence both the evolutionary status and pulsation prop- erties of a star, and a near-surface convective zone has implications for the surface chemical inhomogeneities of Ap stars. Models with higher Z/X tend to have convective cores that last longer than those with a smaller Z/X ratio. Models with a lower Z value tend to be the oldest models in the sample. This is also true for higher X, but the relation is not as strong. The mass of the convective envelope is insignificant in our models. Only a small fraction of the models exhibit convection over a large fraction of the stellar envelope (∼ 1% of the model mass). These models should be more efficient at mixing away patches of chemical inhomogeneities, and as such, we may (potentially) rule out low mass, evolved, and high Z models as candidates Ap stars. Of the models shown in Fig. 4.1, those with a ∆ν ∼ 34 µHz are higher mass models and have Z values greater than the solar value. The change in the heavy metal content of the models produces the largest change on the position of a model on the HR diagram. Many of the models in the errorbox have a mid-range mass of about 1.6 M and a metallicity ranging from Z ∼ 0.014–0.020. Models that seem to best reproduce the observed large spacing are those with a mass & 1.5 M and a near solar composition of Z/X ∼ 0.024. 4.2 Pulsation models The pulsation calculations are carried out using the nonadiabatic pulsation package of David Guenther (JIG, Guenther 1994). JIG solves the six lin- earized, nonadiabatic equations of nonradial stellar oscillations using the 108 4.2. Pulsation models Henyey relaxation method (Henyey et al., 1959). These six complex par- tial differential equations describe the radial dependence of the vertical and horizontal displacement vectors, the Lagrangian perturbations to the en- tropy and the radiative luminosity, as well as the Eulerian perturbations to the gravitational potential and its radial derivative. The time dependence of the perturbed quantities is periodic through the function exp(iσt), where the σ is the complex eigenfrequency. A complete introduction to both the adiabatic and nonadiabatic, nonradial oscillation equations can be found in Unno et al. (1989) or Christensen-Dalsgaard (2003). (See chapter 1 for a short introduction to the subject of non-radial oscillations.) In this thesis we avoid the complications associated with modelling the stability of calculated eigenmodes. As such, we only make use of the adia- batic frequencies calculated by the JIG code. In the adiabatic approximation perturbations to the energy equation (luminosity) are ignored and the sound speed is given by c2s = dp dρ = Γ1 p ρ (4.1) Γ1 = (∂ ln p/∂ ln ρ)S is the first adiabatic exponent. The linear, adiabatic, nonradial stellar pulsation equations represent a fourth-order system of equa- tions with two boundary conditions at the centre of the model and two con- ditions at the models surface. (See § 1.1.1 for the pulsation equations.) The presence of an eigenvalue makes the system nonlinear and a normalization of one of the eigenfunctions (radial displacement ξ) at the surface is used to close the system. We present an alternative method of calculating normal modes of stellar models in § 4.2.1 so, for completeness, we list the adiabatic 109 4.2. Pulsation models equations as given in Unno et al. (1989). Namely, 1 r2 d dr (r2ξ) = gξ c2s − ( 1− `(`+ 1)c 2 s r2σ2 ) p′ ρc2s + `(`+ 1) r2σ2 Φ′ (4.2) 1 ρ dp′ dr = −(N2 − σ2)ξ − g ρc2s p′ − dΦ ′ dr 1 r2 d dr (r2 dΦ′ dr ) = `(`+ 1) r2 Φ′ + 4piGρ ( p′ ρc2s + N2ξ g ) In the above equations primed (´) quantities are Eulerian perturbations (see § 1.1.1), N2 = g (Γ−11 d ln p/dr − d ln ρ/dr) is the Brunt-Väisälä (or bouncy) frequency, and Φ is the gravitational potential. Other symbols have their usual meaning. JIG calculates eigenfunctions defined by the variables first used by Dziembowski (1971) y1 = ξr/r, y2 = 1 gr (p′/ρ+ Φ′) , y3 = 1 gr Φ′, and y4 = 1 g dΦ′ dr (4.3) Using these definitions we arrive at the final form of the adiabatic oscillation equations used in this thesis: r dy1 dr = (Vg − 3) y1 + ( `(`+ 1) c1σ̄2 − Vg ) y2 + Vgy3 (4.4) r dy2 dr = ( c1σ̄ 2 + A∗ ) y1 + (1− U − A∗) y2 + A∗y3 r dy3 dr = (1− U)y3 + y4 r dy4 dr = −UA∗y1 + UVgy2 + (`(`+ 1)− UVg) y3 − Uy4 In the above, the homology relations U = 4piρr3/Mr and Vg ≡ V/Γ1 (with V = GMrρ/rp) are introduced. The measure of the local density to the mean density of the star is given as U and V is the ratio of gravitational energy to the internal energy of the stellar material. A∗ and c1 are defined as 110 4.2. Pulsation models A∗ ≡ rA = −rN2/g and c1 = (r/R)3/(Mr/M). The dimensionless frequency σ̄ = σ √ R3/(GM) is used. The boundary conditions are described in Saio & Cox (1980) and in the books by Unno et al. (1989) and Cox (1980). The outer mechanical boundary condition assumes a finite pressure at the stellar surface. A potential condi- tion matches the perturbed gravitational potential and its derivative at the stellar surface. The two inner boundary conditions require regular solutions of the pulsation variables as the stellar radius goes to 0. Written in terms of the Dziembowski variables, the outer boundary conditions are( 1− 4 + c1σ̄ 2 V ) y1 + ( `(`+ 1) c1σ̄2V − 1 ) y2 + ( 1− `+ 1 V ) y3 = 0 (4.5) and Uy1 + (`+ 1)y3 + y4 = 0 (4.6) The inner boundary conditions are c1σ̄ 2 ` y1 − y2 = 0 (4.7) and `y3 − y4 = 0 (4.8) We calculate frequencies for models of HR 1217 using angular degrees ranging from ` = 0–4 in the frequency range 1900–3100 µHz. This encom- passes the observed oscillations of HR 1217 (see chapter 3). In all there are about 50,000 individual modes (oscillation frequencies) calculated. Example spectra, including magnetic effects, will be discussed in § 4.3 and in chap- ter 5. (See also Fig. 4.4 for example spectra.) Because there are so many modes calculated in a grid of models, we first explore (qualitatively) matches 111 4.2. Pulsation models between quantities like the large and small spacing (Eqs. 1.12 and 1.16). By design our grid of models shows large spacing values appropriate for HR 1217 (∼ 68 µHz). The grid, plotted in Fig. 4.1, shows that increasing Z shifts the models to the lower right of the HR diagram and decreasing Z shifts the models to the upper left. The lowest mass models in our grid that enter our defined error box (highlighting the position of HR 1217 in the HR diagram) have the lowest Z values in our grid. Likewise, the models with the largest mass in our grid that fall in the same error box have the highest Z values as well. Generally, the more luminous the model is, the lower its large spacing is because it tends to be less dense (because of its larger radius). The models with a small spacing of ∼ 2.5 µHz may be important for the interpretation of the frequency analysis of § 3.3. Figure 4.3 shows the small spacing for the models outlined in Tab. 4.1 with the angular degree differences of ` = 2 - ` = 0, ` = 3 - ` = 1, and ` = 4 - ` = 2. The models with the second order spacing of . 3 µHz are either older (compared to other models listed in Tab. 4.1), or have a mass less than ≈ 1.5 M. The metallicity Z of these models also tends to be lower than a solar metallicity. This is in agreement with Cunha et al. (2003) who show the best qualitative model match has a Z value that is less than solar. It is also interesting to note that the second order separations between ` = 4 - ` = 2 are much larger (& 4×) than the observed value of about 2.5 µHz. This suggests that if the observed differences are indeed small spacings, they are separations defined by differences between the lowest degree modes. We will show in § 4.3 that including the magnetic field can dramatically alter the spacings of the models, casting doubt on whether or not we really observe a small 112 4.2. Pulsation models spacing as defined by asymptotic theory. 4.2.1 A new search method for eigenmodes There are a number of numerical methods that have been used to solve the set of Eqs. 4.4. A general discussion of these methods may be found in the lecture notes of Christensen-Dalsgaard (2003). Recently, a number of researchers have compared their software and outline the reduction procedures used to calculate the normal modes of a stellar model (see Moya et al. 2008 and references therein). The most popular methods for calculating oscillation spectra of models are the shooting method and variants of the relaxation (Newton) method (e.g., the Henyey relaxation). In the case of the shooting method, the solution is integrated both inward and outward from the surface and central boundary conditions and then matched at a suitable fitting point for a given eigenfrequency σ2. If the difference between the inward and outward solutions at the fitting point is zero (to some numerical accuracy) then σ2 is an eigenvalue of the problem. Thus, a number of σ2 can be scanned to find the eigenvalue to a prescribed numerical accuracy. The drawback of this method is in the choice of fitting point. This is highly dependent on the mode being investigated and may be placed either in the upper atmosphere of the model or in the deep interior. Experience is required to find a fitting point that yields an accurate and stable numerical solution. Relaxation methods also exist in which all of the eigenfunctions and the eigenvalue are iteratively refined at each mesh point in the model. Examples 113 4.2. Pulsation models Figure 4.3: Small spacings for the models listed in Tab. 4.1 (see legend for colour codes) as a function of mode frequency. The top panel gives the frequency differences of modes with degrees between ` = 4 and 2, the middle panel for degrees between ` = 3 and 1, and the lower panel for degrees between ` = 2 and 0. The potential small spacing (§ 1.1.1, and Eq. 1.16) observed in the MOST data is . 3 µHz (see Tab. 3.2) and is best reproduced by models with lower Z and for small separation between the lowest angular degrees (` = 0 and 2). 114 4.2. Pulsation models of this method are found, for example, in Unno et al. (1989) and Henyey et al. (1964). The JIG code described above uses a relaxation method to find the eigenvalues of a model. This method requires an initial guess that is close to the eigenvalue in order have fast convergence. In addition, the initial setup of this method tends to be complicated. A direct solution to the eigenvalue problem using matrix methods also exists but they tend to be the least efficient (in time and memory) of all the methods (see Christensen-Dalsgaard 2003). A new method of calculating and scanning for eigenmodes was developed by Kobayashi (2007). This method efficiently integrates the set of adiabatic equations using a matrix method and was developed to study normal modes in the Earth’s atmosphere. This method is, in some sense, a cross between a relaxation method and a direct integration. We show the results for two selected models as a consistency check with the modes calculated using JIG. Equations 4.4 are written in matrix form as dyij drj = Bjy i j, i = 1, . . . , 4 and j = 1, . . . ,ℵ (4.9) where B is a 4x4 matrix, defined at the jth shell, with entries filled as the terms on the right hand side of Eqs. 4.4. The model is comprised of ℵ shells with the first shell (j = 1) located at the centre of the model. Equation 4.9 may be rewritten in a finite difference form, relating variables between shells in the model as Cnyn +Dn+1yn+1 = 0 (4.10) with Cn = I + ∆rBn/2, Dn+1 = −I + ∆rBn+1/2, and ∆r = rn+1 − rn. 115 4.2. Pulsation models A Henyey-type banded matrix is formed from the system of Eqs. 4.10 with the following definitions: Pn ≡ [CLn 0]T n = 2, . . . ,ℵ Sn ≡ [DLn CUn ]T n = 2, . . . ,ℵ − 1 and Qn ≡ [0 DUn+1]T n = 1, . . . ,ℵ − 1 The upper indices L and U refer to (respectively) the lower and upper halves of matrices C or D. S1 is a special case which has its upper half as the inner boundary conditions and its lower entries as the upper entries of C1. Likewise, Sℵ is composed of the lower half of Dℵ and the outer boundary conditions. In this banded form, and for a given σ2, the system of equations may be integrated recursively from the centre of the model to its surface using yn = Rnyn+1 where Rn = −X−1n Qn, Xn = PnRn−1 + Sn and starting from R1 = −S−11 Q1. At the surface of the model Xℵyℵ = 0 and σ2 is an eigenvalue only if detX = 0. Integrating the equations in this manner is accomplished in order 116 4.2. Pulsation models ℵ time and with order ℵ memory because storage is only needed for the ℵ R values (required to calculate the eigenfunctions). Along with the speed of this integration method there is also an advantage in the form of the eigenvalue search. Kobayashi (2007) showed that if we expand Xℵyℵ = 0 about a perturbation in the eigenvalue δσ2, we arrive at a small (4 x 4 matrix in the adiabatic case) generalized eigenvalue problem at the surface of the model( Xℵ + δσ2 ∂Xℵ ∂σ2 ) yℵ = 0 (4.11) where the partial derivatives of X are also found through the recursion re- lations used to integrate the system. Given a guess value of σ2 the solution relaxes to the correct eigenvalue by replacing σ2 with σ2 + δσ2 until some predefined stopping criteria is satisfied. We calculate the eigenfrequencies for models A and E of Tab. 4.1 using the method outlined above. Following Roxburgh (2008), we first define the functions Ξ = dΦ′ dr + 4piGρξ (4.12) and rσ2ζ = δp/ρ+ ξg + Φ′ (4.13) where ζ is identified as the horizontal component of the displacement vector. We normalize the stellar structure variables using r = Rx, ρ = M 4piR3 ρ̄, g = GM R2 ḡ, (4.14) c2 = GM R c̄2, p = GM2 4piR4 p̄, σ2 = GM R3 σ̄2 117 4.2. Pulsation models and recast Eqs. 4.2 in the following form: dy1 dx = `(`+ 1) x y0 − 2 x y1 − 1 x y2 (4.15) dy2 dx = −`(`+ 1)ρ̄ḡ xp̄ y0 + ( σ̄2ρ̄ p̄ + 4ḡρ̄ xp̄ ) y1 + ḡρ̄ p̄ y2 − ρ̄ p̄ y4 dy3 dx = −ρ̄y1 + y4 dy4 dx = `(`+ 1)ρ̄ x y0 + `(`+ 1) x2 y3 − 2 x y4 The eigenfunctions of the system are defined by y0 = ζ/R, y1 = ξ/R, y2 = δp p , (4.16) y3 = R GM Φ′, and y4 = R2 GM Ξ Note that in Eqs. 4.15 we use the Lagrangian variation of the pressure δp. The result is that the Brunt-Väisälä (N2) frequency disappears from the linearized pulsation equations. Figure 4.4 shows the comparison between the frequencies calculated using this method and those calculated using JIG for models A and E of Tab. 4.1. The plot shows the (non-dimensional) eigenvalue correction δσ2 as a function of frequency ν (the guess for the eigenvalue of the system) for ` values from 0 to 4. If the correction is positive, our guess eigenvalue is too low and the correction leads us forward (toward the true eigenvalue) by increasing our guess by the increment δσ2. When the correction goes to zero for a given guess frequency, we have found an eigenvalue. Note that not all of the zeroes of the function δσ2(ν) shown in Fig. 4.4 correspond to pulsation eigenvalues for the models A and E. Consider the behaviour of the correction near a zero where the function δσ2(ν) has a positive slope. A positive δσ2 near that zero 118 4.3. The magnetic effects will move our guess further away from the zero. A negative δσ2 will also move us away from the x-intercept if the local slope is positive. If the local slope of δσ2(ν) is negative, δσ2 can be either positive or negative, and the correction will guide our guess towards the zero (or eigenvalue). Therefore, values of the δσ2(ν) function that approach zero with a negative slope (suggesting a decreasing correction) are eigenvalues of the system. The normal modes calculated from JIG are shown as red points distributed along the line δσ2 = 0. The agreement is exact to the precision output by JIG (5 decimal places) and no eigenmodes were missed during the calculation of frequencies used for this thesis. Although only tested in the adiabatic case, this method was originally designed to work for a complex set of non-adiabatic equations used in a geophysical context (Kobayashi, 2007). Traditionally an adiabatic solution is required as an initial guess for both the eigenvalues and the eigenfunctions of the non-adiabatic solution to the stellar pulsation equations. This method offers an efficient alternative to that approach. 4.3 The magnetic effects The effects of the magnetic field on both the oscillation frequencies and the eigenfunctions have been described by a number of authors (see recent re- views by Cunha 2007, Saio 2008 and Shibahashi 2008). The kG strength magnetic fields in roAp stars complicate the interpretation of the observed eigenmodes by introducing deviations from the expected (near asymptot- ically spaced) frequency spectrum. In the outer layers of the roAp stars 119 4.3. The magnetic effects Figure 4.4: A comparison of eigenfrequencies determined by two different methods. The left panels show the non-dimensional eigenfrequency correction δσ2 (normalized by GM/R3) as a function of frequency ν, for ` values of 0 to 4 (from top to bottom panels), for model A of Tab. 4.1. The black dots represent the eigenfrequency corrections described in § 4.2.1 and the red circles are the eigenfrequencies calculated for this thesis with JIG (§ 4.2, and Eqs. 4.4). The right panels are the same as those on the left but use model E of Tab. 4.1. A value of δσ2 = 0 is an eigenvalue of the system of Eqs. 4.15. Modes calculated with JIG agree with those determined using the new method developed by Kobayashi (2007). 120 4.3. The magnetic effects (. 3% of the total stellar mass) the Alfvén speed (∝√B2/ρ) is comparable to, or greater than, the local sound speed (∝√p/ρ). The eigenmodes are no longer purely acoustic and are better described as magnetoacoustic waves. As the stellar density increases inward and the sound speed becomes larger than the Alfvén speed, these magnetoacoustic waves decouple into acoustic and magnetic components. The resultant magnetic slow waves act as an energy sink as they dissipate in the dense stellar interior. It is this loss of energy that results in the shifting of some frequencies and the deviations from the regular asymptotic spacing. There has been recent success in the interpretation of roAp oscillations (e.g., Cunha 2001). Cunha & Gough (2000) (see also Cunha 2006) estimate the magnetic perturbation to the eigenmodes of nonmagnetic stellar models using a vari- ational principle. The method matches a numerical solution of the ideal magneto-hydrodynamic equations in a boundary layer located in the upper envelope and atmosphere of the model to an asymptotic expression for the eigenfunctions of high-overtone, acoustic oscillations in the stellar interior. The phase difference between the numerical and asymptotic expressions for the eigenfunctions is used to estimate the corresponding frequency shifts as a result of magneto-acoustic interactions. A determination of the perturbed eigenmodes using a variational principle, calculated from an estimate of the perturbed eigenfunctions (obtained using a background stellar model that does not include magnetic fields), is an attractive alternative to a pertur- bation analysis or a full scale MHD calculation of magneto-acoustic modes. This method, developed by Cunha & Gough (2000), follows the earlier work of Campbell & Papaloizou (1986). 121 4.3. The magnetic effects To start the modelling procedure, Cunha & Gough (2000) divide the star into a thin outer boundary layer and the interior. In the boundary layer, the Lorentz forces are comparable to, or larger than, the gas pressure, while in the interior the field is essentially force free. Because the magnetic boundary layer is thin, a plane-parallel approximation may be used. It is also assumed that the effects of the magnetic field can be calculated locally at each latitude. For this geometry the field only varies in the vertical direction and has com- ponents B = (Bx, 0, Bz). The magneto-acoustic waves are described using a horizontal wavenumber k = (kx, ky, 0). Using this information, the adia- batic, magnetically non-diffusive, pulsation equations may then be written as (Campbell & Papaloizou 1986, Cunha & Gough 2000 and Cunha 2006) − σ2ρu = i|k|W + (B · ∇)2 u µ0 − kxBx µ0|k| (B · ∇) (∇·ξ) (4.17) − σ2ρv = (B · ∇)2 v µ0 + kyBx µ0|k| (B · ∇) (∇·ξ) (4.18) − σ2ρξz = ∂W ∂z − g∇· (ρξ)− Bz µ0 [(B · ∇) (∇·ξ)] + (B · ∇)2 ξz µ0 (4.19) where W = ξ·∇p+ ( γp+ B2 µ0 ) (∇·ξ)− (B · ∇) (B · ξ) µ0 (4.20) and g = 1 ρ dp dz (4.21) In these equations the displacement vector ξ is decomposed into a vertical component ξz = ξ · ez, a component that is perpendicular to the wavenumber v = ξ·(ez × k)/|k|, and a component that is parallel to the wavenumber u = (ξ · k)/|k|. The local gravitational acceleration is denoted by g and the first adiabatic exponent is γ. All other symbols have their usual meanings. 122 4.3. The magnetic effects In the deep interior a magneto-acoustic mode completely decouples into a pure Alfvénic mode and a pure acoustic mode. This was verified analytically by Roberts & Soward (1983) and numerically by Campbell & Papaloizou (1986). The JWKB approximation is used to describe the magnetic modes in the interior. This approximation to the eigenfunctions takes the functional form (Cunha & Gough 2000; see also Gough 2007) (vmz, umz) v ρ−1/4(C,D) exp [ i ∫ z 0 ( µ0ρσ 2 B2z )2 dz − ikxBxz Bz ] + ρ−1/4(C+, D+) exp [ −i ∫ z 0 ( µ0ρσ 2 B2z )2 dz − ikxBxz Bz ] (4.22) where C, D, C+, and D+ are complex constants. The inward propagating Alfvén waves are expected to dissipate before they are reflected back toward the surface of the star (Roberts & Soward, 1983). With this in mind, the constants C+ and D+ in Eq. 4.22 may then be set to zero to guarantee that no outwardly propagating magnetic waves occur in the interior. In the interior, the vertical component of the uncoupled modes is es- sentially acoustic. The amplitude of this vertical mode may be represented asymptotically by (Cunha & Gough 2000) ξz v Aκ1/2 ρ1/2 cos (∫ z∗ z κdz + δp ) (4.23) were δp is a phase and κ is the vertical acoustic wavenumber. The coordinates z and z∗ represent the depth in the boundary layer and the position of the base of the boundary layer, respectively. To integrate Eqs. 4.17–4.19 there need to be five boundary conditions (Cunha & Gough, 2000) to match the components of the displacement and 123 4.3. The magnetic effects the magnetic field variations. We use a fully reflective mechanical outer boundary condition of the form p∇ · ξ = 0 (4.24) The other outer boundary condition requires that the magnetic field be matched continuously onto a vacuum field. This may be written as (B · ∇)ξ −B∇ · ξ = ∇φm (4.25) where φm is an exponentially decreasing plane wave solution of the Laplace equation. The boundary condition 4.25 can be separated into two conditions: one condition relating the vertical displacement and the variable u (defined above), and the other involving the variable v (defined above; also see Cunha & Gough 2000). The numerical solutions of the system of Eqs. 4.17 to 4.19 are matched to the asymptotic relations 4.22. This provides the two interior boundary conditions. The vertical component of the solution is then matched to Eq. 4.23 at each latitude to obtain values for the magnetic phases δp. The purely acous- tic (numerical) solution; i.e, B = 0 in Eqs. 4.17 to 4.19, is also matched to Eq. 4.23 to obtain the unperturbed phases δp0. Phase shifts ∆δp are cal- culated from the difference between these two phases at each latitude. The matching procedure is outlined in Cunha & Gough (2000) and Cunha (2006). The variational method of Cunha & Gough (2000) is used to estimate the first-order frequency shifts of the eigenmodes caused by a magnetic field. The frequency shifts are calculated from δσM σ0 w − ∆δp σ20 ∫ R∗ r1 c−2s κ−10 dr (4.26) 124 4.3. The magnetic effects where the average of the phase shifts ∆δp over a sphere is ∆δp = ∫ pi 0 ∆δp(Y m ` ) 2 sin θdθ∫ pi 0 (Y m` ) 2 sin θdθ (4.27) The quantities with a subscript of zero are non-perturbed values. A complete description of the numerical procedure used to calculate the frequency shifts is outlined in the appendices of Cunha & Gough (2000) and Cunha (2006). Examples of the magnetic shifts are presented below. There are sudden drops in the frequency corresponding to a maximum of energy loss of that particular mode. These sharp jumps are cyclic and depend on the magnetic field strength, the frequency range explored, and on the mode/magnetic field geometry. It is important to fully explore all relevant frequency ranges, mode geometries and magnetic field strengths if we hope to successfully match a calculated frequency spectrum to observations of roAp stars. For each of the modes calculated from the models presented in § 4.2, a magnetic perturbation is estimated for a dipolar magnetic field with polar strengths running from 1 to 10 kG in steps of 0.1 kG. In total there are 51,795 magnetic models calculated using physical parameters appropriate to the properties of HR 1217. This is the largest grid of magnetic models calculated for the seismic study of any roAp star. The cyclic nature of the magnetic perturbations is shown in Fig. 4.5. Both real and imaginary parts of the magnetic shifts occur regularly as a power-law function (ν × Bn). The index n in the power law is dependant on the structure of the outer layers of the background model. The models in Tab. 4.1 are used in this calculation. A value of n = 0.75 does a good job of phasing together the frequency shifts. This behaviour is discussed in Saio 2008 and references therein. It is important to notice the frequency shifts 125 4.3. The magnetic effects between modes of degrees that differ by two are at least of the same order as the second-order, small spacing (Eq. 1.16). This may hamper efforts to use an observed small spacing as a diagnostic of stellar structure (Dziembowski & Goode, 1996). Figure 4.5: The cyclic behaviour of the real and imaginary magneto-acoustic shifts. The real part of the shifts calculated for models in Tab. 4.1 are presented in the left panels and the imaginary parts are given in the right panels. From top to bottom are the degrees of the modes (from ` = 1 down to ` = 4) and all magnetic fields (B = 1–10 kG) are shown. The position of the shifts is cyclic and can approximately be phased as ν × B0.75. The exponent on B is sensitive to the outer layers of the model. Note that a maximum in the imaginary part corresponds to a sudden jump in the real frequency shift. 126 4.3. The magnetic effects Figure 4.6: The magnetic perturbations for a 5 kG magnetic field using models from Tab. 4.1. The left plot set (consisting of four plots) are for degrees of ` = 1 and 2 and the right plot set is for degrees of 3 and 4. The upper panels show the real frequency shifts as a function of frequency while the lower portions of the figure give the imaginary shifts. Models with frequency shifts of ∼ −15 µHz around 2800 µHz could explain the anomalous frequency (or “missing mode”, see § 1.3) identified by Kurtz et al. (2002). Note that the difference between the magnetic shifts of degrees that differ by 2 can be as large as, or greater than, the small spacing for a main sequence, A-type star (∼ 10 µHz; see Fig. 4.3). The magnetic perturbations to frequencies from 1500 to 3500 µHz are shown in Fig. 4.6 assuming a dipolar magnetic field with a polar strength of 5.0 kG and using the models of Tab. 4.1. This frequency range is consistent with measurements of HR 1217 (§ 3.3) and shows the real and the imaginary 127 4.3. The magnetic effects frequency shifts for modes having degrees of ` = 1 through 4. Note that the difference between real frequency shifts for modes of different degree amounts to a few µHz. The imaginary part of the even ` = 2 and 4 modes are lower than the odd ` counterparts. This was also shown for the case of a polytrope stellar model of Cunha & Gough (2000). In general, if the imaginary part of the frequency is positive, the magneto- acoustic mode loses energy. The smaller amplitude of the imaginary part for the ` = 2 modes, for example, suggest that they are less susceptible to damp- ing from the magnetic field. This effect depends on the geometry of the mode and the magnetic field. Cunha & Gough (2000) and Cunha (2006) show that the maximum energy loss of a magneto-acoustic mode occurs approximately at the latitude where BzBx is a maximum. This occurs at 45 ◦ for a dipo- lar magnetic field. Modes of higher degree ` decrease in amplitude as they approach the equator more quickly than modes of lower degree. When the phase shifts are averaged over a sphere, the net effect is a smaller imaginary frequency shift for modes of larger `. These modes have smaller amplitudes near the latitude where the maximum of energy loss occurs. A typical first step for pulsation modelling of a roAp star is to look at a small number of non-magnetic pulsation models and compare the calcu- lated spacings to those observed in the spectrum of the star. Cunha et al. (2003) did this for HR 1217. They found that models with a sub-solar metal- licity (Z . 0.018) provided a better (qualitative) match to the observed (approximately asymptotic) spacing of HR 1217 (∼ 68 µHz). We extend their analysis by looking at the spacings of models A–E in Tab. 4.1. The large spacings listed in Tab. 4.1 show that models B (∆ν = 66.96 µHz), C 128 4.3. The magnetic effects (∆ν = 71.53 µHz), and D (∆ν = 67.80 µHz) come close to the large spacing of HR 1217. These models have metallicities of Z = 0.008, 0.012, and 0.010, respectively. Each of those metallicities is sub-solar. That being said, only one model (E) in Tab. 4.1 has a near solar metallicity of Z = 0.018. We can not say that only models with a smaller than solar Z value best match the properties of HR 1217. We can only say that the (small number of) models in Tab. 4.1 are in agreement with Cunha et al. (2003). Models with a Z . 0.018 can reproduce the large spacing of HR 1217; but, models with a larger Z can also show such a match (see chapter 5). Perhaps the more important question is: How does the introduction of a magnetic field affect the large spacing of the models? Figure 4.7 shows the large spacing calculated from the model pulsation frequencies (∆ν = νn+1,` − νn,`), as a function of frequency, for the models in Tab. 4.1. The left panel of Fig. 4.7 shows models with B = 0 kG and the right panel shows pulsation models having B = 5 kG. Comparing the models between the two sets of panels shows the large spacing (on average) is not affected by the introduction of a magnetic field. A major exception to this rule occurs when one of the frequencies suddenly jumps in value. At that point (e.g. model A (black line) and B (red line) of Fig. 4.7), the large spacing shows a sharp decrease in value. This decrease can be a large fraction of the value of the large spacing itself. While ∆ν varies slowly as a function of frequency even in the non-magnetic case, the magnetic field perturbation causes an abrupt decreases in ∆ν over one spacing between modes. The inclusion of a magnetic field in the pulsation calculations seems to be the most reasonable way of producing such a sharp, and localized, feature in the 129 4.3. The magnetic effects stellar oscillation spectrum. For HR 1217, the frequencies alternate between ∼ 33.5 and 34.5 µHz over a frequencies ranging from about 2600 to 2800 µHz. That alternating pattern is suddenly cut by about half near the beginning and the end of the observed frequency range (see § 3.3). The introduction of a 5 kG magnetic field to the models in Tab. 4.1 does not show multiple jumps over the range of observed frequencies in the HR 1217 spectrum. The size of the decrease of ∆ν shown in Fig. 4.7 is also too large by a factor of more than two compared to what is observed. In Fig. 4.8 we show the large spacing as a function of magnetic field for the same background models. Each of these background models exhibits a sudden jump in ∆ν for some values of magnetic field. Interestingly, jumps in ∆ν of less than about 20 µHz occur (for some models) at magnetic fields . 5 kG. This is similar behaviour to what is observed in HR 1217. Models with larger magnetic fields tend to show larger jumps for all ` values. 130 4.3. The magnetic effects Figure 4.7: The magnetic perturbations to ∆ν for a 0 and 5 kG magnetic field using models from Tab. 4.1. The left set of four plots show the large spacing ∆ν = νn+1,`− νn,` as a function of frequency νn,` for angular degrees ` = 0 through 4. The right set of plots show that perturbation to the large spacing caused by a 5 kG magnetic field. The inclusion of a magnetic field causes a sharp decrease in ∆ν when there is a jump in frequency (see Fig. 4.6). 131 4.3. The magnetic effects Figure 4.8: The magnetic perturbations to ∆ν for all calculated magnetic field strengths using models from Tab. 4.1. The plots show the large spacing ∆ν = νn+1,` − νn,` as a function of magnetic field strength for ` values identified at the top of each panel. The inclusion of a magnetic field causes a sharp decrease in ∆ν when there is a jump in frequency (see Fig. 4.6). This jump occurs cyclically for varying magnetic field strengths. 132 4.3. The magnetic effects In § 3.3 we observed for the first time a number periodicities with spac- ings that were between ∼ 1 and 3 µHz (Tab. 3.2; spacings 19 through 23). Those spacings are close to what we expect for a second order (small) spac- ing of a main-sequence A-type star. The small spacings for the non-magnetic pulsation models given in Tab. 4.1 are plotted in Fig. 4.3. The properties of the non-magnetic models were previously discussed in § 4.2. If we ob- served small spacings in the MOST data then, based on models A through E, the adjacent modes are most likely even, low-order, ` = 0 and 2 modes. (Once again we note that assessment is based only on the models presented in Tab. 4.1.) Figure 4.9 gives the small spacing for the same models as in Fig. 4.3 with a magnetic perturbation (B = 5 kG) included. The perturba- tion to the small spacing is large and is more reminiscent of what happens to individual frequencies than what happens with the magnetic perturbation to the large spacing. The spacings show a slow increase in value before a sudden decrease (∼ 5–10 µHz). This introduces significant difficulties in dis- tinguishing the magnetic effects from the asymptotic spacings (Dziembowski & Goode, 1996). The small spacings calculated as νn,`−νn−1,`+2 can even be negative. The mode having order n has a greater frequency value than the adjacent mode with order n - 1 in the non-magnetic case. The introduction of a magnetic field can cause this pattern to be reversed so that a frequency labelled n - 1 in the non-magnetic model now has a larger frequency than the mode having order n in the non-magnetic model. In general n is not an observable, but the frequency shifts caused by a magnetic field introduce a large enough perturbation to significantly complicate the analysis of our theoretical spectra. The small spacing perturbations (with an offset added 133 4.3. The magnetic effects for clarity) are given in Fig. 4.10 as a function of magnetic field. The pertur- bation is cyclic and shows the large variation in the magnitude of the second order separations caused by the addition of a magnetic field to the pulsation calculations. 134 4.3. The magnetic effects Figure 4.9: The magnetic perturbations to δν for a 5 kG magnetic field using models from Tab. 4.1. The plots show the small spacing δν = νn,` − νn−1,`+2 as a function of frequency νn,`. The figure labels are the same as the labels in Fig. 4.3 – the small spacings neglecting a magnetic field. Compared to the nonmagnetic case the small spacing can increase or decrease depending on the angular separations. The magnetic perturbation can even cause a frequency (labelled n) with angular degree ` + 2 to have a larger value than the frequency (labelled n - 1) with degree `, resulting in a negative small spacing. This means close (adjacent) frequency pairs (see Fig. 1.3) can appear in the reverse order. 135 4.3. The magnetic effects Figure 4.10: The magnetic perturbations to δν for all calculated magnetic field strengths using models from Tab. 4.1. The panels are labelled with the same ordinates as Fig. 4.3. For clarity, the small spacings for some models are offset by a constant value shown in the legend found in the top panel. The small spacing is highly sensitive to magnetic field strength and the angular degrees of the modes. 136 4.3. The magnetic effects A close inspection of Figs. 4.7 and 4.9 shows a rapid variation in the magnitude of the magnetically perturbed spacings. This rapid variation has an amplitude of a few (∼ 5) µHz over a baseline of about 100 µHz. The small spacings (∼ 3 µHz) observed by MOST seem to vary by about 1 µHz over a similar baseline (see Tab. 3.2). This similarity between the theory and observations is noted with caution. We are unsure if this rapid variation in the spacings is a physical phenomenon or a numerical artifact in our models. A comparison to frequency spacings calculated using another method (e.g., the models of Saio & Gautschy 2004) is necessary to substantiate the reality of this variation. The models for HR 1217 presented in this chapter represents the first time a large grid of realistic stellar models has been calculated for this star that includes the magnetic perturbations to the eigenfrequencies. The gen- eral characteristics of the models was discussed. The following chapter will expand upon this analysis by attempting to find the closest matched model frequencies to the observed MOST periodicities (chapter 3). 137 Chapter 5 Best matched models In the previous chapter we discussed the calculation of a large grid of stellar models of HR 1217 that, for the first time, included the effects of magnetic perturbations to the oscillation modes. The general properties of those mod- els were discussed but a quantitative match to the frequencies extracted from the MOST photometry (chapter 3) was not performed. This chapter outlines a procedure to match the observed periodicities to the calculated magneto- acoustic modes. The properties of models from our grid that best match the observations are described. 5.1 The matching procedure To begin we ask ourselves a simple question: Can models of roAp stars be used to constrain the properties of a roAp star? Mkrtichian et al. (2008) recently studied HD 101065; a roAp star com- monly referred to as Przybylski’s star, and uncovered between 15 and 20 periodicities from their radial velocity data. The authors explored theoreti- cal models with masses ranging from 1.5 to 1.7 M in steps of 0.05 M and with chemical compositions of (X, Z) = (0.7, 0.02) and (0.695, 0.025). Mkr- tichian et al. (2008) included magnetic effects in their pulsation models using the method of Saio (2005) and calculated the magnetic perturbation (for two 138 5.1. The matching procedure or three models in the evolutionary sequences at each mass) for fields with strengths between 2 and 10 kG in steps of 0.2 kG. Their use of a few hundred magnetic models (= 2–3 selected models × 5 available model masses × 40 magnetic field strengths) should be compared to the almost 52,000 models used in this study. They were able to find a mean deviation between the observed oscillations and the theoretical pulsation modes of ∼ 1.3 µHz. (In determining the mean the authors used the logarithm of the radial velocity amplitudes as a weight.) Individual deviations between their best matched modelled modes and their observed frequencies ranged from about 3.3 µHz in the worst match to about 0.1 µHz in the best case. (The observed frequency uncertainties from their study range from 0.004 to 0.47 µHz.) From their matched models they give constraints of M/M = 1.525 ± 0.025, Age = 1.5 ± 0.1 Gyr, log Teff = 3.821 ± 0.006, logL/L = 0.797 ± 0.026, log g = 4.06 ± 0.04, and B = 8.7 ± 0.3 kG on the physical parameters of HD 101065. Mkrtichian et al. (2008) give impressive constraints on the properties of HD 101065 and show the potential for roAp asteroseismology. For HR 1217, Cunha et al. (2003) obtained a qualitative agreement be- tween models and observations. Their conclusions were based on a small number of models that did not include the effects of a magnetic field. These non-magnetic models could not explain deviations from the asymptotic spac- ing that were observed, for example, by Kurtz et al. (2005). Cunha (2001) suggested that the magnetic field was an essential ingredient in the pulsa- tion calculations used to explain the roAp phenomenon and the observed frequency shifts. In this work, the largest grid of magnetically perturbed oscillation frequencies was calculated using the methods outlined in chapter 139 5.1. The matching procedure 4. We illustrated in § 4.3 that the addition of the magnetic perturbation to the pulsation calculations causes large shifts in frequency of certain eigen- modes. The shifts are on the order of what is observed in HR 1217 (∼ 15–20 µHz) and they occur cyclically as a function of magnetic field strength. We also showed that the asymptotic spacings (both large and small) are altered by the magnetic field, with the small spacing showing the largest deviations from the standard, non-magnetic, case. We must now find the model(s) in our grid that best reproduce the observed photometric variations presented in Tab. 3.1. 5.1.1 Seismic observations & models of the Sun Before continuing with the asteroseismic matching of the HR 1217 data to our model grid it is instructive to compare the quality of the frequency matches obtained for a well known star — the Sun. In the introduction chapter (§ 1.1) we alluded to the successes of helioseismology at constraining the physics that makes up a standard solar model (SSM). Detailed reviews of the physics included in the SSM and helioseismic inferences are provided by Christensen-Dalsgaard (2002) and Basu & Antia (2008). We know the mass, radius, composition and age of the Sun from a number of different sources (see Basu & Antia 2008). When we construct a SSM we must meet the constraints provided by those measurements. Typically we calculate the radius and luminosity of the Sun to within 1 part in 105 and match the value of Z/X ' 0.0244 to within about 1 part in 104. The mixing length parameter is used as a scalable parameter that is adjusted to match the SSM radius to the observed solar radius. The best combinations 140 5.1. The matching procedure of constituent physics and scalable parameters arrive at an age estimate of the Sun from solar models of ∼ 4.57 Gyr, in agreement with meteoritic data (Bahcall et al., 1995). Seismic models of the Sun are constructed to constrain the input physics of the SSM (EOS, opacities, diffusion physics, convection physics, composition . . .) with the strict requirement that the other observables mentioned above are also matched. An illustration of the quality of the frequency matches from a SSM and the observations of the Birmingham Solar Oscillations Network (BiSON; Chaplin et al. 1999) is shown in Fig. 5.1. The SSM was calculated with YREC. The EOS and opacity tables used in the construction of this SSM are updated versions of the same tables used in our HR 1217 models and the model atmosphere is constructed using a Krishna Swamy T -τ relation (Krishna Swamy, 1966). Diffusion was not included in this calculation. The relative difference between the adiabatic frequencies calculated with this SSM and the BiSON data is plotted as a function of the BiSON frequencies (Fig. 5.1). The model deviates from the observations by ∼ 10 µHz for the highest overtone (= highest frequency) modes. The calculated solar model frequencies are systematically offset from the observed frequencies.Guenther et al. (1993), for example, discussed the role of diffusion on calculated SSM eigenmodes and showed the inclusion of He diffusion in the SSM model calculation improve the agreement between the model and the observed periodicities by about 1– 5 µHz. The additions of heavy element diffusion, a non-adiabatic frequency calculation, and turbulent pressure corrections from 3D simulations improve the agreement between theory and observation even further. (We again point to the review of Basu & Antia 2008.) Currently, the most accurate solar 141 5.1. The matching procedure models still show frequency differences of about 0.3%, or a few µHz. Even this impressive agreement of a fraction of a percent at high frequencies is still statistically poor given that the individual frequency deviations are many times the frequency uncertainties. The mass, age, radius and composition of the Sun can be matched rea- sonably well by only considering the agreement between model frequencies and observations, even with the systematic offset between the observations and theory at high frequencies.Guenther & Brown (2004) used a χ2 match- ing procedure that calculates the average, weighted square deviation from a set of model frequencies to those observed in the Sun. They found that by matching the frequencies only (they did not take any other measured con- straint into account), they could arrive at a mass of 0.99 M for a model that favoured diffusion and had a composition of Z/X = 0.028 at an age of 4.7 Gyr. Metcalfe et al. (2009) used a genetic algorithm to explore the parameter space for the closest match between the calculated and observed solar modes. Their results are similar to those obtained by Guenther & Brown (2004). A more thorough helioseismic investigation would also include the disc resolved information about the solar oscillations that provides us with mea- surements of millions of modes with identifiable angular degrees starting from ` = 0 and ranging into the thousands. All of the observed solar modes cou- pled with the external constraints on the bulk properties of the Sun allow us to very accurately calculate the internal state of the Sun using inversion methods. In asteroseismology we do not usually have a large number of ob- served oscillation frequencies or a direct measure of the angular degree of those observed light variations. If the matching of the normal modes from 142 5.1. The matching procedure Figure 5.1: Standard Solar model (SSM) frequencies compared to the Birmingham Solar Oscillations Network (BiSON) observations. Shown is the relative difference between the adiabatic frequencies calculated from a SSM (without element diffusion) and those measured from BiSON data as a function of the BiSON frequencies. The model deviates from the observations by ∼ 10 µHz for the highest overtone modes. The addition of Helium and heavy element diffusion in the model, a non-adiabatic frequency calculation, and turbulent pressure corrections from 3D simulations improve the agreement between theory and observation. At this time, the most accurate solar models; including the aforementioned additions, still show frequency differences of about 0.3% (a few µHz). This is still many times the uncertainty of the individual frequency measurements. 143 5.1. The matching procedure a SSM to helioseismic data using the procedures outlined by Guenther & Brown (2004) and Metcalfe et al. (2009) is applicable to other stars, then we expect that accurate frequencies (precise to about 1 µHz or better), will (in the best case) give physical parameters that are accurate to within a few percent. Matching a SSM to the low-` helioseismic observations gives us a rough estimate on the accuracy of the physical parameters we can hope to extract from matching asteroseismic data to models. 5.1.2 Seismic observations & models of HR 1217 Matching observed oscillation frequencies to large grids of modelled pulsa- tion modes is a relatively new extension to the forward problem in asteroseis- mology. The observations are now of sufficiently high photometric quality (thanks to missions like MOST ) to uncover low amplitude oscillations. The extended, continuous, coverage of the observations over long periods of time also gives unprecedented frequency resolution in a Fourier transform that is free of aliasing artifacts. On the theoretical side, the increase in computing speed and memory facilitates the calculation of the large grids of models needed to uncover the parameter combinations that can best reproduce the observations. The models calculated in this thesis will be matched to the MOST obser- vations using a probability model. The matching procedure is similar to that used by Kallinger et al. (2008a). For each model in our grid, the calculated modes are compared to each of the observed frequencies listed in Tab. 3.1. Under the assumption that the residuals between a best-matching model mode and the corresponding observed frequency are normally distributed, 144 5.1. The matching procedure the Gaussian probability for a one frequency match is Pi = f(σν) exp ( −(νmod − νobs) 2 2σ2ν ) (5.1) where f(σν) = 1/ √ 2piσ2ν , σ 2 ν is the observed frequency (νobs) uncertainty (Tab. 3.1), and νmod is the model frequency that is the closest match to the observed frequency. The exponentiated function is essentially a χ2 measure. A probability is calculated for each of the NM observed frequencies and the total probability for the model is lnP = ln NM∏ i=1 Pi = NM∑ i ( ln f(σν)− (νmod − νobs) 2 2σ2ν ) i (5.2) Once a probability is assigned to each of the models in the grid, Eq. 5.2 is normalized by assuming the total grid probability is equal to unity; i.e, each model probability is divided by ∑ P , with the sum taken over all model probabilities within the grid. This method has an immediate advantage over a χ2 matching because it gives a simple way to statistically compare models within the grid and, by adjusting the normalization, models from one grid to those from another. For example, the probability of some model X in our grid divided by the probability of some other model Y in our grid gives a measure of how much more likely model X is compared to model Y. There is also an aesthetic benefit that the probability space around a best matched parameter tends to be magnified (because of the exponentiation) when compared to the same region in χ2 space. Although a χ2 measure or a probability measure will arrive at the same best matched model within a grid, the probability approach shows the location of the matched parameter more readily (Kallinger et al., 2008a). 145 5.1. The matching procedure We match our models to the MOST observations listed in Tab. 3.1 using the 3σ bootstrap uncertainties as the σ2ν input in Eq. 5.1. This is done for reasons discussed in § 3.3. Basically, we are being conservative by using the 3σ bootstrap uncertainties because they tend to be on the same order of mag- nitude as the lowest estimate of the frequency resolution given by Kallinger et al. (2008b); namely 0.25/∆T. The oblique pulsator model described in § 1.2.2 tells us that the observed oscillations in roAp stars consist of the “main” or “true” oscillation mode and a group of surrounding frequencies spaced by the rotation frequency of the star. The number of rotationally split frequencies depends on the degree of the mode and on the geometry of the system. This is observed in the MOST data on HR 1217, where a number of the observed periodicities are spaced by ∼ 0.9 µHz (Tab. 3.2), corresponding to the rotation rate of the star. In the simplest matching case we allow a model frequency to match multiple observed frequencies. Put another way, a frequency and its rotationally spaced sidelobes may all be matched to one mode in a given model. An alternative approach will be discussed in §5.2. We should also note that a proper statistical error should include the model uncertainties as well as the observational uncertainties. For the Sun, deviations between the modelled and the observed frequencies can be a few µHz for the highest order modes. Unfortunately, there is no reliable way to estimate the uncertainties introduced by the addition of a magnetic field in our models. By only using the observed frequency uncer- tainties, we are (dramatically) underestimating the true uncertainty needed to realistically weight our computed probabilities. We will discuss the scale of the uncertainties in more detail below. 146 5.1. The matching procedure Before discussing the results of the matching procedure we will address the scale of the uncertainties and how that scale relates to the presentation of the probabilities of the model properties within our grid. Figure 5.2 shows the matching probabilities in the magnetic field strength B, model mass M, and composition (Z/X) planes for two uncertainty scaling cases: 1. (Left panels of Fig. 5.2) The 3σ bootstrap uncertainties are not scaled, and 2. (Right panels of Fig. 5.2) The 3σ bootstrap uncertainties are scaled by a factor of 1000. Note that the probabilities in Fig. 5.2 are normalized so that the most probable model has a value of one. (The unnormalized probabilities will be given later.) In the first case (with no scaling) the most probable model (to be identified later) is clearly defined, with the next largest probability being, apparently, near zero. Most of the calculated probabilities in this case are essentially zero. This is because the Gaussian probability distribution heavily penalizes frequencies that are not well matched. When the total probability is calculated as a product of small numbers, its value becomes increasingly small. This is mainly caused by the lack of knowledge of a realistic uncertainty associated with our magnetic models. At face value, this graph could mislead a reader into thinking that (B, M/M, Z/X) = (8.8 kG, 1.5, 0.02) is the only reasonable set of parameters that could be assigned to HR 1217 based on the MOST data. We will show later that the situation is more complicated. In the second case, with the uncertainties scaled equally by a factor of 1000, there appears to be no constraints in probability space and no conclusions can be given. If we look for the model with the highest probability in both of the scaling cases, we find that the same model is identified. The equal scaling of uncertainties tends to magnify 147 5.1. The matching procedure regions of lower probability, but the relative uncertainties between observed frequencies results in a clear separation in the quality of the model match. We can arrive at the same result if we scale the uncertainties in a uniform way. This behaviour is obvious from the functional form of Eq. 5.1. We highlight it because, for presentation purposes, we want to find an equal uncertainty scaling that allows us to see how the probability of the model matches behaves in various parameter planes. The behaviour of the probability distribution around a set of parameters gives an indication of how degenerate the stellar parameters are within our model grid. For example, we know from § 4.3 that the magnetic perturbations are cyclic so we might expect that there will be (multiple) groupings of probability in the magnetic field plane showing which field strengths result in a better match to the seismic observations. We also know (as another example) that Z/X causes degeneracies in the position of a star in the HR diagram. So the presentation of the probability spaces within a number of parameter planes is important to identifying where these degeneracies are and also where future modelling should be focused. There is no statistical trickery being used by equally scaling the parameter uncertainties. We are simply finding a set of numbers that are easy to deal with numerically and that, for presentation purposes, helps us get a feel for the matches in various parameter planes. We will clearly state the value of the equal uncertainty scaling being used so that future studies will easily be able to reproduce our results. Some experimentation shows that an uncertainty scaling factor of 50 lets us calculate and properly normalize the model match probabilities, while at the same time allowing enough of a constraint on the models so we can visualize the results. We will further justify the choice of 148 5.1. The matching procedure a scaling factor of 50 at the end of this section. Figure 5.2: Probability distributions for models of HR 1217 using different error scales. All observed frequencies (those associated with both pulsation and rotational modulation) of Tab. 3.1 are used in the model search. Probabilities are assigned using Eq. 5.1. Model frequencies are matched to the observed frequencies with the 3σ bootstrap errors scaled. The left panels use a scale factor of 1 and the right panels scale the observed errors by a factor of 1000. The top panels shows the probability of all models as a function of magnetic field strength, the middle panels are probability as a function of model mass, and the bottom plots give probability as a function of composition (Z/X). Probabilities calculated with unscaled uncertainties show an extreme constraint on the models while those using the highly scaled (1000 ×) uncertainties show no constraint at all. However, because all observed errors are scaled in the same way, the most probable models are similarly ranked in both cases. 149 5.1. The matching procedure Figure 5.3 presents the matching results for a number of parameters in the model grid. The cumulative probability is given in each of the panels. The best matched model in our grid has a magnetic field strength of B = 8.8 kG, a mass of 1.5 M, a composition of Z/X = 0.02, and has completed about half of its main sequence lifetime. Models with magnetic field strengths greater than about 8 kG are preferred. The scaling allows to see that probable models with values near 1 kG do exist. A mass near 1.5 M is favoured, as is a sub-solar composition (Z/X ' 0.0244 for the Sun). Once again, these results are also obtained without scaling the observed uncertainties; but, we see that other parameters combinations should be explored in more detail. Even one poorly matched frequency can potentially drive the total prob- ability of that model close to zero. We mitigated the effect that a poor match has on the graphical analysis of the probability space by uniformly scaling uncertainties. It is also possible to calculate the model probabilities under the assumption that the residuals between the model and observed frequencies follow another distribution function. In Eq. 5.1 we use a Gaus- sian probability function. We also calculate model match probabilities using a Laplace distribution of the form Pi = f(σν) exp ( −|νmod − νobs| σν ) (5.3) where f(σν) = 0.5/σν . The Laplace distribution has fatter tails than a Gaussian distribution. As a result, outlying frequencies are given slightly larger weight when using a Laplace distribution instead of a Gaussian, or Normal, distribution. It is interesting to see how the “close” outliers affect the probability space around different parameter planes. If there is no in- 150 5.1. The matching procedure Figure 5.3: Probability distributions for models with uncertainties scaled by a factor of 50. All observed frequencies (those associated with both pulsation and rotational modulation) of Tab. 3.1 are used in the model search. Probabilities are assigned using Eq. 5.1. The left panels, from top to bottom, show the match probability as a function of magnetic field strength, the model mass, and the composition (Z/X) of the models. The right panels, again listed from top to bottom, show the match probabilities as a function of the per- centage of the main sequence lifetime the model has completed, the surface gravity of the model, and the mass of the model’s convective core in units of total stellar mass. Each panel has the cumulative probabilities plotted with circles and colours (labelled in the top panels) showing the percentage of the total probability. By using this scaling we see more details in the probability distributions than we would have if the uncertainties were not scaled. Magnetic models having B & 8 kG are preferred. 151 5.1. The matching procedure formation about how the residuals between a model and observations are distributed, then a Gaussian distribution is the most conservative choice for the distributed errors (see, e.g., Gregory 2005). Figure 5.4 shows the model match probabilities (labelled as Laplace prob- abilities, and using the same uncertainty scale factor) for the same parameters as in Fig. 5.3. Note that in both cases the same model is identified as the most probable model match. Because outliers are given some weight in the Laplace probabilities, we see a number of peaks in probability that are not seen when using a Gaussian probability model. Are the conclusions the same in both cases? Greater than 60% of the probability (of a model from our grid matching the observations) occurs after a magnetic field strength of 8.8 kG. Between 60 and 90% of the probability contribution occurs for masses between 1.5 and 1.6 M, and the largest contribution from the composition is less than Z/X ∼ 0.02. So each of these (admittedly broad) conclusions re- mains the same if we use either of the Gaussian (Eq. 5.1) or Laplace (Eq. 5.3) probability distributions. A more detailed comparison between the proba- bilities obtained by using the different distributions shows how sensitive the probability space is to frequencies that, in a manner of speaking, do not make the Gaussian cut. 152 5.1. The matching procedure Figure 5.4: Probability distributions for models using a Laplace distribution. All observed frequencies (those associated with both pulsation and rotational modulation) are used in the model search and probabilities are assigned using the Eq. 5.3. Observed uncertainties are scaled by a factor of 50. The above plots have the same format as those in Fig. 5.3. Probability distributions are similar to those shown in Fig. 5.3. By using a Laplace prob- ability distribution rather than a Normal (Gaussian) distribution we allow for a greater probability contribution from model frequency matches that fall in the distribution tails. In this way we can further explore the probability space. At the beginning of this chapter we gave an example of model matches to the observations of the roAp star HD 101065 (Mkrtichian et al., 2008). Mkrtichian et al. (2008) used the mean as a statistic to discriminate between their models of HD 101065 and arrived at a closest matched model having a mean deviation from the observations of about 1.3 µHz. Those authors 153 5.1. The matching procedure used a weighted mean so that poorly fitted frequencies did not skew their mean calculation. In Fig. 5.5 we compare the median frequency differences between our models and the MOST observations of HR 1217 as a function of log probability using both the Gaussian and Laplace distributions. Instead of using a weighted mean like Mkrtichian et al. (2008), we use the median so that we do not bias our average differences toward poorly fitted frequencies. The purpose of Fig. 5.5 is to compare an average difference statistic to the probability method we have presented thus far. We also show in Fig. 5.5 how scaling the observed uncertainties affects the numerical values of our unnormalized probabilities. For each value of probability in Fig. 5.5 there are a number of possible median matches. The Gaussian probability calculation drops off sharply for high probability matches, while the Laplace probability calculation gradu- ally approaches the most probable model(s). The lowest median deviation and the highest probability models both show a small standard deviation be- tween the differences in model and observations. Scaling the observed errors changes the scale and the width of the probability region, so that the models matched to observations with the largest uncertainty values show the small- est range in probabilities. The shape of the probability distribution does not change by (uniformly) scaling the observed errors. Using the mean or median as a discriminant between models and observations is not as effective as using a probability because the probability calculation takes into account the observed uncertainties. Poorly matched models are penalized much more within a probability framework than by using an average statistic. We have to be aware that even one extreme mismatch between an observed and calcu- 154 5.1. The matching procedure lated frequency can dramatically reduce the probability of that model being an acceptable match. In such a case the mean (or median) deviation of the model should be looked at in conjunction with the probability of that model to gauge the match quality. We have introduced a number of relevant matching methodologies but have yet to discuss the quality of the matches or to formally list the physical parameters of the best matched models. In Tab. 5.1 we list a number of properties for the most probable model matches to the MOST observations. The models are listed as the 10 most probable, non-duplicate, models using both the Gaussian and Laplace probability matches. 155 5.1. The matching procedure Figure 5.5: A comparison between the median model frequency matches and the model match probabilities. The left panels show the median of the difference between the obser- vations and the best matched model frequencies for each model in the grid as a function of the natural logarithm of the Gaussian model probabilities. Best matches are located at the lower right in each plot. The top panel highlights the case with no frequency scaling while the middle and bottom panels show the cases with observational uncertainties scaled by a factor of 50 and 1000, respectively. Colour contours show the standard deviation of the matches between the model and the observed frequencies. The right panels are the same as the left except the model probabilities are calculated using the Laplace distri- bution. Note that scaling the observational uncertainties produces similar distributions but the value of the probabilities increases and the range in probabilities decreases with increasing scaling factor. For each probability there are a number of potential median matches. The Gaussian and Laplace probabilities give the same best matched models but the probabilities tend to increase more sharply when using a Gaussian distribution. 156 5.1. T h e m atch in g p ro ced u re Table 5.1: Some properties of the most probable models of HR 1217. The top ten unique models from the Gaussian and Laplace probabilities are listed in decreasing probability order. Deviation refers to the absolute difference between a model frequency and the best matched observed frequency and σ is the standard deviation. The column labelled as probability label refers to the order of the probability from each of the Gaussian or Laplace probabilities. For example, 1 G is the most probable Gaussian probability and 10 L would be the 10th most probable Laplace probability. The probability ratio is defined as the Gaussian probability of the model divided by the Gaussian probability of the most probable model (lnP = −63.1). The models in this table were matched to all frequencies listed in Tab. 3.1 using 50 × 3σ errors. Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio M1 1.50 0.700 0.014 1.05 1.4 0.1434 8.8 0.00 3.8645 0.8826 0.2361 69.25 1866.36 1.18 1.53 1.16 1 G, 1 L 1.000 M2 1.50 0.700 0.014 1.10 1.4 0.1384 8.7 0.00 3.8616 0.8866 0.2438 67.52 1816.38 1.36 1.52 1.29 2 G, 5 L 0.273 M3 1.30 0.700 0.008 2.05 1.4 0.0000 4.8 0.00 3.8680 0.8973 0.2364 64.61 1597.02 1.08 1.31 0.99 2 L, 9 G 0.037 M4 1.50 0.700 0.014 1.00 1.6 0.1434 9.3 0.00 3.8672 0.8787 0.2287 70.99 1915.70 1.23 1.40 0.90 3 G, 4 L 0.183 M5 1.50 0.700 0.014 1.15 1.6 0.1384 9.1 0.00 3.8587 0.8905 0.2517 65.90 1767.56 1.01 1.49 1.35 3 L, 7G 0.055 M6 1.45 0.720 0.010 1.30 1.8 0.1251 1.2 0.00 3.8637 0.8633 0.2281 70.16 1857.29 1.40 1.45 0.95 4 G, 13 L 0.111 M7 1.45 0.720 0.008 1.35 1.6 0.1176 1.0 0.00 3.8739 0.9289 0.2405 66.88 continued on next page 157 5.1. T h e m atch in g p ro ced u re Table 5.1: continued Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio 1703.00 1.37 1.57 1.23 5 G, 12 L 0.091 M8 1.70 0.740 0.020 0.85 1.4 0.1628 8.5 0.00 3.8577 0.9065 0.2617 67.64 1883.37 1.10 1.32 0.89 6 G, 7 L 0.067 M9 1.50 0.700 0.014 1.00 1.6 0.1434 9.2 0.00 3.8672 0.8787 0.2287 70.99 1915.70 1.22 1.22 0.79 6 L, 8 G 0.050 M10 1.50 0.740 0.010 1.20 1.8 0.1264 10.0 0.00 3.8695 0.8728 0.2212 72.78 1931.66 1.05 1.25 0.92 8 L, 24 G 0.004 158 5.1. The matching procedure The most probable model match has a Gaussian probability of lnP = -63.1. That probability was calculated using the uniformly (50×) scaled 3σ frequency errors from Tab. 3.1. For completeness, the probability of that model calculated without scaling the MOST frequency uncertainties is lnP = -1.4 × 104. Clearly, with such low probability, the model match is not a statistically good match. Figure 5.6 shows the deviations between the model and observed frequencies as a function of the MOST frequencies. It is ob- vious that the MOST observations (Tab. 3.1) are determined to a level of precision that is much better than can be matched with our models. How- ever, the mean absolute deviation between the observations and the theory of 1.53 µHz is similar to that obtained by (Mkrtichian et al., 2008) for the roAp star HD 101065. The median absolute deviation between model M1 and the MOST data is 1.18 µHz. An order 1 µHz deviation at frequen- cies of ∼ 2700 µHz constitutes a percentage agreement of 0.04% — A value that is about 10 times better than the current agreement between the high frequencies calculated and observed in the Sun (see Fig. 5.1)! The largest frequency deviation in model M1 is 4.307 µHz between an ` = 3 mode hav- ing ν = 2792.107 µHz and the observed frequency of 2787.799 µHz (ν22 of Tab. 3.1). That frequency is most likely a rotational splitting and not a pulsation mode of HR 1217. Because we allow multiple matches between the model and the observed periodicities, that mode also matches frequencies between 2788.872 and 2792.249 µHz (ν23 through ν27 in Tab. 3.1). Those six frequencies correspond to the “missing modes” (§ 1.3) of Kurtz et al. (2005) and contain; within that frequency cluster, one of the suspected small spac- ings (see Tab. 3.2). The most closely matched frequency is at ν15 = 2720.914 159 5.1. The matching procedure µHz (Tab. 3.1). That is the observed frequency with the largest correspond- ing amplitude. The model mode from M1 that matches that frequency is an ` = 3 mode with ν = 2720.844 µHz. This represents a deviation of 0.070 µHz and is still more than a factor of 10 larger than the observed 3σ uncertainty of 0.005 µHz for that periodicity. Listed in Tab. 3.1 is the newly identified MOST frequency at ν1 = 2603.832 µHz that is spaced by ∼ 15.7 µHz from the next closest mode. A spacing of about 15 µHz deviates from the regular asymptotic spacing of ∼ 33.5 µHz and may represent a magnetically per- turbed mode. The frequency from model M1 that most closely matches ν1 is a mode with angular degree of ` = 1 with ν = 2599.938 µHz. That match deviates from the observed frequency by 3.895 µHz. On average10, model M1 has a large spacing of 69.938 µHz. (This should be compared to the large spacing of 69.25 µHz listed for model M1 in Tab. 5.1 that was calculated us- ing the integral definition of the large spacing given in Eq. 1.12.) The large spacing of model M1 ranges from a minimum of 69.086 µHz to a maximum of 70.449 µHz. The observed large spacing (Tab. 3.2) is, on average, 67.961 ± 0.053µHz. The small spacings (δν2−0) for model M1 range from -4.755 µHz at 2619.780 µHz to -2.100 µHz at 2828.660 µHz. MOST identifies potential small spacings having values between ∼ 1.5 and 2.6 µHz. Figure 5.6 shows that the closest matched modes would be alternating between ` = 0 and ` = 3 and the (potentially) magnetically perturbed frequencies at the smallest and largest observed frequencies have an angular degree of ` = 1. Given that the frequencies show rotational splitting, a radial mode (` = 0) seems an 10The average is taken over all n and ` for frequencies in the range of 2618.600 to 2792.249 µHz. 160 5.1. The matching procedure unlikely candidate for a match. Figure 5.6: A comparison between model M1 (Tab. 5.1) pulsation modes and the MOST frequencies. The relative frequency deviation between theory and observation is plotted against the observed frequencies (Tab. 3.1). The shaded area (± 1.5/∆T ) shows the most conservative frequency resolution discussed in § 2.2. The dotted line is the mean difference between the observations and theory. The dashed line shows the median differ- ence between the model and data points. The matching procedure allows the same model frequency to match multiple observations. 161 5.1. The matching procedure The most probable models listed in Tab. 5.1 show mean deviations from the MOST observations ranging from 1.22 to 1.57 µHz, with a standard error of∼ 0.2 µHz. The median deviation ranges from 1.01 to 1.40 µHz. The model match with the lowest mean deviation is listed as model M9 of Tab. 5.1; however, the model with the lowest median deviation of 0.88 µHz is not one of the most probable models. The properties of the model with the smallest median are listed in Tab. 5.2 and the model is labelled as MM1. Interestingly, the model with the lowest median deviation is extremely improbable and has a mean deviation is about 2.4 µHz. The probability ratio of model MM1 to model M1 of Tab. 5.1 is ∼ 10−27. In Fig. 5.7 we compare the frequency deviations between the modelled modes and observations for models MM1 (Tab. 5.2) and M3 (Tab. 5.1). Model M3 is chosen for comparison because it shows a low mean frequency deviation (1.31 µHz) and because; of all the models listed in Tab. 5.1, its magnetic field strength (4.8 kG) is closest to the observed value of ∼ 2–4.4 kG (Bagnulo et al. 1995, Lüftinger et al. 2008). The frequency deviations of model M3 (left panel of Fig. 5.7) show an alternating pattern of ` = 2 and ` = 1 with the anomalously spaced modes at either end of the observed spectrum identified as ` = 4 modes. There is no distinction between the pulsation, rotation, or small separated modes. That is, the same mode is used to match each frequency group that is separated by about ∆ν/2. Model MM1 has the smallest median deviation of all the models matched using this procedure. Its frequency deviations are shown in the right panel of Fig. 5.7. The modes shown alternate between ` = 3 and ` = 0 modes with some modes; located near those ` = 0 modes, identified as ` = 2 modes. This is what one would expect if a second order 162 5.1. The matching procedure (small) spacing was present in the data — Modes that are closely spaced by about a few µHz from each other that have angular degrees that differ by two. However, it is especially noteworthy that the potentially magnetically perturbed modes at the low and high end of the MOST frequencies are not identified in Fig. 5.7. That is because the modelled modes deviate from the observed modes by more than 10 µHz. The matching of modes from models MM1 and M3 illustrates the interesting result that we can come close to obtaining the magnetically perturbed modes (spaced by about 15 µHz in the MOST data) or the potential small spacings (spaced by about 3 µHz in the MOST data); but, we generally do not match both. None of the most probable models listed in Tab. 5.1 match a 0 kG model. 163 5.1. T h e m atch in g p ro ced u re Table 5.2: Some properties of a selected model of HR 1217. The model with the lowest median deviation between observed and calculated frequencies is listed below. The column labelled as probability label refers to the order of the probability from each of the Gaussian or Laplace probabilities. For example, 1 G is the most probable Gaussian probability and 10 L would be the 10th most probable Laplace probability. The probability ratio is defined as the Gaussian probability of the model divided by the Gaussian probability of the most probable model (see Tab. 5.1). The models in this table were matched to all frequencies listed in Tab. 3.1 using 50 × 3σ errors. Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio MM1 1.60 0.700 0.018 0.900 1.6 0.1634 8.8 0.00 3.8645 0.9243 0.2569 66.68 1809.45 0.88 2.39 4.14 3227 G, 1349 L 1.729×10−27 164 5.1. The matching procedure Figure 5.7: A comparison between the model M3 pulsation modes and the MOST observations (Tab. 3.1) in the left panel. Model MM1 pulsation frequencies (Tab. 5.2) are compared to the MOST observations in the right panel. The labels are the same as those in Fig. 5.6. Model M3 is more probable than model MM1 but the model MM1 frequencies have the smallest median deviation from the MOST observations of all of the models explored. Up to this point we have calculated the probabilities of model matches to all of the observed oscillations extracted from the MOST data. Next we compare the closest matched oscillation spectra to models with the same properties, but having different magnetic field values. These diagrams; that we will refer to as Saio plots (see, e.g., Saio 2005), illustrate how changes in the magnetic field strength can greatly impact the matching of models 165 5.1. The matching procedure to observations. Figure 5.8 shows the magnetically perturbed modes as a function of magnetic field for the two most probable models; M1 (left panel) and M2 (right panel), of Tab. 5.1. The MOST observations are shown as horizontal, dotted, lines and the location of the best fitted magnetic field value is drawn as a solid, vertical line. It is clear that in both cases the magnetic perturbations cause cyclic matches to the observations. In some cases, the magnetic field causes the eigenspectrum to completely miss the observed modes. A small change in magnetic field strength of about 0.1 kG can result in a frequency shift (much) greater than 1 µHz. Frequency jumps are clearly visible and modes tend to cluster in groups of alternating even and odd parity. This is consistent with the MOST observations (chapter 3). A close inspection of the M1 model match shows that this even and odd alternating ` pattern is broken for the two highest frequency groups. Near those frequencies, the alternating pattern switches to an odd-odd ` pattern. The anomalous spacings of ∼ 15 µHz at each end of the observed spectrum are most closely matched by ` = 1 modes (in this case), although the match is not perfect, or even statistically meaningful. There is also evidence that frequency spacings of a few µHz can exist, so that the observed spacings of ∼ 1.5–3 µHz may be explained as magnetic perturbations and not as the traditional asymptotic small spacings. In Fig. 5.9 we show the same type of diagram but for the lowest mean (left panel) and median (right panel) devia- tion models (Tab. 5.2). Those models show similar characteristics; but, in the case of the model with the lowest median deviation from the observations, the anomalous modes at the ends of the spectrum are completely missed. Not all of the models listed in Tab. 5.1 (e.g., model M3, Fig. D.2) match the 166 5.1. The matching procedure lowest and highest of the observed frequencies as odd modes. We also note that there are cases where the modes with ` values having similar parity tend to overlap or even cross each other. This phenomena was discussed at the end of chapter 4 and is the reason that the small spacing (when calculated from νn,` − νn−1,`+2) can become 0 or negative in the presence of a magnetic field. Appendix D provides Saio plots for all of the models presented in the tables in this chapter. The model match probabilities; better described as model plausibilities within our grid, were computed with Eq. 5.1 by scaling our observed uncer- tainties by a factor of 50. At the beginning of this section we stated that this scaling factor, after some experimentation, was chosen because it allowed us to properly visualize and conveniently normalize our probabilities. The lack of a realistic model uncertainty causes most of the models in our grid to get too little weight given our stringent observational uncertainties. We now have enough information, based on the quality of our model matches, to show that a error scaling factor of 50 is reasonable. The average 1σ un- certainty of our observations (Tab. 3.1) is 0.032 µHz and the average 3σ uncertainty is 0.092 µHz. The most precise oscillation frequency identified in the MOST data is ν15 = 2720.914 ± 0.002(1σ)0.005(3σ) µHz. Traditional measures of the frequency resolution when applied to the MOST data range from 0.25/∆T ∼ 0.1 µHz to 1.5/∆T ∼ 0.6 µHz. On average, our 1σ uncertainties would have to multiplied by 18.75 (= 0.6/0.032) to reach the 1.5/∆T ∼ of 0.6 µHz. Our 3σ uncertainties (again on average) would have to be multiplied by 6.52 to reach the same upper resolution limit of 0.6 µHz. Our most precise frequency 167 5.1. The matching procedure Figure 5.8: Magnetically perturbed oscillation frequencies as a function of magnetic field strength for selected models of HR 1217. The panels show the unperturbed frequencies as squares at the far left while magnetically perturbed values are plotted as coloured circles. The angular degrees ` of each mode are coloured according to the labels at the top of each panel. The MOST observations from Tab. 3.1 are plotted as horizontal dotted lines. The left panel shows model M1 of Tab. 5.1 identified as a solid vertical line while the right panel illustrates the match using model M2 of the same table. The best matched models for M1 and M2 have similar properties. Note that the magnetic perturbations produce a cyclic variation with magnetic field strength so that models with a lower magnetic field can also exist for models with physical characteristics like those of M1 and M2; but, these models would not be as statistically significant. 168 5.1. The matching procedure Figure 5.9: Magnetically perturbed oscillation frequencies as a function of magnetic field strength for selected models of HR 1217. Panel labels are the same as those in Fig. 5.8. The solid vertical line in the left panel shows the position of model M9 of Tab. 5.1. This model is selected because it has the smallest mean difference between model and observed frequencies of 1.22 µHz. The right panel identifies the model with the smallest median difference between model and observed frequencies of 0.88 µHz. That properties of that model (labelled MM1) are listed in Tab. 5.2. Although these models have low mean and median values, they are not the most probable matches. Note that the cyclic nature of the magnetic perturbation gives the closest matched modes at both low and high B values. 169 5.1. The matching procedure would have to have its 1σ uncertainty multiplied by 300 (or by 120 for the 3σ uncertainty) to arrive at 0.6 µHz. So, if we were to use the most conservative estimate of frequency resolution (1.5/∆T ) to scale our frequency precision upward, we would have to use a factor that is some where between about 10 to 100. In Fig. 5.5 we showed that the model match probability corre- lates well with the median frequency deviation between the model frequencies and the observed periodicities. The median deviation of our most probable models (Tab. 5.1) is about 1 µHz. If we were to assume that our bootstrap uncertainties should be scaled to match the smallest median deviation (∼ 1 µHz) between our calculated eigenmodes and our observations, our average 1σ uncertainties would have to be multiplied by 31.25 and our average 3σ uncertainties by 10.87. Our smallest frequency uncertainty would have to be multiplied by a few hundred (∼ 200 using the 3σ uncertainties). This would again suggest an uncertainty scale that falls somewhere in the range of a few tens to a few hundreds. When there is some unknown systematic uncertainty, the observed uncertainties can be scaled so that the best matched model has a reduced χ2 = 1. If we do this for model M1 of Tab. 5.1, the observational uncertainties would have to be scaled by 41.06. The frequency scaling of 50 is reasonable given the above scaling scenarios. This is especially true because we are more concerned with the relative probabilities between models within our grid and not the value of the individual probabilities themselves. 170 5.2. Matching the true modes of HR 1217 5.2 Matching the true modes of HR 1217 In the preceding section models were matched to all of the observed period- icities in the MOST data (Tab. 3.1). Different probability distributions and average deviation statistics were used to highlight the differences between the computed magneto-acoustic models and the observed oscillation frequencies of HR 1217. In this section we isolate what we believe are the pulsation modes (as opposed to rotation splittings) in the HR 1217 data and illus- trate what effect using this selection of frequencies has on the most plausible matches. Consider frequencies ν28 and ν29 of Tab. 3.1. They are split by the rota- tion frequency of HR 1217 (∼ 0.9 µHz) but they have the same amplitude to within their 1σ uncertainties. How can we decide which frequency is the ro- tational component and which frequency is the main oscillation mode? This dilemma was also discussed in § 3.3 for frequencies ν8 through ν12 and near ν19 and ν20 (Tab. 3.1). Evidence suggests that there are potential small spac- ings of ∼ 1.5 or 2.5 µHz near these frequency groups, along with rotational splittings, and it is difficult to decide which periodicity is the true oscillation frequency. For the purposes of matching models we select a subset of frequen- cies that we believe to be the main frequency within their respective frequency groupings. These are usually the largest amplitude components within their groupings. In the cases where we are not certain as to which mode should be considered as the main frequency, we select a frequency and adopt an uncertainty that reflects the range in frequency space that is spanned by its closest, approximately rotationally split, neighbours. For example, if we chose ν9 as a main frequency, we adopt a value of [ν10 + σ10 - (ν8 - σ8)]/2.0 171 5.2. Matching the true modes of HR 1217 as its uncertainty. Table 5.3 lists the frequencies that we believe could be the main oscillation modes in the HR 1217 data. With the pulsation modes of HR 1217 identified, we can now assign a unique frequency match to each of the observed frequencies. By assigning a unique frequency match instead of allowing a model frequency to identify itself with multiple observations (as was done in the previous section), we can immediately ignore close model matches to rotational splittings. This is important because a close match to a rotational splitting could easily skew our matching statistics. The unique assignment of a modelled frequency to an observed frequency will also im- mediately give us an indication of how closely individual modes are spaced from each other. 172 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.3: Selected frequencies identified using the MOST data. Both 1 and 3σ uncertainties on the fitted parameters are estimated from 100,000 bootstrap realizations. Frequency uncertainties marked by ∗ are average uncertainties based on the separation from the marked frequency to the nearest (∼ rotationally split) frequency plus its uncertainty (see the text for details). The phases are referenced to the time of the first observations of HR 1217 (= HJD 2451545.00 + 1769.42 days). This table is reproduced in appendix B with frequencies in units of cycles day−1. # ν [µHz] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 2603.832 39 2.22 3.8 ± 0.028 0.080 6 15 0.34 0.93 3 2619.495 125 5.70 9.2 ± 0.015 0.039 6 16 0.11 0.29 6 2652.924 365 5.56 24.2 ± 0.004 0.011 6 16 0.04 0.10 9 2685.659 31 2.83 3.1 ± 0.897∗ 0.951∗ 9 26 0.55 2.18 11 2687.420 467 4.46 29.7 ± 0.004 0.009 6 16 0.03 0.08 15 2720.914 764 1.83 41.6 ± 0.002 0.005 6 16 0.02 0.05 17 2723.469 41 2.08 3.1 ± 0.034 0.097 6 16 0.39 1.10 19 2755.393 118 3.45 8.9 ± 0.015 0.039 6 16 0.17 0.45 21 2756.925 39 5.14 3.1 ± 0.487∗ 0.578∗ 6 17 0.53 1.67 23 2788.872 37 0.24 3.6 ± 0.034 0.096 6 16 0.29 0.80 26 2791.384 82 5.00 6.8 ± 0.033 0.085 7 17 0.29 0.74 continued on next page 173 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.3: continued # ν [µHz] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 28 2805.650 84 1.10 6.9 ± 0.488∗ 0.514∗ 6 15 0.15 0.37 174 5.2. Matching the true modes of HR 1217 As in the last section we show match probabilities assuming both Gaus- sian and Laplace probability distributions. The probabilities calculated using data from Tab. 5.3 are shown in Fig. 5.10. We immediately see that when we compare the probability distribution for the magnetic field strength in Fig. 5.10 to the multiple-match case shown in Figs. 5.3 and 5.4, that the probability peaks for the magnetic field near 8 kG has significantly dropped in comparison to probability peaks near 4 kG. This is an immediate indica- tion that by allowing a single model frequency to match multiple observed periodicities; and, by including the rotational splittings in our data set, we were dramatically altering the weight of the matches. The use of the Gaus- sian or the Laplace distribution once again illustrates how outlying matches near the tails influences the shape of the probability curves. We can draw the same broad conclusions by using either of the Laplace or Gaussian distri- butions but some details of the probability space are exaggerated by choos- ing to match models using the Laplace distribution. The probability space around the mass parameter is broader in Fig. 5.10 than it is in Figs. 5.3 and 5.4. The composition (Z/X) seems to loosely favour values less than 0.02 in Fig. 5.10 but was strongly peaked at about Z/X = 0.02 in Figs. 5.3 and 5.4. In Fig. 5.11 we show the probability space for the percentage of the main sequence lifetime that the models have completed, log10 g, and the models’ convective core mass. In selecting only the pulsation modes in the HR 1217 data, we see that the probability distributions for these parameters are less affected than were the distributions for the magnetic field strength, mass, and composition (see Figs. 5.3 and 5.4). This may not be particularly sur- prising because log10 g varies slowly with position on the HR diagram and 175 5.2. Matching the true modes of HR 1217 the percentage of the main sequence lifetime that a model has traversed is selected by limiting our choice of luminosity and effective temperature (see § 4.1.1). The constraint on which stellar model masses fall in that luminosity and effective temperature range also limits the values of the convective core mass of our models. 176 5.2. Matching the true modes of HR 1217 Figure 5.10: The observed frequencies of Tab. 5.3 are used in the model search with 3σ errors scaled by 50. Probabilities in the left panel are assigned using a Gaussian prob- ability distribution (Eq. 5.1) and the panels on the right show probabilities calculated from a Laplace probability distribution (Eq. 5.3). The top panels show the probability of all models as a function of magnetic field strength, the middle panels are probability as a function of model mass, and the bottom plots give probability as a function of compo- sition (Z/X). The distribution of probabilities for the magnetic field strengths; favouring magnetic fields less than about 5 kG, is a clear contrast from that presented in Fig. 5.3, where larger magnetic fields (∼ 8 kG) are favoured. As in the previous examples, the Laplace distribution tends to highlight more models in probability space. 177 5.2. Matching the true modes of HR 1217 Figure 5.11: Model match probabilities using a subset of the MOST frequency values. The observations are matched in the same way as in Fig. 5.10. The left panels show Gaussian probabilities and the right panels show Laplace probabilities. Listed from top to bottom, the panels show the match probabilities as a function of the percentage of the main sequence lifetime the model has completed, the surface gravity of the model, and the mass of the model’s convective core in units of total stellar mass. These are similar to the probability distributions shown in Figs. 5.3 and 5.4. 178 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.4: Some properties of the most probable models of HR 1217. The top ten unique models from the Gaussian and Laplace probabilities are listed in decreasing probability order. Deviation refers to the absolute difference between a model frequency and the best matched observed frequency and σ is the standard deviation. The column labelled as probability label refers to the order of the probability from each of the Gaussian or Laplace probabilities. For example, 1 G is the most probable Gaussian probability and 10 L would be the 10th most probable Laplace probability. The probability ratio is defined as the Gaussian probability of the model divided by the Gaussian probability of the most probable model (lnP = −31.1). The models in this table were matched to all frequencies listed in Tab. 5.3 using 50 × 3σ errors. Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio U1 1.50 0.720 0.012 1.15 1.6 0.1366 3.1 0.00 3.8657 0.8789 0.2318 70.26 1877.92 2.84 4.51 4.76 1 G, 2 L 1.000 U2 1.45 0.740 0.008 1.50 1.6 0.1132 2.1 0.00 3.8651 0.8888 0.2381 67.60 1749.34 1.88 3.27 3.27 1 L, 4 G 0.483 U3 1.40 0.700 0.008 1.35 1.6 0.1148 1.8 0.00 3.8730 0.9089 0.2322 67.66 1729.36 2.35 3.39 3.15 2 G, 3 L 0.590 U4 1.45 0.720 0.010 1.30 1.8 0.1251 2.2 0.00 3.8637 0.8633 0.2281 70.16 1857.29 1.39 2.92 2.77 3 G, 8 L 0.521 U5 1.60 0.720 0.016 0.90 1.8 0.1595 9.0 0.00 3.8690 0.9114 0.2414 70.20 1899.10 3.07 4.87 5.22 5 G, 5 L 0.406 U6 1.55 0.740 0.012 1.30 1.8 0.1311 9.6 0.00 3.8581 0.9067 0.2609 65.06 1719.39 1.42 4.87 6.27 6 G, 4 L 0.195 U7 1.60 0.720 0.0160 1.05 1.6 0.1545 8.3 0.00 3.8608 0.9232 0.2638 65.20 continued on next page 179 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.4: continued Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio 1754.73 1.89 5.20 6.35 7 G, 12 L 0.190 U8 1.45 0.700 0.0100 1.20 1.4 0.1253 1.9 0.00 3.8670 0.9069 0.2376 67.66 1763.55 2.33 2.99 2.87 8 G, 9 L 0.170 U9 1.55 0.740 0.012 1.25 1.8 0.1366 3.5 0.00 3.8612 0.9030 0.2530 66.69 1766.91 1.09 4.52 5.90 6 L, 12 G 0.104 U10 1.55 0.700 0.016 1.05 1.6 0.1502 8.5 0.00 3.8596 0.9100 0.2595 65.19 1756.98 1.56 5.13 6.38 7 L, 10 G 0.140 180 5.2. Matching the true modes of HR 1217 Table 5.4 lists the properties of the most probable model matches to the MOST frequencies identified in Tab. 5.3. About half of the models listed in Tab. 5.4 have magnetic field strengths between 2 and 5 kG. This is in line with the magnetic field strengths measured in HR 1217 that range from about 2 to 4.5 kG (Bagnulo et al. 1995, Lüftinger et al. 2008). This is in contrast to the most probable model matches given in Tab. 5.1. Those models were matched to all of the frequencies (including rotational splittings) extracted from the HR 1217 data (Tab. 3.1) and gave magnetic field strengths near 1 kG or greater than about 8 kG. The most probable model (U1 of Tab. 5.4) matched using only the “true” pulsation frequencies has a magnetic field strength of B = 3.1 kG. Its prob- ability (unnormalized and calculated using a uniform uncertainty scaling of 50 ×) is lnP = -31.1. As discussed previously, this match probability is low. The MOST observations are measured to a precision that is much better than our modelled frequencies can be matched. Figure 5.12 shows the modelled mode deviations from the MOST frequencies and the Saio plot (magnetically perturbed eigenmodes as a function of magnetic field) for the same model. The mean deviation between the observed frequencies and the model U1 modes is 4.51 µHz with a standard deviation of 4.76 µHz. The median of the frequency deviations is 2.84 µHz. Note again that the matching procedure used in this section is different from the matching procedure used in the pre- vious section. In this section we force a unique mode identification on each of the “true” frequencies measured in the MOST data (Tab. 5.4). Because of this, the mean differences between the theory and observations will tend to be larger than they would have been if we allowed one modelled eigenmode 181 5.2. Matching the true modes of HR 1217 to match multiple observed frequencies. The eigenmodes from model U1 that are the poorest matches are an ` = 2, ν = 2796.694 µHz (matched to ν28 = 2805.650 µHz of Tab. 3.1) mode, an ` = 4, ν = 2783.533 µHz (matched to ν26 = 2791.384 µHz) mode, and a mode having ` = 2 and ν = 2727.779 µHz (matched to ν17 = 2723.469 µHz). All other matches deviate by less than about 1.4 µHz. The best matching mode is at 2720.908 µHz. That mode is an ` = 0 mode and is matched to the most precise of the MOST frequencies (ν15 = 2720.914µHz). The perturbed small spacings (δν 2−0) of model U1 are all negative over the range in frequencies observed in HR 1217. The small spacing has its lowest absolute value of -5.614 µHz at ν = 2653.190 µHz and increases in absolute value to -6.473 µHz at ν = 2859.900 µHz. An average over n and `; between frequencies 2618.600 and 2792.249 µHz, yields a large spacing of 68.792 µHz for model U1. 182 5.2. Matching the true modes of HR 1217 Figure 5.12: (Left) The modelled frequency deviations from the MOST frequencies (Tab. 5.3). The labels are the same as those in Fig. 5.6. Model U1 is the most probable model listed in Tab. 5.4. The average frequency deviation is 4.51 µHz. (Right) Magnet- ically perturbed frequencies as a function of magnetic field strength for the same model. The labels are the same as those in Fig. 5.8. Model U1 has a high probability because it does a reasonable job of matching the most precise frequencies from the MOST data set. The model cannot match frequencies to the high level of frequency precision measured by MOST. The properties of the models having the smallest mean (= 2.32 µHz) and median (= 0.64 µHz) absolute deviation between the theoretical modes and the observed periodicities are listing in Tab. 5.5. The model with the low- est mean deviation (labelled UU1) and the model having the lowest median deviation (UU2) have magnetic field strengths of 1 and 8.4 kG, respectively. 183 5.2. Matching the true modes of HR 1217 Figure 5.13 shows the frequency deviation plot of model UU1. We see that the small spacings (modes spaced by . 3 µHz) are better reproduced than the potentially magnetically perturbed modes (at the highest and lowest fre- quencies) are matched. That being said, the model itself does not constitute a statistically sound match. Saio diagrams shown in Fig. 5.14 for the models (UU1 and UU2) of Tab. 5.5 show that the anomalous, non-asymptotic, fre- quencies located at either end of the observed oscillation spectrum are not well matched. The closest match to those magnetically perturbed modes in model UU1 are matched to ` = 4 modes, and have deviations of about 3.5 µHz at ∼ 2600 µHz and 6.5 µHz at ∼ 2800 µHz. The frequencies of model UU1 tend to cluster in alternating even and odd parity groups, with one no- table exception: Near ∼ 2600 µHz, the ` = 4 mode is separated in frequency from the ` = 0, 2 couple (matched to the next closest mode) by about 11 µHz. On the opposite end of the spectrum (near ∼ 2800 µHz) the last two measured frequencies have associated with them modes with opposite par- ity. This is interesting because the MOST frequencies at the low end of the spectrum could be interpreted as potential small spacings while the high end of the spectrum would not be interpreted as such. Of course, the magnetic perturbations between modes separated in degree by two can be so large that labelling spacings as “small” or second order separations; as calculated in the non-magnetic case, may simply not be appropriate. This behaviour is not apparent in the Saio plot of the model (UU2) with the lowest median frequency deviation (right panel of Fig. 5.14). Saio diagrams for all models listed in the tables of this chapter are given in appendix D. 184 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.5: Some properties of selected models of HR 1217. The models with the lowest mean and median deviations between observed and calculated frequencies are listed (respectively) below. The column labelled as probability label refers to the order of the probability from each of the Gaussian or Laplace probabilities. For example, 1 G is the most probable Gaussian probability and 10 L would be the 10th most probable Laplace probability. The probability ratio is defined as the Gaussian probability of the model divided by the Gaussian probability of the most probable model (see Tab. 5.4). The models in this table were matched to all frequencies listed in Tab. 5.3 using 50 × 3σ errors. Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio UU1 1.40 0.720 0.008 1.50 1.6 0.1092 1.0 0.00 3.8645 0.8679 0.2287 68.61 1782.97 1.92 2.32 1.96 13194 G, 6948 L 3.692× 10−89 UU2 1.55 0.700 0.016 1.05 1.4 0.1502361 8.4 0.00 3.8595 0.9100 0.2598 65.06 1755.53 0.64 4.79 6.31 38 G, 22 L 0.024 185 5.2. Matching the true modes of HR 1217 Figure 5.13: Model UU1 pulsation frequencies (Tab. 5.5) are compared to the MOST observations. The labels are the same as those in Fig. 5.6. Model UU1 is the model with the smallest mean frequency deviation (2.32 µHz) from the MOST frequencies. 186 5.2. Matching the true modes of HR 1217 Figure 5.14: Magnetically perturbed frequencies as a function of magnetic field strength for the models with the lowest mean(model UU1 of Tab. 5.5, left) and median (model UU2 of Tab. 5.5, right) deviation matches. Panel labels are the same as those in Fig. 5.8. Comparing the closest models using the mean or the median as a statistic describing the match to observations can lead to dramatically different “best-fit” models. 5.2.1 Matches to a sub-set of the MOST frequencies In the previous sections (§ 5.1.2 and 5.2) we matched our model grid to the frequencies observed by MOST using all of the frequencies identified in the data (Tab. 3.1) and by using only those modes that we believed to be the pulsation modes of HR 1217 (Tab. 5.4). We were not able to find a model that matched the observations to the level of precision obtained by 187 5.2. Matching the true modes of HR 1217 the MOST mission. That is, we can find models that match the observed frequencies with a mean deviation of a few µHz but the periodicities identified in the MOST time-series have frequency uncertainties that are, on average, hundredths of a µHz. Interestingly, it seems that in the majority of the cases presented thus far, we come closest to matching the frequencies spaced from their neighbours by ∼ 15 µHz (usually referred to as the magnetically perturbed modes) or the frequencies spaced by ∼ 1–3 µHz (usually referred to as the small or second-order-spaced modes); but, we cannot seem to come close to matching both in a given model. In this section we explore this observation further by matching models to two sub-sets of the frequencies listed in Tab. 5.4. In the first sub-set of frequencies we only attempt to match the two anomalous (magnetically perturbed) frequencies and the most precisely de- termined observational frequency. These are located at 2603.832, 2720.914 and 2805.650 µHz. The properties of the models leading to the most probable model match and the model match with the smallest mean deviation from the observations; labelled GS and MS, respectively, are listed in Tab. 5.6. The models have very similar physical characteristics but their matches show very different character (Fig. 5.16). The most probable model match shows the highest and lowest of observed frequencies being matched by either an ` = 0 or 4 mode. Figure 5.15 shows the frequency deviation plot for this model. The model with the smallest mean deviation (from the 3 observed frequencies being considered here) shows the best match to those anomalous, magneti- cally perturbed modes to be an ` = 2 mode. The most probable model in this case matches the lowest two frequency groups with an even-even ` pattern, 188 5.2. Matching the true modes of HR 1217 and then continues with an odd-even alternating pattern. The best mean match, on the other hand, shows an even-odd alternating ` pattern for most of the modes, except near the highest two frequency groups. There, the pat- tern switches to an even-even ` match. This behaviour is illustrated in the Saio plots of models GS and MS (Fig. 5.16). The models GS and MS match well (with averages deviation of 0.23 and 0.09 µHz, respectively) the selected frequencies at 2603.832, 2720.914 and 2805.650 µHz; but, this comes at the expense of poorly matching the other frequencies. Model GS has modes that deviate at most by ∼ 18 µHz while model MS — having a slightly smaller deviation range — has its largest deviation between observations and theory of about 10 µHz. 189 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.6: Some properties of the most probable models of HR 1217. The most probable (using either of the Gaussian or Laplace probability distributions) model (GS) is obtained by matching frequencies at 2603.832, 2720.914 and 2805.650 µHz (see Tab. 5.3). The lowest mean deviation model (MS) is also listed. Deviation refers to the absolute difference between a model frequency and the best matched observed frequency and σ is the standard deviation. Columns with probability labels 1 and 2 refer to the order of the probability from each of the Gaussian or Laplace probabilities. For example, 1 G is the most probable Gaussian probability and 10 L would be the 10th most probable Laplace probability. Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label 1 probability label 2 GS 1.50 0.740 0.010 1.40 1.8 0.1211 6.3 0.00 3.8587 0.8886 0.2507 66.24 1740.21 0.29 0.23 0.19 1 G 1 L MS 1.50 0.740 0.010 1.20 1.6 0.1264 6.8 0.00 3.8694 0.8728 0.2214 72.65 1930.08 0.11 0.09 0.04 138 G 64 L 190 5.2. Matching the true modes of HR 1217 Figure 5.15: Frequency deviations for the most probable model matched to MOST frequencies at 2603.832, 2720.914 and 2805.650 µHz only. The model does a reasonable job of matching frequencies to within about ± 2 µHz. The obvious exception is for the potentially small-spaced frequencies that are spaced by between 1.5–3 µHz from their nearest frequency neighbours. Those frequencies deviate from the observations by between ∼ 4 and 20 µHz. 191 5.2. Matching the true modes of HR 1217 Figure 5.16: The best matched models using a forced match to the magnetically perturbed observations. Panel labels are the same as those in Fig. 5.8. The left panel shows the most probable (using either of the Gaussian or Laplace probability distributions) model obtained by matching frequencies at 2603.832, 2720.914 and 2805.650 µHz only. The right panel shows the model with the lowest mean difference between the theory and observations using only the three frequencies listed above. Model properties are listed in Tab. 5.6. Both models have similar properties. In selecting another sub-set of frequencies from Tab. 5.3 to be matched to our models, we start by choosing only the frequencies identified in both the MOST data and the data of Kurtz et al. (2005). We exclude frequency ν26 = 2791.384 µHz (Tab. 3.1) from our new list. That frequency was common to the MOST observations and to those of Kurtz et al. (2005). It is also one 192 5.2. Matching the true modes of HR 1217 of the so-called “missing modes” of Kurtz et al. (2005). With its elimination, there are no second order (small) spacings present in the data. We add frequency ν1 = 2603.832 µHz to the new listing of frequencies because it is potentially a magnetically perturbed mode. Table 5.7 lists the new frequency sub-set to which our models will be matched. 193 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.7: Selected frequencies identified using the MOST and Kurtz et al. (2005) data. This table gives frequencies found by both MOST and Kurtz et al. (2005) with the omission of all closely (. 3 µHz) spaced frequencies and with the addition of ν1 of Tab. 3.1. A comparison should be made with entries in Tab. 5.3. Both 1 and 3σ uncertainties on the fitted parameters are estimated from 100,000 bootstrap realizations. Frequency uncertainties marked by ∗ are average uncertainties based on the separation from the marked frequency to the nearest (∼ rotationally split) frequency plus its uncertainty (see the text for details). The phases are referenced to the time of the first observations of HR 1217 (= HJD 2451545.00 + 1769.42 days). # ν [µHz] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 2603.832 39 2.22 3.8 ± 0.028 0.080 6 15 0.34 0.93 3 2619.495 125 5.70 9.2 ± 0.015 0.039 6 16 0.11 0.29 6 2652.924 365 5.56 24.2 ± 0.004 0.011 6 16 0.04 0.10 11 2687.420 467 4.46 29.7 ± 0.004 0.009 6 16 0.03 0.08 15 2720.914 764 1.83 41.6 ± 0.002 0.005 6 16 0.02 0.05 19 2755.393 118 3.45 8.9 ± 0.015 0.039 6 16 0.17 0.45 23 2788.872 37 0.24 3.6 ± 0.034 0.096 6 16 0.29 0.80 28 2805.650 84 1.10 6.9 ± 0.488∗ 0.514∗ 6 15 0.15 0.37 194 5.2. Matching the true modes of HR 1217 Figure 5.17 plots the Gaussian probabilities of the most plausible model matches to the observations of Tab. 5.7. We can compare the probability distributions in each parameter plane to those using different super/sub-sets of the identified MOST frequencies. Figures 5.3, 5.4, and 5.10 show probabil- ities for the magnetic field strengths that are broadly distributed around ∼ 4 kG and ∼ 8 kG. This is also true of the probability distribution around the magnetic field strength shown in Fig. 5.17. Of all the frequency combinations explored in this chapter, the most probable magnetic field strengths tend to cluster in groups ranging from 1–5 kG and 8–10 kG. There are no preferred models that have a magnetic field strength of B = 0 kG. Of course, such conclusions come with the caveat that we cannot match the observations to within their measured precision. The magnetic field strength ranges should only be taken as a rough guide and not as statistical confidence limits. The probability space around the mass parameter favours models with masses less than about 1.5 M and the composition (Z/X) seems to loosely favour values less than 0.02. This is true for each of the frequency matches explored. The other parameters common to each of the probability plots; e.g., log g, show little variation when matches are made to different combinations of the frequencies listed in Tab. 3.1. 195 5.2. Matching the true modes of HR 1217 Figure 5.17: Probability distributions for models matched to frequencies in Tab. 5.7. Labels are the same as in Fig. 5.4. Favoured models have magnetic fields that fall within two broad groups. The first probability grouping is between ∼ 1–5 kG and the second range is between about 8–10 kG. This feature is common in all probability plots presented in this chapter. Likewise, the probability distributions for the percentage of main sequence lifetime completed, log10 g, and for the models’ convective core mass are similar in all of the probability plots in this chapter. We label the model with the most probable (lnP = -14.02) match to the observed frequencies of Tab. 5.7 PGS, and list its properties in Tab. 5.8. A plot of its frequency deviations is given in Fig. 5.18 and a Saio plot; showing how the frequencies of model PGS change with magnetic field strength, is given in appendix D (Fig. D.14). The model has a magnetic field strength 196 5.2. Matching the true modes of HR 1217 that falls in the high range of 8.8 kG. The mean deviation between the modelled modes and the observed periodicities is 1.40 µHz. The modes de- viate from the observations by less than 4 µHz. The magnetically perturbed (anomalously-spaced) pulsation modes located at either end of the observed frequency range are best identified as ` = 1 modes (for model PGS). Other modes alternate between ` = 0 and 3. 197 5.2. M atch in g th e tru e m o d es of H R 1217 Table 5.8: Some properties of a selected model of HR 1217. This model has the highest Gaussian probability (lnP = -14.02) of the models matched to the frequencies listed in Tab. 5.7 (using 50 × 3σ errors). Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) PGS 1.50 0.700 0.014 1.05 1.4 0.1434 8.8 0.00 3.8645 0.8826 0.2361 69.25 1866.36 0.57 1.40 1.51 198 5.2. Matching the true modes of HR 1217 Figure 5.18: Deviations between the frequencies of model PGS (Tab. 5.8) and the MOST frequencies. Plot symbols are the same as those in Fig. 5.6. The model PGS has the largest probability (lnP = -14.02) of the models matched to the frequencies listed in Tab. 5.8. The mean deviation between the model frequencies and the MOST observations (as listed in Tab. 5.8) is 1.40 µHz. The model diverges from the observations by 3.89 µHz in the worst case. 199 5.2. Matching the true modes of HR 1217 Two of the best matched model spectra to the frequencies of Tab. 5.7 have already been identified in the multiple-matching section 5.1.2. Model M3 (see Tab. 5.1 for the physical parameters of this model) has the smallest mean deviation between the calculated and observed frequencies. Its mean frequency deviation is 0.640 µHz and the deviations of all modes from the observations are shown in Fig. 5.19. This model matches the anomalously spaced modes at ∼ 2600 and 2800 µHz as ` = 4 modes and associates the other most closely matched modes with an alternating ` = 2 and ` = 1 pattern. Model M3 has the lowest mass (1.3 M) and metallicity (Z = 0.008, Z/X = 0.0114) of all models in our grid. The model has evolved to near the end of its main sequence life and has burned 99.93% of the Hydrogen in its core. The magnetic field matched from this model (= 4.8 kG) is in reasonable agreement with the measured values of ∼ 2–4.5 kG. For comparison we also show the deviations for the second most probable model match to the observed pulsation frequencies listed in Tab. 5.7 in Fig. 5.19. That model, (M6 of Tab. 5.1) matches all of the observed modes to less than 4 µHz and has a mean deviation of 1.29 µHz. M6 has a mass and metallicity that are higher (1.45 M, Z = 0.010, Z/X = 0.0139) than those of model M3 and has a magnetic field strength (B = 1.2 kG) that is smaller than that identified in model M3. As in model M3, the anomalous, non-asymptotically spaced modes are identified as ` = 4 modes. The other most closely matched modes of model M6 alternate with angular degree values of ` = 0 and 1. 200 5.3. On discriminating between models Figure 5.19: The frequency deviations for models M3 and M6 of Tab. 5.1. Plot symbols are the same as those in Fig. 5.6. Model M3 is the model with the smallest mean devia- tion (= 0.640 µHz) between the theoretical eigenfrequencies and the frequencies listed in Tab. 5.8. Model M6 is the second most probable model (lnP = -14.05) when matched to the frequencies in Tab. 5.8. Note that both models were already identified in § 5.1.2. In that section, we allowed a single model frequency to match multiple observations. The exclusion of the potential small spacings (frequencies spaced by ∼ 3 µHz) results in a better (smaller) mean deviation than those obtained earlier this chapter. 5.3 On discriminating between models In chapter 4 we calculated a large grid of pulsation models that included magnetic perturbations. We adopted a large range in metallicity (Z = 0.008– 201 5.3. On discriminating between models 0.022) because the peculiar abundance patterns on the surface of the Ap stars limits a direct inference of their global metallicity. The allowed mass range (M ∼ 1.3–1.8 M) was limited by the external constraints on the luminosity and effective temperature of HR 1217 (see § 4.1.1). A thorough exploration of the range of magnetic field strengths that provided asteroseismically relevant results for HR 1217 had never been performed, so we adopted a wide range in B for our model calculations. We calculated 51,795 models that, as a whole, contain millions of modes to be matched to the frequencies determined from the time-series of MOST photometry. In the preceding sections of this chapter we showed that we were able to match models that have a mean deviation between the theoretically calculated modes and observations of less than a few µHz. Matching models to a fractional accuracy of about 0.05% is an amazing accomplishment considering the complexity of the models and the sensitivity of the pulsation modes to the magnetic perturbations. (The high frequency oscillations in the Sun are matched to a fractional accuracy of about 0.3%). Those models are; however, a poor statistical match because the MOST frequencies are determined to such a high precision. The current state of the observations of roAp star oscillations is so highly precise that the model calculations are being stretched to the limits of their applicability. How well can we distinguish between reasonable models of HR 1217 given the small match probabilities with our current grid? The short answer is that matching modelled frequencies to (ultra-)precise observations (like those from MOST ) is an extremely sensitive tool to dis- criminate between models. In Fig. 5.20 we plot the minimum frequency deviation required to match 50 and 75% (left and right panels of that figure, 202 5.3. On discriminating between models respectively) of the observed frequencies of Tab. 5.3 as a function of magnetic field strength. (The model mass and composition Z/X are also identified in that figure.) Immediately we can see that there are only 142 of 51,795 models that match 50% of the observed frequencies below a 1 µHz threshold. We can rule out 99.7% of the models in our grid by adopting that match crite- ria. In order to match 75% of observations we have to raise our matching threshold to between ∼ 2–3 µHz. In that frequency range, 75% of the most observations are matched by 69 of the 51,795 models. Put another way, 136 models within our grid match between 6 and 8 of the MOST observations (∼ 50–66%) to an accuracy that is less than 1 µHz. Figure 5.21 shows the same type of plot but we require matching 100% of the MOST observations listed in Tab. 5.3. There are 58 models that can match 100% of the observed periodicities with frequency deviations that are not greater than 6 µHz. 203 5.3. On discriminating between models Figure 5.20: Minimum frequency deviations required to match 50 and 75% of the MOST observations as a function of magnetic field strength. The left panel plots the minimum frequency deviation required to match 50% of the MOST observations. The colour bar on the bottom identifies the mass of the model. Symbols shown in the legend identify the composition (Z/X) ranges (∼ sub-solar Z/X, ∼ solar Z/X, ∼ super-solar Z/X) of the models. The right panel shows the minimum frequency deviation required to match 75% of the most frequencies. A total of 142 models (= 0.3% of the 51,795 models) match 50% of the MOST frequencies below 1 µHz and 69 models (= 0.1% of the models in the grid) match 75% of the MOST periodicities with deviations between 2–3 µHz. 204 5.3. On discriminating between models Figure 5.21: Minimum frequency deviations required to match 100% of the MOST ob- servations. The labels are the same as those of Fig. 5.20. There are a total of 58 models (= 0.1% of the 51,795 models) that match all of the MOST observations with frequency deviations between 4.7–6 µHz. The model (labelled LPC) with the smallest frequency deviation required 205 5.3. On discriminating between models to match 100% of the MOST observations (Tab. 5.3) has properties that are listed in Tab. 5.9. The frequency deviations for model LPC are shown in Fig. 5.22 and a Saio plot for that same model is given in appendix D (Fig. D.14). In this model, the closest matches to the anomalous frequencies at about 2600 and 2800 µHz are ` = 0 and 2 modes, respectively. The modes alternate in a pattern that would suggest that the closely (small or second order) spaced modes are in ` = 1, 3 and ` = 0, 4 pairs. Of course, the match in model LPC gives a small spacing that is between 2–3 times what is observed (. 3 µHz). The magnetic field of model LPC is 2.4 kG and is consistent with the lower end of the Lüftinger et al. (2008) measurement (B between 2–4.4 kG). 206 5.3. O n d iscrim in atin g b etw een m o d els Table 5.9: Some properties of a selected model of HR 1217. This model has the smallest frequency deviation (= 4.759 µHz) required to match all of the MOST frequencies. The column labelled as probability label refers to the order of the probability from each of the Gaussian or Laplace probabilities. For example, 1 G is the most probable Gaussian probability and 10 L would be the 10th most probable Laplace probability. The probability ratio is defined as the Gaussian probability of the model divided by the Gaussian probability of the most probable model (see Tab. 5.4). The models in this table were matched to all frequencies listed in Tab. 5.3 using 50 × 3σ errors. Model ID M/M X Z Age (Gyr) α Mconv.core/M? B (kG) Mconv.env./M? Log10 (Teff ) Log10 (L/L) Log10 (R/R) ∆ν (µHz) cut-off (µHz) median deviation (µHz) mean deviation (µHz) σ deviation (µHz) probability label probability ratio LPC 1.75 0.740 0.022 0.65 1.8 0.1746 2.4 0.00 3.866 0.9222 0.2535 70.50 1966.12 2.55 2.43 1.60 3460 G, 1740 L 7.476 × 10−35 207 5.3. On discriminating between models Figure 5.22: Deviations between the frequencies of model LPC (Tab. 5.9) and the MOST frequencies. The model LPC has the smallest frequency deviation (∼ 4.7 µHz) of all the models that is needed to match 100% of the MOST observations. This model is still highly improbable and is 10−35 times less likely than model U1 (Tab. 5.4). Given the range in frequency that is required to match all of the MOST 208 5.3. On discriminating between models observations, we next see if our observed variations are matched more often to a mode with one particular angular degree over another. Figures 5.23, 5.24, and 5.25 show histograms of the number of matched modes for each of the angular degrees (` = 0–4). A match occurs if the best fit model frequencies fall within the 3σ uncertainties for the periodicities listed in Tab. 5.3. The histograms showing the matched ` values for each of the observed frequencies are presented in the left panels of Fig. 5.23 for all of the magnetic models. The right panels show only B = 0 kG models. There appears to be no statistically preferred ` value for any of the observed frequencies. This is also true for magnetic models with B = 2 and 4 kG (Fig. 5.24) and B = 6 and 8 kG (Fig. 5.25). The modelled frequency matches tend to alternate between even and odd modes. If we assume that for a particular frequency an odd or even mode match is equally likely, we could immediately eliminate half of the models in our grid by applying an external constraint on the mode parity for that match. 209 5.3. On discriminating between models Figure 5.23: Histograms of matched model frequencies. A match occurs if the best fit model frequencies fall within the 3σ uncertainties for the periodicities listed in Tab. 5.3. The left panels present histograms of the matched frequencies in all of the magnetic and non-magnetic models. The right panels show matches in cases where B = 0 kG. Each column shows modes with different angular degrees, increasing from ` = 0 at the top to ` = 4 at the bottom. There are no preferred matches for ` values. Matches near 2650 µHz seem to be matched less often for all ` values when compared to the other frequencies. 210 5.3. On discriminating between models Figure 5.24: Histograms of matched model frequencies. The format is the same as that in Fig. 5.23. The left panels now show only models with B = 2 kG, and the right panels only those models with B = 4 kG. Model frequencies near 2650 µHz are again under-matched, especially for the ` = 0, B = 4 kG case. 211 5.3. On discriminating between models Figure 5.25: Histograms of matched model frequencies. The format is the same as that in Fig. 5.23. The left panels now show only models with B = 6 kG, and the right panels only those models with B = 8 kG. The ` = 0 and 2 modes tend to be matched more often near 2800 µHz in the 8 kG models. As in the cases with different magnetic fields, models matched to the observed frequencies near 2650 µHz are less common. 212 5.3. On discriminating between models We have used the largest grid of magneto-acoustic modes to date to search for a match to the most precise photometric data ever measured on the roAp star HR 1217. We are able to match models to within (on average) a few µHz. This represents an impressive accomplishment and gives hope that fu- ture advancements in theoretical modelling of roAp stars will yield accurate and meaningful constraint on the physics of these unique stellar structures. The matches explored are not perfect and local variations in the probability space of our modelled parameters occur for each of the matching scenarios discussed in this chapter. The magnetic field perturbations are cyclic in na- ture, and as such, leads to parameter degeneracies when matching models to observations. It is important to note that by using a small grid of models, one could, conceivably, find a reasonable match (with say a mean deviation of 1–2 µHz) to the observations. Such a match may not be (and is most likely not) the best global match to the data. Likewise, the large span of Z used in our model grid introduces degeneracies that allow a large range in model mass and evolutionary state to enter the luminosity-effective temperature er- ror box into which HR 1217 falls (§ 4.1.1). In terms of probabilities, Figs. 5.3, 5.4, 5.10, and 5.17 show probabilities for the magnetic field strengths that are broadly distributed in groups ranging from 1–5 kG and 8–10 kG. There are no preferred models in our grid that have a magnetic field strength of B = 0 kG. The probability space around the mass parameter favours models with masses less than about 1.5 M and the composition (Z/X) seems to loosely favour values less than 0.02. If the mean is used as a discriminant between models, there are no preferred parameter combinations. The mean or median frequency deviation; when used as the only matching statistic, 213 5.3. On discriminating between models does not provide enough of a constraint on the models to draw even broad conclusions. The ranges in parameters we give based on our probability distributions should be used as guidelines for future modelling efforts and should not be taken as strict statistical limits on the physical parameters of HR 1217. We do not place confidence limits on any of the physical pa- rameters of HR 1217 from our modelling efforts. The lack of a statistically meaningful match, and complications arising from the degeneracies in some of the model parameters, makes claims of precise parameter limits dubious at best. A model independent measure of the parity of the observed oscillations would greatly constrain the matched model properties. We will discuss the quality of the matches and suggest alternate modelling avenues that should be explored in the (following) concluding chapter. 214 Chapter 6 Summary, conclusions, & future work The roAp star HR 1217 exhibits a rich eigenspectrum that obeys the regular asymptotic (§ 1.1.1) spacing between its oscillation modes. The observed frequencies are spaced by an alternating pattern of ∼ 33.5 and 34.5 µHz — values that are interpreted as half of the large spacing ∆ν — with an im- portant deviation. A global, multisite, photometric campaign (Kurtz et al., 1989) revealed an anomalous frequency spacing of ∼ 50 µHz that could not be explained by asymptotic theory. Much later, Cunha (2001) theorized that the strong magnetic field (∼ 2–5 kG) of HR 1217 could significantly perturb its normal oscillation modes. She predicted the existence of a “missing mode” that would explain the anomalous increase in the regular spacings. At nearly the same time as Cunha’s prediction there was another (global) photometric campaign organized by the Whole Earth Telescope (WET). Those new data (having a duty cycle of 36%, see Kurtz et al. 2002 and 2005) found a number of new periodicities that agreed well with the predictions of Cunha. A non- asymptotic, potentially magnetically perturbed, frequency spacing of ∼ 15 µHz was found at about 2785 µHz along with a frequency group that showed a spacing of 2.5 µHz. This was cautiously interpreted as a potential second 215 Chapter 6. Summary, conclusions, & future work order separation, or small spacing (§ 1.1.1), that could possibly constrain the age of HR 1217. It was shown for the first time that the magnetic field plays a direct role in the pulsation spacings of a roAp star. The photomet- ric variations identified in these earlier ground-based campaigns are shown schematically in Fig. 1.7. Motivated by these earlier results, MOST observed HR 1217 near con- tinuously as a Fabry target (§ 2.1) for about 30 days in late 2004. The satellite collected more than 68,000 photometric data points with a duty cy- cle of 95%. These data represent the ultimate photometric measurements on HR 1217. We used CAPER (Cameron et al., 2006, 2008); a tool developed specifically for time-series analysis of MOST data (§ 2.2), to determine the oscillation frequencies of HR 1217. The Fourier parameters extracted from the MOST light curve are summarized in Tab. 3.1. The uncertainties in our fitted parameters were determined using 100,000 bootstrap samples. In total there were 29 frequencies identified, each having a S/N ≥ 3.5 (& 2.5σ detection) to within uncertainties. (The reduction is described in full detail in chapter 3.) The most precise amplitude determined from the MOST data is 6 µmag, making this the most precise data set ever collected on HR 1217. We now address each of the observational objectives put forth in chapter 1 (§ 1.4) below: 1. It has been known since the early 1980s that HR 1217 is a multiperiodic variable star with near equally spaced modes, alternately spaced by ∼ 33.5 and 34.5 µHz. The most recent confirmation of this was the analysis of the Whole Earth Telescope data by Kurtz et al. (2005), that achieved a duty cycle of 36% and reached an unprecedented precision 216 Chapter 6. Summary, conclusions, & future work of 14 µmag for a ground-based photometric study. We will be able to unambiguously identify the spacings in the MOST data because of its near continuous coverage and ultra-high precision. Are the already identified periodicities constant over time for all observations on HR 1217, or do they vary over time, pointing to a selective mode damping mechanism or some nonlinear interaction? 2. The strange, apparently non-asymptotic, spacing that was observed be- tween the highest two frequencies in HR 1217 remained a mystery until the recent results of Kurtz et al. (2005, 2002). Those authors identified a previously unidentified frequency near 2790 µHz that could only be explained as a magnetic perturbation Cunha (2001). Explaining that frequency (or lack of that frequency) has driven the theory of roAp star oscillations for more than two decades. Is this frequency identified in the MOST data on HR 1217 and is it stable since the last observations of Kurtz et al. (2005)? The average difference between the frequencies extracted from the data of Kurtz et al. (2005) and those from the MOST data (§ 3.3) is 0.08 ± 0.09 µHz (3σ uncertainty), indicating that all frequencies identified by both the ground- based and space-based campaigns are consistent with each other. Those peri- ods common to each of the data sets are stable since the initial ground-based collaboration of Kurtz et al. (1989), even though there are some differences in the amplitudes of the dominant peaks (e.g., § 3.3). Frequencies near the “missing mode” at ∼ 2790 µHz (present in the WET data of Kurtz et al. 2005, but were not found in the Kurtz et al. 1989 data) are identified in the MOST photometry. This suggests that those frequencies are stable over a 217 Chapter 6. Summary, conclusions, & future work baseline of at least 4 years. It should be noted that these frequencies show the largest deviations (∼ 0.2 µHz) from those determined by Kurtz et al. (2005) because they are located near a harmonic of the orbital period of the MOST satellite (16 × 165 µHz = 2640 µHz and 17 ×165 µHz = 2805 µHz) and are potentially influenced by the stray light (§ 2.1). That being said, the deviations between the MOST frequencies near 2790 µHz and the corre- sponding Kurtz et al. (2005) frequencies are near their 3σ uncertainties. Any deviations between the MOST and Kurtz et al. (2005) frequencies are well below the resolution of 1/T = 0.4 µHz. Because the MOST and WET data show common frequencies, the alternating values of ∼ 33.5 and ∼ 34.5 µHz remain the same. Those values are consistent with acoustic modes that are alternating between even and odd angular degree ` and provide an astero- seismic parallax that is consistent with the Hipparcos parallax for HR 1217 (Matthews et al. 1999, see also § 1.3). 3. We expect to observe a number of new periodicities in the data because of the high level of photometric precision and the near contiguous data sampling of the MOST satellite. Can new periodicities be identified and, if so, can they be used to constrain the physics of this magnetic pulsator? 4. A recent spectroscopic study on HR 1217 by Mkrtichian & Hatzes (2005) identified new frequencies at ∼ 2553 and 2585 µHz. These fre- quencies are interesting because they approximately match the alter- nating ∼ 33.5 and 34.5 µHz pattern previously observed in HR 1217, but were not identified in the photometry of Kurtz et al. (2005). Does the MOST data uncover those potentially new periodicities? 218 Chapter 6. Summary, conclusions, & future work There were 7 new frequencies identified in the MOST data (§ 3.3). Recur- ring frequency spacings of∼ 1.5 and∼ 2.5 µHz are evident in the MOST data. Main sequence models of A-type stars (e.g., § 4.2) have small spacings (§ 1.1.1) that are about 3 µHz and higher. This small spacing can potentially constrain the age of the star because its value is most sensitive to changes in the composition of the stellar core as the star evolves. The magnetic per- turbations to the frequencies may be of the same order, or larger than, the small spacing (e.g., Dziembowski & Goode 1996). This was also illustrated in § 4.3. We must be cautious in interpreting these spacings as small sepa- rations. Some frequencies in the MOST data are spaced in such a way that it is not immediately obvious whether or not the frequencies are rotational splittings or “small spacings”. A repeating pattern of ∼ 1.5 and ∼ 2.5 µHz is evident if ν9 is chosen as the main (not a rotational splitting) frequency in the grouping. However, the uncertainties suggest that the difference between ν10 and ν11 is not a rotational splitting. One would expect the frequency set ν10, ν11, and ν12 to be a rotational triplet based on the alternating ∼ ∆ν/2. It is possible that magnetic effects are causing a non-repeating pattern of ∼ 1.5 and ∼ 2.5 µHz, or that the closely spaced modes are convolved in such a way that a stable least squares frequency solution exists where ν10 and ν11 are spaced close to, but not at (within the 3σ errors) the rotation frequency of the star. Because a spacing of ∼ 2.5 µHz had been observed by Kurtz et al. (2005), and there is more than one example of this spacing in the MOST data, we believe that these spacings are not related to unresolved rotational splittings. It should be noted here that the rotation frequency observed in the MOST data (0.924 ± 0.047 µHz, see § 3.2 and 3.3) is de- 219 Chapter 6. Summary, conclusions, & future work termined to a level of precision that allows us to compare it to the observed “small separations”, but it is not determined to a high enough level of preci- sion to allow us to favour one of the previously measured rotation frequency measurements (§ 3.2). A new MOST frequency at ν1 = 2603.832 µHz is inconsistent with the new frequencies at ∼ 2553 and 2585 µHz found in radial velocity measure- ments of Mkrtichian & Hatzes (2005). Recent results by Kurtz et al. (2006) suggest that there can be pulsations detected in radial velocity data that are not present in photometric data. This was illustrated for the case of the roAp star HD 134214 (Cameron et al. 2006, § 2.3.1) where MOST photometry identified a single periodicity during a short observing run while Kurtz et al. (2006) identified up to 6 periodicities in their radial velocity data (not ob- served synchronously with MOST ). It is possible that there is some unknown selection effect in the atmospheres of roAp stars that reflects certain modes at different depths, causing results from spectroscopy to sometimes be dis- crepant with photometric measurements. 5. In order to determine the reliability and uniqueness of the periodicities in a multiperiodic variable, we need to develop tools that can test the precision and resolution of our data set. Techniques we have developed for other MOST targets will be applied to the HR 1217 data. The res- olution of our data set and the assertion that closely spaced frequencies are spaced by the rotation rate of the star will be tested. The bootstrap technique described in § 2.2 provides a means to test if we are extracting resolved (or unresolved) frequencies in the data. In § 3.3 220 Chapter 6. Summary, conclusions, & future work we were able to show that the addition of a number of closely space frequen- cies (spaced by approximately half of the rotation frequency of HR 1217) causes the fit to become unstable and the new frequencies (Tab. 3.3) clearly interact with the previously identified, resolved, parameters of Tab. 3.1 (see Fig. 3.8). Because of this, we are confident that we have extracted only resolved frequencies. The ultra-precise photometric measurements of the MOST satellite reveal a number of new periodicities that are spaced close to what is expected for an asymptotic small spacing (. 3 µHz) and one new frequency that is spaced by an amount (∼ 15 µHz) that is reminiscent of the magnetically perturbed modes identified by Kurtz et al. (2005). We next summarize the modelling efforts we used to describe the observed oscillations in HR 1217. In recent years there have been a number of studies that have attempted to model specific roAp stars. Cunha et al. (2003) examined a small number of stellar models for HR 1217 in hopes of determining its evolutionary status. They did not include the effects of a magnetic field and were not able to re- produce the anomalous spacings. Gruberbauer et al. (2008) and Huber et al. (2008) attempted to constrain the properties of the roAp stars γ Equ and 10 Aql using ∼ 10 A-star models (including magnetic perturbations using the method of Saio & Gautschy 2004) and data collected by MOST. HD101065; a.k.a. Przybylski’s star, was studied by Mkrtichian et al. (2008), who also use only a few stellar models (again using the method of Saio & Gautschy 2004) to explore the physics of that star. Their results were discussed in more detail in chapter 5. Brandao et al. (2008) also use only a few (non-magnetic) models to study the spectrum of α Cir that was obtained from photometry 221 Chapter 6. Summary, conclusions, & future work collected with the WIRE (NASA’s Wide Field Infrared Explorer) satellite (Bruntt et al., 2009; Buzasi, 2002). Each of the above studies places some limits on the properties of the star they were studying, even without a thor- ough parameter space search. Interestingly, the modelling efforts referenced above that included magnetic effects arrived at “best matched” magnetic field strengths that were about a factor of 2 larger than what was measured from spectropolarimetric observations. Our study explores the magnetic effects on the pulsation frequencies for models of HR 1217 for the first time and significantly extends the number of models and the breadth of the parameter space explored when compared to earlier asteroseismic studies on HR 1217, or any other roAp star. The high quality MOST data and this extensive grid of state-of-the-art models of magneto-acoustic modes for HR 1217 is a first, and is unlikely to be surpassed in the near future. Our grid covers the observable characteristics of HR 1217 (Teff = 7400 +100 −200 K, L = 7.8 ± 0.7 L, B ∼ 2–5 kG), and includes magnetic perturbations to the calculated normal modes of the models. These models are outlined in detail in chapter 4. This grid extensively covered the parameter space (§ 4.1.1) in stellar mass (1.3 to 1.8 M in 0.05 M steps), composition (Z = 0.008 to 0.022 in steps of 0.002 and X = 0.700 to 0.740 in steps of 0.020) and frequency range (` = 0–4 over the range 1900–3100 µHz.) appropriate for HR 1217. Mixing length parameters of α = 1.4, 1.6, and 1.8 are adopted and the models in the grid have a large spacing (∆ν ∼ (M/R3)1/2) that ranges from ∼ 57–81 µHz. The pulsation mode calculation was tested using a new method developed by Kobayashi (2007, § 4.2.1) for terrestrial seismology. The modes calculated using the JIG 222 Chapter 6. Summary, conclusions, & future work code were confirmed by comparison to modes calculated using Kobayashi’s method. (Showing the consistency between the modes calculated using the two different pulsation codes was the last of the modelling goals outlined in § 1.3) Our model grid, composed of 569 A-star models that fall in the HR 1217 luminosity–effective temperature error bounds, is the most extensive grid of models used to study the magnetic effects on the pulsation modes of roAp star to date. The magnetic perturbations to the eigenfrequencies of the models were estimated using the variational method of Cunha & Gough (2000); see § 4.3, assuming a dipolar magnetic field, with polar strengths ranging from 1 to 10 kG in steps of 0.1 kG. This brings the total number of magneto-acoutic pulsation models to 51,795. The modelling goals outlined in § 1.3 have a common thread: Namely, how closely can we match the frequencies (and or spacings) observed by MOST, and what constraints can we place on the physical parameters of the star? In chapter 5 we showed that we are able to match models to within (on average) a few µHz. A mean match of a few µHz represents a fractional accu- racy of about (∼ 1.5 µHz / 2700 µHz =) 0.05%. For comparison, the high fre- quency oscillations in the Sun are matched to a fractional accuracy of about 0.3%. This is an extraordinary accomplishment considering the complexity of our models and the large perturbations introduced by the inclusion of a mag- netic field. The matches obtained are not perfect; but, matching modelled frequencies to the MOST frequencies has proven to be an extremely sensitive tool to discriminate between models. For example, 136 models within our grid match between 6 and 8 (∼ 50–66%) of the MOST observations to an 223 Chapter 6. Summary, conclusions, & future work accuracy that is less than 1 µHz. That represents only (100 × 136/51,795 =) 0.3% of the models in our grid. We were able to show that there are 58 models that can match 100% of the observed periodicities with frequency deviations that are not greater than 6 µHz (§ 5.3). There are about (100 × 17579/51796 =) 33% of the models in our grid that match the observed large spacing of HR 1217 (∼ 68 µHz) to within about 1.25 µHz. (To arrive at a value for the large spacing we calculated an average over all n and ` for fre- quencies in the range of 2618.600 to 2792.249 µHz.) Using the large spacing as a discriminant between models is less effective than matching individual frequencies. The matching of the observed frequencies to our modelled modes eliminates a large percentage of parameter space that is allowed within the luminosity-effective temperature error bounds for HR 1217. We cannot find a model that matches both the magnetically perturbed modes (spaced by ∼ 15 µHz) and the oscillations spaced by the small spacing (. 3 µHz) simulta- neously. This presents a significant challenge to the modelling of roAp stars and remains an outstanding problem. The probability of our models matching the MOST frequencies was thor- oughly explored in chapter 5. In general, the probability of a match is very low. While we achieve average deviations of a few µHz between our measured periodicities and the modelled eigenmodes, the measured precision on the MOST frequencies is on the order of 0.01 µHz. The magnetic field perturba- tions are cyclic in nature, and as such, there are parameter degeneracies when matching models to observations. The large span of Z used in our model grid introduces an indeterminacy that allow a large range in model mass and evo- lutionary state to enter the luminosity-effective temperature error box into 224 Chapter 6. Summary, conclusions, & future work which HR 1217 falls (§ 4.1.1). The magnetic field of HR 1217 was estimated by Bagnulo et al. (1995) to be 3.9 kG. More recently, Lüftinger et al. (2008) measured a field variation over the rotation period of HR 1217 of 2.2 to 4.4 kG. We have illustrated in chapter 5 that probabilities of a model match for the magnetic field strengths broadly distributed in groups ranging from 1–5 kG and 8–10 kG. In previous studies on roAp stars (e.g., Gruberbauer et al. 2008, and the others mentioned above), the modelled magnetic field strengths were about a factor of 2 larger than the spectroscopically measured values. The range in probability for a match to the magnetic field strength of HR 1217 in our models are consistent with both the spectroscopically determined values and twice those values. We conclude that there are probably magnetic field matches for those other roAp stars that are consistent with external mea- sures, provided the model parameter space is explored in more detail. There are no preferred models in our grid that have a magnetic field strength of B = 0 kG. The probability space around the mass parameter favours models with masses less than about 1.5 M and the composition (Z/X) seems to loosely favour values less than 0.02. There is no meaningful constraint on the mixing length parameter α. If the mean is used as a discriminant between models, there are no preferred parameter combinations because there is a wide range of models that have low mean deviation matches. The ranges in parameters we give based on our probability distributions should be used as guidelines for future modelling efforts and should not be taken a strict statistical limits on the physical parameters of HR 1217. We do not place confidence limits on any of the physical parameters of HR 1217 from our models. The lack of a statistically meaningful match, and complications arising from the degenera- 225 Chapter 6. Summary, conclusions, & future work cies in some of the model parameters, makes claims of precise limits dubious at best. The current state of the observations of roAp star oscillations is so highly precise that the model calculations are being stretched to the limits of their applicability. In stellar seismology we usually look for the closest matched model and then explore different physics to see how the models can be improved. Stellar models do not generally correspond to simple analytical models where one parameter can be tweaked to match an observation without having to change other parameters as well. That is simply why we have explored a large coverage in the parameter space relevant for models of HR 1217. As an aside, most of the modes observed in the Sun could not be matched in the 80s and early 90s. The best helioseismic agreement at high frequencies still show a discrepancy that is many standard deviations from the observations. Our grid of magnetically perturbed oscillation models represents a unique first step to try to constrain the properties of a roAp star. The models we calculate use standard physics and focus on the addition of a magnetic perturbation to the oscillation modes as an important (first) non-standard addition. Calculating our models came at a significant computational cost so we focused on as wide of a parameter space as we possibly could. This comes at the expense of fine grid resolution. We found that our current grid resolution in age and mass is not sufficient to be able to properly constrain the properties of HR 1217. Take, for example, model E of Tab. 4.1. Over its corresponding evolutionary sequence the large spacing changes from 80 to 68 µHz. There were 11 models with detailed seismic information output. The change in ∆ν is not equally spaced along an evolutionary track and ranges 226 Chapter 6. Summary, conclusions, & future work by about 0.2 µHz near 68 µHz to about 1 µHz near 80 µHz. We expect a percentage change in ∆ν to roughly correspond to the same percentage change for a given mode along an evolving set of models. That means the frequencies between the models in our grid (that are adjacent in age) change from about 5 to 40 µHz near 2700 µHz. A model that matches frequencies to within about 5 µHz can potentially be changed in age slightly to get a better match — at least in the non-magnetic case. The behaviour of the modes in the magnetic models is more difficult to predict because of the non-linear coupling between frequencies, magnetic field strength and the structure of the outer layers of the model. So where should future modelling efforts be focused? Recent, non-standard, models were used to explore the excitation physics of the roAp stars (Gautschy et al. 1998,Balmforth et al. 2001,Saio 2005, and Théado et al. 2005,2009). Al- though some of those models excited oscillations in a very specific frequency range, or over specific temperature and luminosity ranges, none were success- ful in calculating a globally applicable excitation model. This leaves us with a number of unsolved problems in our understanding of the roAp star pulsa- tions that should also be addressed. These include: 1) the understanding of what excites the observed periodicities in these stars, 2) how the pulsation modes interact with the chemical gradients in the upper stellar atmosphere, 3) why some roAp stars show frequencies above the theoretical acoustic cut- off frequency, and 4) what the effects of alternate magnetic geometries are on the pulsation modes. We point to recent reviews by Kurtz & Martinez (2000), Cunha (2007), Saio (2008) and Shibahashi (2008) to highlight the current efforts within the asteroseismic community to address these issues. 227 Chapter 6. Summary, conclusions, & future work Because previous studies using non-standard model physics focused on the excitation of the calculated modes, they did not highlight what affect their model alterations had on the individually calculated model frequen- cies and spacings. This is necessary to define which non-standard additions should be added to future model grids to improve the matches to observa- tions. A fiducial model sequence should be defined so that modellers within the roAp community can easily compare and contrast the addition of new physics to a standard set of models. To increase the efficiency of grid cal- culations, we suggest that a study be undertaken that defines the minimum spacing in the basic stellar parameters (M, Z/X, ∆ν, B) that is needed to interpolate modelled frequencies to within a sub-µHz accuracy. Interpolation within a small grid of models having a relatively large spread in parameters is generally untrustworthy. 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Monograph series on nonthermal phenomena in stellar atmospheres 244 Appendix A HD 127756 and HD 217543 parameter tables Tables of fit parameters identified in the slowly pulsating Be (SPBe) stars HD 127756 and HD 217543 by the MOST team (Cameron et al., 2008). 245 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.1: Frequencies identified in the star HD 127756 by the MOST team (Cameron et al., 2008) # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 0.0335 7.2 3.60 8.68 ± 0.0009 0.0023 0.2 0.6 0.11 0.30 2 0.0739 2.2 0.59 6.97 ± 0.0020 0.0055 0.1 0.4 0.22 0.58 3 0.1300 8.8 3.65 9.50 ± 0.0007 0.0017 0.1 0.4 0.07 0.19 4 0.1664 7.3 1.27 8.31 ± 0.0004 0.0010 0.2 0.4 0.06 0.16 5 0.1957 5.0 2.64 8.71 ± 0.0007 0.0018 0.2 0.5 0.10 0.24 6 0.2530 1.8 4.11 5.12 ± 0.0031 0.0078 0.2 0.4 0.32 0.81 7 0.2827 1.4 4.27 3.89 ± 0.0032 0.0079 0.2 0.4 0.34 0.88 8 0.7108 2.3 3.31 5.74 ± 0.0015 0.0038 0.2 0.6 0.15 0.37 9 0.7393 1.7 5.22 4.80 ± 0.0016 0.0040 0.2 0.5 0.19 0.52 10 0.7740 5.1 4.80 9.04 ± 0.0005 0.0013 0.2 0.5 0.06 0.17 11 0.8035 6.7 4.24 8.16 ± 0.0005 0.0013 0.3 0.7 0.03 0.08 12 0.8280 7.2 4.13 8.22 ± 0.0005 0.0015 0.3 0.7 0.03 0.09 13 0.8702 2.4 2.57 5.17 ± 0.0015 0.0048 0.2 0.4 0.15 0.45 14 0.9149 10.2 4.18 11.12 ± 0.0004 0.0011 0.1 0.4 0.04 0.10 15 0.9595 6.5 2.67 9.80 + 0.0006 0.0015 0.1 0.4 0.06 0.17 continued on next page246 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.1: continued # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ . . . . . . 0.9595 6.5 2.67 9.80 − 0.0006 0.0015 0.1 0.4 0.06 0.16 16 1.0236 4.2 0.31 8.74 ± 0.0007 0.0021 0.1 0.4 0.08 0.22 17 1.0595 2.3 0.06 5.00 ± 0.0015 0.0042 0.1 0.4 0.15 0.41 18 1.1297 1.9 4.55 4.00 ± 0.0024 0.0064 0.2 0.5 0.28 0.72 19 1.1418 6.2 1.65 8.24 ± 0.0007 0.0016 0.2 0.5 0.08 0.21 20 1.5301 1.2 4.16 4.04 ± 0.0021 0.0051 0.1 0.3 0.22 0.56 21 1.7761 1.8 5.56 5.65 ± 0.0017 0.0047 0.1 0.4 0.19 0.46 22 1.8208 4.8 4.63 8.97 + 0.0009 0.0024 0.2 0.5 0.10 0.35 . . . . . . 1.8208 4.8 4.63 8.97 − 0.0009 0.0024 0.2 0.5 0.10 0.30 23 1.8441 5.1 4.67 9.11 ± 0.0010 0.0027 0.2 0.7 0.09 0.24 24 1.8685 2.1 4.81 5.08 ± 0.0019 0.0054 0.2 0.5 0.18 0.49 25 1.9066 4.3 3.26 8.53 + 0.0007 0.0019 0.2 0.4 0.08 0.23 . . . . . . 1.9066 4.3 3.26 8.53 − 0.0007 0.0019 0.2 0.4 0.08 0.19 26 2.0171 1.1 3.44 3.87 ± 0.0028 0.0076 0.1 0.4 0.32 0.79 27 2.0512 6.1 1.91 9.38 + 0.0006 0.0015 0.1 0.4 0.08 0.29 . . . . . . 2.0512 6.1 1.91 9.38 − 0.0006 0.0015 0.1 0.4 0.08 0.19 continued on next page 247 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.1: continued # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 28 2.1160 1.8 0.11 5.10 ± 0.0014 0.0036 0.1 0.4 0.15 0.40 29 2.7814 1.5 2.77 5.22 ± 0.0018 0.0049 0.1 0.4 0.21 0.54 30 2.8771 1.0 6.18 3.55 + 0.0029 0.0072 0.1 0.3 0.34 1.01 . . . . . . 2.8771 1.0 6.18 3.55 − 0.0029 0.0086 0.1 0.3 0.34 1.01 248 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.2: Frequencies identified in the star HD 217543 by the MOST team (Cameron et al., 2008) # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 0.0269 12.9 1.01 10.80 + 0.0003 0.0035 0.2 1.2 0.07 0.58 . . . . . . 0.0269 12.9 1.01 10.80 − 0.0003 0.0066 0.2 0.7 0.07 0.30 2 0.0806 7.1 0.09 5.48 + 0.0013 0.0058 0.3 0.7 0.14 0.43 . . . . . . 0.0806 7.1 0.09 5.48 − 0.0013 0.0031 0.3 0.7 0.14 0.43 3 0.1201 13.3 1.76 8.28 ± 0.0012 0.0043 0.2 0.7 0.10 0.36 4 0.1806 5.7 3.03 4.59 ± 0.0016 0.0071 0.3 0.8 0.12 0.57 5 0.2454 4.6 3.97 5.10 ± 0.0017 0.0074 0.3 0.9 0.12 0.62 6 0.2904 1.9 2.03 3.15 ± 0.0042 0.0121 0.2 1.7 0.27 1.01 7 0.3133 3.1 4.67 4.40 ± 0.0024 0.0099 0.3 2.2 0.19 0.75 8 0.3987 2.0 5.78 3.97 ± 0.0022 0.0053 0.2 0.6 0.19 0.59 9 0.5147 1.6 2.80 3.32 ± 0.0034 0.0091 0.2 0.6 0.31 0.82 10 0.5668 1.7 4.37 3.17 ± 0.0034 0.0091 0.2 0.6 0.32 0.79 11 0.6154 2.0 5.61 3.09 ± 0.0026 0.0062 0.2 0.5 0.23 0.56 12 0.7772 1.3 1.87 3.11 ± 0.0026 0.0068 0.2 0.5 0.23 0.61 13 1.0006 2.1 2.68 3.67 ± 0.0019 0.0051 0.2 0.5 0.16 0.42 continued on next page249 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.2: continued # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 14 1.5826 2.4 0.72 4.36 ± 0.0016 0.0048 0.2 0.5 0.16 0.44 15 1.7164 2.2 1.85 4.21 ± 0.0020 0.0053 0.2 0.5 0.19 0.52 16 1.7948 2.6 0.45 4.16 + 0.0019 0.0054 0.2 0.5 0.17 0.69 . . . . . . 1.7948 2.6 0.45 4.16 − 0.0019 0.0074 0.2 0.5 0.17 0.46 17 1.8881 6.0 1.81 5.66 + 0.0011 0.0038 0.3 1.6 0.10 0.55 . . . . . . 1.8881 6.0 1.81 5.66 − 0.0011 0.0038 0.3 0.8 0.10 0.44 18 1.9237 12.7 3.51 8.17 + 0.0006 0.0014 0.2 0.7 0.05 0.48 . . . . . . 1.9237 12.7 3.51 8.17 − 0.0006 0.0056 0.2 0.7 0.05 0.18 19 1.9643 4.9 4.44 5.07 + 0.0009 0.0209 0.4 3.6 0.08 0.19 . . . . . . 1.9643 4.9 4.44 5.07 − 0.0009 0.0027 0.4 1.8 0.08 1.28 20 1.9971 7.6 6.22 6.80 + 0.0008 0.0029 0.3 4.7 0.05 0.17 . . . . . . 1.9971 7.6 6.22 6.80 − 0.0008 0.0029 0.3 0.8 0.05 0.41 21 2.0335 23.8 2.25 14.25 + 0.0002 0.0007 0.3 0.9 0.02 0.39 . . . . . . 2.0335 23.8 2.25 14.25 − 0.0002 0.0036 0.3 5.3 0.02 0.02 22 2.0704 7.1 5.30 6.26 + 0.0010 0.0031 0.3 3.9 0.07 1.18 . . . . . . 2.0704 7.1 5.30 6.26 − 0.0010 0.0102 0.3 0.7 0.07 0.28 continued on next page 250 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.2: continued # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 23 2.1237 9.9 2.50 7.28 + 0.0006 0.0066 0.2 1.1 0.05 0.13 . . . . . . 2.1237 9.9 2.50 7.28 − 0.0006 0.0012 0.2 0.7 0.05 0.52 24 2.1766 9.4 4.20 6.80 + 0.0006 0.0081 0.3 0.7 0.06 0.38 . . . . . . 2.1766 9.4 4.20 6.80 − 0.0006 0.0038 0.3 0.9 0.06 0.53 25 2.2288 6.5 6.03 5.68 + 0.0010 0.0037 0.2 0.6 0.08 0.39 . . . . . . 2.2288 6.5 6.03 5.68 − 0.0010 0.0037 0.2 1.0 0.08 0.39 26 2.2985 2.8 1.60 4.06 + 0.0018 0.0047 0.2 0.5 0.18 0.48 . . . . . . 2.2985 2.8 1.60 4.06 − 0.0018 0.0047 0.2 0.5 0.18 0.42 27 2.3674 3.0 1.59 4.27 ± 0.0016 0.0046 0.2 0.6 0.15 0.40 28 3.7755 2.0 4.00 4.00 ± 0.0019 0.0049 0.2 0.5 0.18 0.51 29 3.8947 2.3 0.17 4.33 + 0.0019 0.0045 0.2 0.5 0.18 0.56 . . . . . . 3.8947 2.3 0.17 4.33 − 0.0019 0.0045 0.2 0.5 0.18 0.42 30 4.0017 3.4 2.13 4.52 ± 0.0014 0.0037 0.2 0.5 0.13 0.35 31 4.0646 4.2 1.18 5.20 + 0.0015 0.0045 0.2 0.6 0.15 0.56 . . . . . . 4.0646 4.2 1.18 5.20 − 0.0015 0.0048 0.2 0.6 0.15 0.37 32 4.1079 2.6 1.03 4.08 ± 0.0025 0.0068 0.2 0.6 0.20 0.52 continued on next page 251 A p p en d ix A . H D 127756 an d H D 217543 p aram eter tab les Table A.2: continued # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 33 4.1886 5.9 4.84 6.01 + 0.0011 0.0031 0.2 0.6 0.12 0.73 . . . . . . 4.1886 5.9 4.84 6.01 − 0.0011 0.0060 0.2 0.7 0.12 0.24 34 4.2350 4.9 0.84 5.83 + 0.0017 0.0044 0.2 0.7 0.16 1.00 . . . . . . 4.2350 4.9 0.84 5.83 − 0.0017 0.0080 0.2 0.6 0.16 0.35 35 4.2665 2.3 3.05 4.15 + 0.0027 0.0081 0.2 0.6 0.27 1.16 . . . . . . 4.2665 2.3 3.05 4.15 − 0.0027 0.0085 0.2 0.6 0.27 0.73 36 4.4017 3.6 2.01 4.91 + 0.0012 0.0033 0.2 0.5 0.14 0.51 . . . . . . 4.4017 3.6 2.01 4.91 − 0.0012 0.0033 0.2 0.5 0.14 0.36 37 4.4695 2.1 4.02 4.20 + 0.0019 0.0055 0.2 0.5 0.20 0.57 . . . . . . 4.4695 2.1 4.02 4.20 − 0.0019 0.0055 0.2 0.5 0.20 0.41 38 4.5406 2.4 5.24 4.26 + 0.0020 0.0056 0.2 0.5 0.18 0.55 . . . . . . 4.5406 2.4 5.24 4.26 − 0.0020 0.0056 0.2 0.5 0.18 0.52 39 4.6210 1.6 5.40 3.38 ± 0.0029 0.0072 0.2 0.5 0.27 0.73 40 4.7226 2.6 4.63 4.47 + 0.0016 0.0042 0.2 0.5 0.17 0.47 . . . . . . 4.7226 2.6 4.63 4.47 − 0.0016 0.0042 0.2 0.5 0.17 0.46 252 Appendix B Frequency lists in cycles day−1 It is more common to use units of µHz when presenting both theoretical and observational results on roAp stars. However, there are still cases when units of cycles day−1 are used. The latter units are particularly useful when describing the fine (rotational) spacings observed in roAp stars. In this ap- pendix we reproduce the tables in the main text with frequencies in units of cycles day−1. 253 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.1: Table 3.1 with frequencies in units of cycles day−1 # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 224.9711 39 2.22 3.8 ± 0.0025 0.0069 6 15 0.34 0.93 2 226.2470 50 3.29 4.2 ± 0.0021 0.0057 6 16 0.27 0.72 3 226.3244 125 5.70 9.2 ± 0.0013 0.0034 6 16 0.11 0.29 4 226.4061 68 1.68 6.3 ± 0.0024 0.0064 6 16 0.28 0.74 5 229.1326 96 3.36 7.5 ± 0.0015 0.0039 6 16 0.14 0.35 6 229.2127 365 5.56 24.2 ± 0.0004 0.0010 6 16 0.04 0.10 7 229.2955 92 2.07 7.8 ± 0.0011 0.0028 6 16 0.10 0.27 8 231.9695 43 2.02 3.7 ± 0.0047 0.0125 6 16 0.38 1.10 9 232.0409 31 2.83 3.1 ± 0.0078 0.0209 9 26 0.55 2.18 10 232.1187 123 1.26 8.9 ± 0.0010 0.0025 7 17 0.10 0.26 11 232.1931 467 4.46 29.7 ± 0.0003 0.0008 6 16 0.03 0.08 12 232.2747 139 1.13 10.9 ± 0.0008 0.0020 6 16 0.07 0.18 13 234.9202 51 4.50 5.8 ± 0.0044 0.0118 6 16 0.25 0.68 14 235.0043 319 6.04 23.3 ± 0.0005 0.0012 6 16 0.05 0.13 15 235.0870 764 1.83 41.6 ± 0.0002 0.0005 6 16 0.02 0.05 continued on next page254 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.1: continued # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 16 235.1654 177 4.59 12.4 ± 0.0009 0.0023 6 16 0.10 0.26 17 235.3077 41 2.08 3.1 ± 0.0029 0.0084 6 16 0.39 1.10 18 237.9862 110 1.81 8.5 ± 0.0014 0.0037 6 16 0.17 0.45 19 238.0660 118 3.45 8.9 ± 0.0013 0.0034 6 16 0.17 0.45 20 238.1214 67 2.75 6.2 ± 0.0033 0.0093 6 16 0.36 1.05 21 238.1983 39 5.14 3.1 ± 0.0039 0.0138 6 17 0.53 1.67 22 240.8659 24 4.72 2.7 ± 0.0110 0.0274 6 16 0.90 2.59 23 240.9585 37 0.24 3.6 ± 0.0030 0.0083 6 16 0.29 0.80 24 241.0277 33 5.56 3.3 ± 0.0041 0.0136 7 16 0.38 1.16 25 241.0911 37 2.85 4.4 ± 0.0063 0.0169 7 18 0.45 1.24 26 241.1755 82 5.00 6.8 ± 0.0028 0.0073 7 17 0.29 0.74 27 241.2503 57 1.47 5.6 + 0.0047 0.0178 9 24 0.35 1.06 . . . . . . 241.2503 57 1.47 5.6 − 0.0047 0.0154 9 25 0.35 1.06 28 242.4082 84 1.10 6.9 ± 0.0013 0.0033 6 15 0.15 0.37 29 242.4896 81 2.09 7.1 ± 0.0016 0.0041 6 16 0.16 0.44 255 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.2: Table 3.2 with frequencies in units of cycles day−1. Columns with headers νi - νj use i and j to denote the frequency number from Table B.1. The average rotation separation (spacings 9 . . . 18 & 27 . . . 35) is 0.0798 ± 0.0041 µHz (3σ = 0.0112 µHz). # νi - νj δ † [µHz] σδ† 3σδ† # νi - νj δ† [µHz] σδ† 3σδ† 1 6 - 3 2.8883 0.0013 0.0035 19 11 - 8 0.2236 0.0047 0.0125 2 11 - 6 2.9804 0.0005 0.0012 20 11 - 9 0.1522 0.0079 0.0209 3 15 - 11 2.8939 0.0003 0.0009 21 17 - 15 0.2208 0.0029 0.0084 4 19 - 15 2.9790 0.0013 0.0035 22 21 - 19 0.1324 0.0041 0.0142 5 23 - 19 2.8926 0.0032 0.0090 23 26 - 23 0.2170 0.0041 0.0111 6 3 - 1 1.3533 0.0028 0.0077 24 28 - 26 1.2326 0.0031 0.0080 7 17 - 8 3.3382 0.0055 0.0150 25 17 - 9 3.2669 0.0084 0.0225 8 21 - 17 2.8906 0.0048 0.0162 26 26 - 21 2.9773 0.0048 0.0156 9 3 - 2 0.0773 0.0026 0.0069 27 16 - 15 0.0785 0.0009 0.0026 10 4 - 3 0.0816 0.0026 0.0069 28 19 - 18 0.0797 0.0017 0.0052 11 6 - 5 0.0800 0.0017 0.0043 29 20 - 19 0.0555 0.0035 0.0095 12 7 - 6 0.0828 0.0009 0.0026 30 21 - 20 0.0769 0.0052 0.0164 13 9 - 8 0.0715 0.0095 0.0242 31 23 - 22 0.0926 0.0112 0.0285 continued on next page256 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.2: continued # νi - νj δ † [µHz] σδ† 3σδ† # νi - νj δ† [µHz] σδ† 3σδ† 14 10 - 9 0.0778 0.0078 0.0207 32 24 - 23 0.0693 0.0052 0.0156 15 11 - 10 0.0743 0.0009 0.0026 33 26 - 25 0.0845 0.0069 0.0181 16 12 - 11 0.0816 0.0009 0.0017 34 27 - 26 0.0748 0.0052 0.0173 17 14 - 13 0.0841 0.0043 0.0121 35 29 - 28 0.0814 0.0017 0.0052 18 15 - 14 0.0827 0.0009 0.0017 257 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.3: Table 3.3 with frequencies in units of cycles day−1. Frequencies numbered 30 and higher are the unresolved components. # ν [cycles day−1] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ 1 224.9710 38 2.22 ± 0.0023 5 0.26 0.0067 16 0.11 2 226.2480 45 3.25 ± 0.0020 5 0.21 0.0060 17 0.50 3 226.3196 106 5.96 ± 0.0021 12 0.20 0.0047 15 0.75 4 226.4012 64 2.07 ± 0.0037 5 0.34 0.0072 24 1.42 5 229.1339 95 3.20 ± 0.0012 5 0.16 0.0042 17 0.36 6 229.2125 364 5.59 ± 0.0003 5 0.04 0.0011 16 0.08 7 229.2938 81 2.13 ± 0.0017 9 0.14 0.0052 17 0.31 8 231.9733 43 1.67 ± 0.0038 5 0.35 0.0073 30 0.95 9 232.0429 40 3.02 ± 0.0042 10 0.44 0.0133 16 0.11 10 232.1167 111 1.37 ± 0.0018 10 0.10 0.0037 36 0.12 11 232.1927 475 4.50 ± 0.0003 9 0.05 0.0007 16 0.23 12 232.2726 131 1.30 ± 0.0015 8 0.17 0.0031 16 0.88 13 234.9166 54 5.18 ± 0.0036 8 0.52 0.0099 16 0.50 14 235.0042 320 6.11 ± 0.0003 5 0.06 0.0010 28 0.58 15 235.0875 761 1.81 + 0.0004 8 0.02 0.0012 36 0.12 continued on next page 258 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.3: continued # ν [cycles day−1] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ . . . . . . 235.0875 761 1.81 − 0.0004 8 0.02 0.0012 36 0.12 16 235.1680 184 4.45 ± 0.0012 6 0.13 0.0026 16 0.23 17 235.3059 31 2.33 ± 0.0024 9 0.27 0.0073 16 0.88 18 237.9839 108 1.99 + 0.0019 5 0.18 0.0042 16 0.50 . . . . . . 237.9839 108 1.99 − 0.0019 5 0.18 0.0043 16 0.45 19 238.0632 112 4.02 ± 0.0024 6 0.22 0.0057 28 0.58 20 238.1313 53 1.75 + 0.0038 12 0.40 0.0356 25 1.29 . . . . . . 238.1313 53 1.75 − 0.0038 12 0.40 0.0098 25 1.29 21 238.1960 33 5.35 ± 0.0027 6 0.29 0.0099 16 1.24 22 240.8660 24 4.73 ± 0.0041 5 0.44 0.0123 15 1.37 23 240.9585 36 0.19 ± 0.0026 5 0.28 0.0077 15 0.84 24 241.0280 34 5.48 ± 0.0031 5 0.31 0.0093 15 0.95 25 241.0901 35 3.02 ± 0.0029 5 0.30 0.0086 16 0.87 26 241.1750 82 5.04 ± 0.0013 5 0.14 0.0039 16 0.41 27 241.2498 58 1.50 ± 0.0022 5 0.22 0.0067 16 0.69 28 242.4082 84 1.10 ± 0.0011 5 0.12 0.0032 15 0.36 continued on next page 259 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.3: continued # ν [cycles day−1] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ 29 242.4896 82 2.09 ± 0.0011 5 0.13 0.0034 15 0.37 30 226.3359 36 4.63 + 0.0135 12 0.72 0.0247 24 1.65 . . . . . . 226.3359 36 4.63 − 0.0135 12 0.72 0.0247 24 4.38 31 226.4391 31 5.19 ± 0.0085 6 0.54 0.0164 17 1.50 32 229.2147 10 2.08 + 0.0927 8 2.00 0.1700 15 4.13 . . . . . . 229.2147 10 2.08 − 0.0927 8 2.00 0.1700 10 2.08 33 229.2515 51 3.47 ± 0.0029 5 0.31 0.0098 17 0.99 34 232.0707 57 2.84 ± 0.0029 8 0.27 0.0120 24 1.20 35 232.2356 55 2.10 ± 0.0037 7 0.47 0.0086 18 0.92 36 234.9088 25 3.63 + 0.0096 8 1.22 0.0289 27 2.63 . . . . . . 234.9088 25 3.63 − 0.0096 8 1.22 0.0289 25 2.73 37 235.0711 47 2.06 + 0.0056 16 0.51 0.0168 47 1.68 . . . . . . 235.0711 47 2.06 − 0.0056 16 0.51 0.0168 44 1.68 38 235.1226 77 3.06 ± 0.0030 10 0.20 0.0073 21 0.76 39 238.0214 48 3.48 ± 0.0051 6 0.49 0.0119 18 1.28 40 238.0867 55 6.07 + 0.0074 17 0.49 0.0251 42 1.76 continued on next page 260 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.3: continued # ν [cycles day−1] A [µmag] φ [rad] σν σA σφ 3σν 3σA 3σφ . . . . . . 238.0867 55 6.07 − 0.0074 17 0.49 0.0251 42 2.57 41 241.2817 28 0.19 ± 0.0042 5 0.46 0.0128 15 1.39 261 A p p en d ix B . F req u en cy lists in cy cles d ay − 1 Table B.4: Table 5.3 with frequencies in units of cycles day−1. Frequency uncertainties marked by ∗ are average uncertainties based on the separation from the marked frequency to the nearest (∼ rotationally split) frequency plus its uncertainty. # ν [cycles day−1] A [µmag] φ [rad] ∼ S/N σν 3σν σA 3σA σφ 3σφ 1 224.9711 39 2.22 3.8 ± 0.0024 0.0069 6 15 0.34 0.93 3 226.3244 125 5.7 9.2 ± 0.0013 0.0034 6 16 0.11 0.29 6 229.2126 365 5.56 24.2 ± 0.0003 0.0010 6 16 0.04 0.1 9 232.0409 31 2.83 3.1 ± 0.0775 ∗ 0.0822∗ 9 26 0.55 2.18 11 232.1931 467 4.46 29.7 ± 0.0003 0.0008 6 16 0.03 0.08 15 235.0870 764 1.83 41.6 ± 0.0002 0.0004 6 16 0.02 0.05 17 235.3077 41 2.08 3.1 ± 0.0029 0.0084 6 16 0.39 1.1 19 238.0660 118 3.45 8.9 ± 0.0013 0.0034 6 16 0.17 0.45 21 238.1983 39 5.14 3.1 ± 0.0421∗ 0.0499∗ 6 17 0.53 1.67 23 240.9585 37 0.24 3.6 ± 0.0029 0.0083 6 16 0.29 0.8 26 241.1756 82 5 6.8 ± 0.0029 0.0073 7 17 0.29 0.74 28 242.4082 84 1.1 6.9 ± 0.0422∗ 0.0444∗ 6 15 0.15 0.37 262 Appendix C Bootstrap distributions All bootstrap distributions for the fit parameters given in Tab. 3.1 are pre- sented below. In each figure the top panels show the phase (φ) in radians, the middle panels are the amplitudes (A) in mmag, and lower panels are fre- quencies (ν) in µHz. Increasing parameter numbers from Tab. 3.1 are read left to right for each figure. Thus, distributions for parameter set (ν1, A1, φ1) are shown in the left panels while distributions for the set (ν2, A2, φ2) are shown on the right. In each plot the green and red lines show the 1 and 2 σ uncertainties calculated by counting 68% and 95% of the realizations, cen- tred on the best fit parameter, respectively. The pink lines show the best fit parameter with the standard deviation calculated from the analytic formula and the blue dots are the mean of the distribution with the standard error on that mean. (Note: The phases are calculated by referencing to the time of the first MOST observation, HJD 2451545.00 + 1769.42 days.) 263 Appendix C. Bootstrap distributions Figure C.1: Bootstrap distributions for fit parameters 1 and 2 in Tab. 3.1. 264 Appendix C. Bootstrap distributions Figure C.2: Bootstrap distributions for fit parameters 3 and 4 in Tab. 3.1. 265 Appendix C. Bootstrap distributions Figure C.3: Bootstrap distributions for fit parameters 5 and 6 in Tab. 3.1. 266 Appendix C. Bootstrap distributions Figure C.4: Bootstrap distributions for fit parameters 7 and 8 in Tab. 3.1. 267 Appendix C. Bootstrap distributions Figure C.5: Bootstrap distributions for fit parameters 9 and 10 in Tab. 3.1. 268 Appendix C. Bootstrap distributions Figure C.6: Bootstrap distributions for fit parameters 11 and 12 in Tab. 3.1. 269 Appendix C. Bootstrap distributions Figure C.7: Bootstrap distributions for fit parameters 13 and 14 in Tab. 3.1. 270 Appendix C. Bootstrap distributions Figure C.8: Bootstrap distributions for fit parameters 15 and 16 in Tab. 3.1. 271 Appendix C. Bootstrap distributions Figure C.9: Bootstrap distributions for fit parameters 17 and 18 in Tab. 3.1. 272 Appendix C. Bootstrap distributions Figure C.10: Bootstrap distributions for fit parameters 19 and 20 in Tab. 3.1. 273 Appendix C. Bootstrap distributions Figure C.11: Bootstrap distributions for fit parameters 21 and 22 in Tab. 3.1. 274 Appendix C. Bootstrap distributions Figure C.12: Bootstrap distributions for fit parameters 23 and 24 in Tab. 3.1. 275 Appendix C. Bootstrap distributions Figure C.13: Bootstrap distributions for fit parameters 25 and 26 in Tab. 3.1. 276 Appendix C. Bootstrap distributions Figure C.14: Bootstrap distributions for fit parameters 27 and 28 in Tab. 3.1. 277 Appendix C. Bootstrap distributions Figure C.15: Bootstrap distributions for fit parameters 29 in Tab. 3.1. 278 Appendix D Magnetic perturbation (Saio) plots Plots of magnetically perturbed pulsation modes are presented below. The plots give perturbed frequencies as a function of magnetic field (Saio plots) for degrees of ` ranging from 0 to 4. (See the legend at the top of each plot for degree identifications.) The caption below each plot identifies the model using the same labelling system used in Tables 5.1, 5.2, 5.4, 5.5, 5.9, 5.6, and 5.8. Each plot identifies (with a solid vertical line) the closest matched model listed in the aforementioned tables found in chapter 5. The observed frequencies from the HR 1217 data (Tab. 3.1) are shown as horizontal black dots in each plot. 279 Appendix D. Magnetic perturbation (Saio) plots Figure D.1: Saio plots showing the best fit models M1 (top) and M2 (bottom) in Tab. 5.1. 280 Appendix D. Magnetic perturbation (Saio) plots Figure D.2: Saio plots showing the best fit models M3 (top) and M4 (bottom) in Tab. 5.1. 281 Appendix D. Magnetic perturbation (Saio) plots Figure D.3: Saio plots showing the best fit models M5 (top) and M6 (bottom) in Tab. 5.1. 282 Appendix D. Magnetic perturbation (Saio) plots Figure D.4: Saio plots showing the best fit models M7 (top) and M8 (bottom) in Tab. 5.1. 283 Appendix D. Magnetic perturbation (Saio) plots Figure D.5: Saio plots showing the best fit models M9 (top) and M10 (bottom) in Tab. 5.1. 284 Appendix D. Magnetic perturbation (Saio) plots Figure D.6: Saio plots showing the best fit models M9 (top) and MM1 (bottom) in Tables 5.1 and 5.2. 285 Appendix D. Magnetic perturbation (Saio) plots Figure D.7: Saio plots showing the best fit models U1 (top) and U2 (bottom) in Tab. 5.4. 286 Appendix D. Magnetic perturbation (Saio) plots Figure D.8: Saio plots showing the best fit models U3 (top) and U4 (bottom) in Tab. 5.4. 287 Appendix D. Magnetic perturbation (Saio) plots Figure D.9: Saio plots showing the best fit models U5 (top) and U6 (bottom) in Tab. 5.4. 288 Appendix D. Magnetic perturbation (Saio) plots Figure D.10: Saio plots showing the best fit models U7 (top) and U8 (bottom) in Tab. 5.4. 289 Appendix D. Magnetic perturbation (Saio) plots Figure D.11: Saio plots showing the best fit models U9 (top) and U10 (bottom) in Tab. 5.4. 290 Appendix D. Magnetic perturbation (Saio) plots Figure D.12: Saio plots showing the best fit models UU1 (top) and UU2 (bottom) in Tab. 5.5. 291 Appendix D. Magnetic perturbation (Saio) plots Figure D.13: Saio plots showing the best fit models GS (top) and MS (bottom) in Tab. 5.6. 292 Appendix D. Magnetic perturbation (Saio) plots Figure D.14: Saio plots showing the best fit models PGS (top) and LPC (bottom). Model properties are given in Tabs. 5.8 and Tab. 5.9, respectively. 293
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