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Asteroseismic tuning of the magnetic star HR 1217 : understanding magnetism and stellar structure through.. Cameron, Christopher J. 2010

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Asteroseismic Tuning of theMagnetic Star HR 1217Understanding magnetism and stellarstructure through MOST spacebasedphotometrybyChristopher J. CameronB. Sc., Saint Mary’s University, 2001M. Sc., University of British Columbia, 2004A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Astronomy)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August, 2010c Christopher J. Cameron 2010AbstractThe chemically peculiar A (Ap) stars show extreme examples of astrophys-ical processes that have only recently been studied in detail in one otherstar | the Sun. These stars exhibit spectral anomalies caused by di usionof some ionic species in a stellar atmosphere threaded by a strong ( kG),organized magnetic  eld. A subset of the Ap stars rapidly oscillate (roAp)with periods ranging from 5 to 25 minutes. One of these roAp stars, HR1217, is well studied with data from two global (ground-based) photometriccampaigns that led to asteroseismic evidence of magnetically perturbed os-cillation modes. This was the motivation to make HR 1217 a MOST spacemission target.Our analysis of the almost 30 days of near-continuous MOST photometryon HR 1217 reveals a number of new periodicities that show spacings of 15,2.5, and 1.5  Hz. These new frequencies can be interpreted as magneticallyperturbed oscillations and potentially second order spacings that could con-strain the age and the magnetic interior of the star for the  rst time. Thesedata are collected with a 95% duty cycle and reach a precision of 6  mag,making this by far the best photometric data set on HR 1217.In addition, we present a grid of almost 52,000 stellar pulsation modelsincluding a large range of magnetic dipole  eld strengths (1{10 kG). ThisiiAbstractis the largest grid of stellar pulsation models of any Ap star to date andis critical to the interpretation of the MOST photometry. Our models canmatch the MOST observations to a fractional accuracy of about 0.05% witha mean deviation between theory and observation of a few  Hz. A uniquemodel match to the MOST observations could not be found. The resultshighlight the sensitivity to physics that has not been usually incorporatedin Ap interior models, and the complex nature of the interaction of globallyorganized magnetic  elds with stellar pulsation eigenmodes.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviStatement of Collaboration . . . . . . . . . . . . . . . . . . . . . . xvii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The sounding of stellar interiors . . . . . . . . . . . . . . . . 21.1.1 Nonradial stellar oscillations . . . . . . . . . . . . . . 41.2 The chemically peculiar A stars . . . . . . . . . . . . . . . . . 161.2.1 Magnetic  elds, di usion, & Ap stars . . . . . . . . . 171.2.2 Stellar seismology & Ap stars . . . . . . . . . . . . . . 211.3 The roAp star HR 1217 . . . . . . . . . . . . . . . . . . . . . 291.4 Thesis goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34ivTable of Contents2 The MOST mission: Data reduction techniques . . . . . . . 382.1 An overview of the MOST mission . . . . . . . . . . . . . . . 382.2 Photometry & time-series analysis . . . . . . . . . . . . . . . 402.3 Applications of CAPER to other MOST science . . . . . . . 462.3.1 The roAp star HD 134214 . . . . . . . . . . . . . . . . 472.3.2 The B supergiant HD 163899 . . . . . . . . . . . . . . 482.3.3 The SPBe stars HD 127756 & HD 217543 . . . . . . . 493 MOST & HR 1217 . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1 The light curve reduction . . . . . . . . . . . . . . . . . . . . 623.2 On the rotation period of HR 1217 . . . . . . . . . . . . . . . 663.3 The frequency analysis . . . . . . . . . . . . . . . . . . . . . . 754 Modelling stellar oscillations . . . . . . . . . . . . . . . . . . . 984.1 Stellar evolution models . . . . . . . . . . . . . . . . . . . . . 984.1.1 De ning the parameter space . . . . . . . . . . . . . . 1004.2 Pulsation models . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2.1 A new search method for eigenmodes . . . . . . . . . . 1134.3 The magnetic e ects . . . . . . . . . . . . . . . . . . . . . . . 1195 Best matched models . . . . . . . . . . . . . . . . . . . . . . . . 1385.1 The matching procedure . . . . . . . . . . . . . . . . . . . . . 1385.1.1 Seismic observations & models of the Sun . . . . . . . 1405.1.2 Seismic observations & models of HR 1217 . . . . . . 1445.2 Matching the true modes of HR 1217 . . . . . . . . . . . . . . 1715.2.1 Matches to a sub-set of the MOST frequencies . . . . 1875.3 On discriminating between models . . . . . . . . . . . . . . . 201vTable of Contents6 Summary, conclusions, & future work . . . . . . . . . . . . . 215Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230AppendicesA HD 127756 and HD 217543 parameter tables . . . . . . . . 245B Frequency lists in cycles day 1 . . . . . . . . . . . . . . . . . . 253C Bootstrap distributions . . . . . . . . . . . . . . . . . . . . . . 263D Magnetic perturbation (Saio) plots . . . . . . . . . . . . . . . 279viList of Tables3.1 Frequency model parameters identi ed using the MOST data. 803.2 The frequency separations  y identi ed in the MOST data. . . 863.3 Unresolved frequencies identi ed in the MOST data. . . . . . 934.1 Some properties of selected models. . . . . . . . . . . . . . . . 1075.1 Properties of the most probable models of HR 1217 | M series.1575.2 Properties of model MM1. . . . . . . . . . . . . . . . . . . . . 1645.3 Selected frequencies from the MOST data | A. . . . . . . . . 1735.4 Properties of the most probable models of HR 1217 | U series.1795.5 Properties of models UU1 and UU2. . . . . . . . . . . . . . . . 1855.6 Properties of models GS and MS. . . . . . . . . . . . . . . . . 1905.7 Selected frequencies from the MOST data | B . . . . . . . . 1945.8 Properties of model PGS. . . . . . . . . . . . . . . . . . . . . 1985.9 Properties of model LPC. . . . . . . . . . . . . . . . . . . . . 207A.1 Frequencies identi ed in the star HD 127756 . . . . . . . . . . 246A.2 Frequencies identi ed in the star HD 217543 . . . . . . . . . . 249B.1 Table 3.1 with frequencies in units of cycles day 1 . . . . . . . 254B.2 Table 3.2 with frequencies in units of cycles day 1 . . . . . . . 256viiList of TablesB.3 Table 3.3 with frequencies in units of cycles day 1 . . . . . . . 258B.4 Table 5.3 with frequencies in units of cycles day 1 . . . . . . . 262viiiList of Figures1.1 A schematic representation of surface spherical harmonics. . . 81.2 Eigenfunctions of a low-order g-mode and a high-order p-modefrom a solar model. . . . . . . . . . . . . . . . . . . . . . . . . 101.3 A schematic showing frequency spacings. . . . . . . . . . . . . 141.4 Schematic diagram of the oblique rotator model. . . . . . . . . 191.5 A theoretical HR diagram showing the approximate locationof roAp stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.6 Variation in the pulsation amplitude and the magnetic  eldstrength of HR 1217. . . . . . . . . . . . . . . . . . . . . . . . 251.7 Schematic amplitude spectra of HR 1217 from past data. . . . 322.1 MOST light curve of HD 127756 . . . . . . . . . . . . . . . . 532.2 The Fourier amplitude spectrum of the HD 127756 light curve 542.3 Zoomed region around the largest peak in the DFT of the HD127756 light curve. . . . . . . . . . . . . . . . . . . . . . . . . 552.4 A comparison of bootstrap distributions | SPBe star HD 127756 562.5 MOST light curve of HD 217543 . . . . . . . . . . . . . . . . 582.6 Fourier amplitude spectrum of the HD 217543 light curve. . . 592.7 A comparison of bootstrap distributions | SPBe star HD 217543 60ixList of Figures3.1 Photometric reduction steps using the MOST data. . . . . . . 653.2 The reduced HR 1217 light curve. . . . . . . . . . . . . . . . . 673.3 Likelihood ratio for the rotation period of HR 1217. . . . . . . 713.4 Bootstrap distributions of the rotation frequency of HR 1217. 743.5 The Fourier amplitude spectrum of the HR 1217 light curve. . 793.6 Normalized pulsation frequency amplitudes as a function oftime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.7 Time-frequency plot of the HR 1217 data. . . . . . . . . . . . 903.8 Comparison between resolved and unresolved frequencies. . . . 924.1 HR diagram showing the extremes of the model parameters. . 1054.2 Histograms of model properties. . . . . . . . . . . . . . . . . . 1064.3 Small spacings for the models listed in Tab. 4.1. . . . . . . . . 1144.4 Eigenfrequencies calculated using two di erent methods. . . . 1204.5 The cyclic behaviour of the magnetic perturbations. . . . . . . 1264.6 The magnetic perturbations for a 5 kG magnetic  eld. . . . . 1274.7 The magnetic perturbations to   for B = 0 kG and 5 kG. . . 1314.8 The magnetic perturbations to   for all B values. . . . . . . 1324.9 The magnetic perturbations to   for B = 5 kG. . . . . . . . . 1354.10 The magnetic perturbations to   for all B values. . . . . . . . 1365.1 Standard Solar model (SSM) frequencies compared to the Birm-ingham Solar Oscillations Network (BiSON) observations. . . . 1435.2 Probability plots with  scaled by factors of 1 and 1000. . . . 1495.3 Probability plots with  scaled by a factor of 50. . . . . . . . . 1515.4 Probability plots using the Laplace distribution. . . . . . . . . 153xList of Figures5.5 The median frequency deviation vs model probability . . . . . 1565.6  M1 -  MOST vs.  MOST. . . . . . . . . . . . . . . . . . . . . . 1615.7  M3 -  MOST and  MM1 -  MOST vs.  MOST . . . . . . . . . . . 1655.8 Saio plots for models M1 and M2. . . . . . . . . . . . . . . . . 1685.9 Saio plots for models M9 and MM1. . . . . . . . . . . . . . . . 1695.10 Probability plots for models matched to  MOST (Tab. 5.3) | A.1775.11 Probability plots for models matched to  MOST (Tab. 5.3) | B.1785.12  U1 -  MOST vs  MOST and a Saio plot for model U1. . . . . . 1835.13  UU1 -  MOST vs  MOST . . . . . . . . . . . . . . . . . . . . . . 1865.14 Saio plots for models UU1 and UU2. . . . . . . . . . . . . . . 1875.15  GS -  MOST vs.  MOST . . . . . . . . . . . . . . . . . . . . . . 1915.16 Saio plots for models GS and MS. . . . . . . . . . . . . . . . . 1925.17 Probability plot for models matched to  MOST (Tab. 5.7). . . . 1965.18  PGS -  MOST vs.  MOST. . . . . . . . . . . . . . . . . . . . . . 1995.19  M3 -  MOST and  M6 -  MOST vs.  MOST. . . . . . . . . . . . 2015.20 Minimum frequency deviations required to match 50 and 75%of the MOST observations. . . . . . . . . . . . . . . . . . . . . 2045.21 Minimum frequency deviations required to match 100% of theMOST observations. . . . . . . . . . . . . . . . . . . . . . . . 2055.22  LPC -  MOST vs.  MOST. . . . . . . . . . . . . . . . . . . . . . 2085.23 Histograms of matched frequencies | all B values and B = 0. 2105.24 Histograms of matched frequencies | B = 2 kG and B = 4 kG.2115.25 Histograms of matched frequencies | B = 6 kG and B = 8 kG.212C.1 Bootstrap distributions for  t parameters 1 and 2. . . . . . . . 264C.2 Bootstrap distributions for  t parameters 3 and 4. . . . . . . . 265xiList of FiguresC.3 Bootstrap distributions for  t parameters 5 and 6. . . . . . . . 266C.4 Bootstrap distributions for  t parameters 7 and 8. . . . . . . . 267C.5 Bootstrap distributions for  t parameters 9 and 10. . . . . . . 268C.6 Bootstrap distributions for  t parameters 11 and 12. . . . . . 269C.7 Bootstrap distributions for  t parameters 13 and 14. . . . . . 270C.8 Bootstrap distributions for  t parameters 15 and 16. . . . . . 271C.9 Bootstrap distributions for  t parameters 17 and 18. . . . . . 272C.10 Bootstrap distributions for  t parameters 19 and 20. . . . . . 273C.11 Bootstrap distributions for  t parameters 21 and 22. . . . . . 274C.12 Bootstrap distributions for  t parameters 23 and 24. . . . . . 275C.13 Bootstrap distributions for  t parameters 25 and 26. . . . . . 276C.14 Bootstrap distributions for  t parameters 27 and 28. . . . . . 277C.15 Bootstrap distributions for  t parameters 29. . . . . . . . . . . 278D.1 Saio plots showing the best  t models M1 and M2. . . . . . . 280D.2 Saio plots showing the best  t models M3 and M4. . . . . . . 281D.3 Saio plots showing the best  t models M5 and M6. . . . . . . 282D.4 Saio plots showing the best  t models M7 and M8. . . . . . . 283D.5 Saio plots showing the best  t models M9 and M10. . . . . . . 284D.6 Saio plots showing the best  t models M9 and MM1. . . . . . 285D.7 Saio plots showing the best  t models U1 and U2. . . . . . . . 286D.8 Saio plots showing the best  t models U3 and U4. . . . . . . . 287D.9 Saio plots showing the best  t models U5 and U6. . . . . . . . 288D.10 Saio plots showing the best  t models U7 and U8. . . . . . . . 289D.11 Saio plots showing the best  t models U9 and U10. . . . . . . 290D.12 Saio plots showing the best  t models UU1 and UU2. . . . . . 291xiiList of FiguresD.13 Saio plots showing the best  t models GS and MS. . . . . . . 292D.14 Saio plots showing the best  t models PGS and LPC. . . . . . 293xiiiAcknowledgementsThis thesis would not have been possible without the support of my PhDsupervisor Jaymie Matthews. He has gone beyond the call of duty as asupervisor and a mentor to make sure that I was exposed to cutting edgeresearch, and that my work was made known to other researchers in our eld. I learned a lot from him. There are times in a person’s life when they nd out who their real friends are | Jaymie Matthews is a real friend. Mysincerest thanks.Thank you to the MOST science team for giving me the opportunityto work with such high quality data. I can only hope that you will be aspleased with the results as I am. Margarida Cunha, thank you for spending asigni cant amount of your time to teach me about your research on the roApstars. I have bene ted greatly from scienti c discussions with Hideyuki Saio.He has read many versions of this thesis and has encouraged my research atevery 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 listed tomy many rants over the years. To my other cohort in the MOST lab, JasonRowe: Our science discussions have spurred many great ideas. You are oneof the few people that can interpret my ramblings as I try to work out thexivAcknowledgementsdetails of my new projects. Your friendship and honest advice are alwayswelcome and appreciated.I was fortunate to meet many interesting people while studying at UBC.The science discussions with other graduate students were an important partof my growth as a researcher; but, I also managed to have some fun timesoutside of the academic environment ::: I would especially like to thankKelsey Ho man, Mark Huber, Jason Kalirai, Thomas Pfrommer, and JasonRowe 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 everythingI have ever done. Without them I would not have been able to  nish thisproject. 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 theother, to Europe, and back again. Her devotion and patients mean the worldto me. There would be no thesis without your loving motivation. Thank youfor taking this journey with me.xvDedicationFor my son, Owen.xviStatement of CollaborationThis following is intended to notify the reader of publications in which workfrom this thesis appears. It also serves to identify people who have collabo-rated on various aspects of this work.In chapter 2 (x2.2 and 2.3) the time-series analysis techniques and modelinterpretations 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 (seethe bibliography) should be credited as collaborators on the work presentedin chapter 2. My contribution to the above publications was primarily thefrequency analysis, light curve analysis, some model interpretation and writ-ing of the manuscripts. Data and models from this thesis have been presentedand 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 providedthe software used to calculate the magnetic perturbations to the oscillationfrequencies in chapters 4 and 5. I performed the model calculations and in-terpreted the results presented in those chapters. Dr. J. F. Rowe should becredited with the preliminary photometric reduction of the HR 1217 data.This thesis has bene ted from the revisions suggested by my PhD ex-xviiStatement of Collaborationamining committee members: Drs. Hiromoto Shibahashi, Michael Bostock,and Jeremy Heyl. My supervisory committee, Drs. Jaymie Matthews, NeilBalmforth, Brett Gladman, and Scott Oser have helped guide and re ne the nal format and content of this thesis.xviiiChapter 1IntroductionThis 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 andinterpreting the many physical processes that must be explored to fully de-scribe the observed properties of HR 1217 is a daunting task. With thisin mind, we compare the unique observations of the pulsations in HR 1217obtained by the MOST satellite (chapter 2) to pulsation models that focuson the magnetic e ects that in uence the properties of the calculated, andobserved, pulsation modes.There have been many comprehensive reviews on the subject of stellarseismology, and in the past few years there has been a growing number ofreviews focusing on the variable, chemically peculiar A stars as well. Thesewill be referenced extensively as necessary sources of background informationon both the theory and measured properties that detail the rapidly oscillatingAp (roAp) star phenomenon.In this introductory chapter we brie y outline stellar seismology, the pecu-liar A stars, and the history of the observations of HR 1217 that are relevantto this work. Further emphasis on the details of data acquisition, reductionmethods and on the modelling of stellar oscillations (in the context of HR11.1. The sounding of stellar interiors1217) are found in the later chapters of this thesis.1.1 The sounding of stellar interiorsObservations of stars are typically only skin deep. Spectroscopic measure-ments of star light give a glimpse of a razor-thin layer of the stellar surfacefrom 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 anestimate of its temperature is available.In the early 1960s Leighton et al. (1962) noticed there were spatiallyincoherent, wave-like disturbances on the surface of the Sun with periodsnear 5 minutes. This oscillation period should be compared to the longerdynamical timescale1 of approximately one hour that can be estimated for theSun. Ulrich (1970) and Leibacher & Stein (1971) were the  rst to interpretthese oscillations as sound waves produced in the solar interior that resonatein acoustic cavities de ned by changes in the physical state, e.g., pressure anddensity, of the solar material with increasing depth. These sound waves, alsoknown as p-modes because pressure is the restoring force, produce standingwave patterns as they propagate and provide the possibility of seismically1The dynamical period of a self gravitating sphere is inversely proportional to thesquare-root of the mean density of the material (pR3=GM) and represents the time itwould take for a particle on the surface of the sphere to free-fall to the centre. Theexpression for this ‘free-fall’ time contains the mass M and the radius R of the sphere andthe physical dimensions of the problem are scaled by the gravitational constant G.21.1. The sounding of stellar interiorsmeasuring the variation of sound speed throughout the recesses of the Sun.Seismology of the Sun (helioseismology) was then possible and astronomerswere at last able to lift the optically thick veil that hid the interior of astar. Since these early beginnings, millions of p-modes have been identi edin the Sun and measurements of its internal rotation rate, the depth of itsconvective zone, constraints on element di usion, and the run of sound speedhave all been inferred. Recent reviews on helioseismology are provided byChristensen-Dalsgaard (2002) and Basu & Antia (2008).The stellar (non-solar) equivalent of helioseismology is asteroseismology.Many stars show both photometric and spectroscopic variations that areconsistent with p-modes and longer period g-modes (buoyancy waves withgravity taking the role of the restoring force). Some examples include:  Scuti stars (oscillation periods of P  0.5{2 h), RR Lyrae stars (P  1.5{24h),  Cephei stars (P  3{6 h), variable white dwarfs (P  100{1600 s) andsupergiants (P ranges from hours to days). Cox (1980) gives a detailed de-scription of primarily radial (spherically symmetric) stellar oscillations whilethe general, nonradial, case of oscillations with both horizontal and radialdisplacements is discussed in detail by Unno et al. (1989). An introductionto many types of variable stars and the theory of stellar oscillations can befound in Christensen-Dalsgaard (2003), and review articles by Gautschy &Saio (1995, 1996), Cunha et al. (2007), and Aerts et al. (2008) also providegeneral asteroseismic discussions and some recent results. Below I summarizethe properties of stellar oscillations that are important for the interpretationof the rapidly oscillating Ap (roAp) stars (P  5{30 min) | A particulartype of variable star that is described in more detail in x 1.2.2. One roAp31.1. The sounding of stellar interiorsstar, HR 1217, is the focus of this research project.1.1.1 Nonradial stellar oscillationsThe properties of stellar oscillations are introduced in this section along withthe asymptotic relations that connect the properties of individual oscillationmodes, frequencies, and spacings to the mean properties of a star. Thenotation 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 concisereviews are provided by Gautschy & Saio (1995), Shibahashi (2005), andBasu & Antia (2008).Stellar pulsations are described by the usual  uid equations. They arethe continuity equation, the momentum equation (a.k.a. the equation of mo-tion or the Euler equation), an energy equation that details the  ow andtransfer of energy in the  uid, and Poisson’s equation relating the gravita-tional potential,  , of the  uid to its internal density,  . These equations arerepresented, in the order presented above, by@ @t =  r ( v) (1.1)  @@t + v r v =  rp  r @p@t + v rp = c2s @ @t + v r  r2 = 4 G where the gas pressure and  uid velocity at a given time t are given by p andv, respectively. All variables are functions of depth within the model, i.e.,41.1. The sounding of stellar interiorsradius r, or mass shell m(r). The energy equation (the third equation in theset of Eqs. 1.1) is given in the adiabatic approximation so that no heat gainsor losses occur within the  uid over an oscillation cycle. In this approximationthe adiabatic sound speed cs is related to the gas pressure and density byp 1p= , with the  rst adiabatic exponent de ned as  1 = (@lnp=@ln )S.The adiabatic approximation is commonly used to model stellar oscillations.In this thesis we focus on the magnetic perturbation to the acoustic oscillationspectrum of variable stars and the matching of those perturbed modes tohigh quality data (see chapters 4 and 5). Because we are not exploring theexcitation mechanism of these magnetic oscillation modes, any non-adiabatice ects will, for simplicity, be ignored.A non-rotating, non-magnetic star has spherical symmetry. This is alsoapproximately true for slow rotation and if a small magnetic  eld is present.Assuming that oscillations about this static, symmetric, reference state havesmall 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 themback into Eqs. 1.1, and keeping only  rst order terms (see, for example, Unnoet al. 1989). We do this so we can describe a complex oscillation pattern onthe 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(i t), with (angular) oscillation frequency  , andthat the radial and angular components are separable, the pulsation variables(for example, pressure) can be expressed asf(r; ; ;t) = f0(r) +f0(r)Yml ( ; ) exp(i t) (1.2)The perturbation f0(r) to the spherically symmetric background state f0(r)51.1. The sounding of stellar interiorsin Eq. 1.2 is evaluated at a  xed radius r. This is called an Eulerian pertur-bation. We can also de ne the Lagrangian perturbation as a perturbationcalculated in a reference frame moving along with the  uid. If a  uid elementmoves a small distance  =  r from the reference position r0, the Lagrangianperturbation can be written as f(r) = f(r0 + ) f0(r0) (1.3)By expanding f(r0 +  ) to  rst order about the background state, Eq. 1.3becomes f(r) = f(r0) +  rf0 f0(r0) (1.4)The di erence f(r0) f0(r0) is just the Eulerian perturbation (f0) evaluatedat the radius r0. Placing f0(r0) into Eq. 1.4, we arrive at f(r) = f0(r0) +  rf0 (1.5)This expression relates the Lagrangian and Eulerian perturbations. Eachof the pulsation variables can be expressed in either of the Eulerian or La-grangian reference frames. The choice is usually one of mathematical ornumerical convenience. The linearized version of Eqs. 1.1 (and a discussionon the boundary conditions needed to complete the solution set) are foundin x 4.2 (see Eqs. 4.2).The angular coordinates in Eq. 1.2 are described by spherical harmonicsYml ( ; ). The index ‘ is commonly called the degree of the mode and dividesthe 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 contributions61.1. The sounding of stellar interiorsfrom low-‘ modes because the contribution from modes with ‘&3 averagesout when the light from the stellar disc is summed. The index m is theazimuthal order of the mode and represents the number of longitudinal nodallines on the stellar surface. Because the spherical harmonics are proportionalto exp(im ), the phase velocity is @ @t phase=  m (1.6)The sign of m indicates the direction in which the wave travels. A standingwave pattern can be formed on the surface of the star by superimposing twowaves travelling in opposite directions because the background state is, to rst 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 arelines of latitude (zonal modes) and if m = ‘ they are all lines of longitude(sectoral modes). If m takes on values between these extremes, the modesare known as tesseral modes and of the ‘ nodal lines, ‘ jmj are lines oflatitude. Examples are drawn schematically in Fig. 1.1 for the cases ‘ = 3and m = 0, 1, 2, and 3. Because the modes are degenerate with respectto m, an agent such as rotation is needed to de ne an axis that breaks thespherical symmetry and lifts the 2‘+ 1 degeneracy.71.1. The sounding of stellar interiorsFigure 1.1: A schematic representation of di erent surface spherical harmonics. The up-per left sphere shows harmonics having an angular degree of ‘ = 3 and an azimuthalorder of m = 0 while the upper right sphere illustrates ‘ = 3 and m = 1. Thelower 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 ‘ jmj lines of latitude on each sphere. (Thisimage was taken from the wikipedia website http://en.wikipedia.org/wiki/File:Harmoniques_spheriques_positif_negatif.png and is available under the GNU FreeDocumentation License.)Along with ‘ and m, there is another index that describes the character ofan individual component of the linear analysis outlined above. The equationsdescribing the nonradial stellar oscillations can be reduced to a fourth order81.1. The sounding of stellar interiorsdi erential equation with the oscillation frequency (squared) as an eigenvalueof the system. The eigenvalues and eigenfunctions of the system are orderedinto two groups (p and g-modes) that are indexed by n. Each frequency2 hasassociated with it a displacement vector (or eigenfunction) with components n;‘;m =  r(r)Yml ; h(r)@Yml@ ; h(r)sin @Yml@  exp(i t) (1.7)The index n is the radial order of the oscillations and corresponds to thenumber of radial nodes (zeros) in the eigenfunction  n;‘;m. In general, p-modes (or acoustic waves) have frequencies that increase with increasing n.Their frequencies are greater than the dynamical frequency pGM=R3. Thep-modes have their largest radial displacement near the surface of a star. Theg-modes (or buoyancy waves), on the other hand, have periods that increasewith decreasing n and are longer than the dynamical period pR3=(GM).The amplitude of the radial eigenfunction is largest near the centre of a starfor g-modes. This behaviour is illustrated in Fig. 1.2 for a p and g-modecalculated using a model of the Sun. The total displacement on the surfaceof a variable star can, in principle be described by summing Eq. 1.7 over allvalues of n, ‘, and m associated with the eigenfrequencies that most closelymatch 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.91.1. The sounding of stellar interiorsFigure 1.2: Eigenfunctions (presented here as the radial displacement  normalized byradius r) of a low-order g-mode (red) and a high order p-mode (black) calculated using amodel 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. (Negativen values are used for g-modes by convention.) The g-mode has its largest amplitude nearthe centre of the model while the high-order p-mode has its largest amplitude near thesurface.Using the terminology de ned above we will now consider the properties101.1. The sounding of stellar interiorsof p-modes in more detail because they are important to the interpretationof the data and models presented in chapters 4 and 5.It can be shown (see Christensen-Dalsgaard 2003) that the radial wavenumber, kr, of a p-mode is approximatelyk2r =  2c2s  ‘(‘+ 1)r2 (1.8)The second term in the above is identi ed as the horizontal wave number,kh. As a wave with an oscillation frequency  travels inward from the surfaceof a star toward its centre, the local sound speed increases with increasingtemperature. The horizontal wavenumber begins to dominate, causing thewave front to slowly bend until the wave refracts upward again. At thisturning radius, rt, the radial wave number kr goes to zero and the wave hasonly a horizontal displacement. Put another wayr2t = c2s(rt)‘(‘+ 1) 2 (1.9)Thus, for a given frequency and angular degree, a wave will bend di erentlyfrom star to star (or from model to model) depending on how the soundspeed changes as a function of radius within the star (or model). This is theessence of stellar seismology | A method by which the interior structure ofa star can be estimated by observing oscillations having di erent  and ‘.When studying the properties of stellar oscillations it is useful to focus onsubsets of the calculated oscillation spectra that have special properties. Forexample, modes with a high angular degree behave di erently than modeswith 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 are111.1. The sounding of stellar interiorsessentially radial and have vertical wave numbers of kr   =cs near thestellar surface. These modes tend to have oscillation frequencies of a fewmHz (a period of 5 min corresponds to a frequency of 3.3 mHz) and areconsistent with oscillations observed in the Sun.The number of nodes a given eigenfunction has between the centre andsurface of a star can be estimated by adding up the number of nodes perwavelength there are in that radius range (see Hansen & Kawaler 1994).That is,n/Z R02dr=  Z R0krdr (1.10)This relation gives a hint that the frequency of a given mode is related to thenumber of nodes the eigenfunction has because kr/ . A careful asymptoticanalysis of high-overtone, low-degree, p-modes by Tassoul (1980, 1990) gives n;‘ = n+ ‘2 +      A(‘+ 1=2)2     2 n;‘ (1.11)with  = 2Z R0drcs(r)  1(1.12)andA = 14 2     Z R0dcsdrdlnr (1.13)where we now make the transition between angular frequency  and theobserved oscillation frequencies  =  =2 . In this asymptotic form (seealso Christensen-Dalsgaard 2003) constants  ; , and  are small and dependon the surface structure of the star. Equation 1.11 is valid for adiabaticoscillations about a spherically symmetric star and for p-modes with n ‘.This asymptotic relation holds when n=‘ 10 or greater.121.1. The sounding of stellar interiorsEquation 1.11 tells us that frequencies of the same degree and with ordersthat di er by one are approximately equally spaced by   (=  n+1;‘  n;‘).The quantity   is known as the large spacing and it will be used extensivelyin this thesis. If the modes have orders that di er from each other by oneand ‘ values that alternate between even and odd values, Eq. 1.11 providesa frequency spacing of   =2. If, however, consecutive ‘ values are either alleven or odd, the frequencies are spaced by about   . Another spacing thathas diagnostic importance is the second-order, or small spacing,   . Modeswith ‘ values di ering by two and n values di ering by one ( n;‘  n 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, forexample, in the Sun is shown. (For completeness, a solar model yields valuesof    135  Hz and    10  Hz.) Frequencies generally increase invalue as n and ‘ increase. Figure 1.3 presents both large and small spacings,shown between consecutive overtones, depending on the degree di erencebetween adjacent modes. It is important to note that both the large andsmall spacings are functions of both frequency and degree and are not strictlyconstant over a large frequency range for a given star or model.131.1. The sounding of stellar interiors   νν(n,lscript)ν(n,lscript+1)ν(n,lscript+2)ν(n,lscript+3)ν(n+1,lscript−1)ν(n+1,lscript)ν(n+1,lscript+1)∆νδνFigure 1.3: A schematic showing frequency spacings. Frequencies increase to the right inthis diagram. Modes with common radial overtone n are shown as either solid or dashedvertical lines. Both the large and small spacings (  =  n+1;‘  n;‘ and   =  n;‘  n 1;‘+2, respectively) are shown. In general, increasing either n or ‘ causes an increasein frequency and the larger the degree of the mode, the lower its observed amplitude.The spacing parameters can easily be isolated in a calculated oscillationspectra by noting the radial overtones and the degrees of the modes. Moreimportant is the physical interpretation of these parameters and what thespacing between observed oscillation periods can tell us about the star.Physically the large spacing   is the inverse of the time it takes soundto cross the diameter of the star. The sound speed can be approximated aspp= and is dimensionally the same as pGM=R. Placing this into Eq. 1.12shows the large spacing is on the same order as the inverse of the free-fall (ordynamical) 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 the141.1. The sounding of stellar interiorslarge spacing is approximately  = (0:205 0:011) GMR3 1=2Hz (1.14)with G, M, and R given in SI units. Matthews et al. (1999) rewrote thelarge spacing in terms of a star’s e ective temperature Te and luminosity Lto give  = (6:64 0:36) 10 16M1=2T3e L 3=4 Hz (1.15)In this form L and M are in solar units and Te in degrees Kelvin. If a regularfrequency spacing is observed in an oscillating star it can be used to determinethe star’s luminosity independent of other distance determination methodslike trigonometric parallax. This method of asteroseismically determining astar’s luminosity assumes there is some constraint on the star’s mass and ameasurement its e ective 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 toprobe the deep stellar interior, while at the same time cancelling out thesmall contributions of ‘ near the stellar surface, motivates the isolation ofthe small spacing. Computing the frequency di erence  n;‘  n 1;‘+2 in Eq.1.11 shows  /Z R01rdcsdrdr (1.16)Close to the centre of a star, the leading contribution to the small spacingcomes from the 1=r factor in the integrand of Eq. 1.16. As a star evolves, itsinterior composition changes through nuclear burning (increasing the amountof 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 molecular151.2. The chemically peculiar A starsweight in its core increases causing a decrease in both the local sound speedand in the small spacing. Therefore, the small spacing can be exploitedas 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 starsWe next discuss a subset of A-type stars that have a number of distinguishingcharacteristics that strongly in uence 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 lateB to early F spectral type with a temperature range of about 7;000 K. Te .14;000 K. The peculiarities in these stars are observed as spectral line strengthanomalies that are signi cantly di erent from the majority of other stars andare interpreted as abundance enhancements or depletions on the stellar sur-face. The peculiar line strengths correspond to photospheric overabundancesup to 105 times the corresponding solar value and under abundances downto 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 starson the main sequence (North et al., 2008). Each group is de ned by a spe-ci c temperature range, magnetic  eld strength, spectral line anomalies, andtheir pulsational properties. As an example, the hotter Ap stars, with tem-161.2. The chemically peculiar A starsperatures ranging from about 10,000{14,000 K, can exhibit either Si or Hgand Mn peculiarities in their spectra. The Ap Si stars have a detectablemagnetic  eld while the Ap HgMn stars generally do not. The cooler starswith Te .10;000 K are classi ed as Ap SrCrEu stars and show very strongline strengths for the rare earth elements (La to Er) Sr, Cr, and Eu. Bothhot 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 Wol (1983) and Kurtz& Martinez (2000).1.2.1 Magnetic  elds, di usion, & Ap starsThe Ap stars show magnetic  elds that appear to be global, predominatelydipolar, and have strengths ranging from about 0:3 kG to 30 kG (Landstreet,1992a). These magnetic  elds 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  elds in stars is given by Mathys (1989),Landstreet (1992a; 1992b and 1993), and more recently by Wade (2006).In the presence of a magnetic  eld, spectral lines having a wavelength, , can show splitting caused by the Zeeman e ect. For magnetic  eldstrengths up to a few tens of kG, the separation between the split spec-tral line components is / 2B (Landstreet, 1992b). Competing e ects likethermal and Doppler broadening of spectral lines means that only a smallsubset of stars with B & 10 kG and small projected (rotational) velocitiesvsin(i) . 10 km s 1 show signi cant broadening from the Zeeman e ect(Landstreet, 1992b). Examples of this average magnetic  eld modulus deter-mination are given by Mathys et al. (1997).171.2. The chemically peculiar A starsWhen Zeeman components are not resolved it is possible to exploit thepolarization properties of the magnetically perturbed lines. If a spectralline is observed in both left and right-circularly polarized (LCP and RCPrespectively) light, the contribution to the spectral line from each of theunresolved, split components is di erent. The di erence in the position of themean wavelength of the line in RCP and LCP light provides a measurementof the line-of-sight component of the magnetic  eld averaged over the stellardisc. This measurement is known as the mean longitudinal  eld strengthand is represented by Bl. For most Ap stars, Bl, and its variation withrotation of the star, are the only magnetic observations available because theyare relatively easy to obtain (Landstreet, 1993)|The measurement is mostsensitive to modest  elds with simple structures and no apriori informationabout the line pro le is needed.Bl varies periodically through rotation phase  asBl( )/Bp (cos cosi+ sin sinicos ) (1.17)Here Bp is the polar  eld strength, assumed to be a centred dipole, i isthe angle between the rotation axis and the line of sight, and  is the anglebetween the rotation and magnetic axis. The zero point of the rotation phase( = 0) is de ned at the time of magnetic maximum. The geometry of thismodel, 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-sightchanges, leading to a modulation of Bl with the rotation of the star.181.2. The chemically peculiar A starsβφiFigure 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 rotatesthrough a phase angle  (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  eld vary in strength.The aspect of the magnetic axis varies because it is inclined to the star’s rotation axis byan angle  and to the observer’s line-of-sight by an angle i. See the discussion in x 1.2.1for more details.If there is a strong  eld (&1 kG) we may also obtain information aboutits geometry from the transverse  eld component. In this case, the central,unsplit, spectral line component will saturate before the split componentsand the integrated line pro le will have a net linear polarization (e.g., Leroyet al., 1993). From this net linear polarization one can, in some cases, obtain191.2. The chemically peculiar A starsunique values for i,  and Bp de ned in Eq. 1.17. The technique that exploitsthis characteristic of the Zeeman components is known as broadband linearpolarization.The separation of elements through gravitational settling and radiativelifting of some ionic species was used to explain the observed abundancepatches 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 (mainlyHydrogen) mixture will sink under the in uence of gravity. Exceptions occurfor elements that have absorption lines at the wavelengths near the local  uxmaximum. In these cases, the elements may be levitated upward toward thestellar surface if the radiative forces are greater than the gravitational force.Once the elements reach an equilibrium position between the gravitationaland radiative forces they may accumulate in su cient amounts, resulting inabundance anomalies at these locations. General reviews of di usion theorymay be found in Vauclair & Vauclair (1982), Michaud & Pro tt (1993) andmore 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 ofthese stars that exceed the di usion velocities of a few cm s 1 the abundanceanomalies will simply be mixed away (Michaud, 1976). While the main se-quence A stars do not have large convective envelopes, the A giants do. Thisexplains why the A-type giants lose their abundance peculiarities. For thedi usion mechanism in these stars to be e cient there must be a stabilizingmechanism against turbulence. It is believed that the magnetic  eld providesthis stabilizing mechanism (e.g., Gough & Tayler 1966 and Balmforth et al.201.2. The chemically peculiar A stars2001). A magnetic  eld also in uences the distribution of elements, caus-ing horizontally distributed abundances on the stellar surface near regions ofhorizontal magnetic  eld lines (Michaud et al., 1981). The e ect of a mag-netic  eld on element di usion is described in Babel & Michaud (1991) andMichaud (1996). For a recent discussion on the spectroscopic determinationof abundance variations with depth in Ap star atmospheres see Ryabchikova(2008).1.2.2 Stellar seismology & Ap starsBefore the late 1970s observations of variable stars did not show p-modeswith 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 stellarrotation period; ranging from days to years), and also show rapid oscillationswith periods ranging from about 5 to 30 minutes. The amplitudes of theoscillations are generally small, with semi-amplitudes being less than about10 mmag when seen through a Johnson B  lter (Johnson & Morgan, 1953).The roAp stars tend to be the coolest members of the Ap class (usuallyshowing enhancements in one or all of Sr, Cr, and Eu) and have masses ofless than about 3.0 M . The approximate position of the roAp stars onthe Hertzsprung-Russell (HR) diagram is shown in Fig. 1.5. Some of thesestars are multi-periodic and exhibit asymptotic (seex1.1.1) spacings indica-tive of the large spacing. While some oscillate with periods that seem to bestable over many years, other members of the class may pulsate with only211.2. The chemically peculiar A starsone periodicity or have frequencies that vary over a timescale as short asa few days. To date there are about 40 known roAp stars. Some detailedreviews 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  eld strengthvariations. The magnetic  elds of the Ap stars seem to play a critical role inboth the micro- and macroscopic physics that in uences these unique stel-lar structures. In the outer layers of the Ap stars the strong magnetic  eldproduces pressures that are greater than the local gas pressure and it is ex-pected that this results in an observable e ect on the rapid oscillations ofthe roAp stars (see, e.g., Cunha 2007). In fact, the magnetic  eld in uencesthe observed oscillation spectrum of these stars in a number of ways. One ofwhich is a ‘point-of-view’ e ect where the observed frequencies are split intomultiple components (akin to the Zeeman splitting of spectral lines) spacedby the rotation period of the star. Another is a physical perturbation to thedisplacements of the rapidly oscillating medium, producing frequency shifts(and spacings) that cannot be predicted by models that do not include mag-netic  elds. These two examples are discussed below, with more details onthe latter of these e ects being provided in chapter 4.It was noted inx1.2 and 1.2.1 that the magnetic  eld and the accumulatedpatches of certain elements on the surface of an Ap star were linked. As anAp star rotates, the magnetic  eld strength varies as the magnetic pole comesin-and-out of an observer’s  eld-of-view. The spotty abundance patches onthe star’s surface tend to lower the stellar  ux, causing a decrease in the lightan observer sees, while the spot (or a portion of it) is in view. Interestingly,221.2. The chemically peculiar A starsFigure 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 x 4.1. Vertical dashed lines show the upper and lower bounds ofe ective temperature (6;000 K .Te . 9;000 K) where the roAp phenomenon occurs.The roAp stars are main-sequence (stars that burn Hydrogen in their cores) variables andhave a mass range of 1:3 M .M.3:0 M .231.2. The chemically peculiar A starsthe amplitude of the rapid stellar oscillations of the roAp stars is generallyat maximum during this minimum of the mean light of the star. Each of thelong-term variability (or mean light variations), the rapid variability, andthe magnetic  eld strength are related to each other, and are functions ofrotation phase. Figure 1.6 compares the variation in the amplitude of therapid pulsation ( 6 minute period) to that of the magnetic  eld strength as afunction 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 withthe magnetic  eld. (Later, in Fig. 3.6 of x 3.3, we show that the mean lightamplitude; associated with the spotty abundance patches, and the pulsationamplitudes vary in anti-phase as HR 1217 rotates.) The modulation of themagnetic  eld strength is described well by the oblique rotator model ofStibbs (1950, seex1.2.1). Following this model, Kurtz (1982) suggested thatthe amplitudes of the rapid variations may be modulated in a similar way,through what is now called the oblique pulsator model.241.2. The chemically peculiar A starsFigure 1.6: (Upper panel) The variation in the pulsation amplitude of HR 1217 (through aB  lter) as a function of the rotation phase (referenced to a rotation period of about 12.5days) of the star. The magnetic  eld variation de ned by Bl (shown as H e in the  gure) inx1.2.1 as a function of rotation phase is presented in the lower panel. The pulsations ( 6minute period) have amplitudes that vary synchronously with the magnetic  eld strength.Pulsation data and magnetic  eld data were taken from Kurtz (1982) and Preston (1972)respectively. This  gure was obtained with permission from Matthews (1991).In this model, the pulsation axis of the star is aligned with the magneticaxis and both are inclined to the rotation axis. As the star rotates theaspect of the pulsation and magnetic axis varies, modulating the pulsationamplitudes. The geometry is the same as that shown in Fig. 1.4 with themagnetic and pulsation axis being the same. Consider an axisymmetric (m251.2. The chemically peculiar A stars= 0) pulsation mode oscillating with frequency  . Kurtz (1982) showed theluminosity varies with rotation like LL /Pm‘ (cos ) cos [ t+’p] (1.18)wherePm‘ is the associated Legendre polynomial, ’p is an arbitrary oscillationphase, and  is the rotation phase angle zeroed at the time of pulsation ormagnetic maximum. For a dipole pulsation mode ‘ = 1 and the Legendrepolynomial is equal to cos . Equation 1.18 may be expanded as LL  A0 cos ( t+’p) +A1 [cos (f +  gt+’p) + cos (f   gt+’p)](1.19)where  was replaced by the product of the rotation rate and time,  t,and the two amplitude functions are de ned as A0 = cosicos and A1 =(sinisin )=2. This model predicts that a single dipole mode is split intoa triplet exactly spaced by the rotation period of the star. Generally, amode with degree ‘ is split into 2‘ + 1 frequencies. The roAp stars doshow this  ne structure in their oscillation spectra (see, e.g., chapter 3).The shortcoming of this simple model put forth by Kurtz (1982) is thatit does not explain the amplitude asymmetry between the split frequencycomponents observed in some roAp stars. Equation 1.19 shows us that thesplit frequencies should have the same amplitudeA1. The equal amplitudes ofthe rotationally split components is not observed in the Ap stars’ oscillationspectra. This is resolved if the e ects of both rotation and the magnetic  eldon the eigenfunctions (through the Coriolis and Lorentz forces) are taken intoaccount. The most recent contribution to the oblique pulsator model comesfrom Bigot & Dziembowski (2002). They used a non-perturbative approach261.2. The chemically peculiar A starsto show that the centrifugal force is important in determining the frequencysplitting while the Coriolis force is dominant in determining the amplitudeasymmetries. Those authors also show that the pulsation, rotation, andmagnetic axes are all inclined to each other, i.e., the magnetic and pulsationaxis are not necessarily aligned. The geometry of roAp oscillations is reviewedby Cunha (2005).The e ects of the strong magnetic  eld on the rapidly varying normalmodes (i.e., those eigenfrequencies calculated in the case of no magnetic eld or rotation) will be discussed in chapter 4. Recent theoretical reviewsdiscussing the interaction between the magnetic  eld and the pulsation fre-quencies are given by Cunha (2007), Saio (2008) and Shibahashi (2008). Inthe 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 becomesigni cant enough to strongly modify the motions of the local plasma. Withthis in mind, a number of authors have attempted to model the e ect ofa strong magnetic  eld on high overtone stellar pulsations that have theirlargest amplitudes near the stellar surface. Roberts & Soward (1983) de-scribed an analytic matching procedure used to link the surface e ects tothose of pulsations in the stellar interior where the magnetic  eld can be ne-glected. Numerical experiments by Campbell & Papaloizou (1986) expandedupon 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 eld modi es the surface of an Ap star to such a high degree that a singlespherical harmonic cannot describe the displacement, which has a signi cant271.2. The chemically peculiar A starshorizontal component. Each of the investigations above recognize that in thedeep interior of the star, the magnetic and acoustic components of the dis-placement decouple and the magnetic component dissipates as it propagatesinward, draining energy from the pulsation. The above studies conclude thatthere is a signi cant observable e ect on the oscillation spectrum of an Apstar; however, there are some di erences 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 di ermainly 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 causedby the presence of a magnetic  eld. The method is described in x 4.3 andhas been successfully tested against observations of HR 1217 (discussed inthe following section).Finally, it should be noted that a number of authors have attempted tomodify stellar evolution models in such a way as to excite the frequenciesthat are observed in roAp stars. Recent e orts include modi cations tothe atmospheric temperature strati cation (Gautschy et al., 1998) to re ectpulsation modes, the inclusion of helium gradients and convective suppressionof the stellar envelope (Balmforth et al., 2001), a magnetic and radiativedamping (non-adiabatic) study of oscillations Saio (2005), and the inclusionof the e ects of stellar winds and metal gradients in the stellar envelope(Th eado et al. 2005,2009). While there have been some successes in excitingthe high overtone modes in Ap stars, none of the studies is able to provide281.3. The roAp star HR 1217a globally applicable excitation mechanism for these stars. In this study wefocus solely on adiabatic results; i.e, we ignore excitation, so that we can, forthe  rst time, attempt to constrain the physical parameters of a roAp starusing a large grid of magnetically perturbed pulsation models.1.3 The roAp star HR 1217This thesis focuses on observations and models of the roAp star HR 1217(a.k.a HD 24712; DO Eri; B = 6.3 mag;  = -12 50 56:0078;  = 03 hr 55 min16:s128 ). This was one of the  rst Ap stars to be identi ed as a roAp star byKurtz (1982) and has since become one of the most studied. Recently, high-quality data has provided information about the abundance, magnetic andphotometric characteristics of HR 1217. The most recent photometric datais analyzed in this thesis and is, in many ways, an unrivalled photometricdata set on HR 1217 (chapter 3). A review of the other recent observationsof HR 1217 is presented below.The magnetic  eld of HR 1217 was measured by Bagnulo et al. (1995).They were able to model the magnetic  eld geometry for HR 1217 and givevalues of 137 , 150 , and 3:9 kG for i,  and Bp respectively. Estimateduncertainties on the above angular measurements are 2{3 and the uncer-tainly of the polar  eld strength is about 5%. The mean longitudinal  eld,Bl, for HR 1217 varies between  0:5 and 1:5 kG (Preston, 1972), and isillustrated in the lower panel of Fig. 1.6. Kochukhov & Wade (2007) discusscurrent magnetic and rotation properties of HR 1217. The most recent mag-netic  eld measurements are given by L uftinger et al. (2008). They measure291.3. The roAp star HR 1217a magnetic  eld variation of 2.2 to 4.4 kG (no uncertainty given) based on aDoppler imaging analysis.The  rst thorough abundance analysis of HR 1217 was performed byRyabchikova et al. (1997). Their results are consistent with the idea thatthe abundance enhancements on the surfaces of these stars are patchy. Inparticular they  nd the mean chemical abundances vary with the magneticand rotational phase of the star. When compared to solar values, the rareearth elements are the most overabundant and show the largest variationover 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 1217based on the work of Ryabchikova et al. (1997) is presented in x 4.1.1.The  rst global photometric campaign to observe HR 1217 was headedby Don Kurtz and Jaymie Matthews (Kurtz et al., 1989). The goal of thoseobservations was to achieve as much continuous coverage of the star as pos-sible so that gaps in their data would not a ect 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-dayperiod. The results of the campaign are shown schematically in the top panelof Fig. 1.7. There exists an ambiguity in identifying   from the observedfrequency spacing (x 1.1.1). If the modes are alternating between even andodd ‘ values we would expect to see a    68  Hz. In fact, the alternat-ing spacing of 33 and 34  Hz is consistent with models having alternatingeven 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 thesame. Each of the frequencies in Fig. 1.7 are actually multiplets having fre-301.3. The roAp star HR 1217quencies spaced by approximately 0.9  Hz (the rotational frequency of thestar). The unexpected frequency spacing ( 50  Hz) between the last twofrequencies (between  2750{2800  Hz) is not consistent with asymptotictheory and the observed large spacings. The gap in the frequency spectrumof HR 1217 observed by Kurtz et al. (1989) is usually said to be a \missingmode".Almost a decade after Kurtz et al. (1989) released their results the truelarge spacing value of  68  Hz was unambiguously determined. UsingEq. 1.15, Matthews et al. (1999) were able to calculate a parallax for HR1217 based on an inferred large spacing of 68  Hz. Their predicted parallaxof  = 19:23 0:54 mas was shown to be consistent with the recent Hipparcosparallax (Perryman et al., 1997) of  = 20:41 0:84 mas (x 4.1.1).311.3. The roAp star HR 1217Figure 1.7: Schematic amplitude spectra of HR 1217 oscillations from previous obser-vations. The upper panel shows the frequencies identi ed in the 1986 global observationcampaign of HR 1217 (Kurtz et al., 1989). Frequencies are regularly spaced (see discussionon asymptotic spacing in x 1.1), alternating between  33.5 (green) and 34.5 (blue)  Hz.The exception to the regular spacing is a spacing of 50  Hz represented by a red line inthe top panel. The middle panel gives the frequencies found in the more recent 2000 globalcampaign (Kurtz et al., 2005). This campaign uncovered a \missing" frequency near 2800 Hz that  t with the previously identi ed asymptotic spacing of 33.5  Hz (representedby the red line in the middle panel). In addition to this discovery, Kurtz et al. (2005)discovered a frequency spacing of  2.5  Hz (the di erence between the red and greenlines) that is consistent with a second order asymptotic spacing for a main sequence A star.The furthest frequency spacing (blue line) of  14  Hz is inconsistent with the observedasymptotic spacings and may be explained by magnetic perturbations (see Cunha 2001and x 4.3). The lower panel shows the results from recent spectroscopic observations ofMkrtichian & 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)  Hz may exist at the low frequency end of the oscillation spectrum.321.3. The roAp star HR 1217A success in the interpretation of the frequency spacings of HR 1217 wasprovided by Cunha (2001). She predicted that magnetic damping could bethe cause of the missing frequency in the 1986 data set. She also showedthat some frequencies may be shifted by approximately 10{20  Hz becauseof magnetic  eld e ects (Cunha & Gough 2000, and Cunha 2001; furtherdiscussion is provided inx4.3). In 2000, HR 1217 was selected to be observedin another global campaign (organized by the Whole Earth Telescope, WET;see Kleinman et al. 1996 and Kurtz & Martinez 2000) and a preliminarydata reduction for this data set did  nd the missing frequency at  2795 Hz (Kurtz et al., 2002). Although it was long suspected, this was the rst time it was shown that the in uence of the magnetic  eld along withstellar rotation and surface chemical gradients plays a critical role in thepulsation characteristics of the roAp stars. These unique properties of theroAp stars could allow us the opportunity to model the magneto-acousticpulsation modes and study in detail an environment that is signi cantlydi erent 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 themiddle panel of Fig. 1.7. They collected at total of 35 days of data with aduty cycle of 36% and reached an amplitude precision of 14  mag (one ofthe most precise ground based photometric studies to date). Along with thediscovery of the missing mode, they also discovered a closely spaced frequencygrouping of about 2.5  Hz that could not be explained by the oblique pulsatormodel. This \small" spacing is on the order of magnitude that one wouldexpect for a second order asymptotic spacing discussed in x 1.1.1 for a main331.4. Thesis goalssequence A star. This could be the  rst time such a small spacing wasobserved in a star other than the Sun. That being said, the diagnostic powerof the small separation may be limited because the magnetic perturbationsto the frequencies may be of the same order, or larger than, the small spacing(Dziembowski & Goode, 1996). This is further illustrated in x 4.3.A recent radial velocity study by Mkrtichian & Hatzes (2005) has alsouncovered some new frequencies (shown in the lower panel of Fig. 1.7). Thoseobservations covered about a month of time (from 22 Dec, 1997 to 30 Jan1998 with two additional days in early Feb 1998) giving about 5 hours ofobservations per night for 9 nights over the month. They  nd new frequenciesat 2553 and 2585  Hz. These frequencies were only identi ed in a subset ofthe authors’ data and are, admittedly, of low signi cance and of low frequencyresolution.1.4 Thesis goalsProgress 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 andtheoretical tools. In this thesis we attempt to tie together the most recentphotometric observations (chapter 3) on the roAp star HR 1217 (obtainedby the MOST satellite, which is described in chapter 2) to theoretical stellarevolution and pulsation models that include the e ects of a magnetic  eld(chapter 4).The data collected by the MOST mission has obtained both ultra-highprecision and time sampling that has never been achieved before, on this341.4. Thesis goalstarget (HR 1217), and for a vast majority of asteroseimic targets. With thisdata in hand, we have the following observational goals and questions:1. It has been known since the early 1980s that HR 1217 is a multiperiodicvariable with near equally spaced modes, alternating between  33.5and 34.5  Hz. The most recent con rmation of this was the analysis ofWhole Earth Telescope data by Kurtz et al. (2005), who achieved a dutycycle of 36% and reached an unprecedented precision of 14  mag fora ground-based photometric study. We will be able to unambiguouslyidentify the spacings in the MOST data because of its near continuouscoverage and ultra-high precision. Are the already identi ed periodici-ties constant over time for all observations on HR 1217, or do they varyover time, pointing to a selective mode damping mechanism or somenonlinear interaction?2. The strange, apparently non-asymptotic, spacing that was observed be-tween the highest two frequencies in HR 1217 remained a mystery untilthe recent results of Kurtz et al. (2005, 2002). Those authors identi eda previously unidenti ed frequency near 2790  Hz that could only beexplained as a magnetic perturbation Cunha (2001). Explaining thatfrequency (or lack of that frequency) has driven the theory of roAp staroscillations for more than two decades. Is this frequency identi ed inthe MOST data on HR 1217 and is it stable since the last observationsof Kurtz et al. (2005)?3. We expect to observe a number of new periodicities in the data becauseof the high level of photometric precision and the near contiguous data351.4. Thesis goalssampling of the MOST satellite. Can new periodicities be identi edand, if so, can they be used to constrain the physics of this magneticpulsator?4. A recent spectroscopic study on HR 1217 by Mkrtichian & Hatzes(2005) identi ed new frequencies at  2553 and 2585  Hz. These fre-quencies are interesting because they approximately match the alter-nating  33.5 and 34.5  Hz pattern previously observed in HR 1217,but were not identi ed in the photometry of Kurtz et al. (2005). Doesthe MOST data uncover those potentially new periodicities?5. In order to determine the reliability and uniqueness of the periodicitiesin a multiperiodic variable, we need to develop tools that can test theprecision and resolution of our data set. Techniques we have developedfor other MOST targets will be applied to the HR 1217 data. The res-olution of our data set and the assertion that closely spaced frequenciesare 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 perturbedpulsation modes are calculated and are matched to the observed frequenciesof HR 1217 (chapter 5) for the  rst time. The scope of this model grideclipses any previous attempts to constrain the properties of HR 1217, orany other roAp star. The following theoretical questions will be answered inthis thesis:1. For the  rst time, a realistic stellar model grid, covering a vast swathof parameter space appropriate for matching the observable character-361.4. Thesis goalsistics of HR 1217, is constructed that includes the magnetic perturba-tions to the calculated normal modes of the models. Is the magneticperturbation a necessary ingredient for the matching of a model to theobservations?2. Are the spacings observed by the MOST mission reproducible with thephysics 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 anobserved frequency?5. We test the calculations of the pulsation modes in our grid by usinga new method developed by Kobayashi (2007) for the calculation ofnormal modes in a geophysical context.The methodology and the results of the observational and theoreticalanalysis of the HR 1217 observations from the MOST mission are outlinedand discussed throughout the following chapters. The results are summarizedin chapter 6 of this thesis.37Chapter 2The MOST mission: Datareduction techniquesThe HR 1217 data presented in chapter 3 of this thesis was collected by theMOST satellite and analyzed using techniques developed over a period ofabout 5 years by the MOST team and its collaborators; including the authorof this thesis. In this chapter we give an overview of the MOST mission, thephotometric and time-series methods that are relevant to the HR 1217 datareduction, and  nish with some examples of other MOST results that usedthe described data analysis methods.2.1 An overview of the MOST missionThe Microvariability & Oscillations of STars (MOST) mission is Canada’s rst space telescope3. Launched into a sun-synchronous orbit at an altitudeof  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 typicalsuitcase) functions as an ultra-precise light meter, that routinely obtains near3Being a Canadian space mission it is also appropriate that the acronym MOST worksequally well in French ::: Microvariabilit e et Oscillations STellaire382.1. An overview of the MOST missioncontinuous4, micro-magnitude, photometric measurements of oscillating starsand extrasolar planets.The instrument is a 15/17.3 cm Rumak-Maksutov telescope that directslight through a custom broadband  lter, covering a wavelength range of 350{700 nm,5 to two CCDs: One used for tracking and the other for scienti cmeasurements. The experiment is outlined in Matthews et al. (2000), withmore technical details provided by Walker et al. (2003).Originally, MOST produced scienti c data in Fabry and direct imagingmodes. 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 e ect of individual pixelvariations (caused, for example, by radiation). The direct imaging modeprovides in-focus photometry of fainter (&6 mag) stars on the science CCD.The satellite, through software updates, was eventually (after about a yearof operation) able to achieve sub-arcsecond pointing, allowing scienti c datato be gathered on guide stars from the tracking CCD, greatly increasing thescienti c output of the mission. MOST functioned in this way until early2006, when the tracking CCD suddenly failed. It is suspected that a par-ticle hit damaged the CCD. Since that time, both science and tracking areperformed on the science CCD.The  rst science results from MOST were published by Matthews et al.4MOST has a Continuous Viewing Zone that ranges in declination from +36     18 . Stars in this declination range can, at most, be viewed continuously for up to 60days.5A Johnson B  lter (Johnson & Morgan, 1953) is centred at about 445 nm with a fullwidth at half maximum of  94 nm (see Binney & Merri eld 1998)392.2. 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 scienti c accom-plishments of the MOST instrument to date. Data on binary stars, rapidlyrotating stars, stars with spots, magnetic stars, exoplanets, and even stablestars are providing new insights into exotic and, in many cases, otherwiseuntestable physical scenarios. With more than 1000 stars observed|about aquarter of which show detectable light variations | the scienti c discoveriesare on-going.2.2 Photometry & time-series analysisReducing MOST photometry is a challenging task because of stray-lightcontamination caused by Earth shine illuminating the CCDs. This parasiticaddition to the stellar signal depends on the orbital phase and orientation ofthe satellite and varies with season. From one orbital period ( 101 min) toanother, this contaminant changes its character, making it very di cult tomodel.The result is a superposition of orbital components ( 14 cycles day 1),along with their harmonics and aliases, and the stellar signal in the Fourieramplitude spectra (discussed below) of a target’s light curve. Methods haveevolved since the launch of MOST to combat the stray-light e ect. These arediscussed in detail in Rowe et al. (2006b, 2006c), Reegen et al. (2006), andRowe (2008). Typical photometric data techniques such as sky backgroundremoval (estimated from the dimmest  200 pixels),  at- elding, and PSFmodelling (using  150 of the brightest pixels) are also discussed in those402.2. Photometry & time-series analysispapers. Photometric errors are estimated from Poisson statistics and aredescribed, for example, in Everett & Howell (2001).The asteroseismic results are determined from relative amplitude  uctu-ations in the star’s light curve. Absolute magnitude determinations are notnecessary, so limiting the stray-light contribution to the data is more impor-tant than applying mean corrections to the counts. In the work presentedin this thesis the stray-light was suppressed by subtracting a smoothed pro- le 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 orbitalbins used to  nd the correction, the less aggressive is the reduction of theorbital 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  nding the correction to data phased over a small number of satelliteorbital periods, we remove orbital signal averaged over shorter timescales,essentially over-smoothing the orbital component. The main advantage ofthis method is that no prior information about the structure of the orbitalcontamination is assumed. Any remaining long term trends can be removedby either subtracting a low-order polynomial or a spline  t to binned data.The application of this method is further discussed in x 3.1 (see Fig. 3.1 foran example of the stray-light contamination to the HR 1217 light curve).The frequency analysis of the reduced time-series is carried out usingCAPER6(Cameron et al., 2006). CAPER is a collection of Fortran driverroutines that use a Discrete Fourier Transform (DFT) as a frequency andamplitude estimation tool and nonlinear least squares  tting to re ne the6Controlled or Automated Period Estimation and Re nement412.2. Photometry & time-series analysisidenti ed parameters. The software package has been used in a number ofrecent 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 thesame 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 fromthe DFT) is de ned byAn( ) = 2Tpa2n +b2n (2.1)where an and bn are the imaginary and real parts of the Fourier transformand are calculated froman =TXj=1f(tj) sin(2  tj) and bn =TXj=1f(tj) cos(2  tj) (2.2)In these expressions f(tj) is the magnitude (or  ux value) at time tj and  is the current frequency of interest. The amplitude spectrum is calculatede ciently over a range of  values using the algorithm proposed by Kurtz(1985). This algorithm is a modi ed version of the Deeming algorithm forcomputing the DFT of data with sampling gaps (Deeming 1975; also seeMatthews & Wehlau 1985). Once the largest peak in the amplitude spectrumis identi ed, its phase can be calculated as  = arctan(an=bn).The set of parameters ( , A,  ) obtained from the DFT are next re nedby nonlinear least squares  tting using the Levenberg-Marquardt method(Press et al., 1992)7 which minimizes  2 between a  tting function and thedata. A sinusoidal  tting function of the formFfit(t) = Asin(2  t+ ) (2.3)7The name of the Numerical Recipes routine used by CAPER is mrqmin422.2. Photometry & time-series analysisis used by CAPER.Once Ffit is optimized, it is subtracted from the original data set. AFourier spectrum of the residuals is calculated and a new parameter set isidenti ed. These new parameters, in conjunction with the previous set(s), arere- tted to the original time-series and the process is iterated to a prede nedstopping criterion, usually set as a signal-to-noise (S/N) threshold. The  tis improved at each step based on the original data until there are no longerany meaningful changes in the residuals from the  t.A S/N threshold of  3:5 is commonly used in the asteroseismic com-munity as a lower limit to the signi cance of a detectable peak in the DFT.Peaks in the DFT with a S/N greater than this represent a detection &2.5 (Breger et al. 1993 and Kuschnig et al. 1997). This is adopted as alower limit to the signi cance of the extracted periodicities in this work. TheS/N for each recovered signal component is calculated (before prewhitening8that component) as the mean amplitude in a box around the identi ed peak,sigma clipped until the mean converges. The clipping is done so that addi-tional high amplitude peaks near the identi ed frequency do not skew thelocal mean amplitude (noise) value. This method of calculating the S/N hastwo potential sources of uncertainty: 1. The width of the box used to av-erage the amplitude spectrum (the noise) and 2. Uncertainties in the  ttedamplitude. The uncertainty in the noise calculation is estimated by varyingthe width of the averaging box in small steps and then calculating an averageof each noise value along with a standard error on that average noise. Once8Prewhitening is a term used to describe the subtraction of a sinusoidal function fromthe data.432.2. Photometry & time-series analysisamplitude uncertainties are assessed from a bootstrap analysis (describedbelow), we combine both to arrive at the  nal uncertainty in S/N.The uncertainties we calculate for time-series parameters derived fromnonlinear least-squares methods depend on the noise of the data (which maybe a combination of instrumental and random processes) and on the timesampling. It is also known that  tted phase and frequency parameters arecorrelated, leading to underestimated uncertainties in these parameters whencalculated from a covariance matrix (see, e.g., Montgomery & O’Donoghue1999). A bootstrap analysis is an ideal way to assess the uncertainties in tted parameters. The procedure has recently been used in a number ofMOST 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, Cameronet al. 2008, Rowe 2008, Rowe et al. 2008a, and Rowe et al. 2008b) andwas used by Clement et al. (1992) to estimate the uncertainties in Fourierparameters they derived for RR Lyrae stars.The bootstrap (see Wall & Jenkins 2003) produces a distribution foreach calculated parameter by constructing a large number of light curves(usually > 10;000) from the original data. No assumptions need to be madeabout noise properties of the data and individual photometric errors are notneeded. Each new light curve is assembled by randomly selecting N pointsfrom the original light curve (also containing N points) with the possibilityof replacement. In this way, the new synthetic light curves preserve the noiseproperties of the original data. The  t procedure is repeated for each newlight curve, eventually building distributions in each of the  t parameters.Recently, Breger (2007) discussed the di erence between frequency resolu-442.2. Photometry & time-series analysistion and the precision of  t parameters to time-series data. Traditionally oneassigns 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 fromthe amplitude spectrum. This roughly corresponds to the spacing betweenthe main peak of the window function9 and the peak of its  rst sidelobe.Lower, and arguably more realistic, estimates are used by Kurtz (1981), whoestimates frequency resolution as 0:5 T 1 (approximately the half-width athalf-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 numberof simulated data sets. Ultimately, the resolution of frequencies in Fourierspace is a function of S/N (or signi cance) of each individual peak; see, forexample, Kallinger et al. (2008b), and the above criteria are only estimatesused as an average frequency resolution for all frequencies of a given dataset. The precision of  tted parameters, on the other hand, can be estimatedby re tting identi ed parameters to a large number of data sets created bysampling the  tted 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 di ers from such a Monte Carlo procedure in thatthere 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 canbe estimated because the windowing (determined from the temporal sampling9The 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.452.3. Applications of CAPER to other MOST scienceof the data) of each of the re-sampled data sets is randomly changed. Thus,the uncertainties on each of the  tted parameters take into account both theinherent noise of the data and the sampling of the data as well. Put anotherway, the robustness of our  t parameters to the variations in the data (dueto any source) is estimated.2.3 Applications of CAPER to otherMOST scienceThe following sections review a number of published examples of MOST ob-servations that use the reduction methods described above, and that wereused for the reduction of the HR 1217 observations collected by MOST. (Inparticular, each subsection outlines the results of a Fourier analysis using theCAPER software and a combined bootstrapping error analysis.) The resultsfor 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 foundin the references provided. Although the following analyses are common tothose undertaken in this thesis, the results are independent of this work andreader may wish to proceed to the analysis of the HR 1217 data in chapter3.462.3. Applications of CAPER to other MOST science2.3.1 The roAp star HD 134214The roAp star HD 134214 was observed by MOST as a trial Direct Imagingtarget on 1 May 2006 and the results were published by Cameron et al. (2006).The star was in one of two target  elds observed during each MOST satelliteorbit (period = 101.413 min), so data were collected 47% of the observingtime. 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 of2948.97 0.55  Hz. This is consistent with earlier ground based photometriccampaigns (e.g. Kreidl et al. 1994).Recent results by Kurtz et al. (2006) suggest that HD 134214 is oscillatingspectroscopically in up to 6 modes. Our observations can rule out additionalphotometric oscillations at a 2 detection level of 0.36 mmag, based on S/Nestimations described in x 2.2. Our estimated DFT noise level of  0.12mmag 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 fromfour 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 modesseen spectroscopically: 1. Is it a case of radial velocities of certain elementsand ionization stages being more sensitive to degrees of higher ‘? 2. If thesensitivity of the photometry can be improved su ciently, will these modesalso appear? 3. Is the broadband photometry just averaging over too widean extent of the pulsating atmosphere? or 4. Might there be new physics inthe upper atmospheres of (some) roAp stars to account for the di erencesobserved?472.3. Applications of CAPER to other MOST scienceKurtz et al. (2007) presented new photometric measurements on HD134214 and con rm some of the frequencies seen spectroscopically by Kurtzet al. (2006). The authors suggest that the additional frequencies may bebelow 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 Cameronet al. (2006) with over a month of data. The results are currently beinganalyzed and will hopefully shed new light on the questions raised by theseworks.2.3.2 The B supergiant HD 163899The MOST satellite observed the B supergiant HD 163899 (B2 Ib/II) as aguide star and detected 48 frequencies . 2.8 cycles day 1 with amplitudesof a few mmag and less (Saio et al., 2006). HD 163899 was observed as oneof 20 guide stars used during the photometry measurements of WR 103 (HD164270). Observations were made from 2005 June 14 to July 21 for a totalof 36.6 days. Because the WR 103  eld is outside the MOST ContinuousViewing 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 generallythought that no g-modes are excited in less luminous B supergiants becausestrong radiative damping is expected in the core. Theoretical models calcu-lated by Saio and Gautschy (Saio et al., 2006) show that such g-modes areexcited in massive post-main-sequence stars. Excitation by the Fe-bump inopacity is possible because g-modes can be partially re ected at a convectivezone associated with the hydrogen-burning shell, which signi cantly reducesradiative damping in the core. The MOST light curve on HD 163899 shows482.3. Applications of CAPER to other MOST sciencethat such a re ection of g-modes actually occurs, and reveals the existenceof a previously unrecognized type of variable, slowly pulsating B supergiants(SPBsg). Such g-modes have great potential for asteroseismology. HD 163899is the  rst member of the SPBsg family, showing both g-modes and hybrid Cephei-type (p-mode) behaviour.2.3.3 The SPBe stars HD 127756 & HD 217543MOST discovered SPBe (Slowly Pulsating Be) oscillations in the stars HD127756 (B1/B2 Vne) and HD 217543 (B3 Vpe). The results are publishedin Cameron et al. (2008). HD 127756 and HD 217543 join the stars HD163868 (Walker et al., 2005) and  CMi (Saio et al., 2007) as SPBe starsdiscovered by MOST. High radial order g-modes with pulsation frequenciesin the co-rotating frame that are much smaller than the rotation frequencyappear in groups that depend on the azimuthal ordermin a DFT. Theoreticalmodels (see, for example, models calculated by Saio in Cameron et al. 2008)indicate that, in rapidly rotating stars, high-order g-modes are excited nearthe Fe opacity bump as in SPB stars. One di erence from slowly rotatingSPB stars is the fact that among the high-order g-modes, prograde modesare predominantly excited. These modes have frequencies of  jmj inthe observer’s frame with  being the rotation frequency of the star. Form =  1 and  2, expected frequencies are grouped at  and 2 consistentwith 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 bymatching the observed frequency groups to models of rapidly rotating stars.492.3. Applications of CAPER to other MOST scienceHD 127756 was observed for a total of 30.7 days by MOST . The target eld was outside the MOST CVZ so there is a gap during part of each 101.4min satellite orbit resulting in a duty cycle of 30.7% (Note that our datasampling ( 20 s), even with gaps every orbit, thoroughly covers the timescale of the intrinsic variations ( 1 day period) of HD 127756, giving us ane ective duty cycle of near 100%). The light curve is presented in Fig. 2.1and shows clear variations with periods near 1 day and 0:5 day. Table A.1lists the frequencies, amplitudes, phases, the 1 and 3 uncertainties from thebootstrap analysis (using 100,000 realizations), and the S/N of the 30 mostsigni cant periodicities. The 1 and 3 uncertainties are estimated for eachparameter as the width of the region, centred on the parameter in question,that contains 68 and 99% of the bootstrap realizations, respectively. The  tis shown superimposed over two zoomed sections of the light curve labelledA and B in the lower panels of Fig. 2.1.The amplitude spectrum of the data along with the  tted points and theresiduals from the  t are shown in the top panel of Fig. 2.2. Most of thefrequencies 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 theSPBe star HD 163868 (Walker et al., 2005). The lower panel of Fig. 2.2 plotsthe S/N for each of the identi ed periodicities and the window function ofthe data. Among the frequencies listed in Tab. A.1,  1 = 0:0335 cycles day 1and  2 = 0:0739 cycles day 1 have the fewest observed cycles (close to thelength of the run at 1=30:7 = 0:0326 cycles day 1) and are included to reducethe scatter in the residuals from our  t. They may not be genuine stellaroscillation frequencies but it should be noted that observational artifacts502.3. Applications of CAPER to other MOST scienceassociated with the baselines of other MOST observations, especially withsuch a relatively large amplitude of 7 mmag as in the case of  1 here, havenot been observed.A comparison of the closely spaced frequencies near  1 cycles day 1 tothe window function and to our  t is given in Fig. 2.3. Notice that the peakwith the largest amplitude has an asymmetric component that is wider thanthe window function. When that frequency ( 14) is prewhitened, signi cantpower remains near that asymmetry and is  tted as  13 (shown as the datapoint with the smallest amplitude in Fig. 2.3). These frequencies are spacedby  0:04 cycles day 1 which is greater than the Rayleigh criterion for ourdata ( 0:03 cycles day 1). The points are clearly separated in frequencywithin their respective 3 errorbars. The peak labelled as Ax ( 15 in TableA.1) is spaced from  14 at nearly the resolution limit suggested by Loumos& Deeming (1978) ( 0:06 cycles day 1). This peak is clearly resolved from 14 and has an amplitude that is  4 times that of the  rst side lobe of thewindow function (labelled as Ay). Although the amplitude of this peak maybe in uenced by windowing of the data, the evidence suggests the frequen-cies are resolved. Frequencies  18 and  19 (shown circled in Fig. 2.3) have thesmallest frequency separation and are barely resolved within their 3 error-bars with a separation of  0:004 cycles day 1. The bootstrap distributionsfor these frequencies are plotted in Fig. 2.4 and shows the parameters arenormally distributed and the frequency distributions of  18 and  19 nearlyoverlap.We suggest, based on our bootstrap distributions, that frequencies spacedby less than 0:5 T 1  0:0163 cycles day 1 (within their 3 errorbars) are at512.3. Applications of CAPER to other MOST sciencethe resolution limit of our data set. Using this resolution criterion, frequencypairs  6 and  7;  18 and  19; and  23 and  24 may not be resolved frequencies.Cameron et al. (2008) show that the resolution of these frequencies doesn’ta ect the physical interpretation of the data.MOST observed HD 217543 as a guide star for a total of 26.1 days witha duty cycle of  33% (like HD 127756 the e ective duty cycle is closerto 100%). Figure 2.5 shows the light curve with clear periods of  0:5and  0:25 days with modulations characteristic of more complex multi-periodicity. The  t to the 40 most signi cant frequencies (see Tab. A.2) isshown 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 to3.38. These are below the S/N  3:5 limit described above and represent &2 detections (Kuschnig et al., 1997). They are included to illustrate thatwithin the S/N errors plotted in the lower panel of Fig. 2.6 all identi edfrequencies reach the S/N  3:5 limit. These periodicities do not adverselyin uence the  t and do not change the physical interpretations of Cameronet al. (2008).Figure 2.6 shows an amplitude spectrum of HD 217543 in the top paneland the S/N of the identi ed frequencies and the window function of thedata in the lower panel. As with HD 127756, most frequencies are groupedaround 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 thanthe corresponding ones of HD 127756. Frequencies  1 = 0:0269 cycles day 1522.3. Applications of CAPER to other MOST scienceFigure 2.1: MOST light curve of HD 127756. The top panel shows the entire light curvespanning a total of 30.7 days. The middle and the bottom panels are expanded light curvesfor the portions A and B, respectively, indicated in the top panel. Solid lines indicate the t of the 30 signi cant frequencies (Tab. A.1) from the frequency analysis of the full lightcurve. The short-term variability seen in the middle panel is a consequence of stray Earthshine modulated with the MOST satellite orbital period of  101:4 min.532.3. Applications of CAPER to other MOST scienceFigure 2.2: (Top panel) The Fourier amplitude spectrum of the light curve of HD 127756.Filled (blue) circles with 3 error bars are the  tted parameters (see Tab. A.1). Theinverted dash{dot line is the residual amplitude spectrum obtained after the  t was sub-tracted from the light curve. (Lower panel) The S/N of the identi ed periodicities with 3 uncertainties estimated from both the  tted amplitudes and frequencies and the mean ofthe amplitude spectrum (see x 2.2 for details). The light grey line represents the windowfunction of the data centred on the frequency with the largest amplitude and scaled to themaximum S/N for clarity.542.3. Applications of CAPER to other MOST scienceFigure 2.3: The zoomed region around the largest peak in the amplitude spectrum of HD127756. The window function is shown as the inverted, dotted line and the  t is shown aspoints with 3 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  rst sidelobe of the windowfunction (labelled Ay) is 4 times smaller than the second largest peak in the amplitudespectrum at Ax. The resolution of frequencies  18 and  19 (both circled) is discussed insection 2.2.552.3. Applications of CAPER to other MOST scienceFigure 2.4: A comparison of bootstrap distributions for parameter sets ( 18, A18,  18)and ( 19, A19,  19) [see Tab. A.1] for 100,000 realizations of the HD 127756 light curve.The top panels are distributions for the  tted phase ( ) while the middle and lower panelsshow distributions for the amplitude (A) and frequency ( ) parameters, respectively. Ineach panel symbols are shown (from top to bottom) for the 1 (F) and 2 ( ) errorintervals containing 68 and 95 % of the realizations (note that Tab. A.1 lists the 3 , or99%, error interval) centred on the  tted parameter. Below those symbols in each panelare the 1 ( ) errorbars obtained from the formula de nition of standard deviation andthe mean ( ) of the distribution with the standard error on the mean. These distributionsare shown because the frequencies are the closest to each other.562.3. Applications of CAPER to other MOST scienceand  2 = 0:0806 cycles day 1 are close to the length of run (1=26:1 = 0:0383cycles day 1) but were included to reduce the residuals in the light curve.They may not be intrinsic stellar pulsations.Frequencies  6 and  7 of Table A.2 overlap within their 3 uncertainties.The bootstrap distributions are given in Fig. 2.7 and show all parameters arenormally distributed like those in Fig. 2.4 for HD 127756. However, the longtails on the frequency distributions suggest that these frequencies are notfully 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 their3 errorbars) are at (or below) the resolution limit of our data set. Thismeans frequency sets  6 and  7;  19 and  20; and  34 and  35 may not be fullyresolved.The SPBe stars provide an opportunity to determine the rotation fre-quency without referring to vsini (Cameron et al., 2008). It now seemspossible to determine the rotation rates of these stars by matching the ob-served frequency groups to models of rapidly rotating stars. This is only apossibility now because of the high quality of the MOST data. Cameronet al. (2008) provide a detailed modelling e ort for the SPBe stars alreadypublished by MOST team. However, even without presenting detailed mod-els here, we can say that HD 217543 rotates about twice as fast as HD127756 based solely on the spacings between the frequency groups observedin Figs. 2.2 and 2.6. Unfortunately, detailed g-mode asteroseismolgy is notpossible at this time because of the complexities involved in modelling theserapidly rotating stars and the long observation runs needed to resolve the572.3. Applications of CAPER to other MOST scienceFigure 2.5: MOST light curve of HD 217543. (Top panel) The full light curve for a total of26.1 days. The middle and the bottom panels show expanded light curves for the portionsA and B (respectively) indicated in the top panel. Solid lines indicate the  t of the 40 mostsigni cant frequencies from the frequency analysis of the full light curve (see Tab. A.2).582.3. Applications of CAPER to other MOST scienceFigure 2.6: Fourier amplitude spectrum of the light curve of HD 217543 and the identi edfrequency parameters from Table A.2. The panels and the meaning of the symbols aredescribed in Fig. 2.2.592.3. Applications of CAPER to other MOST scienceFigure 2.7: A comparison of bootstrap distributions for parameter sets ( 6, A6,  6) and ( 7,A7,  7) [see Tab. A.2] for 100,000 realizations of the HD 217543 light curve. Symbols arethe same as those in Fig. 2.4. These distributions are shown because the  tted frequenciesare the closest to each other. In this case, the long tails on the frequency distributionssuggest that these frequencies are not fully resolved.602.3. Applications of CAPER to other MOST sciencemany g-modes excited in these stars.61Chapter 3MOST & HR 1217The data analysis methods discussed in chapter 2 are applied to the HR1217 photometry collected by the MOST satellite. This chapter begins withan outline of the photometric reduction, followed by a discussion on thedetermination of the rotation period of HR 1217, and then  nishes with adetailed frequency analysis of the photometric time-series. The results fromthis chapter will be interpreted, for the  rst time, using a dense grid of stellarmodels outlined in the following chapters.3.1 The light curve reductionMOST observed HR 1217 near continuously as a Fabry target (x 2.1) fromNov. 5 to Dec. 4, 2004. Integration times for the target were 30 sec and morethan 72,000 data points were collected. The preliminary light curve reductionwas performed (by Dr. Jason Rowe) using methods described inx2.2. 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 therunning mean trend from data (e.g.,Rucinski et al. 2004). Residual trends inthe data were removed by subtracting a mean value, determined by splining0.005 day bins, while at the same time, making sure that no periodicitieswere introduced or subtracted near the periods of interest. The interesting623.1. The light curve reductionfrequency range is the range where oscillations were previously identi ed (x1.3) in HR 1217, and approximately covers 2500 to 2800  Hz. Extremeoutliers 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 describedbelow, and are presented in Fig. 3.1. There are two  0.2 day gaps in thedata caused by a particle hit on the CCD and a satellite software upgradeduring the observations. The resulting duty cycle of the data, based on thosegaps, is 95%. If the duty cycle is calculated as the total number of potentialdata points that could have been taken every 30 sec over 29 days, the resultwould be closer to 82%. Because the oscillation periods HR 1217 are about 6minutes, our time sampling and coverage warrant the use of the 95% e ectiveduty cycle.Along with stray-light, the long-term, rotational modulation | with aperiod near 12.5 days (x 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  nallight curve for HR 1217 plotted in Fig. 3.2. There are three steps in thelight curve reduction. The  rst step, shown in the top panel (labelled A),shows the light curve with only a  rst sigma clip performed to remove theoutliers. The light curve clearly shows repeating spikes in brightness thatare superimposed over the 12.5 day, near sinusoidal, stellar rotation trend inthe 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 thesubtraction of the mean trend at each orbital phase, results in a light curve633.1. The light curve reductionshown in the middle left panel (labelled B). At this step, the light curve onlyshows the rotational modulation in the HR 1217 data. There are a total ofthree mean-light maximums and two minimums covered over the 29 days ofobservations. The third, and  nal, 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 ofthe light variations that remain in the data; i.e., the light variations causedby the rapid,  6 min, oscillations (x 1.2.2, 1.3), are less than 1 mmag. The nal light curve is shown in the lower, left panel (labelled C), and illustrateswhat looks like a beat pattern in the data. During mean light minimum, therapid oscillations reach their maximum amplitude. This will be shown, anddiscussed, later in this chapter.We next check that only the stray-light and rotation trends are removedfrom the data. In the right panels of Fig. 3.1, the DFTs of the correspondinglight curves in the left panels are plotted. In order to show the reductionof the stray-light amplitude over a large range in frequency space, we havephased the DFT to the orbital frequency ( 14 cycles day 1) for each ofthese reduction steps and showed only a window in phase space that is  6cycles day 1 wide and centred on the Nth orbital harmonic. (The value of Nis indicated by the colour bar below the right panels in the  gure.) The fre-quency range where pulsations occur in HR 1217 is inverted in those panels.The stray-light components, and their associated aliases, are dramaticallyreduced in amplitude by the  nal reduction step. Because of the phase win-dow used in the right panels of Fig. 3.1, the frequency peak associated with643.1. The light curve reductionFigure 3.1: Photometric reduction steps using the MOST data of HR 1217. The leftpanels show the light curves of HR 1217 in three separate reductions phases described inx3.1. The  rst 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  nal,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 about12.5 days. The right panels show the phased DFTs for each of the reduction steps, foldedat the orbital frequency of the satellite ( 14 cycles day 1). Each panel shows a phasewindow  6 cycles day 1 wide centred on the Nth orbital harmonic. The Nth harmonicsare colour coded, with the colour legend below the lowest panel. The amplitude of thetop right panel is set to the amplitude of the peak in the DFT that is associated withthe rotation of HR 1217. The DFTs are inverted for frequencies in the range of 220 to250 cycles day 1, where the known oscillation frequencies of HR 1217 are observed. Notethe reduction of the orbital components and the daily aliases (vertical dotted lines) frompanels A to C.653.2. On the rotation period of HR 1217the rotation period of HR 1217 is not shown. (It is approximately 14 cyclesday 1 away from the  rst stray-light peak.) The amplitude of the  rst, upperright, 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 thanthe stray-light and pulsation amplitudes.The  nal, zoomed, version of the time-series is presented in Fig. 3.2. Thetop panel shows the light curve with an apparent beat pattern having a pe-riod of about 12.5 days (x 3.2) and the lower panels are zoomed portions(about 0.6 hr) of the data. Over-layed on those zoomed panels is the  ttedsinusoidal function (parameters are found in Tab. 3.1) obtained using theCAPER software (x 2.2). The errorbars on the photometric data are esti-mated from Poisson statistics and are about 0.1 mmag. The  t is discussedin more detail below.3.2 On the rotation period of HR 1217The oscillations observed in roAp stars are described well by the obliquepulsator model (x 1.2.2). As a result, the DFTs of most roAp star lightcurves show a number of periodicities that are spaced from each other bythe stars’ rotation frequencies. (This is illustrated later in this chapter, inFig. 3.8, for HR 1217.) The rotation period of HR 1217 has been measured bya number of authors. Kurtz & Marang (1987) determine a rotation periodof 12.4572  0.0003 days from photometric data collected from a varietyof sources. A slightly longer period of 12.4610 days (no uncertainty given)was determined by Mathys (1991) based on the variation of magnetic  eld663.2. On the rotation period of HR 1217Figure 3.2: The reduced HR 1217 light curve. The entire ( 29 day) time-series is shown inthe top panel after being reduced using procedures outlined inx3.1. The lower two panelsare expanded views of sections labelled A and B in the top panel, respectively, and eachcover approximately 0.6 hr of data. The nonlinear least squares  t using a multi-sinusoidfunction (with parameters listed in Tab. 3.1) is over-plotted on the data points for sectionsA and B. Those  t parameters are described in x 3.3. Errorbars represent Poisson errorsand are about 0.1 mmag. The duty cycle for the entire light curve is about 95%.673.2. On the rotation period of HR 1217strength over time. More recent measurements by Leone & Catanzaro (2004)and Ryabchikova et al. (2005) arrive at values of 12.4582 0.0006 days and12.45877 0.00016 days, respectively. Assuming that the magnetic  eld andphotometric variations should yield the same rotation period; i.e., there areno zero-point shifts that have to be taken into account, these rotation perioddeterminations seem to be statistically inconsistent. The MOST data on HR1217 covers about 29 days, corresponding to  2.3 rotation periods. Overthat 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 rotationperiods observed does not allow us to determine the rotation period of HR1217 to the level of precision quoted in the above works. As a comparisonbetween the completeness of the coverage of the MOST observations to thecoverage of the data used in the works cited above, we note that Ryabchikovaet al. (2005) { having the most precise period determination { arrive at arotation period using 39 data points collected over about 50 years of magnetic eld measurements.A single sinusoid is the simplest model that gives information about am-plitude, frequency, and phase. This model was used to estimate the rotationperiod of HR 1217 in the studies described above. A  t to a single sinusoidfunction using the MOST data, with the unbinned light curve ( 70,000 datapoints), and without the rotation trend removed (middle panel of Fig. 3.1),yields the following  t parameters: 1. Amplitude, Am = 1.1634 10 2 mag,2. Rotation frequency,  m = 0.0807 cycles day 1 (period = 12.3916 days),and 3. Phase,  m = 3.9481 rad. (The phase is referenced to the time of683.2. On the rotation period of HR 1217the  rst data point of the observations = HJD 2451545.00 + 1769.42 days.)These are the minimum  2 parameters calculated by CAPER (x 2.2). The t 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 ofthe star given that there are only a few cycles observed. However, with areduced  2  17:5 the quality of the single sinusoid  t is not satisfactory.The residuals of the  t have a standard deviation of 2.7 mmag compared tothe individual point uncertainties of about 1 mmag. The sharpness of theminimums is illustrated through the unprecedented time coverage of the HR1217 photometry. Introducing additional Fourier components to our  t willimprove the overall quality of the  tted parameters but our single frequency t is precise enough to determine which variations are associated with therotation of the star, given that only a few cycles are observed.The rotation period determined from our  t is shorter than the valuesmeasured by the sources listed above. We illustrate the di erence betweenthe MOST rotation period and previously determined rotation periods bylooking at the likelihood of the  ts. The top panel of Fig. 3.3 shows the HR1217 light curve with the single sinusoid  t plotted over the data (green line).(It should be noted again that the peaks at the light minimums are muchsharper than those predicted from a single sinusoidal model but this simplemodel gives us a rotation period that can be used to identify the e ects ofrotation in our oscillation spectrum.) By  xing the amplitude of our  t atAm = 1.1634 10 2 mag and then varying frequency and phase parameters,we can see how changing those quantities (frequency and phase) a ects thelikelihood of the model  t. For each (Am,  ,  ) parameter set we construct693.2. On the rotation period of HR 1217a  2 measure from 2(d;Am; ; ; ) =NXi=1(Am sin[2  ti + ] d(ti))22 2 (3.1)where each of the i = 1:::N photometric points, measured at time ti, arerepresented by d. A similar  2 measure can be constructed for the single,best- t, set of parameters (Am,  m,  m). Labelling this  2m, we construct alikelihood ratio asL=L0 = exp    2 (3.2)where   2 =  2   2m. The individual photometric errors ( in Eq. 3.1)are taken to be unity so we can explore the variation of parameters per uniterror. This is done for clarity of presentation. Individual point uncertainties(  1 mmag) can be used and result is a very sharply peaked likelihoodfunction near our best  tted parameters. Figure 3.3 (lower panel) shows theresults of this analysis. The MOST  t and the previously determine valuesare represented by white dots in the centre of the contour plot. They areconsistent with each other for this ideal case of unit residual error. The largerthe   2 value is, the further the  t solution is from the minimum  2m value.As discussed in x 2.2, the uncertainty in the rotation frequency can rangefrom about 0:25= T = 0:009 cycles day 1 to 1:5= T = 0:05 cycles day 1.Using our best  t value of  m = 0.0807 cycles day 1, and those estimates offrequency 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 precisionon the measured rotation period of HR 1217 (using the MOST data) thatthe previous studies achieve (four decimal places).703.2. On the rotation period of HR 1217Figure 3.3: Likelihood ratio for the rotation period of HR 1217. The top panel shows thetime-series with the  t of a single sinusoid over-plotted in green. The parameters of the tted sinusoid represent the minimum  2 solution using CAPER. The lower panel presentsthe contours for the likelihood ratio in frequency-phase ( - ) space. The likelihood ratiois de ned by Eqs. 3.1 and 3.2. The white dots near the centre of contour plot showthe separation between the MOST determination of the rotation frequency (0.0807 cyclesday 1) compared to the previous measurements discussed in this in this section ( 0.0803cycles day 1).713.2. On the rotation period of HR 1217The simplest explanation for the di erence in the measured precision fromall sources is that each data set spans a di erent length of time. For example,the magnetic data used by Ryabchikova et al. (2005) covers a period of about50 yrs while the MOST data covers about 30 days. But why is the  ttedrotation period so inconsistent given that we all  t a similar model to ourdata sets?Each group used observations that were spread (unevenly) over time.As a result, the sharp (non-sinusoidal) amplitude minimums seen in theMOST data (Fig. 3.3) were essentially smoothed out in their observations.To explore what a ect the time sampling has on the  tted rotation periodfrom the MOST data, we bin the data in di erent bin sizes (0 bins, 25 bins,and 100 bins), re t the parameters, and perform a bootstrap analysis using25,000 realizations. By binning we increase the time between individual ob-servations and essentially smooth the sharp minimums observed in the lightcurve. Figure 3.4 gives the results of this experiment. The lower panel ofFig. 3.4 shows the the rotation frequency for all of the binned data. Thepreviously determined values of the rotation frequency are over-plotted (witherrors smaller than the point sizes), as is the approximate resolution (x 2.2)of 0:25= T = 0.009 cycles day 1 (shown as the horizontal dashed line). Inthe  rst, 0 data bin case, the  t is determined to be 0.08073 0.00008 cyclesday 1. The 100 data bin case yields a rotation frequency of 0.081  0.002cycles 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 perioddetermination is with those that were previously determined. This experi-723.2. On the rotation period of HR 1217ment is not used to improve agreement between our observations and thosepreviously determined values. The precisions quoted in each study are sim-ply a function of the length of the observations. The inconsistencies in thecalculated values comes primarily from the di erences in the completenessof the time sampling between the various data sets. Binning the data givesa quick illustration of the sensitivity of rotation period determination to thesampling of the data. The high precision obtained for the  t using all of theMOST data re ects the fact that the  t parameters are insensitive to smallvariations in the data. This can be thought of as a standard error on a meandetermination | the more data you have, the smaller the error you measurefor that mean. However, in our case, a single sinusoid does not properlymodel the sharp minimums of the light curve and overstating the precisionof the measurement without acknowledging the pitfalls of the model couldbe misleading.733.2. On the rotation period of HR 1217Figure 3.4: Bootstrap distributions of the rotation frequency of HR 1217. The panelsshow (from top to bottom) the bootstrap (using 25,000 realizations) distributions for thephase  , amplitude A, and frequency  parameters of the single sinusoid  t (shown in thelegend of the top panel). The  lled, dashed distributions represent the light curve of HR1217 (Fig. 3.2) binned into 25 segments. The distributions with no  ll are for the lightcurve with 100 equal bins and the black  lled distributions is the  t using all of the data.Symbols in the lower panel are for previously determined rotation frequency values (seediscussion in x 3.2). Note that the frequency resolution estimated from the length of ourdata set (given by  0:25= T) is shown as the dashed horizontal line in the lower paneland is much larger than the errors obtained for any of the rotation frequencies presented.743.3. The frequency analysisWe will show in the next section that determining the rotation frequencyof HR 1217 to better than 10 3  Hz (9 10 5 cycles day 1) does not in- uence our asteroseismic analysis because our pulsation frequencies are notdetermined to better than that level of precision.3.3 The frequency analysisThe identi cation and re nement of the rapid variations in the HR 1217 datais done using the CAPER software that was described in chapter 2. Thelargest peaks in the DFT are sequentially identi ed, are used to determinean appropriate set of amplitude, frequency and phase parameters. A set ofsinusoids characterized by those parameters is then subtracted from the lightcurve. Parameters are re ned as they are identi ed by nonlinear least squares tting to the original time-series until the identi ed parameters have a signal-to-noise (S/N) of less than about 3.5. This corresponds to a detection limitof & 2.5 (Breger et al. 1993, Kuschnig et al. 1997).An analysis of the MOST photometry of HR 1217 yields a series of 29periodicities that are listed in Tab. 3.1 and are shown plotted over a DFT inFig 3.5. (Note that appendix B has all tables presented in this section withfrequencies listed in cycles day 1 rather than in  Hz.) Uncertainties for theidenti ed parameters are determined from 100,000 bootstrap samples (seex 2.2). Appendix C shows all of the bootstrap distributions for the  ttedparameters. In all cases the distributions are normal (Gaussian) in natureand all of the  tted amplitudes have a S/N > 3.5 within the determined un-certainties. The uncertainties we calculate for time-series parameters using753.3. The frequency analysisnonlinear least-squares methods depends on the noise of the data, correla-tions between parameters, and on the time sampling. For example, it isknown that  tted phase and frequency parameters are correlated, leading tounderestimated uncertainties in these parameters when calculated from a co-variance matrix (see, e.g., Montgomery & O’Donoghue 1999). Typically theuncertainties obtained by using only the diagonal elements of the covariancematrix are underestimated by a factor of 2 or more. The uncertainties de-termined for the parameters identi ed 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 Hz. The frequencies extracted from the MOST data have an average (1 )uncertainty of 0.032  Hz. We have checked that the uncertainties of the pa-rameters calculated using the MOST data are about the same as those fromthe parameters identi ed from the WET data if the covariance matrix is usedto calculate the uncertainties. Therefore, we are con dent that our bootstrapanalysis gives realistic error estimates that take into account correlations be-tween our frequency and phase parameters. The 1 frequency uncertaintiesare listed in all data tables along with the 3 values. As discussed in x2.2, the uncertainty (commonly calculated as the frequency resolution of thedata) on extracted periodicities from a time-series can range from 0.25/ Tto 1.5/ T, depending on the preference of the investigators. The bootstrapanalysis gives a measure of the precision of the  tted parameters and gives anindication of the frequency resolution of our data. To be conservative we willgenerally make reference to our 3 bootstrap uncertainties when discussingthe  tted frequency parameters found in this thesis.763.3. The frequency analysisThe MOST mission has uncovered the following new frequencies in thespectrum of HR 1217 (Tab. 3.1):  1 = 2603:832  Hz,  8 = 2684:832  Hz, 9 = 2685:659  Hz,  13 = 2718:984  Hz,  17 = 2723:469  Hz,  20 =2756:035  Hz, and  21 = 2756:925  Hz. The average di erence between thefrequencies observed by Kurtz et al. (2005) for HR 1217 and the MOST resultsis 0.08  Hz. The average 3 error on that di erence is  0:09  Hz. Thisindicates that all frequencies identi ed by both the ground-based and space-based campaigns are consistent with each other and shows that those periodscommon 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 Hz (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 identi ed in theMOST data. ( 22 through  27 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 byMOST and Kurtz et al. (2005) is not possible because the data sets weremeasured using di erent photometric  lters that average over di erent partsof the visible spectrum. However, the MOST data shows the frequency ( 2720  Hz) having the largest corresponding amplitude ( 760  mag) is stillthe 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 frequencyof 2687.60  0.06  Hz. Although amplitude di erences have been shownbetween the campaigns of Kurtz et al. (1989) and Kurtz et al. (2005), thenet power (sum of the square amplitudes) is conserved. The precision of773.3. The frequency analysisthe amplitude determinations from the WET data was 14  mag. The mostprecise amplitude determined from the MOST data is 6  mag; a value that isless than half of the quoted precision of the WET data, even without takinginto account the di erences in the precision calculations.783.3. The frequency analysisFigure 3.5: (Top Panel) The Fourier amplitude spectrum of the HR 1217 light curve. Filledcircles with 3 error bars are the  tted parameters (see Tab. 3.1). The inverted dash{dotline is the residual amplitude spectrum obtained after the  t was subtracted from thelight curve. (Lower Panel) The signal{to{noise (S/N) of the identi ed periodicities with3 uncertainties estimated from both the  tted amplitudes and the mean of the amplitudespectrum. The light grey line represents the window function of the data centred on thefrequency with the largest amplitude and scaled to the maximum S/N for clarity.793.3.ThefrequencyanalysisT able 3.1: F requ e n c y mo del parameters iden ti ed using the MOST data. Both 1 and 3  uncertain ties on the  tted parametersare estimated f rom 100,000 b o otstrap realizations. The phases are reference d to the time of the  rst observ ations of HR 1217(= HJD 2451545.00 + 1769.42 da ys). See app endix B for frequencies in units of c y c les da y  1 .#  [  Hz] A [  mag]  [rad]  S/N   3    A 3  A   3   1 2603.832 39 2.22 3.8  0.028 0.080 6 15 0.34 0.932 2618.600 50 3.29 4.2  0.024 0.066 6 16 0.27 0.723 2619.495 125 5.70 9.2  0.015 0.039 6 16 0.11 0.294 2620.440 68 1.68 6.3  0.028 0.074 6 16 0.28 0.745 2651.998 96 3.36 7.5  0.017 0.045 6 16 0.14 0.356 2652.924 365 5.56 24.2  0.004 0.011 6 16 0.04 0.107 2653.883 92 2. 07 7.8  0.012 0.033 6 16 0.10 0.278 2684.832 43 2. 02 3.7  0.054 0.145 6 16 0.38 1.109 2685.659 31 2. 83 3.1  0.091 0.241 9 26 0.55 2.1810 2686.560 123 1.26 8.9  0.011 0.029 7 17 0.10 0.2611 2687.420 467 4.46 29.7  0.004 0.009 6 16 0.03 0.0812 2688.364 139 1.13 10.9  0.009 0.023 6 16 0.07 0.1813 2718.984 51 4.50 5.8  0.051 0.137 6 16 0.25 0.6814 2719.957 319 6.04 23.3  0.006 0.014 6 16 0.05 0.13c ontinue d on next p age803.3.ThefrequencyanalysisT able 3.1: c ontinue d#  [  Hz] A [  mag]  [rad]  S/N   3    A 3  A   3   15 2720.914 764 1.83 41.6  0.002 0.005 6 16 0.02 0.0516 2721.822 177 4.59 12.4  0.011 0.027 6 16 0.10 0.2617 2723.469 41 2.08 3.1  0.034 0.097 6 16 0.39 1.1018 2754.470 110 1.81 8.5  0.016 0.043 6 16 0.17 0.4519 2755.393 118 3.45 8.9  0.015 0.039 6 16 0.17 0.4520 2756.035 67 2.75 6.2  0.038 0.107 6 16 0.36 1.0521 2756.925 39 5.14 3.1  0.045 0.159 6 17 0.53 1.6722 2787.799 24 4.72 2.7  0.127 0.317 6 16 0.90 2.5923 2788.872 37 0.24 3.6  0.034 0.096 6 16 0.29 0.8024 2789.673 33 5.56 3.3  0.047 0.157 7 16 0.38 1.1625 2790.406 37 2.85 4.4  0.073 0.196 7 18 0.45 1.2426 2791.384 82 5.00 6.8  0.033 0.085 7 17 0.29 0.7427 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 1.0628 2805.650 84 1.10 6.9  0.015 0.038 6 15 0.15 0.37c ontinue d on next p age813.3.ThefrequencyanalysisT able 3.1: c ontinue d#  [  Hz] A [  mag]  [rad]  S/N   3    A 3  A   3   29 2806.592 81 2.09 7.1  0.018 0.048 6 16 0.16 0.44823.3. The frequency analysisTable 3.2 lists the calculated di erences between the various frequenciesidenti ed in the MOST data. We adopt the symbol  y as a general spacingbetween two frequencies. There are 19 potential frequency spacings havinga value of  y  0.9  Hz. In the previous section (x 3.2) we identify therotation period of HR 1217 to be  12.5 days, corresponding to a rotationfrequency of 0.08 cycles day 1 or 0.93  Hz. The average of those 19 poten-tially rotationally split frequencies is 0.924  0.047  Hz, and is consistentwith the observed rotation rate of HR 1217. As stated at the end ofx3.2, therotationally split frequencies measured in the MOST data do not reach thesame level of precision (.10 3  Hz) as that on the rotation rate quoted byother authors (seex3.2). Using one author’s value for the rotation rate overanother’s does not change the frequencies determined from the MOST data.Frequency spacings  y1 through  y5,  y8 and  y26 of Tab. 3.2 show alternatingvalues of  33.5 and  34.5  Hz. These spacings were also identi ed inprevious studies of HR 1217 (e.g.,Kurtz et al. 2005) and are interpreted ashalf of the large separation,   . In chapter 1 (x 1.1.1) we state that thelarge spacing is related to the mean density of the star by Eq. 1.14. Becauseof this, Matthews et al. (1999) was able estimate the asteroseismic parallaxof HR 1217 and showed that it was consistent with the measured Hipparcosparallax (x1.3). These consistent measurements on the position of HR 1217in the HR diagram will be used to de ne the relevant model parameter spacein x 4.1.1. The alternating values of  33.5 and  34.5  Hz suggests thatthe observed modes are alternating between even and odd angular degree.Another set of recurring frequency spacings are 1.5 and 2.5  Hz ( y19through  y23). Main sequence models of A-type stars (e.g., x 4.2) have small833.3. The frequency analysisspacings (x 1.1.1) that are about 3  Hz and higher. This small spacing canpotentially constrain the age of the star because its value is most sensitiveto changes in the composition of the stellar core as the star evolves.Are these observed spacings actually rotational splittings? Consider theaverage rotation frequency derived from all of the (near rotationally split)frequency spacings in Tab. 3.2. Its value is 0.924 0.047 Hz and multiplyingit by 2 and 3 yields 1.848  0.094  Hz and 2.772  0.141  Hz, respectively.This suggests that  y19 and  y23 are potentially rotational components in theset of frequencies  8 through  12 (Tab. 3.1). However, the frequency spacingbetween  10 and  11 ( y15) is not consistent with a rotational spacing, evenusing 3 uncertainties. Likewise, the frequency spacing between  19 and  20( y29) is the most inconsistent with a rotational splitting in Tab. 3.2. Soif these are related to rotation there are other anomalies between adjacentfrequencies that need to be explained. That, coupled with observations ofKurtz et al. 2005, who also see  y23  2.5  Hz, leads us to believe that theseadditional spacings cannot be explained away as rotational components in theoscillation spectrum. Because of the low amplitudes associated with thosepotential small separations, including them in our nonlinear  tting routinedoes not in uence the other parameters in the  t. It should also be notedthat interpreting these spacings as small separations may not be appropriatein strongly magnetic stars (e.g., x 1.3 and 4.3).Kurtz et al. (2005) discovered a frequency spacing of  14  Hz that wasexplained by Cunha (2001) as a potentially magnetically perturbed mode.This spacing has also been identi ed in the MOST photometry along withanother possible magnetic spacing of 15.663  Hz ( y6) at the opposite end843.3. The frequency analysisof the spectrum. These modes spacings do not conform to the alternatingasymptotic pattern that is observed between the other periodicities in theMOST data and will be modelled in the following chapters of this thesis.853.3.ThefrequencyanalysisT able 3.2: The frequency separations  y of the frequencies iden ti ed in the MOST data. Columns with headers  i -  j use iand j to denote the frequency n um b er from T ab. 3.1. The a v erage rotation separation (spacings 9 : : : 18 & 27 : : : 35) is 0.924 0.047  Hz (3  = 0.130  Hz). See app end ix B for frequencies in units of cycles da y  1 .#  i -  j  y [  Hz]   y 3   y #  i -  j  y [  Hz]   y 3   y1 6 - 3 33.429 0.015 0.041 19 11 - 8 2.588 0.054 0.1452 11 - 6 34.495 0.006 0.014 20 11 - 9 1.761 0.091 0.2423 15 - 11 33.494 0.004 0.010 21 17 - 15 2.555 0.034 0.0974 19 - 15 34.479 0.015 0.040 22 21 - 19 1.532 0.047 0.1645 23 - 19 33.479 0.037 0.104 23 26 - 23 2.512 0.047 0.1286 3 - 1 15.663 0.032 0.089 24 28 - 26 14.266 0.036 0.0937 17 - 8 38.637 0.064 0.174 25 17 - 9 37.811 0.097 0.2608 21 -17 33.456 0.056 0.187 26 26 - 21 34.459 0.056 0.1819 3 - 2 0.895 0.029 0.077 27 16 - 15 0.908 0.011 0.02810 4 - 3 0.945 0.032 0.084 28 19 - 18 0.923 0.022 0.05811 6 - 5 0.926 0.018 0.046 29 20 - 19 0.642 0.041 0.11412 7 - 6 0.958 0.013 0.035 30 21 - 20 0.890 0.059 0.19213 9 - 8 0.827 0.106 0.281 31 23 - 22 1.072 0.131 0.332c ontinue d on next p age863.3.ThefrequencyanalysisT able 3.2: c ontinue d#  i -  j  y [  Hz]   y 3   y #  i -  j  y [  Hz]   y 3   y14 10 - 9 0.901 0.091 0.243 32 24 - 23 0.802 0.058 0.18415 11 - 10 0.860 0.012 0.030 33 26 - 25 0.978 0.080 0.21316 12 - 11 0.945 0.010 0.025 34 27 - 26 0.866 0.063 0.19717 14 - 13 0.973 0.051 0.138 35 29 - 28 0.942 0.023 0.06118 15 - 14 0.957 0.006 0.015873.3. The frequency analysisIn x 1.2.2 it was noted that the roAp stars show amplitude modulationthat 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 theline-of-sight of the observer. The amplitude modulation of the sinusoidalcomponents identi ed in the MOST data is shown in Fig. 3.6. The time-series is divided into 30 segments and the amplitudes and phases are re ttedat a  xed frequency for the modes identi ed in Tab. 3.1. Rotation modu-lation is clearly seen for the periodicities with the three largest amplitudesbut the other frequencies have corresponding amplitudes that are too smallto properly see the variation. In general, all modes follow the apparent\beat" period of  12.5 days in the data and show maximum amplitudes atminimum brightness (maximum magnitude) of the light curve. This is alsocon rmed by a time-frequency analysis presented in Fig. 3.7. In that  gure,the DFT is calculated for a 4 day segment of the data and then a movingaverage DFT is made by advancing the data segment by 0.1 days and thenrecalculating the DFT. There is clear substructure surrounding the highestamplitude frequency. Because the time coverage of the MOST data is near100%, we can see the amplitude modulations in the light curve of HR 1217in an unprecedented way.883.3. The frequency analysisFigure 3.6: Normalized pulsation frequency amplitudes as a function of time. The HR1217 time-series is broken into 30 segments and the amplitudes of the frequencies listed inthe  gure are then re tted for each data segment. Plotted are the  tted amplitudes per 3 error. The light curves of the HR 1217 data with and without the rotational modulationremoved are shown above the computed amplitudes using an arbitrary scaling. Note thatthe amplitude maxima correspond to the maximum magnitudes (minimum brightness) ofthe light curve with the rotation modulation and also to the maximum \beat" amplitudeof the light curve with the rotation modulation removed.893.3. The frequency analysisFigure 3.7: Time-frequency plot of the HR 1217 data. The top panel is the DFT of theHR 1217 data with the red line representing the median noise in the DFT over the datarange presented above. The lower plot is a moving box average DFT of the data. The boxwidth is 4 days and it is moved at 0.1 day intervals. The contours represent S/N of themoving box.903.3. The frequency analysisFinally, we discuss the resolution of the observed periodicities in theMOST data and the potential for the identi cation of other, closely spacedfrequencies. During the frequency analysis there were a number of frequen-cies identi ed that were spaced from their neighbour frequencies by aboutone half of the rotation frequency of HR 1217. To assess the resolution ofthese frequencies we perform a bootstrap analysis (using 100,000 realiza-tions) on the data but keep these frequencies as part of the  tted solution.In order to converge the  t with these additional frequencies we had to lowerthe convergence criteria on the  2  tting routine in CAPER by 2 orders ofmagnitude, allowing the  t to relax to its  nal solution using a lower tol-erance. The  t with all of the components is given in Tab. 3.3 (frequencieshaving numbers greater than 30 are the unresolved components). The resultsfor the top 5 most prominent (largest amplitude) frequency groups is shownin Fig. 3.8 with the  ts using both the resolved and unresolved frequenciesover-plotted on the DFT. The addition of the extra, unresolved frequenciescauses the  t to become unstable and the new frequencies clearly interactwith the previously identi ed, resolved, parameters of Tab. 3.1.The identi cations of the unresolved frequencies that are about half of therotation period of the star are highly suspect given the resolution of the dataset. While there may be independent frequencies still to be discovered in thedata, 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).913.3. The frequency analysisFigure 3.8: Comparison between resolved and unresolved frequencies. A bootstrap analysisusing 100,000 realizations is shown for the  ve largest amplitude components in the HR1217 data. Black points are for the  nal frequency solution (Tab. 3.1) and the red pointsare the unresolved frequency components found in the data (Tab. 3.3). Green dots in thetop panel show the bootstrap points. Note that the frequencies common to both Tabs. 3.1and 3.3 agree within the uncertainties while the additional frequencies tend to adverselya ect the  t, and in some cases overlap each other.923.3.ThefrequencyanalysisT able 3.3: Unresolv ed frequency mo del parameters iden ti ed using the MOST data. Both 1 and 3  uncertain ties on the  ttedparameters are estimated from 100,000 b o otstrap realizations. F requencies with n um b ers greater than 30 are the unresolv edcomp onen ts. The phases are referenced to the time of the  rst observ ations of HR 1217 (= HJD 2451545.00 + 1769.42 da ys).See app endix B for frequencies in units of cycles da y  1 .#  [  Hz] A [  mag]  [rad]    A   3   3  A 3   1 2603.831 38 2.22  0.027 5 0.26 0.077 16 0.112 2618.611 45 3.25  0.023 5 0.21 0.070 17 0.503 2619.440 106 5.96  0.025 12 0.20 0.055 15 0.754 2620.384 64 2.07  0.042 5 0.34 0.083 24 1.425 2652.012 95 3.20  0.014 5 0.16 0.049 17 0.366 2652.923 364 5.59  0.004 5 0.04 0.013 16 0.087 2653.863 81 2.13  0.020 9 0.14 0.060 17 0.318 2684.876 43 1.67  0.044 5 0.35 0.084 30 0.959 2685.682 40 3.02  0.049 10 0.44 0.153 16 0.1110 2686.536 111 1.37  0.020 10 0.10 0.042 36 0.1211 2687.416 475 4.50  0.004 9 0.05 0.009 16 0.2312 2688.340 131 1.30  0.018 8 0.17 0.036 16 0.8813 2718.943 54 5.18  0.042 8 0.52 0.115 16 0.50c ontinue d on next p age933.3.ThefrequencyanalysisT able 3.3: c ontinue d#  [  Hz] A [  mag]  [rad]    A   3   3  A 3   14 2719.956 320 6.11  0.004 5 0.06 0.012 28 0.5815 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 0.1216 2721.852 184 4.45  0.014 6 0.13 0.030 16 0.2317 2723.448 31 2.33  0.028 9 0.27 0.085 16 0.8818 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 0.4519 2755.361 112 4.02  0.028 6 0.22 0.066 28 0.5820 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 1.2921 2756.898 33 5.35  0.031 6 0.29 0.115 16 1.2422 2787.801 24 4.73  0.047 5 0.44 0.143 15 1.3723 2788.871 36 0.19  0.030 5 0.28 0.089 15 0.8424 2789.676 34 5.48  0.036 5 0.31 0.107 15 0.9525 2790.395 35 3.02  0.034 5 0.30 0.100 16 0.8726 2791.377 82 5.04  0.015 5 0.14 0.045 16 0.41c ontinue d on next p age943.3.ThefrequencyanalysisT able 3.3: c ontinue d#  [  Hz] A [  mag]  [rad]    A   3   3  A 3   27 2792.243 58 1.50  0.025 5 0.22 0.078 16 0.6928 2805.651 84 1.10  0.013 5 0.12 0.038 15 0.3629 2806.593 82 2.09  0.013 5 0.13 0.039 15 0.3730 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 4.3831 2620.823 31 5.19  0.099 6 0.54 0.190 17 1.5032 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 2.0833 2653.374 51 3.47  0.034 5 0.31 0.113 17 0.9934 2686.004 57 2.84  0.033 8 0.27 0.139 24 1.2035 2687.912 55 2.10  0.042 7 0.47 0.100 18 0.9236 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 2.7337 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 1.6838 2721.327 77 3.06  0.035 10 0.20 0.084 21 0.76c ontinue d on next p age953.3.ThefrequencyanalysisT able 3.3: c ontinue d#  [  Hz] A [  mag]  [rad]    A   3   3  A 3   39 2754.877 48 3.48  0.059 6 0.49 0.138 18 1.2840 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 2.5741 2792.612 28 0.19  0.049 5 0.46 0.148 15 1.39963.3. The frequency analysisThe frequencies identi ed 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, wewere able to identify a number of additional frequencies that need to beinterpreted in the context of models of magnetic stars. The following twochapters describe the modelling procedure used in this work and the e ortsput forth to theoretically match and interpret the MOST observations. Likethe MOST data, the parameter space appropriate to the oscillations of HR1217 is explored in an unequalled, and unique, way.97Chapter 4Modelling stellar oscillationsIn this chapter a grid of evolution and pulsation models of A-type stars iscalculated exploring a variety of parameters, constrained by the observedproperties of HR 1217. The e ect of a magnetic  eld on the resulting pul-sation frequencies is estimated, for the  rst time, using an extensive grid ofstellar models. The matching of models to the observed periodicities of HR1217 is described in chapter 5.4.1 Stellar evolution modelsStellar models are calculated using the Yale Rotating Evolution Code (YREC)in its non-rotating con guration. 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 andKippenhahn & Weigert 1994). These equations are standard and may foundin a number of texts on stellar structure (e.g, Kippenhahn & Weigert 1994).The nuclear cross sections, element di usion, the equation of state and opac-ities available to YREC were discussed recently by Demarque et al. (2008).The code is  exible enough to allow automatic computation of a number ofstellar evolutionary tracks with the convective treatment or atmospheric pa-rameters of one’s choice. For example, Fig. 1.5 shows a number of evolution984.1. Stellar evolution modelstracks on an HR diagram that were computed using YREC. Tracks of stellarmodels with parameters suited to those observed in HR 1217 are shown inFig. 4.1 and are discussed in x 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 structureequations 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 obtainthe 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 e ects that di erent EOS tables have on thecalculated solar pulsation frequencies. They  nd the e ect is less than about1  Hz. This type of detailed comparison between oscillation frequencies anddi erent input physics is most fruitful for helioseismic investigations becausethe global parameters of the Sun are well known. In asteroseismic inves-tigations we make the assumption that the physics included and tested inhelioseismic investigations are valid for other stars.The energy transport equations of stellar structure require informationabout the opacity of the stellar plasma. These are interpolated from twoseparate tables depending on the local temperature in the model. If thetemperature of a mass shell in a given model is greater than logT = 4.1, theOPAL opacities (Iglesias & Rogers, 1996) are used. The low-temperature(molecular) opacity tables of Alexander & Ferguson (1994) are used to obtainthe opacity for temperatures less than logT = 4.1. Linear interpolation isused to smoothly transition between the two tables.994.1. Stellar evolution modelsThe models calculated in this work are evolved from the zero age mainsequence (ZAMS) to a point near the end of the main sequence. Each modelgenerated has 3000 shells evenly distributed over the interior, envelope andthe 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- ) relation.YREC is used in its non-rotating con guration even though real stars dorotate. Because the roAp stars are generally moderate rotators, and becausethe observed pulsation modes are high-order, rotation has little a ect onthe frequencies. Shibahashi & Saio (1985) estimate the e ect of rotationon the calculated pulsation modes to be < 10 3 (frequency units) so theadditional complexity of adding rotation to the calculations would not yielduseful information.4.1.1 De ning the parameter spaceStellar models are calculated for a given mass, luminosity, e ective temper-ature and element composition (speci ed as a mass average). In most casesspectroscopic observations give information about the e ective temperatureand the heavy metal (elements with atomic numbers > 2) content of a star.The luminosity can be estimated for stars using parallax measurements andthe mass is only tightly constrained if the star belongs to a binary fromwhich an orbital solution can be derived. There is also the potential to useasteroseismic observations to further constrain the evolutionary status of astar through parameters like the large spacing   described in chapter 1 (see1004.1. Stellar evolution modelsMatthews et al. 1999). Even with information on all observable quantities,the standard treatment of convective energy transport (mixing length theoryof B ohm-Vitense 1958) adds a free parameter known as the mixing lengthparameter  . This parameter sets the number of pressure scale-heights aconvective element rises before releasing its heat to the surrounding plasma.Traditionally,  is calibrated by calculating a solar model, having the sameinput physics as the model grid that is being calculated, that reproduces thesolar radius and luminosity. For example, a value of  = 1:6 is appropriatefor a solar model without element di usion or overshoot (Guenther, 2002).Cunha et al. (2003) examined a set of stellar models for HR 1217 in hopesof determining its evolutionary status. Their results are complementary tothose of this thesis and there is an overlap in the choice of model parameterspresented below. The model physics described above was also chosen to besimilar to their choices for an easier comparison. This study explores themagnetic e ects on the pulsation frequencies for models of HR 1217 for the rst time and signi cantly extends the number of models and the breadth ofthe parameter space explored when compared to earlier asteroseismic studieson HR 1217.The mass of HR 1217 was estimated previously by a number of authorsusing a variety of assumptions. Shibahashi & Saio (1985) estimated a massof 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 HR1217 is a rigid rotator and the recent results of Kochukhov & Bagnulo (2006) nd 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 that1014.1. Stellar evolution modelscome close to the observed luminosity and e ective 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 e ective temperature of HR 1217 is estimated from two sources.Ryabchikova et al. (1997) use their spectral synthesis code to estimate ane ective temperature of Te = 7250 K (with no uncertainty provided). Theother estimate comes from the Str omgren photometry of Moon & Dworetsky(1985). Matthews et al. (1999) used those measurements to estimate ane ective temperature of 7400  100 K. We adopt a value of Te = 7400+100 200K to cover the span in temperature determined in those earlier works. Thisis identical to the e ective temperature estimate used by Cunha et al. (2003)and is consistent with a more recent estimate of Te = 7350 K (L uftingeret al., 2008) using the same methods as Ryabchikova et al. (1997).The luminosity of HR 1217 is determined from its Hipparcos (Perrymanet al., 1997) parallax. Matthews et al. (1999) derive a luminosity of 7.8 0.7L from a parallax of 20.41 0.84 mas (milli-arc seconds). An asteroseismicparallax can also be estimated from Eq. 1.15. Using the above estimateson Te and mass, and assuming a large spacing of  68  Hz we arrive ata luminosity of 8:2+1:6 1:5 L . If half the large spacing ( 34  Hz) is used,the asteroseismic luminosity is 20:7+4:0 3:8 L . The latter value is inconsistentwith the more accurate Hipparcos value. All physical parameters used inour model grid are also consistent with those determined by Kochukhov &Bagnulo (2006).A di culty arises when determining the metallicity for the Ap stars be-cause the metal content estimated from spectroscopic observations can dra-1024.1. Stellar evolution modelsmatically change over the magnetic (rotation) phase of the star. There is alsono reason to believe that the interior metal content of an Ap star is re ectedby the observed, patchy, surface values. The spectroscopy of Ryabchikovaet al. (1997) indicates [Fe/H]  0.32  16% (an average of Fe I and II linesat 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 itis (naively) assumed that the average of all of the metal abundances fromRyabchikova et al. (1997) for each of the magnetic maximum and minimumphases represent the interior heavy metal content of HR 1217, we estimatea value of Z  0:022. Thus, the inferred interior Z for HR 1217 can varyover a large range depending on how the surface metallicity is used. Forcompleteness we use Z = 0.008 to 0.022 in steps of 0.002 in our evolutionarymodels and adopt a hydrogen mass fraction of X = 0.700 to 0.740 in steps of0.020. The large extent of our parameter space in composition encompassesthe uncertainties from the observed [Fe/H] above.Finally, mixing length parameters of  = 1:4; 1:6; and 1.8 are used inthe model grid. If a star possesses a convective envelope,  may be used toset the adiabatic temperature gradient at the base of the convective zone.Luminosity is a function of temperature and radius so varying  will slightlychange the radius of the model because it changes the temperature at thebase of the convective zone (Guenther et al., 1992). Because the envelopesof A stars are essentially radiative, any structural changes from the di erentvalues of  are small when compared to the e ects introduced by varying theother model parameters.All models are plotted on the theoretical Hertzsprung-Russell (HR) di-1034.1. Stellar evolution modelsagram in Fig. 4.1. The Hipparcos luminosity and the e ective temperaturedetermination discussed above constrain the location of HR 1217 in the HRdiagram. The error box outlining that area in the HR diagram is shown as ablack box in Fig. 4.1. There were approximately 105 stellar models generatedin the model grid. Detailed models (containing information necessary for thepulsation analysis; see x 4.2) were output in age steps of 0.05 Gyr from theZAMS, decreasing the number of models by an order of magnitude. Of thosemodels, a total of 569 fall within the HR 1217 error box. Those 569 modelsare used in the pulsation and magnetic perturbation (x 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 moremodels at 1.65 M . Composition is also nearly uniform in its distribution.For comparison, in the Grevesse & Noels (1993) mixture (Z/X) = 0.0245and the metallicity is Z = 0.0175. The lower panel of Fig. 4.2 plots themean sound speed of the models, o set by the mixing length parameter  ,as a function of Z/X. All models are split evenly for each of the mixinglength values used in the calculations ( = 1.4, 1.6, and 1.8). The largespacing of the models (   (M/R3)1=2) ranges from  57{81  Hz. Somephysical characteristics of the models that are closest to the four corners ofthe 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 andwill be used in the following sections as examples illustrating the propertiesof the oscillation spectra and magnetic perturbations to those model spectra.1044.1. Stellar evolution modelsFigure 4.1: A theoretical HR diagram showing the extremes of the explored parameterspace. The black error box represents the Hipparcos luminosity and the e ective temper-ature appropriate for HR 1217 (see x 4.1.1 for details). Contours show lines of constant  and  ve evolutionary tracks are selected showing how position on the HR diagram isa ected by changing composition and mass. The metallicity has the greatest e ect on theevolutionary status of the stellar model. A track shifts toward the lower right of the HRdiagram (making the stars cooler and less luminous) by increasing Z.1054.1. Stellar evolution modelsFigure 4.2: Histograms of model properties. The three smaller panels at the top of the gure 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 plus the ratio of the average dynamical speed squared to that of the sun as a functionof composition. That quantity is plotted to illustrate that the models are nearly equallysplit in mixing length values ( = 1.4, 1.6 and 1.8).1064.1.StellarevolutionmodelsT able 4.1: Some prop erties of selected mo dels. Mo dels A and B ha v e luminosities and e ectiv e temp eratures that are theclosest to the upp er left and righ t corners (resp ectiv ely) of the error b o x sho wn in Fig. 4.1. Mo del C is the closes t to thecen tre of the error b o x while mo dels D and E are nearest to the lo w er left and righ t corners, resp ectiv ely .# M = M  X Z Age (Gyr)  M conv :cor e = M ?M conv :env := M ? log 10 ( T e ) log 10 ( L=L  ) log 10 ( R =R  )   (  Hz) Acoustic cut-o (  Hz)A 1.30 0.700 0.008 2.15 1.8 0.0000.0 3.8577 0.9284 0.2727 57.60 1393.52B 1.45 0.720 0.008 1.35 1.8 0.1180.0 3.8740 0.9289 0.2403 66.96 1703.88C 1.55 0.740 0.012 1.10 1.8 0.1420.0 3.8691 0.8912 0.2312 71.53 1908.77D 1.40 0.700 0.010 1.40 1.8 0.1190.0 3.8576 0.8518 0.2346 67.80 1790.08E 1.65 0.740 0.018 0.45 1.8 0.1480.0 3.8730 0.8523 0.2039 80.76 2280.421074.2. Pulsation modelsConvection can in uence both the evolutionary status and pulsation prop-erties of a star, and a near-surface convective zone has implications for thesurface chemical inhomogeneities of Ap stars. Models with higher Z/X tendto 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. Thisis also true for higher X, but the relation is not as strong. The mass of theconvective envelope is insigni cant in our models. Only a small fraction ofthe models exhibit convection over a large fraction of the stellar envelope( 1% of the model mass). These models should be more e cient at mixingaway 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    34  Hz are highermass models and have Z values greater than the solar value. The change inthe heavy metal content of the models produces the largest change on theposition of a model on the HR diagram. Many of the models in the errorboxhave 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 spacingare those with a mass & 1.5 M and a near solar composition of Z/X Pulsation modelsThe pulsation calculations are carried out using the nonadiabatic pulsationpackage of David Guenther (JIG, Guenther 1994). JIG solves the six lin-earized, nonadiabatic equations of nonradial stellar oscillations using the1084.2. Pulsation modelsHenyey relaxation method (Henyey et al., 1959). These six complex par-tial di erential equations describe the radial dependence of the vertical andhorizontal displacement vectors, the Lagrangian perturbations to the en-tropy and the radiative luminosity, as well as the Eulerian perturbations tothe gravitational potential and its radial derivative. The time dependenceof the perturbed quantities is periodic through the function exp(i t), wherethe  is the complex eigenfrequency. A complete introduction to both theadiabatic and nonadiabatic, nonradial oscillation equations can be found inUnno et al. (1989) or Christensen-Dalsgaard (2003). (See chapter 1 for ashort introduction to the subject of non-radial oscillations.)In this thesis we avoid the complications associated with modelling thestability of calculated eigenmodes. As such, we only make use of the adia-batic frequencies calculated by the JIG code. In the adiabatic approximationperturbations to the energy equation (luminosity) are ignored and the soundspeed is given byc2s = dpd =  1p (4.1) 1 = (@lnp=@ln )S is the  rst 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 x 1.1.1 for the pulsation equations.) Thepresence of an eigenvalue makes the system nonlinear and a normalizationof one of the eigenfunctions (radial displacement  ) at the surface is used toclose the system. We present an alternative method of calculating normalmodes of stellar models in x 4.2.1 so, for completeness, we list the adiabatic1094.2. Pulsation modelsequations as given in Unno et al. (1989). Namely,1r2ddr(r2 ) = g c2s   1 ‘(‘+ 1)c2sr2 2 p0 c2s +‘(‘+ 1)r2 2  0 (4.2)1 dp0dr =  (N2  2)  g c2sp0 d 0dr1r2ddr(r2d 0dr ) =‘(‘+ 1)r2  0 + 4 G  p0 c2s +N2 g In the above equations primed ( ) quantities are Eulerian perturbations (seex 1.1.1), N2 = g   11 dlnp=dr dln =dr is the Brunt-V ais al a (or bouncy)frequency, and  is the gravitational potential. Other symbols have theirusual meaning. JIG calculates eigenfunctions de ned by the variables  rstused by Dziembowski (1971)y1 =  r=r; y2 = 1gr (p0= +  0); y3 = 1gr 0; and y4 = 1gd 0dr (4.3)Using these de nitions we arrive at the  nal form of the adiabatic oscillationequations used in this thesis:rdy1dr = (Vg 3)y1 + ‘(‘+ 1)c1 2  Vg y2 +Vgy3 (4.4)rdy2dr =  c1 2 +A  y1 + (1 U A )y2 +A y3rdy3dr = (1 U)y3 +y4rdy4dr =  UA y1 +UVgy2 + (‘(‘+ 1) UVg)y3 Uy4In the above, the homology relations U = 4  r3=Mr and Vg V= 1 (withV = GMr =rp) are introduced. The measure of the local density to themean density of the star is given as U and V is the ratio of gravitationalenergy to the internal energy of the stellar material. A and c1 are de ned as1104.2. Pulsation modelsA  rA = rN2=g and c1 = (r=R)3=(Mr=M). The dimensionless frequency =  pR3=(GM) is used.The boundary conditions are described in Saio & Cox (1980) and in thebooks by Unno et al. (1989) and Cox (1980). The outer mechanical boundarycondition assumes a  nite pressure at the stellar surface. A potential condi-tion matches the perturbed gravitational potential and its derivative at thestellar surface. The two inner boundary conditions require regular solutionsof the pulsation variables as the stellar radius goes to 0. Written in terms ofthe Dziembowski variables, the outer boundary conditions are 1 4 +c1 2V y1 + ‘(‘+ 1)c1 2V  1 y2 + 1 ‘+ 1V y3 = 0 (4.5)andUy1 + (‘+ 1)y3 +y4 = 0 (4.6)The inner boundary conditions arec1 2‘ y1 y2 = 0 (4.7)and‘y3 y4 = 0 (4.8)We calculate frequencies for models of HR 1217 using angular degreesranging from ‘ = 0{4 in the frequency range 1900{3100  Hz. This encom-passes the observed oscillations of HR 1217 (see chapter 3). In all there areabout 50,000 individual modes (oscillation frequencies) calculated. Examplespectra, including magnetic e ects, will be discussed in x 4.3 and in chap-ter 5. (See also Fig. 4.4 for example spectra.) Because there are so manymodes calculated in a grid of models, we  rst explore (qualitatively) matches1114.2. Pulsation modelsbetween 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 HR1217 ( 68  Hz). The grid, plotted in Fig. 4.1, shows that increasing Z shiftsthe models to the lower right of the HR diagram and decreasing Z shifts themodels to the upper left. The lowest mass models in our grid that enter ourde ned 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 largestmass in our grid that fall in the same error box have the highest Z values aswell. Generally, the more luminous the model is, the lower its large spacingis because it tends to be less dense (because of its larger radius).The models with a small spacing of  2:5  Hz may be important forthe interpretation of the frequency analysis of x 3.3. Figure 4.3 shows thesmall spacing for the models outlined in Tab. 4.1 with the angular degreedi erences of ‘ = 2 - ‘ = 0, ‘ = 3 - ‘ = 1, and ‘ = 4 - ‘ = 2. The modelswith the second order spacing of . 3  Hz are either older (compared toother models listed in Tab. 4.1), or have a mass less than  1.5 M . Themetallicity 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 qualitativemodel match has a Z value that is less than solar. It is also interesting tonote that the second order separations between ‘ = 4 - ‘ = 2 are muchlarger (&4 ) than the observed value of about 2.5  Hz. This suggests thatif the observed di erences are indeed small spacings, they are separationsde ned by di erences between the lowest degree modes. We will show inx 4.3 that including the magnetic  eld can dramatically alter the spacingsof the models, casting doubt on whether or not we really observe a small1124.2. Pulsation modelsspacing as de ned by asymptotic theory.4.2.1 A new search method for eigenmodesThere are a number of numerical methods that have been used to solve the setof Eqs. 4.4. A general discussion of these methods may be found in the lecturenotes of Christensen-Dalsgaard (2003). Recently, a number of researchershave compared their software and outline the reduction procedures used tocalculate the normal modes of a stellar model (see Moya et al. 2008 andreferences therein). The most popular methods for calculating oscillationspectra 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 inwardand outward from the surface and central boundary conditions and thenmatched at a suitable  tting point for a given eigenfrequency  2. If thedi erence between the inward and outward solutions at the  tting point iszero (to some numerical accuracy) then  2 is an eigenvalue of the problem.Thus, a number of  2 can be scanned to  nd the eigenvalue to a prescribednumerical accuracy. The drawback of this method is in the choice of  ttingpoint. This is highly dependent on the mode being investigated and may beplaced either in the upper atmosphere of the model or in the deep interior.Experience is required to  nd a  tting point that yields an accurate andstable numerical solution.Relaxation methods also exist in which all of the eigenfunctions and theeigenvalue are iteratively re ned at each mesh point in the model. Examples1134.2. Pulsation modelsFigure 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 di erences of modeswith degrees between ‘ = 4 and 2, the middle panel for degrees between ‘ = 3 and 1, andthe lower panel for degrees between ‘ = 2 and 0. The potential small spacing (x1.1.1, andEq. 1.16) observed in the MOST data is . 3  Hz (see Tab. 3.2) and is best reproducedby models with lower Z and for small separation between the lowest angular degrees (‘ =0 and 2).1144.2. Pulsation modelsof this method are found, for example, in Unno et al. (1989) and Henyeyet al. (1964). The JIG code described above uses a relaxation method to nd the eigenvalues of a model. This method requires an initial guess that isclose to the eigenvalue in order have fast convergence. In addition, the initialsetup of this method tends to be complicated.A direct solution to the eigenvalue problem using matrix methods alsoexists but they tend to be the least e cient (in time and memory) of all themethods (see Christensen-Dalsgaard 2003).A new method of calculating and scanning for eigenmodes was developedby Kobayashi (2007). This method e ciently integrates the set of adiabaticequations using a matrix method and was developed to study normal modesin the Earth’s atmosphere. This method is, in some sense, a cross betweena relaxation method and a direct integration. We show the results for twoselected models as a consistency check with the modes calculated using JIG.Equations 4.4 are written in matrix form asdyijdrj = Bjyij; i = 1;:::;4 and j = 1;:::;@ (4.9)where B is a 4x4 matrix, de ned at the jth shell, with entries  lled as theterms on the right hand side of Eqs. 4.4. The model is comprised of@shellswith the  rst shell (j = 1) located at the centre of the model. Equation 4.9may be rewritten in a  nite di erence form, relating variables between shellsin the model asCnyn +Dn+1yn+1 = 0 (4.10)with Cn = I +  rBn=2, Dn+1 = I +  rBn+1=2, and  r = rn+1 rn.1154.2. Pulsation modelsA Henyey-type banded matrix is formed from the system of Eqs. 4.10with the following de nitions:Pn [CLn 0]T n = 2;:::;@Sn [DLn CUn ]T n = 2;:::;@ 1andQn [0 DUn+1]T n = 1;:::;@ 1The upper indices L and U refer to (respectively) the lower and upper halvesof matrices C or D. S1 is a special case which has its upper half as theinner boundary conditions and its lower entries as the upper entries of C1.Likewise, S@ is composed of the lower half of D@ and the outer boundaryconditions.In this banded form, and for a given  2, the system of equations may beintegrated recursively from the centre of the model to its surface usingyn = Rnyn+1whereRn = X 1n Qn;Xn = PnRn 1 +Snand starting from R1 = S 11 Q1.At the surface of the model X@y@ = 0 and  2 is an eigenvalue only ifdetX = 0. Integrating the equations in this manner is accomplished in order1164.2. Pulsation models@time and with order@memory because storage is only needed for the@Rvalues (required to calculate the eigenfunctions).Along with the speed of this integration method there is also an advantagein the form of the eigenvalue search. Kobayashi (2007) showed that if weexpand X@y@ = 0 about a perturbation in the eigenvalue   2, we arrive at asmall (4 x 4 matrix in the adiabatic case) generalized eigenvalue problem atthe surface of the model X@ +  2@X@@ 2 y@ = 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  2 the solutionrelaxes to the correct eigenvalue by replacing  2 with  2 +   2 until someprede ned stopping criteria is satis ed.We calculate the eigenfrequencies for models A and E of Tab. 4.1 usingthe method outlined above. Following Roxburgh (2008), we  rst de ne thefunctions = d 0dr + 4 G  (4.12)andr 2 =  p= + g +  0 (4.13)where  is identi ed as the horizontal component of the displacement vector.We normalize the stellar structure variables usingr = Rx;  = M4 R3   ; g = GMR2  g; (4.14)c2 = GMR  c2; p = GM24 R4  p;  2 = GMR3  21174.2. Pulsation modelsand recast Eqs. 4.2 in the following form:dy1dx =‘(‘+ 1)x y0 2xy1 1xy2 (4.15)dy2dx =  ‘(‘+ 1)  gx p y0 +  2   p +4 g x p y1 +  g  p y2   py4dy3dx =    y1 +y4dy4dx =‘(‘+ 1) x y0 +‘(‘+ 1)x2 y3 2xy4The eigenfunctions of the system are de ned byy0 =  =R; y1 =  =R; y2 =  pp ; (4.16)y3 = RGM 0; and y4 = R2GM Note that in Eqs. 4.15 we use the Lagrangian variation of the pressure  p.The result is that the Brunt-V ais al a (N2) frequency disappears from thelinearized pulsation equations.Figure 4.4 shows the comparison between the frequencies calculated usingthis method and those calculated using JIG for models A and E of Tab. 4.1.The plot shows the (non-dimensional) eigenvalue correction   2 as a functionof frequency  (the guess for the eigenvalue of the system) for ‘ values from0 to 4. If the correction is positive, our guess eigenvalue is too low and thecorrection leads us forward (toward the true eigenvalue) by increasing ourguess by the increment   2. When the correction goes to zero for a givenguess frequency, we have found an eigenvalue. Note that not all of the zeroesof the function   2( ) shown in Fig. 4.4 correspond to pulsation eigenvaluesfor the models A and E. Consider the behaviour of the correction near a zerowhere the function   2( ) has a positive slope. A positive   2 near that zero1184.3. The magnetic e ectswill move our guess further away from the zero. A negative   2 will also moveus away from the x-intercept if the local slope is positive. If the local slope of  2( ) is negative,   2 can be either positive or negative, and the correctionwill guide our guess towards the zero (or eigenvalue). Therefore, values ofthe   2( ) function that approach zero with a negative slope (suggestinga decreasing correction) are eigenvalues of the system. The normal modescalculated from JIG are shown as red points distributed along the line   2 =0. The agreement is exact to the precision output by JIG (5 decimal places)and no eigenmodes were missed during the calculation of frequencies usedfor this thesis.Although only tested in the adiabatic case, this method was originallydesigned to work for a complex set of non-adiabatic equations used in ageophysical context (Kobayashi, 2007). Traditionally an adiabatic solutionis required as an initial guess for both the eigenvalues and the eigenfunctionsof the non-adiabatic solution to the stellar pulsation equations. This methodo ers an e cient alternative to that approach.4.3 The magnetic e ectsThe e ects of the magnetic  eld on both the oscillation frequencies and theeigenfunctions have been described by a number of authors (see recent re-views by Cunha 2007, Saio 2008 and Shibahashi 2008). The kG strengthmagnetic  elds in roAp stars complicate the interpretation of the observedeigenmodes by introducing deviations from the expected (near asymptot-ically spaced) frequency spectrum. In the outer layers of the roAp stars1194.3. The magnetic e ectsFigure 4.4: A comparison of eigenfrequencies determined by two di erent methods.The left panels show the non-dimensional eigenfrequency correction   2 (normalized byGM=R3) as a function of frequency  , 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 describedinx4.2.1 and the red circles are the eigenfrequencies calculated for this thesis with JIG (x4.2, and Eqs. 4.4). The right panels are the same as those on the left but use model E ofTab. 4.1. A value of   2 = 0 is an eigenvalue of the system of Eqs. 4.15. Modes calculatedwith JIG agree with those determined using the new method developed by Kobayashi(2007).1204.3. The magnetic e ects(.3% of the total stellar mass) the Alfv en speed (/pB2= ) is comparableto, or greater than, the local sound speed (/pp= ). The eigenmodes are nolonger purely acoustic and are better described as magnetoacoustic waves.As the stellar density increases inward and the sound speed becomes largerthan the Alfv en speed, these magnetoacoustic waves decouple into acousticand magnetic components. The resultant magnetic slow waves act as anenergy sink as they dissipate in the dense stellar interior. It is this loss ofenergy that results in the shifting of some frequencies and the deviationsfrom the regular asymptotic spacing. There has been recent success in theinterpretation of roAp oscillations (e.g., Cunha 2001).Cunha & Gough (2000) (see also Cunha 2006) estimate the magneticperturbation to the eigenmodes of nonmagnetic stellar models using a vari-ational principle. The method matches a numerical solution of the idealmagneto-hydrodynamic equations in a boundary layer located in the upperenvelope and atmosphere of the model to an asymptotic expression for theeigenfunctions of high-overtone, acoustic oscillations in the stellar interior.The phase di erence between the numerical and asymptotic expressions forthe eigenfunctions is used to estimate the corresponding frequency shifts asa result of magneto-acoustic interactions. A determination of the perturbedeigenmodes using a variational principle, calculated from an estimate of theperturbed eigenfunctions (obtained using a background stellar model thatdoes not include magnetic  elds), 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 workof Campbell & Papaloizou (1986).1214.3. The magnetic e ectsTo start the modelling procedure, Cunha & Gough (2000) divide the starinto a thin outer boundary layer and the interior. In the boundary layer, theLorentz forces are comparable to, or larger than, the gas pressure, while inthe interior the  eld is essentially force free. Because the magnetic boundarylayer is thin, a plane-parallel approximation may be used. It is also assumedthat the e ects of the magnetic  eld can be calculated locally at each latitude.For this geometry the  eld only varies in the vertical direction and has com-ponents B = (Bx;0;Bz). The magneto-acoustic waves are described usinga horizontal wavenumber k = (kx;ky;0). Using this information, the adia-batic, magnetically non-di usive, pulsation equations may then be writtenas (Campbell & Papaloizou 1986, Cunha & Gough 2000 and Cunha 2006)  2 u = ijkjW + (B r)2 u 0 kxBx 0jkj(B r) (r  ) (4.17)  2 v = (B r)2 v 0+ kyBx 0jkj(B r) (r  ) (4.18)  2  z = @W@z  gr (  ) Bz 0[(B r) (r  )] + (B r)2  z 0(4.19)whereW =   rp+  p+ B2 0 (r  ) (B r) (B  ) 0(4.20)andg = 1 dpdz (4.21)In these equations the displacement vector  is decomposed into a verticalcomponent  z =   ez, a component that is perpendicular to the wavenumberv =   (ez k)=jkj, and a component that is parallel to the wavenumberu = (  k)=jkj. The local gravitational acceleration is denoted by g and the rst adiabatic exponent is  . All other symbols have their usual meanings.1224.3. The magnetic e ectsIn the deep interior a magneto-acoustic mode completely decouples into apure Alfv enic mode and a pure acoustic mode. This was veri ed analyticallyby Roberts & Soward (1983) and numerically by Campbell & Papaloizou(1986). The JWKB approximation is used to describe the magnetic modesin the interior. This approximation to the eigenfunctions takes the functionalform (Cunha & Gough 2000; see also Gough 2007)(vmz;umz)v  1=4(C;D) exp"iZ z0  0  2B2z 2dz ikxBxzBz#+  1=4(C+;D+) exp" iZ z0  0  2B2z 2dz ikxBxzBz#(4.22)where C, D, C+, and D+ are complex constants. The inward propagatingAlfv en waves are expected to dissipate before they are re ected back towardthe surface of the star (Roberts & Soward, 1983). With this in mind, theconstants C+ and D+ in Eq. 4.22 may then be set to zero to guarantee thatno 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 representedasymptotically by (Cunha & Gough 2000) zvA{1=2 1=2 cos Z z z{dz + p (4.23)were  p is a phase and { is the vertical acoustic wavenumber. The coordinatesz and z represent the depth in the boundary layer and the position of thebase of the boundary layer, respectively.To integrate Eqs. 4.17{4.19 there need to be  ve boundary conditions(Cunha & Gough, 2000) to match the components of the displacement and1234.3. The magnetic e ectsthe magnetic  eld variations. We use a fully re ective mechanical outerboundary condition of the formpr  = 0 (4.24)The other outer boundary condition requires that the magnetic  eld bematched continuously onto a vacuum  eld. This may be written as(B r)  Br  =r m (4.25)where  m is an exponentially decreasing plane wave solution of the Laplaceequation. The boundary condition 4.25 can be separated into two conditions:one condition relating the vertical displacement and the variable u (de nedabove), and the other involving the variable v (de ned above; also see Cunha& Gough 2000). The numerical solutions of the system of Eqs. 4.17 to 4.19are matched to the asymptotic relations 4.22. This provides the two interiorboundary conditions.The vertical component of the solution is then matched to Eq. 4.23 ateach latitude to obtain values for the magnetic phases  p. The purely acous-tic (numerical) solution; i.e, B = 0 in Eqs. 4.17 to 4.19, is also matched toEq. 4.23 to obtain the unperturbed phases  p0. Phase shifts   p are cal-culated from the di erence between these two phases at each latitude. Thematching procedure is outlined in Cunha & Gough (2000) and Cunha (2006).The variational method of Cunha & Gough (2000) is used to estimatethe  rst-order frequency shifts of the eigenmodes caused by a magnetic  eld.The frequency shifts are calculated from  M 0 w   p 20RR r1 c 2s { 10 dr (4.26)1244.3. The magnetic e ectswhere the average of the phase shifts   p over a sphere is  p =R 0   p(Ym‘ )2 sin d R 0 (Ym‘ )2 sin d (4.27)The quantities with a subscript of zero are non-perturbed values. A completedescription of the numerical procedure used to calculate the frequency shiftsis outlined in the appendices of Cunha & Gough (2000) and Cunha (2006).Examples of the magnetic shifts are presented below. There are suddendrops in the frequency corresponding to a maximum of energy loss of thatparticular mode. These sharp jumps are cyclic and depend on the magnetic eld strength, the frequency range explored, and on the mode/magnetic  eldgeometry. It is important to fully explore all relevant frequency ranges, modegeometries and magnetic  eld strengths if we hope to successfully match acalculated frequency spectrum to observations of roAp stars. For each of themodes calculated from the models presented inx4.2, a magnetic perturbationis estimated for a dipolar magnetic  eld with polar strengths running from1 to 10 kG in steps of 0.1 kG. In total there are 51,795 magnetic modelscalculated using physical parameters appropriate to the properties of HR1217. This is the largest grid of magnetic models calculated for the seismicstudy 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 apower-law function (  Bn). The index n in the power law is dependanton the structure of the outer layers of the background model. The modelsin Tab. 4.1 are used in this calculation. A value of n = 0.75 does a good jobof phasing together the frequency shifts. This behaviour is discussed in Saio2008 and references therein. It is important to notice the frequency shifts1254.3. The magnetic e ectsbetween modes of degrees that di er by two are at least of the same order asthe second-order, small spacing (Eq. 1.16). This may hamper e orts to usean 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. Thereal part of the shifts calculated for models in Tab. 4.1 are presented in the left panels andthe imaginary parts are given in the right panels. From top to bottom are the degrees ofthe modes (from ‘ = 1 down to ‘ = 4) and all magnetic  elds (B = 1{10 kG) are shown.The position of the shifts is cyclic and can approximately be phased as   B0:75. Theexponent on B is sensitive to the outer layers of the model. Note that a maximum in theimaginary part corresponds to a sudden jump in the real frequency shift.1264.3. The magnetic e ectsFigure 4.6: The magnetic perturbations for a 5 kG magnetic  eld using models fromTab. 4.1. The left plot set (consisting of four plots) are for degrees of ‘ = 1 and 2 and theright plot set is for degrees of 3 and 4. The upper panels show the real frequency shiftsas a function of frequency while the lower portions of the  gure give the imaginary shifts.Models with frequency shifts of  15  Hz around 2800  Hz could explain the anomalousfrequency (or \missing mode", see x 1.3) identi ed by Kurtz et al. (2002). Note that thedi erence between the magnetic shifts of degrees that di er by 2 can be as large as, orgreater than, the small spacing for a main sequence, A-type star ( 10  Hz; see Fig. 4.3).The magnetic perturbations to frequencies from 1500 to 3500  Hz areshown in Fig. 4.6 assuming a dipolar magnetic  eld with a polar strength of5.0 kG and using the models of Tab. 4.1. This frequency range is consistentwith measurements of HR 1217 (x3.3) and shows the real and the imaginary1274.3. The magnetic e ectsfrequency shifts for modes having degrees of ‘ = 1 through 4. Note that thedi erence between real frequency shifts for modes of di erent degree amountsto a few  Hz. The imaginary part of the even ‘ = 2 and 4 modes are lowerthan the odd ‘ counterparts. This was also shown for the case of a polytropestellar 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 forthe ‘ = 2 modes, for example, suggest that they are less susceptible to damp-ing from the magnetic  eld. This e ect depends on the geometry of the modeand the magnetic  eld. Cunha & Gough (2000) and Cunha (2006) show thatthe maximum energy loss of a magneto-acoustic mode occurs approximatelyat the latitude where BzBx is a maximum. This occurs at 45 for a dipo-lar magnetic  eld. Modes of higher degree ‘ decrease in amplitude as theyapproach the equator more quickly than modes of lower degree. When thephase shifts are averaged over a sphere, the net e ect is a smaller imaginaryfrequency shift for modes of larger ‘. These modes have smaller amplitudesnear the latitude where the maximum of energy loss occurs.A typical  rst step for pulsation modelling of a roAp star is to look ata 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  Hz). We extendtheir analysis by looking at the spacings of models A{E in Tab. 4.1. Thelarge spacings listed in Tab. 4.1 show that models B (  = 66:96  Hz), C1284.3. The magnetic e ects(  = 71:53  Hz), and D (  = 67:80  Hz) come close to the large spacingof 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, onlyone model (E) in Tab. 4.1 has a near solar metallicity of Z = 0.018. We cannot say that only models with a smaller than solar Z value best match theproperties of HR 1217. We can only say that the (small number of) models inTab. 4.1 are in agreement with Cunha et al. (2003). Models with a Z.0:018can reproduce the large spacing of HR 1217; but, models with a larger Z canalso show such a match (see chapter 5).Perhaps the more important question is: How does the introduction ofa magnetic  eld a ect the large spacing of the models? Figure 4.7 showsthe large spacing calculated from the model pulsation frequencies (  = n+1;‘  n;‘), as a function of frequency, for the models in Tab. 4.1. Theleft panel of Fig. 4.7 shows models with B = 0 kG and the right panelshows pulsation models having B = 5 kG. Comparing the models betweenthe two sets of panels shows the large spacing (on average) is not a ectedby the introduction of a magnetic  eld. A major exception to this ruleoccurs 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 spacingshows a sharp decrease in value. This decrease can be a large fraction ofthe value of the large spacing itself. While   varies slowly as a functionof frequency even in the non-magnetic case, the magnetic  eld perturbationcauses an abrupt decreases in   over one spacing between modes. Theinclusion of a magnetic  eld in the pulsation calculations seems to be themost reasonable way of producing such a sharp, and localized, feature in the1294.3. The magnetic e ectsstellar oscillation spectrum. For HR 1217, the frequencies alternate between 33.5 and 34.5  Hz over a frequencies ranging from about 2600 to 2800  Hz.That alternating pattern is suddenly cut by about half near the beginningand the end of the observed frequency range (see x 3.3). The introductionof a 5 kG magnetic  eld to the models in Tab. 4.1 does not show multiplejumps over the range of observed frequencies in the HR 1217 spectrum. Thesize of the decrease of   shown in Fig. 4.7 is also too large by a factor ofmore than two compared to what is observed. In Fig. 4.8 we show the largespacing as a function of magnetic  eld for the same background models. Eachof these background models exhibits a sudden jump in   for some values ofmagnetic  eld. Interestingly, jumps in   of less than about 20  Hz occur(for some models) at magnetic  elds . 5 kG. This is similar behaviour towhat is observed in HR 1217. Models with larger magnetic  elds tend toshow larger jumps for all ‘ values.1304.3. The magnetic e ectsFigure 4.7: The magnetic perturbations to   for a 0 and 5 kG magnetic  eld usingmodels from Tab. 4.1. The left set of four plots show the large spacing   =  n+1;‘  n;‘as a function of frequency  n;‘ for angular degrees ‘ = 0 through 4. The right set of plotsshow that perturbation to the large spacing caused by a 5 kG magnetic  eld. The inclusionof a magnetic  eld causes a sharp decrease in   when there is a jump in frequency (seeFig. 4.6).1314.3. The magnetic e ectsFigure 4.8: The magnetic perturbations to   for all calculated magnetic  eld strengthsusing models from Tab. 4.1. The plots show the large spacing   =  n+1;‘  n;‘ as afunction of magnetic  eld strength for ‘ values identi ed at the top of each panel. Theinclusion of a magnetic  eld causes a sharp decrease in   when there is a jump infrequency (see Fig. 4.6). This jump occurs cyclically for varying magnetic  eld strengths.1324.3. The magnetic e ectsIn x 3.3 we observed for the  rst time a number periodicities with spac-ings that were between  1 and 3  Hz (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-magneticpulsation models given in Tab. 4.1 are plotted in Fig. 4.3. The propertiesof the non-magnetic models were previously discussed in x 4.2. If we ob-served small spacings in the MOST data then, based on models A throughE, 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 presentedin Tab. 4.1.) Figure 4.9 gives the small spacing for the same models as inFig. 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 happensto individual frequencies than what happens with the magnetic perturbationto the large spacing. The spacings show a slow increase in value before asudden decrease ( 5{10  Hz). This introduces signi cant di culties in dis-tinguishing the magnetic e ects from the asymptotic spacings (Dziembowski& Goode, 1996). The small spacings calculated as  n;‘  n 1;‘+2 can even benegative. The mode having order n has a greater frequency value than theadjacent mode with order n - 1 in the non-magnetic case. The introductionof a magnetic  eld can cause this pattern to be reversed so that a frequencylabelled n - 1 in the non-magnetic model now has a larger frequency thanthe mode having order n in the non-magnetic model. In general n is notan observable, but the frequency shifts caused by a magnetic  eld introducea large enough perturbation to signi cantly complicate the analysis of ourtheoretical spectra. The small spacing perturbations (with an o set added1334.3. The magnetic e ectsfor clarity) are given in Fig. 4.10 as a function of magnetic  eld. The pertur-bation is cyclic and shows the large variation in the magnitude of the secondorder separations caused by the addition of a magnetic  eld to the pulsationcalculations.1344.3. The magnetic e ectsFigure 4.9: The magnetic perturbations to   for a 5 kG magnetic  eld using modelsfrom Tab. 4.1. The plots show the small spacing   =  n;‘  n 1;‘+2 as a function offrequency  n;‘. The  gure labels are the same as the labels in Fig. 4.3 { the small spacingsneglecting a magnetic  eld. Compared to the nonmagnetic case the small spacing canincrease or decrease depending on the angular separations. The magnetic perturbationcan even cause a frequency (labelled n) with angular degree ‘ + 2 to have a larger valuethan 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.1354.3. The magnetic e ectsFigure 4.10: The magnetic perturbations to   for all calculated magnetic  eld strengthsusing 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 o set by a constant value shown inthe legend found in the top panel. The small spacing is highly sensitive to magnetic  eldstrength and the angular degrees of the modes.1364.3. The magnetic e ectsA close inspection of Figs. 4.7 and 4.9 shows a rapid variation in themagnitude of the magnetically perturbed spacings. This rapid variation hasan amplitude of a few ( 5)  Hz over a baseline of about 100  Hz. Thesmall spacings ( 3  Hz) observed by MOST seem to vary by about 1  Hzover a similar baseline (see Tab. 3.2). This similarity between the theory andobservations is noted with caution. We are unsure if this rapid variation inthe 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 realityof this variation.The models for HR 1217 presented in this chapter represents the  rsttime a large grid of realistic stellar models has been calculated for this starthat includes the magnetic perturbations to the eigenfrequencies. The gen-eral characteristics of the models was discussed. The following chapter willexpand upon this analysis by attempting to  nd the closest matched modelfrequencies to the observed MOST periodicities (chapter 3).137Chapter 5Best matched modelsIn the previous chapter we discussed the calculation of a large grid of stellarmodels of HR 1217 that, for the  rst time, included the e ects of magneticperturbations to the oscillation modes. The general properties of those mod-els were discussed but a quantitative match to the frequencies extracted fromthe MOST photometry (chapter 3) was not performed. This chapter outlinesa procedure to match the observed periodicities to the calculated magneto-acoustic modes. The properties of models from our grid that best match theobservations are described.5.1 The matching procedureTo begin we ask ourselves a simple question: Can models of roAp stars beused 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 20periodicities 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 andwith chemical compositions of (X, Z) = (0.7, 0.02) and (0.695, 0.025). Mkr-tichian et al. (2008) included magnetic e ects in their pulsation models usingthe method of Saio (2005) and calculated the magnetic perturbation (for two1385.1. The matching procedureor three models in the evolutionary sequences at each mass) for  elds withstrengths between 2 and 10 kG in steps of 0.2 kG. Their use of a few hundredmagnetic models (= 2{3 selected models  5 available model masses  40magnetic  eld strengths) should be compared to the almost 52,000 modelsused in this study. They were able to  nd a mean deviation between theobserved oscillations and the theoretical pulsation modes of  1.3  Hz. (Indetermining the mean the authors used the logarithm of the radial velocityamplitudes as a weight.) Individual deviations between their best matchedmodelled modes and their observed frequencies ranged from about 3.3  Hz inthe worst match to about 0.1  Hz in the best case. (The observed frequencyuncertainties from their study range from 0.004 to 0.47  Hz.) From theirmatched models they give constraints of M/M = 1.525 0.025, Age = 1.5 0.1 Gyr, logTeff = 3.821 0.006, logL=L = 0.797 0.026, logg = 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 HD101065 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 smallnumber of models that did not include the e ects of a magnetic  eld. Thesenon-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  eld was an essential ingredient in the pulsa-tion calculations used to explain the roAp phenomenon and the observedfrequency shifts. In this work, the largest grid of magnetically perturbedoscillation frequencies was calculated using the methods outlined in chapter1395.1. The matching procedure4. We illustrated in x 4.3 that the addition of the magnetic perturbation tothe 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 Hz) and they occur cyclically as a function of magnetic  eld strength. Wealso showed that the asymptotic spacings (both large and small) are alteredby the magnetic  eld, with the small spacing showing the largest deviationsfrom the standard, non-magnetic, case. We must now  nd the model(s) inour grid that best reproduce the observed photometric variations presentedin Tab. Seismic observations & models of the SunBefore continuing with the asteroseismic matching of the HR 1217 data to ourmodel grid it is instructive to compare the quality of the frequency matchesobtained for a well known star | the Sun. In the introduction chapter(x 1.1) we alluded to the successes of helioseismology at constraining thephysics that makes up a standard solar model (SSM). Detailed reviews ofthe physics included in the SSM and helioseismic inferences are provided byChristensen-Dalsgaard (2002) and Basu & Antia (2008).We know the mass, radius, composition and age of the Sun from a numberof di erent sources (see Basu & Antia 2008). When we construct a SSMwe must meet the constraints provided by those measurements. Typicallywe calculate the radius and luminosity of the Sun to within 1 part in 105and match the value of Z/X ’ 0.0244 to within about 1 part in 104. Themixing length parameter is used as a scalable parameter that is adjusted tomatch the SSM radius to the observed solar radius. The best combinations1405.1. The matching procedureof constituent physics and scalable parameters arrive at an age estimate ofthe Sun from solar models of  4.57 Gyr, in agreement with meteoriticdata (Bahcall et al., 1995). Seismic models of the Sun are constructed toconstrain the input physics of the SSM (EOS, opacities, di usion physics,convection physics, composition :::) with the strict requirement that theother observables mentioned above are also matched.An illustration of the quality of the frequency matches from a SSM and theobservations of the Birmingham Solar Oscillations Network (BiSON; Chaplinet al. 1999) is shown in Fig. 5.1. The SSM was calculated with YREC. TheEOS and opacity tables used in the construction of this SSM are updatedversions of the same tables used in our HR 1217 models and the modelatmosphere is constructed using a Krishna Swamy T- relation (KrishnaSwamy, 1966). Di usion was not included in this calculation. The relativedi erence between the adiabatic frequencies calculated with this SSM and theBiSON data is plotted as a function of the BiSON frequencies (Fig. 5.1). Themodel deviates from the observations by  10  Hz for the highest overtone(= highest frequency) modes. The calculated solar model frequencies aresystematically o set from the observed frequencies.Guenther et al. (1993), forexample, discussed the role of di usion on calculated SSM eigenmodes andshowed the inclusion of He di usion in the SSM model calculation improvethe agreement between the model and the observed periodicities by about 1{5  Hz. The additions of heavy element di usion, a non-adiabatic frequencycalculation, and turbulent pressure corrections from 3D simulations improvethe agreement between theory and observation even further. (We again pointto the review of Basu & Antia 2008.) Currently, the most accurate solar1415.1. The matching proceduremodels still show frequency di erences of about 0.3%, or a few  Hz. Eventhis impressive agreement of a fraction of a percent at high frequencies is stillstatistically poor given that the individual frequency deviations are manytimes 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 frequenciesand observations, even with the systematic o set between the observationsand theory at high frequencies.Guenther & Brown (2004) used a  2 match-ing procedure that calculates the average, weighted square deviation from aset of model frequencies to those observed in the Sun. They found that bymatching 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 thatfavoured di usion and had a composition of Z/X = 0.028 at an age of 4.7Gyr. Metcalfe et al. (2009) used a genetic algorithm to explore the parameterspace 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 discresolved information about the solar oscillations that provides us with mea-surements of millions of modes with identi able 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 allowus to very accurately calculate the internal state of the Sun using inversionmethods. In asteroseismology we do not usually have a large number of ob-served oscillation frequencies or a direct measure of the angular degree ofthose observed light variations. If the matching of the normal modes from1425.1. The matching procedureFigure 5.1: Standard Solar model (SSM) frequencies compared to the Birmingham SolarOscillations Network (BiSON) observations. Shown is the relative di erence betweenthe adiabatic frequencies calculated from a SSM (without element di usion) and thosemeasured from BiSON data as a function of the BiSON frequencies. The model deviatesfrom the observations by  10  Hz for the highest overtone modes. The addition ofHelium and heavy element di usion in the model, a non-adiabatic frequency calculation,and turbulent pressure corrections from 3D simulations improve the agreement betweentheory and observation. At this time, the most accurate solar models; including theaforementioned additions, still show frequency di erences of about 0.3% (a few  Hz).This is still many times the uncertainty of the individual frequency measurements.1435.1. The matching procedurea SSM to helioseismic data using the procedures outlined by Guenther &Brown (2004) and Metcalfe et al. (2009) is applicable to other stars, thenwe expect that accurate frequencies (precise to about 1  Hz or better), will(in the best case) give physical parameters that are accurate to within a fewpercent. Matching a SSM to the low-‘ helioseismic observations gives us arough estimate on the accuracy of the physical parameters we can hope toextract from matching asteroseismic data to models.5.1.2 Seismic observations & models of HR 1217Matching 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 su ciently high photometric quality(thanks to missions like MOST) to uncover low amplitude oscillations. Theextended, continuous, coverage of the observations over long periods of timealso gives unprecedented frequency resolution in a Fourier transform that isfree of aliasing artifacts. On the theoretical side, the increase in computingspeed and memory facilitates the calculation of the large grids of modelsneeded to uncover the parameter combinations that can best reproduce theobservations.The models calculated in this thesis will be matched to the MOST obser-vations using a probability model. The matching procedure is similar to thatused by Kallinger et al. (2008a). For each model in our grid, the calculatedmodes are compared to each of the observed frequencies listed in Tab. 3.1.Under the assumption that the residuals between a best-matching modelmode and the corresponding observed frequency are normally distributed,1445.1. The matching procedurethe Gaussian probability for a one frequency match isPi = f(  ) exp  ( mod  obs)22 2  (5.1)where f(  ) = 1=p2  2 ,  2 is the observed frequency ( obs) uncertainty(Tab. 3.1), and  mod is the model frequency that is the closest match to theobserved frequency. The exponentiated function is essentially a  2 measure.A probability is calculated for each of the NM observed frequencies and thetotal probability for the model islnP = lnNMYi=1Pi =NMXi lnf(  ) ( mod  obs)22 2  i(5.2)Once a probability is assigned to each of the models in the grid, Eq. 5.2 isnormalized by assuming the total grid probability is equal to unity; i.e, eachmodel probability is divided by PP, with the sum taken over all modelprobabilities within the grid. This method has an immediate advantageover a  2 matching because it gives a simple way to statistically comparemodels within the grid and, by adjusting the normalization, models fromone grid to those from another. For example, the probability of some modelX in our grid divided by the probability of some other model Y in our gridgives a measure of how much more likely model X is compared to model Y.There is also an aesthetic bene t that the probability space around a bestmatched parameter tends to be magni ed (because of the exponentiation)when compared to the same region in  2 space. Although a  2 measure ora probability measure will arrive at the same best matched model within agrid, the probability approach shows the location of the matched parametermore readily (Kallinger et al., 2008a).1455.1. The matching procedureWe match our models to the MOST observations listed in Tab. 3.1 usingthe 3 bootstrap uncertainties as the  2 input in Eq. 5.1. This is done forreasons discussed in x 3.3. Basically, we are being conservative by using the3 bootstrap uncertainties because they tend to be on the same order of mag-nitude as the lowest estimate of the frequency resolution given by Kallingeret al. (2008b); namely 0.25/ T. The oblique pulsator model described inx 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 frequenciesspaced by the rotation frequency of the star. The number of rotationallysplit frequencies depends on the degree of the mode and on the geometryof the system. This is observed in the MOST data on HR 1217, where anumber of the observed periodicities are spaced by  0.9  Hz (Tab. 3.2),corresponding to the rotation rate of the star. In the simplest matching casewe allow a model frequency to match multiple observed frequencies. Putanother way, a frequency and its rotationally spaced sidelobes may all bematched to one mode in a given model. An alternative approach will bediscussed in x5:2. We should also note that a proper statistical error shouldinclude the model uncertainties as well as the observational uncertainties.For the Sun, deviations between the modelled and the observed frequenciescan be a few  Hz for the highest order modes. Unfortunately, there is noreliable way to estimate the uncertainties introduced by the addition of amagnetic  eld in our models. By only using the observed frequency uncer-tainties, we are (dramatically) underestimating the true uncertainty neededto realistically weight our computed probabilities. We will discuss the scaleof the uncertainties in more detail below.1465.1. The matching procedureBefore discussing the results of the matching procedure we will addressthe scale of the uncertainties and how that scale relates to the presentation ofthe probabilities of the model properties within our grid. Figure 5.2 shows thematching probabilities in the magnetic  eld strength B, model mass M, andcomposition (Z/X) planes for two uncertainty scaling cases: 1. (Left panelsof Fig. 5.2) The 3 bootstrap uncertainties are not scaled, and 2. (Rightpanels of Fig. 5.2) The 3 bootstrap uncertainties are scaled by a factor of1000. Note that the probabilities in Fig. 5.2 are normalized so that the mostprobable model has a value of one. (The unnormalized probabilities will begiven later.) In the  rst case (with no scaling) the most probable model(to be identi ed later) is clearly de ned, with the next largest probabilitybeing, apparently, near zero. Most of the calculated probabilities in this caseare essentially zero. This is because the Gaussian probability distributionheavily penalizes frequencies that are not well matched. When the totalprobability is calculated as a product of small numbers, its value becomesincreasingly small. This is mainly caused by the lack of knowledge of arealistic 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 beassigned to HR 1217 based on the MOST data. We will show later thatthe situation is more complicated. In the second case, with the uncertaintiesscaled equally by a factor of 1000, there appears to be no constraints inprobability space and no conclusions can be given. If we look for the modelwith the highest probability in both of the scaling cases, we  nd that thesame model is identi ed. The equal scaling of uncertainties tends to magnify1475.1. The matching procedureregions of lower probability, but the relative uncertainties between observedfrequencies results in a clear separation in the quality of the model match. Wecan 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 highlightit because, for presentation purposes, we want to  nd an equal uncertaintyscaling that allows us to see how the probability of the model matches behavesin various parameter planes. The behaviour of the probability distributionaround a set of parameters gives an indication of how degenerate the stellarparameters are within our model grid. For example, we know from x 4.3that the magnetic perturbations are cyclic so we might expect that there willbe (multiple) groupings of probability in the magnetic  eld plane showingwhich  eld strengths result in a better match to the seismic observations.We also know (as another example) that Z/X causes degeneracies in theposition of a star in the HR diagram. So the presentation of the probabilityspaces within a number of parameter planes is important to identifying wherethese degeneracies are and also where future modelling should be focused.There is no statistical trickery being used by equally scaling the parameteruncertainties. We are simply  nding a set of numbers that are easy to dealwith numerically and that, for presentation purposes, helps us get a feel forthe matches in various parameter planes. We will clearly state the value of theequal uncertainty scaling being used so that future studies will easily be ableto reproduce our results. Some experimentation shows that an uncertaintyscaling factor of 50 lets us calculate and properly normalize the model matchprobabilities, while at the same time allowing enough of a constraint on themodels so we can visualize the results. We will further justify the choice of1485.1. The matching procedurea scaling factor of 50 at the end of this section.Figure 5.2: Probability distributions for models of HR 1217 using di erent 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. Modelfrequencies are matched to the observed frequencies with the 3 bootstrap errors scaled.The left panels use a scale factor of 1 and the right panels scale the observed errors bya factor of 1000. The top panels shows the probability of all models as a function ofmagnetic  eld 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). Probabilitiescalculated with unscaled uncertainties show an extreme constraint on the models whilethose 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 aresimilarly ranked in both cases.1495.1. The matching procedureFigure 5.3 presents the matching results for a number of parameters inthe model grid. The cumulative probability is given in each of the panels.The best matched model in our grid has a magnetic  eld strength of B =8.8 kG, a mass of 1.5 M , a composition of Z/X = 0.02, and has completedabout half of its main sequence lifetime. Models with magnetic  eld strengthsgreater than about 8 kG are preferred. The scaling allows to see that probablemodels with values near 1 kG do exist. A mass near 1.5 M is favoured, asis a sub-solar composition (Z/X ’ 0.0244 for the Sun). Once again, theseresults are also obtained without scaling the observed uncertainties; but, wesee 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 e ect that a poormatch has on the graphical analysis of the probability space by uniformlyscaling uncertainties. It is also possible to calculate the model probabilitiesunder the assumption that the residuals between the model and observedfrequencies follow another distribution function. In Eq. 5.1 we use a Gaus-sian probability function. We also calculate model match probabilities usinga Laplace distribution of the formPi = f(  ) exp  j mod  obsj   (5.3)where f(  ) = 0:5=  . The Laplace distribution has fatter tails than aGaussian distribution. As a result, outlying frequencies are given slightlylarger weight when using a Laplace distribution instead of a Gaussian, orNormal, distribution. It is interesting to see how the \close" outliers a ectthe probability space around di erent parameter planes. If there is no in-1505.1. The matching procedureFigure 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 leftpanels, from top to bottom, show the match probability as a function of magnetic  eldstrength, 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 themodel, and the mass of the model’s convective core in units of total stellar mass. Eachpanel has the cumulative probabilities plotted with circles and colours (labelled in the toppanels) showing the percentage of the total probability. By using this scaling we see moredetails in the probability distributions than we would have if the uncertainties were notscaled. Magnetic models having B& 8 kG are preferred.1515.1. The matching procedureformation about how the residuals between a model and observations aredistributed, then a Gaussian distribution is the most conservative choice forthe 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 parametersas in Fig. 5.3. Note that in both cases the same model is identi ed as themost probable model match. Because outliers are given some weight in theLaplace probabilities, we see a number of peaks in probability that are notseen when using a Gaussian probability model. Are the conclusions the samein both cases? Greater than 60% of the probability (of a model from ourgrid matching the observations) occurs after a magnetic  eld strength of 8.8kG. Between 60 and 90% of the probability contribution occurs for massesbetween 1.5 and 1.6 M , and the largest contribution from the compositionis 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 di erent distributions shows how sensitive theprobability space is to frequencies that, in a manner of speaking, do not makethe Gaussian cut.1525.1. The matching procedureFigure 5.4: Probability distributions for models using a Laplace distribution. All observedfrequencies (those associated with both pulsation and rotational modulation) are used inthe model search and probabilities are assigned using the Eq. 5.3. Observed uncertaintiesare 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 greaterprobability 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 matchesto the observations of the roAp star HD 101065 (Mkrtichian et al., 2008).Mkrtichian et al. (2008) used the mean as a statistic to discriminate betweentheir models of HD 101065 and arrived at a closest matched model havinga mean deviation from the observations of about 1.3  Hz. Those authors1535.1. The matching procedureused a weighted mean so that poorly  tted frequencies did not skew theirmean calculation. In Fig. 5.5 we compare the median frequency di erencesbetween our models and the MOST observations of HR 1217 as a function oflog probability using both the Gaussian and Laplace distributions. Insteadof using a weighted mean like Mkrtichian et al. (2008), we use the median sothat we do not bias our average di erences toward poorly  tted frequencies.The purpose of Fig. 5.5 is to compare an average di erence statistic to theprobability method we have presented thus far. We also show in Fig. 5.5how scaling the observed uncertainties a ects the numerical values of ourunnormalized probabilities.For each value of probability in Fig. 5.5 there are a number of possiblemedian matches. The Gaussian probability calculation drops o sharply forhigh probability matches, while the Laplace probability calculation gradu-ally approaches the most probable model(s). The lowest median deviationand the highest probability models both show a small standard deviation be-tween the di erences in model and observations. Scaling the observed errorschanges the scale and the width of the probability region, so that the modelsmatched to observations with the largest uncertainty values show the small-est range in probabilities. The shape of the probability distribution doesnot change by (uniformly) scaling the observed errors. Using the mean ormedian as a discriminant between models and observations is not as e ectiveas using a probability because the probability calculation takes into accountthe observed uncertainties. Poorly matched models are penalized much morewithin a probability framework than by using an average statistic. We haveto be aware that even one extreme mismatch between an observed and calcu-1545.1. The matching procedurelated frequency can dramatically reduce the probability of that model beingan acceptable match. In such a case the mean (or median) deviation of themodel should be looked at in conjunction with the probability of that modelto gauge the match quality.We have introduced a number of relevant matching methodologies buthave yet to discuss the quality of the matches or to formally list the physicalparameters of the best matched models. In Tab. 5.1 we list a number ofproperties for the most probable model matches to the MOST observations.The models are listed as the 10 most probable, non-duplicate, models usingboth the Gaussian and Laplace probability matches.1555.1. The matching procedureFigure 5.5: A comparison between the median model frequency matches and the modelmatch probabilities. The left panels show the median of the di erence between the obser-vations and the best matched model frequencies for each model in the grid as a functionof the natural logarithm of the Gaussian model probabilities. Best matches are located atthe lower right in each plot. The top panel highlights the case with no frequency scalingwhile the middle and bottom panels show the cases with observational uncertainties scaledby a factor of 50 and 1000, respectively. Colour contours show the standard deviation ofthe matches between the model and the observed frequencies. The right panels are thesame as the left except the model probabilities are calculated using the Laplace distri-bution. Note that scaling the observational uncertainties produces similar distributionsbut the value of the probabilities increases and the range in probabilities decreases withincreasing scaling factor. For each probability there are a number of potential medianmatches. The Gaussian and Laplace probabilities give the same best matched models butthe probabilities tend to increase more sharply when using a Gaussian distribution.1565.1.ThematchingprocedureT able 5.1: Some prop erties of the most probable mo dels of HR 1217. The top ten unique mo dels from th e Gaussian andLaplace probabilities are listed in decreasing probabilit y order. Deviati on refers to the absolute di erence b et w een a mo delfrequency and the b es t matc hed observ ed frequency and  is the standar d deviation. The column lab elled as prob abilit y lab elrefers to the order of the prob abilit y f rom eac h of the G au s sian or Laplace probabilities. F or example, 1 G is the most probableGaussian prob abilit y and 10 L w ould b e the 10th most probable Laplace probabilit y . The probabilit y ratio is de ned as theGaussian probabilit y of the mo del divided b y the Gaussian probabilit y of the most pr obable mo del ( ln P =  63 : 1). The mo delsin this table w ere matc hed to all frequencies listed in T ab. 3.1 using 50  3  errors.Mo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratioM1 1.50 0.700 0. 014 1.05 1.4 0.14348.8 0.00 3.8645 0.8826 0.2361 69.251866.36 1. 18 1.53 1.16 1 G, 1 L 1.000M2 1.50 0.700 0. 014 1.10 1.4 0.13848.7 0.00 3.8616 0.8866 0.2438 67.521816.38 1.36 1.52 1.29 2 G, 5 L 0.273M3 1.30 0.700 0. 008 2.05 1.4 0.00004.8 0.00 3.8680 0.8973 0.2364 64.611597.02 1.08 1.31 0.99 2 L, 9 G 0.037M4 1.50 0.700 0. 014 1.00 1.6 0.14349.3 0.00 3.8672 0.8787 0.2287 70.991915.70 1.23 1.40 0.90 3 G, 4 L 0.183M5 1.50 0.700 0. 014 1.15 1.6 0.13849.1 0.00 3.8587 0.8905 0.2517 65.901767.56 1.01 1.49 1.35 3 L, 7G 0.055M6 1.45 0.720 0. 010 1.30 1.8 0.12511.2 0.00 3.8637 0.8633 0.2281 70.161857.29 1.40 1.45 0.95 4 G, 13 L 0.111M7 1.45 0.720 0. 008 1.35 1.6 0.11761.0 0.00 3.8739 0.9289 0.2405 66.88c ontinue d on next p age1575.1.ThematchingprocedureT able 5.1: c ontinue dMo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratio1703.00 1. 37 1.57 1.23 5 G, 12 L 0.091M8 1.70 0.740 0. 020 0.85 1.4 0.16288.5 0.00 3.8577 0.9065 0.2617 67.641883.37 1. 10 1.32 0.89 6 G, 7 L 0.067M9 1.50 0.700 0. 014 1.00 1.6 0.14349.2 0.00 3.8672 0.8787 0.2287 70.991915.70 1.22 1.22 0.79 6 L, 8 G 0.050M10 1.50 0.740 0.010 1.20 1.8 0.126410.0 0.00 3.8695 0.8728 0.2212 72.781931.66 1. 05 1.25 0.92 8 L, 24 G 0.0041585.1. The matching procedureThe most probable model match has a Gaussian probability of lnP =-63.1. That probability was calculated using the uniformly (50 ) scaled 3 frequency errors from Tab. 3.1. For completeness, the probability of thatmodel calculated without scaling the MOST frequency uncertainties is lnP= -1.4  104. Clearly, with such low probability, the model match is not astatistically good match. Figure 5.6 shows the deviations between the modeland 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 ofprecision that is much better than can be matched with our models. How-ever, the mean absolute deviation between the observations and the theoryof 1.53  Hz is similar to that obtained by (Mkrtichian et al., 2008) for theroAp star HD 101065. The median absolute deviation between model M1and the MOST data is 1.18  Hz. An order 1  Hz deviation at frequen-cies of  2700  Hz constitutes a percentage agreement of 0.04% | A valuethat is about 10 times better than the current agreement between the highfrequencies calculated and observed in the Sun (see Fig. 5.1)! The largestfrequency deviation in model M1 is 4.307  Hz between an ‘ = 3 mode hav-ing  = 2792.107  Hz and the observed frequency of 2787.799  Hz ( 22 ofTab. 3.1). That frequency is most likely a rotational splitting and not apulsation mode of HR 1217. Because we allow multiple matches between themodel and the observed periodicities, that mode also matches frequenciesbetween 2788.872 and 2792.249  Hz ( 23 through  27 in Tab. 3.1). Those sixfrequencies correspond to the \missing modes" (x1.3) 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  15 = 2720.9141595.1. The matching procedure Hz (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  = 2720.844  Hz. This represents a deviation of 0.070  Hzand is still more than a factor of 10 larger than the observed 3 uncertaintyof 0.005  Hz for that periodicity. Listed in Tab. 3.1 is the newly identi edMOST frequency at  1 = 2603.832  Hz that is spaced by  15:7  Hz fromthe next closest mode. A spacing of about 15  Hz deviates from the regularasymptotic spacing of  33.5  Hz and may represent a magnetically per-turbed mode. The frequency from model M1 that most closely matches  1is a mode with angular degree of ‘ = 1 with  = 2599.938  Hz. That matchdeviates from the observed frequency by 3.895  Hz. On average10, model M1has a large spacing of 69.938  Hz. (This should be compared to the largespacing of 69.25  Hz listed for model M1 in Tab. 5.1 that was calculated us-ing the integral de nition of the large spacing given in Eq. 1.12.) The largespacing of model M1 ranges from a minimum of 69.086  Hz to a maximum of70.449  Hz. The observed large spacing (Tab. 3.2) is, on average, 67.961  0.053 Hz. The small spacings (  2 0) for model M1 range from -4.755  Hzat 2619.780  Hz to -2.100  Hz at 2828.660  Hz. MOST identi es potentialsmall spacings having values between  1.5 and 2.6  Hz. Figure 5.6 showsthat the closest matched modes would be alternating between ‘ = 0 and ‘= 3 and the (potentially) magnetically perturbed frequencies at the smallestand largest observed frequencies have an angular degree of ‘ = 1. Given thatthe frequencies show rotational splitting, a radial mode (‘ = 0) seems an10The average is taken over all n and ‘ for frequencies in the range of 2618.600 to2792.249  Hz.1605.1. The matching procedureunlikely candidate for a match.Figure 5.6: A comparison between model M1 (Tab. 5.1) pulsation modes and theMOST frequencies. The relative frequency deviation between theory and observationis plotted against the observed frequencies (Tab. 3.1). The shaded area ( 1:5= T) showsthe most conservative frequency resolution discussed inx2.2. The dotted line is the meandi erence between the observations and theory. The dashed line shows the median di er-ence between the model and data points. The matching procedure allows the same modelfrequency to match multiple observations.1615.1. The matching procedureThe most probable models listed in Tab. 5.1 show mean deviations fromthe MOST observations ranging from 1.22 to 1.57  Hz, with a standard errorof 0.2  Hz. The median deviation ranges from 1.01 to 1.40  Hz. The modelmatch 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  Hz is not oneof the most probable models. The properties of the model with the smallestmedian 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 hasa mean deviation is about 2.4  Hz. The probability ratio of model MM1to model M1 of Tab. 5.1 is  10 27. In Fig. 5.7 we compare the frequencydeviations between the modelled modes and observations for models MM1(Tab. 5.2) and M3 (Tab. 5.1). Model M3 is chosen for comparison becauseit shows a low mean frequency deviation (1.31  Hz) and because; of allthe models listed in Tab. 5.1, its magnetic  eld strength (4.8 kG) is closestto the observed value of  2{4.4 kG (Bagnulo et al. 1995, L uftinger et al.2008). The frequency deviations of model M3 (left panel of Fig. 5.7) showan alternating pattern of ‘ = 2 and ‘ = 1 with the anomalously spacedmodes at either end of the observed spectrum identi ed as ‘ = 4 modes.There is no distinction between the pulsation, rotation, or small separatedmodes. That is, the same mode is used to match each frequency group thatis separated by about   =2. Model MM1 has the smallest median deviationof all the models matched using this procedure. Its frequency deviations areshown 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,identi ed as ‘ = 2 modes. This is what one would expect if a second order1625.1. The matching procedure(small) spacing was present in the data | Modes that are closely spaced byabout a few  Hz from each other that have angular degrees that di er bytwo. However, it is especially noteworthy that the potentially magneticallyperturbed modes at the low and high end of the MOST frequencies are notidenti ed in Fig. 5.7. That is because the modelled modes deviate from theobserved modes by more than 10  Hz. The matching of modes from modelsMM1 and M3 illustrates the interesting result that we can come close toobtaining the magnetically perturbed modes (spaced by about 15  Hz inthe MOST data) or the potential small spacings (spaced by about 3  Hz inthe MOST data); but, we generally do not match both. None of the mostprobable models listed in Tab. 5.1 match a 0 kG model.1635.1.ThematchingprocedureT able 5.2: Some prop erties of a selected mo del of HR 1217. The mo del with the lo w est median deviation b et w een observ ed andcalculated frequencies is listed b elo w. The column lab elled as probabilit y lab el refers to the order of the probabilit y from eac hof the Gaussian or Laplace probabilities. F or example, 1 G is the most probable Gaussian probabilit y and 1 0 L w ould b e the10th most probable Laplace probabilit y . The probabilit y ratio is de ned as the Gaussian prob abilit y of the mo del divided b ythe Gaussian probabilit y of the m ost probable mo del (see T ab. 5 . 1). The mo dels in this table w ere matc hed to all frequencieslisted in T ab. 3.1 using 50  3  errors.Mo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratioMM1 1.60 0.700 0.018 0.900 1. 6 0.16348.8 0.00 3.8645 0.9243 0.2569 66.681809.45 0. 88 2.39 4.14 3227 G, 1349 L 1.729  10  271645.1. The matching procedureFigure 5.7: A comparison between the model M3 pulsation modes and theMOST 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 arethe same as those in Fig. 5.6. Model M3 is more probable than model MM1 but the modelMM1 frequencies have the smallest median deviation from the MOST observations of allof the models explored.Up to this point we have calculated the probabilities of model matchesto all of the observed oscillations extracted from the MOST data. Next wecompare the closest matched oscillation spectra to models with the sameproperties, but having di erent magnetic  eld values. These diagrams; thatwe will refer to as Saio plots (see, e.g., Saio 2005), illustrate how changesin the magnetic  eld strength can greatly impact the matching of models1655.1. The matching procedureto observations. Figure 5.8 shows the magnetically perturbed modes as afunction of magnetic  eld for the two most probable models; M1 (left panel)and M2 (right panel), of Tab. 5.1. The MOST observations are shown ashorizontal, dotted, lines and the location of the best  tted magnetic  eldvalue is drawn as a solid, vertical line. It is clear that in both cases themagnetic perturbations cause cyclic matches to the observations. In somecases, the magnetic  eld causes the eigenspectrum to completely miss theobserved modes. A small change in magnetic  eld strength of about 0.1 kGcan result in a frequency shift (much) greater than 1  Hz. Frequency jumpsare clearly visible and modes tend to cluster in groups of alternating evenand odd parity. This is consistent with the MOST observations (chapter3). A close inspection of the M1 model match shows that this even and oddalternating ‘ pattern is broken for the two highest frequency groups. Nearthose frequencies, the alternating pattern switches to an odd-odd ‘ pattern.The anomalous spacings of  15  Hz at each end of the observed spectrumare most closely matched by ‘ = 1 modes (in this case), although the matchis not perfect, or even statistically meaningful. There is also evidence thatfrequency spacings of a few  Hz can exist, so that the observed spacings of 1.5{3  Hz may be explained as magnetic perturbations and not as thetraditional asymptotic small spacings. In Fig. 5.9 we show the same type ofdiagram but for the lowest mean (left panel) and median (right panel) devia-tion models (Tab. 5.2). Those models show similar characteristics; but, in thecase 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 the1665.1. The matching procedurelowest and highest of the observed frequencies as odd modes. We also notethat there are cases where the modes with ‘ values having similar parity tendto overlap or even cross each other. This phenomena was discussed at theend of chapter 4 and is the reason that the small spacing (when calculatedfrom  n;‘  n 1;‘+2) can become 0 or negative in the presence of a magnetic eld. Appendix D provides Saio plots for all of the models presented in thetables in this chapter.The model match probabilities; better described as model plausibilitieswithin 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 thisscaling factor, after some experimentation, was chosen because it allowedus to properly visualize and conveniently normalize our probabilities. Thelack of a realistic model uncertainty causes most of the models in our gridto get too little weight given our stringent observational uncertainties. Wenow have enough information, based on the quality of our model matches,to show that a error scaling factor of 50 is reasonable. The average 1 un-certainty of our observations (Tab. 3.1) is 0.032  Hz and the average 3 uncertainty is 0.092  Hz. The most precise oscillation frequency identi ed inthe MOST data is  15 = 2720.914 0:002(1 )0:005(3 )  Hz. Traditional measures of thefrequency resolution when applied to the MOST data range from 0:25= T  0.1  Hz to 1:5= T  0.6  Hz. On average, our 1 uncertainties would haveto multiplied by 18.75 (= 0.6/0.032) to reach the 1:5= T  of 0.6  Hz. Our3 uncertainties (again on average) would have to be multiplied by 6.52 toreach the same upper resolution limit of 0.6  Hz. Our most precise frequency1675.1. The matching procedureFigure 5.8: Magnetically perturbed oscillation frequencies as a function of magnetic  eldstrength for selected models of HR 1217. The panels show the unperturbed frequencies assquares 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 eachpanel. The MOST observations from Tab. 3.1 are plotted as horizontal dotted lines. Theleft panel shows model M1 of Tab. 5.1 identi ed as a solid vertical line while the rightpanel illustrates the match using model M2 of the same table. The best matched modelsfor M1 and M2 have similar properties. Note that the magnetic perturbations produce acyclic variation with magnetic  eld strength so that models with a lower magnetic  eldcan also exist for models with physical characteristics like those of M1 and M2; but, thesemodels would not be as statistically signi cant.1685.1. The matching procedureFigure 5.9: Magnetically perturbed oscillation frequencies as a function of magnetic  eldstrength 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. Thismodel is selected because it has the smallest mean di erence between model and observedfrequencies of 1.22  Hz. The right panel identi es the model with the smallest mediandi erence between model and observed frequencies of 0.88  Hz. That properties of thatmodel (labelled MM1) are listed in Tab. 5.2. Although these models have low mean andmedian values, they are not the most probable matches. Note that the cyclic nature of themagnetic perturbation gives the closest matched modes at both low and high B values.1695.1. The matching procedurewould have to have its 1 uncertainty multiplied by 300 (or by 120 for the 3 uncertainty) to arrive at 0.6  Hz. So, if we were to use the most conservativeestimate of frequency resolution (1:5= T) to scale our frequency precisionupward, we would have to use a factor that is some where between about10 to 100. In Fig. 5.5 we showed that the model match probability corre-lates well with the median frequency deviation between the model frequenciesand the observed periodicities. The median deviation of our most probablemodels (Tab. 5.1) is about 1  Hz. If we were to assume that our bootstrapuncertainties should be scaled to match the smallest median deviation ( 1 Hz) between our calculated eigenmodes and our observations, our average1 uncertainties would have to be multiplied by 31.25 and our average 3 uncertainties by 10.87. Our smallest frequency uncertainty would have to bemultiplied by a few hundred ( 200 using the 3 uncertainties). This wouldagain suggest an uncertainty scale that falls somewhere in the range of a fewtens to a few hundreds. When there is some unknown systematic uncertainty,the observed uncertainties can be scaled so that the best matched model hasa reduced  2 = 1. If we do this for model M1 of Tab. 5.1, the observationaluncertainties would have to be scaled by 41.06. The frequency scaling of 50 isreasonable given the above scaling scenarios. This is especially true becausewe are more concerned with the relative probabilities between models withinour grid and not the value of the individual probabilities themselves.1705.2. Matching the true modes of HR 12175.2 Matching the true modes of HR 1217In the preceding section models were matched to all of the observed period-icities in the MOST data (Tab. 3.1). Di erent probability distributions andaverage deviation statistics were used to highlight the di erences between thecomputed magneto-acoustic models and the observed oscillation frequenciesof HR 1217. In this section we isolate what we believe are the pulsationmodes (as opposed to rotation splittings) in the HR 1217 data and illus-trate what e ect using this selection of frequencies has on the most plausiblematches.Consider frequencies  28 and  29 of Tab. 3.1. They are split by the rota-tion frequency of HR 1217 ( 0.9  Hz) but they have the same amplitude towithin their 1 uncertainties. How can we decide which frequency is the ro-tational component and which frequency is the main oscillation mode? Thisdilemma was also discussed in x 3.3 for frequencies  8 through  12 and near 19 and  20 (Tab. 3.1). Evidence suggests that there are potential small spac-ings of  1.5 or 2.5  Hz near these frequency groups, along with rotationalsplittings, and it is di cult to decide which periodicity is the true oscillationfrequency. For the purposes of matching models we select a subset of frequen-cies that we believe to be the main frequency within their respective frequencygroupings. These are usually the largest amplitude components within theirgroupings. In the cases where we are not certain as to which mode shouldbe considered as the main frequency, we select a frequency and adopt anuncertainty that re ects the range in frequency space that is spanned byits closest, approximately rotationally split, neighbours. For example, if wechose  9 as a main frequency, we adopt a value of [ 10 +  10 - ( 8 -  8)]/2.01715.2. Matching the true modes of HR 1217as its uncertainty. Table 5.3 lists the frequencies that we believe could be themain oscillation modes in the HR 1217 data. With the pulsation modes ofHR 1217 identi ed, we can now assign a unique frequency match to each ofthe observed frequencies. By assigning a unique frequency match instead ofallowing a model frequency to identify itself with multiple observations (aswas done in the previous section), we can immediately ignore close modelmatches to rotational splittings. This is important because a close match toa rotational splitting could easily skew our matching statistics. The uniqueassignment of a modelled frequency to an observed frequency will also im-mediately give us an indication of how closely individual modes are spacedfrom each other.1725.2.MatchingthetruemodesofHR1217T able 5.3: Selected frequencies iden ti ed using the MOST data. Both 1 and 3  uncertain ties on the  tted parameters areestimated from 100,000 b o otstrap realizations. F requency uncertain ties mark ed b y  are a v erage uncertain ties based on theseparation from the mark ed frequency to the nearest (  rotationally split) frequency plus its un c ertain t y (see the text fordetails). The phases are referenced to th e time of the  r s t observ ations of HR 1217 (= HJD 2451545.00 + 1769.42 da ys). Thistable is repro duced in app endix B with frequencies in units of cycles da y  1 .#  [  Hz] A [  mag]  [rad]  S/N   3    A 3  A   3   1 2603.832 39 2.22 3.8  0.028 0.080 6 15 0.34 0.933 2619.495 125 5.70 9.2  0.015 0.039 6 16 0.11 0.296 2652.924 365 5.56 24.2  0.004 0.011 6 16 0.04 0.109 2685.659 31 2.83 3.1  0.897  0.951  9 26 0.55 2.1811 2687.420 467 4.46 29.7  0.004 0.009 6 16 0.03 0.0815 2720.914 764 1.83 41.6  0.002 0.005 6 16 0.02 0.0517 2723.469 4 1 2.08 3.1  0.034 0.097 6 16 0.39 1.1019 2755.393 118 3.45 8.9  0.015 0.039 6 16 0.17 0.4521 2756.925 3 9 5.14 3.1  0.487  0.578  6 17 0.53 1.6723 2788.872 3 7 0.24 3.6  0.034 0.096 6 16 0.29 0.8026 2791.384 8 2 5.00 6.8  0.033 0.085 7 17 0.29 0.74c ontinue d on next p age1735.2.MatchingthetruemodesofHR1217T able 5.3: c ontinue d#  [  Hz] A [  mag]  [rad]  S/N   3    A 3  A   3   28 2805.650 8 4 1.10 6.9  0.488  0.514  6 15 0.15 0.371745.2. Matching the true modes of HR 1217As in the last section we show match probabilities assuming both Gaus-sian and Laplace probability distributions. The probabilities calculated usingdata from Tab. 5.3 are shown in Fig. 5.10. We immediately see that whenwe compare the probability distribution for the magnetic  eld strength inFig. 5.10 to the multiple-match case shown in Figs. 5.3 and 5.4, that theprobability peaks for the magnetic  eld near 8 kG has signi cantly droppedin comparison to probability peaks near 4 kG. This is an immediate indica-tion that by allowing a single model frequency to match multiple observedperiodicities; and, by including the rotational splittings in our data set, wewere dramatically altering the weight of the matches. The use of the Gaus-sian or the Laplace distribution once again illustrates how outlying matchesnear the tails in uences the shape of the probability curves. We can drawthe 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 spacearound the mass parameter is broader in Fig. 5.10 than it is in Figs. 5.3 and5.4. The composition (Z/X) seems to loosely favour values less than 0.02 inFig. 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 mainsequence lifetime that the models have completed, log10g, and the models’convective core mass. In selecting only the pulsation modes in the HR 1217data, we see that the probability distributions for these parameters are lessa ected than were the distributions for the magnetic  eld strength, mass,and composition (see Figs. 5.3 and 5.4). This may not be particularly sur-prising because log10g varies slowly with position on the HR diagram and1755.2. Matching the true modes of HR 1217the percentage of the main sequence lifetime that a model has traversed isselected by limiting our choice of luminosity and e ective temperature (seex4.1.1). The constraint on which stellar model masses fall in that luminosityand e ective temperature range also limits the values of the convective coremass of our models.1765.2. Matching the true modes of HR 1217Figure 5.10: The observed frequencies of Tab. 5.3 are used in the model search with 3 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 calculatedfrom a Laplace probability distribution (Eq. 5.3). The top panels show the probability ofall models as a function of magnetic  eld strength, the middle panels are probability asa 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  eld strengths; favouringmagnetic  elds less than about 5 kG, is a clear contrast from that presented in Fig. 5.3,where larger magnetic  elds ( 8 kG) are favoured. As in the previous examples, theLaplace distribution tends to highlight more models in probability space.1775.2. Matching the true modes of HR 1217Figure 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 showGaussian probabilities and the right panels show Laplace probabilities. Listed from topto bottom, the panels show the match probabilities as a function of the percentage of themain sequence lifetime the model has completed, the surface gravity of the model, and themass of the model’s convective core in units of total stellar mass. These are similar to theprobability distributions shown in Figs. 5.3 and 5.4.1785.2.MatchingthetruemodesofHR1217T able 5.4: Some prop erties of the most probable mo dels of HR 1217. The top ten unique mo dels from th e Gaussian andLaplace probabilities are listed in decreasing probabilit y order. Deviati on refers to the absolute di erence b et w een a mo delfrequency and the b es t matc hed observ ed frequency and  is the standar d deviation. The column lab elled as prob abilit y lab elrefers to the order of the prob abilit y f rom eac h of the G au s sian or Laplace probabilities. F or example, 1 G is the most probableGaussian prob abilit y and 10 L w ould b e the 10th most probable Laplace probabilit y . The probabilit y ratio is de ned as theGaussian probabilit y of the mo del divided b y the Gaussian probabilit y of the most pr obable mo del ( ln P =  31 : 1). The mo delsin this table w ere matc hed to all frequencies listed in T ab. 5.3 using 50  3  errors.Mo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratioU1 1.50 0.720 0.012 1.15 1.6 0.13663.1 0.00 3.8657 0.8789 0.2318 70.261877.92 2. 84 4.51 4.76 1 G, 2 L 1.000U2 1.45 0.740 0.008 1.50 1.6 0.11322.1 0.00 3.8651 0.8888 0.2381 67.601749.34 1. 88 3.27 3.27 1 L, 4 G 0.483U3 1.40 0.700 0.008 1.35 1.6 0.11481.8 0.00 3.8730 0.9089 0.2322 67.661729.36 2. 35 3.39 3.15 2 G, 3 L 0.590U4 1.45 0.720 0.010 1.30 1.8 0.12512.2 0.00 3.8637 0.8633 0.2281 70.161857.29 1. 39 2.92 2.77 3 G, 8 L 0.521U5 1.60 0.720 0.016 0.90 1.8 0.15959.0 0.00 3.8690 0.9114 0.2414 70.201899.10 3. 07 4.87 5.22 5 G, 5 L 0.406U6 1.55 0.740 0.012 1.30 1.8 0.13119.6 0.00 3.8581 0.9067 0.2609 65.061719.39 1. 42 4.87 6.27 6 G, 4 L 0.195U7 1.60 0.720 0.0160 1.05 1.6 0.15458.3 0.00 3.8608 0.9232 0.2638 65.20c ontinue d on next p age1795.2.MatchingthetruemodesofHR1217T able 5.4: c ontinue dMo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratio1754.73 1. 89 5.20 6.35 7 G, 12 L 0.190U8 1.45 0.700 0.0100 1.20 1.4 0.12531.9 0.00 3.8670 0.9069 0.2376 67.661763.55 2. 33 2.99 2.87 8 G, 9 L 0.170U9 1.55 0.740 0.012 1.25 1.8 0.13663.5 0.00 3.8612 0.9030 0.2530 66.691766.91 1. 09 4.52 5.90 6 L, 12 G 0.104U10 1.55 0.700 0.016 1.05 1. 6 0.15028.5 0.00 3.8596 0.9100 0.2595 65.191756.98 1. 56 5.13 6.38 7 L, 10 G 0.1401805.2. Matching the true modes of HR 1217Table 5.4 lists the properties of the most probable model matches to theMOST frequencies identi ed in Tab. 5.3. About half of the models listed inTab. 5.4 have magnetic  eld strengths between 2 and 5 kG. This is in linewith the magnetic  eld strengths measured in HR 1217 that range from about2 to 4.5 kG (Bagnulo et al. 1995, L uftinger et al. 2008). This is in contrastto the most probable model matches given in Tab. 5.1. Those models werematched to all of the frequencies (including rotational splittings) extractedfrom the HR 1217 data (Tab. 3.1) and gave magnetic  eld strengths near 1kG 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  eld strength of B = 3.1 kG. Its prob-ability (unnormalized and calculated using a uniform uncertainty scaling of50 ) is lnP = -31.1. As discussed previously, this match probability is low.The MOST observations are measured to a precision that is much better thanour modelled frequencies can be matched. Figure 5.12 shows the modelledmode deviations from the MOST frequencies and the Saio plot (magneticallyperturbed eigenmodes as a function of magnetic  eld) for the same model.The mean deviation between the observed frequencies and the model U1modes is 4.51  Hz with a standard deviation of 4.76  Hz. The median of thefrequency deviations is 2.84  Hz. Note again that the matching procedureused in this section is di erent from the matching procedure used in the pre-vious section. In this section we force a unique mode identi cation on eachof the \true" frequencies measured in the MOST data (Tab. 5.4). Because ofthis, the mean di erences between the theory and observations will tend tobe larger than they would have been if we allowed one modelled eigenmode1815.2. Matching the true modes of HR 1217to match multiple observed frequencies. The eigenmodes from model U1 thatare the poorest matches are an ‘ = 2,  = 2796.694  Hz (matched to  28 =2805.650  Hz of Tab. 3.1) mode, an ‘ = 4,  = 2783.533  Hz (matched to 26 = 2791.384  Hz) mode, and a mode having ‘ = 2 and  = 2727.779  Hz(matched to  17 = 2723.469  Hz). All other matches deviate by less thanabout 1.4  Hz. The best matching mode is at 2720.908  Hz. That mode isan ‘ = 0 mode and is matched to the most precise of the MOST frequencies( 15 = 2720.914 Hz). The perturbed small spacings (  2 0) of model U1 areall negative over the range in frequencies observed in HR 1217. The smallspacing has its lowest absolute value of -5.614  Hz at  = 2653.190  Hz andincreases in absolute value to -6.473  Hz at  = 2859.900  Hz. An averageover n and ‘; between frequencies 2618.600 and 2792.249  Hz, yields a largespacing of 68.792  Hz for model U1.1825.2. Matching the true modes of HR 1217Figure 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 probablemodel listed in Tab. 5.4. The average frequency deviation is 4.51  Hz. (Right) Magnet-ically perturbed frequencies as a function of magnetic  eld strength for the same model.The labels are the same as those in Fig. 5.8. Model U1 has a high probability because itdoes 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 byMOST.The properties of the models having the smallest mean (= 2.32  Hz) andmedian (= 0.64  Hz) absolute deviation between the theoretical modes andthe observed periodicities are listing in Tab. 5.5. The model with the low-est mean deviation (labelled UU1) and the model having the lowest mediandeviation (UU2) have magnetic  eld strengths of 1 and 8.4 kG, respectively.1835.2. Matching the true modes of HR 1217Figure 5.13 shows the frequency deviation plot of model UU1. We see thatthe small spacings (modes spaced by . 3  Hz) are better reproduced thanthe potentially magnetically perturbed modes (at the highest and lowest fre-quencies) are matched. That being said, the model itself does not constitutea 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 notwell matched. The closest match to those magnetically perturbed modes inmodel UU1 are matched to ‘ = 4 modes, and have deviations of about 3.5 Hz at  2600  Hz and 6.5  Hz at  2800  Hz. The frequencies of modelUU1 tend to cluster in alternating even and odd parity groups, with one no-table exception: Near 2600  Hz, the ‘ = 4 mode is separated in frequencyfrom the ‘ = 0, 2 couple (matched to the next closest mode) by about 11 Hz. On the opposite end of the spectrum (near  2800  Hz) the last twomeasured frequencies have associated with them modes with opposite par-ity. This is interesting because the MOST frequencies at the low end of thespectrum could be interpreted as potential small spacings while the high endof the spectrum would not be interpreted as such. Of course, the magneticperturbations between modes separated in degree by two can be so largethat labelling spacings as \small" or second order separations; as calculatedin the non-magnetic case, may simply not be appropriate. This behaviouris not apparent in the Saio plot of the model (UU2) with the lowest medianfrequency deviation (right panel of Fig. 5.14). Saio diagrams for all modelslisted in the tables of this chapter are given in appendix D.1845.2.MatchingthetruemodesofHR1217T able 5.5: Some prop erties of selected mo dels of HR 1217. The mo de l s with the lo w est me an and median deviations b et w eenobserv ed and calculated frequencies are listed (resp ectiv ely) b elo w. The column lab elled as prob abilit y lab el refers to theorder of the probabilit y from eac h of the Gaussian or Laplace probabilities. F or example, 1 G is the most probable Gaussianprobabilit y and 10 L w ould b e the 10th most probable Laplace probabilit y . The probabilit y ratio is de ned as the Gaussianprobabilit y of the mo del divided b y the Gaussian probabilit y of the most pr o b able mo del (see T ab. 5.4). The mo dels in thistable w ere matc hed to all frequencies listed in T ab. 5.3 using 50  3  errors.Mo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratioUU1 1.40 0.720 0.008 1.50 1.6 0.10921.0 0.00 3.8645 0.8679 0.2287 68.611782.97 1. 92 2.32 1.96 13194 G, 6948 L 3 : 692  10  89UU2 1.55 0.700 0.016 1.05 1.4 0.15023618.4 0.00 3.8595 0.9100 0.2598 65.061755.53 0. 64 4.79 6.31 38 G, 22 L 0.0241855.2. Matching the true modes of HR 1217Figure 5.13: Model UU1 pulsation frequencies (Tab. 5.5) are compared to theMOST observations. The labels are the same as those in Fig. 5.6. Model UU1 is the modelwith the smallest mean frequency deviation (2.32  Hz) from the MOST frequencies.1865.2. Matching the true modes of HR 1217Figure 5.14: Magnetically perturbed frequencies as a function of magnetic  eld strengthfor the models with the lowest mean(model UU1 of Tab. 5.5, left) and median (modelUU2 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 thematch to observations can lead to dramatically di erent \best- t" models.5.2.1 Matches to a sub-set of the MOST frequenciesIn the previous sections (x 5.1.2 and 5.2) we matched our model grid tothe frequencies observed by MOST using all of the frequencies identi ed inthe data (Tab. 3.1) and by using only those modes that we believed to bethe pulsation modes of HR 1217 (Tab. 5.4). We were not able to  nd amodel that matched the observations to the level of precision obtained by1875.2. Matching the true modes of HR 1217the MOST mission. That is, we can  nd models that match the observedfrequencies with a mean deviation of a few  Hz but the periodicities identi edin the MOST time-series have frequency uncertainties that are, on average,hundredths of a  Hz. Interestingly, it seems that in the majority of thecases presented thus far, we come closest to matching the frequencies spacedfrom their neighbours by  15  Hz (usually referred to as the magneticallyperturbed modes) or the frequencies spaced by  1{3  Hz (usually referredto as the small or second-order-spaced modes); but, we cannot seem to comeclose to matching both in a given model. In this section we explore thisobservation further by matching models to two sub-sets of the frequencieslisted in Tab. 5.4.In the  rst sub-set of frequencies we only attempt to match the twoanomalous (magnetically perturbed) frequencies and the most precisely de-termined observational frequency. These are located at 2603.832, 2720.914and 2805.650  Hz. The properties of the models leading to the most probablemodel match and the model match with the smallest mean deviation from theobservations; labelled GS and MS, respectively, are listed in Tab. 5.6. Themodels have very similar physical characteristics but their matches show verydi erent character (Fig. 5.16). The most probable model match shows thehighest 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 frequenciesbeing considered here) shows the best match to those anomalous, magneti-cally perturbed modes to be an ‘ = 2 mode. The most probable model in thiscase matches the lowest two frequency groups with an even-even ‘ pattern,1885.2. Matching the true modes of HR 1217and then continues with an odd-even alternating pattern. The best meanmatch, on the other hand, shows an even-odd alternating ‘ pattern for mostof the modes, except near the highest two frequency groups. There, the pat-tern switches to an even-even ‘ match. This behaviour is illustrated in theSaio plots of models GS and MS (Fig. 5.16). The models GS and MS matchwell (with averages deviation of 0.23 and 0.09  Hz, respectively) the selectedfrequencies at 2603.832, 2720.914 and 2805.650  Hz; but, this comes at theexpense of poorly matching the other frequencies. Model GS has modes thatdeviate at most by  18  Hz while model MS | having a slightly smallerdeviation range | has its largest deviation between observations and theoryof about 10  Hz.1895.2.MatchingthetruemodesofHR1217T able 5.6: Some prop erties of the most probable mo dels of HR 1217. The most p robable (using either of the G au s sian orLaplace probabilit y distributions) mo del (GS) is obtained b y matc hing frequencies at 2603.832, 2720.914 and 2805.650  Hz(see T ab. 5.3). The lo w est mean deviation mo del (MS) is also listed. Deviation refers to the absolute di erence b et w een amo del frequency and the b es t matc hed observ ed frequency and  is the standard deviation. Columns with probabilit y lab els1 and 2 refer to the order of the probabilit y from eac h of the Gaussian or Laplace probabilities. F or example, 1 G is the mostprobable Gaussian prob abilit y and 10 L w ould b e the 10th most probable Laplace probabilit y .Mo d e l ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el 1 probabilit y lab el 2GS 1.50 0.740 0. 010 1.40 1.8 0. 12116.3 0.00 3.8587 0.8886 0.2507 66.241740.21 0.29 0. 23 0.19 1 G 1 LMS 1.50 0.740 0.010 1.20 1.6 0.12646.8 0.00 3.8694 0.8728 0.2214 72.651930.08 0.11 0. 09 0.04 138 G 64 L1905.2. Matching the true modes of HR 1217Figure 5.15: Frequency deviations for the most probable model matched toMOST frequencies at 2603.832, 2720.914 and 2805.650  Hz only. The model does areasonable job of matching frequencies to within about  2  Hz. The obvious exceptionis for the potentially small-spaced frequencies that are spaced by between 1.5{3  Hz fromtheir nearest frequency neighbours. Those frequencies deviate from the observations bybetween  4 and 20  Hz.1915.2. Matching the true modes of HR 1217Figure 5.16: The best matched models using a forced match to the magnetically perturbedobservations. Panel labels are the same as those in Fig. 5.8. The left panel shows the mostprobable (using either of the Gaussian or Laplace probability distributions) model obtainedby matching frequencies at 2603.832, 2720.914 and 2805.650  Hz only. The right panelshows the model with the lowest mean di erence between the theory and observationsusing 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 toour models, we start by choosing only the frequencies identi ed in both theMOST data and the data of Kurtz et al. (2005). We exclude frequency  26= 2791.384  Hz (Tab. 3.1) from our new list. That frequency was commonto the MOST observations and to those of Kurtz et al. (2005). It is also one1925.2. Matching the true modes of HR 1217of 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 addfrequency  1 = 2603.832  Hz to the new listing of frequencies because it ispotentially a magnetically perturbed mode. Table 5.7 lists the new frequencysub-set to which our models will be matched.1935.2.MatchingthetruemodesofHR1217T able 5.7: Selected frequencies iden ti ed using the MOST and Kurtz et al. (2005) data. T h is table giv es frequencies fou nd b yb oth MOST and Kurtz et al. (2005) with the omission of all closely ( . 3  Hz) spaced frequencies and with the addition of 1 of T ab. 3.1. A comparison should b e made with en tries in T ab. 5.3. Both 1 and 3  uncertain ties on the  tted parametersare e stimated from 100,000 b o otstrap realizations. F requency uncertain ties mark ed b y  are a v erage uncertain ti e s based onthe separation from the mark e d frequency to the nearest (  rotationally split) frequency plus its uncertain t y (see the text fordetails). The phases are referenced to the time of the  rst observ ations of HR 1217 (= HJD 2451545.00 + 1769.42 da ys).#  [  Hz] A [  mag]  [rad]  S/N   3    A 3  A   3   1 2603.832 39 2.22 3.8  0.028 0.080 6 15 0.34 0.933 2619.495 125 5.70 9.2  0.015 0.039 6 16 0.11 0.296 2652.924 365 5.56 24.2  0.004 0.011 6 16 0.04 0.1011 2687.420 467 4.46 29.7  0.004 0.009 6 16 0.03 0.0815 2720.914 764 1.83 41.6  0.002 0.005 6 16 0.02 0.0519 2755.393 118 3.45 8.9  0.015 0.039 6 16 0.17 0.4523 2788.872 3 7 0.24 3.6  0.034 0.096 6 16 0.29 0.8028 2805.650 8 4 1.10 6.9  0.488  0.514  6 15 0.15 0.371945.2. Matching the true modes of HR 1217Figure 5.17 plots the Gaussian probabilities of the most plausible modelmatches to the observations of Tab. 5.7. We can compare the probabilitydistributions in each parameter plane to those using di erent super/sub-setsof the identi ed MOST frequencies. Figures 5.3, 5.4, and 5.10 show probabil-ities for the magnetic  eld strengths that are broadly distributed around 4kG and  8 kG. This is also true of the probability distribution around themagnetic  eld strength shown in Fig. 5.17. Of all the frequency combinationsexplored in this chapter, the most probable magnetic  eld strengths tend tocluster in groups ranging from 1{5 kG and 8{10 kG. There are no preferredmodels that have a magnetic  eld strength of B = 0 kG. Of course, suchconclusions come with the caveat that we cannot match the observations towithin their measured precision. The magnetic  eld strength ranges shouldonly be taken as a rough guide and not as statistical con dence limits. Theprobability space around the mass parameter favours models with massesless than about 1.5 M and the composition (Z/X) seems to loosely favourvalues 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., logg,show little variation when matches are made to di erent combinations of thefrequencies listed in Tab. 3.1.1955.2. Matching the true modes of HR 1217Figure 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  elds that fall withintwo broad groups. The  rst probability grouping is between  1{5 kG and the secondrange is between about 8{10 kG. This feature is common in all probability plots presentedin this chapter. Likewise, the probability distributions for the percentage of main sequencelifetime completed, log10g, and for the models’ convective core mass are similar in all ofthe probability plots in this chapter.We label the model with the most probable (lnP = -14.02) match to theobserved frequencies of Tab. 5.7 PGS, and list its properties in Tab. 5.8. Aplot of its frequency deviations is given in Fig. 5.18 and a Saio plot; showinghow the frequencies of model PGS change with magnetic  eld strength, isgiven in appendix D (Fig. D.14). The model has a magnetic  eld strength1965.2. Matching the true modes of HR 1217that falls in the high range of 8.8 kG. The mean deviation between themodelled modes and the observed periodicities is 1.40  Hz. The modes de-viate from the observations by less than 4  Hz. The magnetically perturbed(anomalously-spaced) pulsation modes located at either end of the observedfrequency range are best identi ed as ‘ = 1 modes (for model PGS). Othermodes alternate between ‘ = 0 and 3.1975.2.MatchingthetruemodesofHR1217T able 5.8: Some prop erties of a selected mo del of HR 1217. This mo del has the highest Gaussian probabil it y (ln P = -14.02)of the mo dels matc hed to the frequencies listed in T ab. 5.7 (using 50  3  errors).Mo del ID M/M  X Z Age ( Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz)PGS 1.50 0. 700 0.014 1.05 1.4 0. 14348.8 0.00 3. 8645 0.8826 0.2361 69.251866.36 0.57 1.40 1.511985.2. Matching the true modes of HR 1217Figure 5.18: Deviations between the frequencies of model PGS (Tab. 5.8) and theMOST frequencies. Plot symbols are the same as those in Fig. 5.6. The model PGS hasthe largest probability (lnP = -14.02) of the models matched to the frequencies listed inTab. 5.8. The mean deviation between the model frequencies and the MOST observations(as listed in Tab. 5.8) is 1.40  Hz. The model diverges from the observations by 3.89  Hzin the worst case.1995.2. Matching the true modes of HR 1217Two of the best matched model spectra to the frequencies of Tab. 5.7have already been identi ed in the multiple-matching section 5.1.2. ModelM3 (see Tab. 5.1 for the physical parameters of this model) has the smallestmean deviation between the calculated and observed frequencies. Its meanfrequency deviation is 0.640  Hz and the deviations of all modes from theobservations are shown in Fig. 5.19. This model matches the anomalouslyspaced modes at  2600 and 2800  Hz as ‘ = 4 modes and associates theother most closely matched modes with an alternating ‘ = 2 and ‘ = 1pattern. 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 evolvedto near the end of its main sequence life and has burned 99.93% of theHydrogen in its core. The magnetic  eld matched from this model (= 4.8kG) is in reasonable agreement with the measured values of 2{4.5 kG. Forcomparison we also show the deviations for the second most probable modelmatch 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 than4  Hz and has a mean deviation of 1.29  Hz. M6 has a mass and metallicitythat are higher (1.45 M , Z = 0.010, Z/X = 0.0139) than those of model M3and has a magnetic  eld strength (B = 1.2 kG) that is smaller than thatidenti ed in model M3. As in model M3, the anomalous, non-asymptoticallyspaced modes are identi ed as ‘ = 4 modes. The other most closely matchedmodes of model M6 alternate with angular degree values of ‘ = 0 and 1.2005.3. On discriminating between modelsFigure 5.19: The frequency deviations for models M3 and M6 of Tab. 5.1. Plot symbolsare the same as those in Fig. 5.6. Model M3 is the model with the smallest mean devia-tion (= 0.640  Hz) between the theoretical eigenfrequencies and the frequencies listed inTab. 5.8. Model M6 is the second most probable model (lnP = -14.05) when matchedto the frequencies in Tab. 5.8. Note that both models were already identi ed in x 5.1.2.In that section, we allowed a single model frequency to match multiple observations. Theexclusion of the potential small spacings (frequencies spaced by  3  Hz) results in abetter (smaller) mean deviation than those obtained earlier this chapter.5.3 On discriminating between modelsIn chapter 4 we calculated a large grid of pulsation models that includedmagnetic perturbations. We adopted a large range in metallicity (Z = 0.008{2015.3. On discriminating between models0.022) because the peculiar abundance patterns on the surface of the Ap starslimits 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 luminosityand e ective temperature of HR 1217 (seex4.1.1). A thorough exploration ofthe range of magnetic  eld strengths that provided asteroseismically relevantresults for HR 1217 had never been performed, so we adopted a wide rangein B for our model calculations. We calculated 51,795 models that, as awhole, contain millions of modes to be matched to the frequencies determinedfrom the time-series of MOST photometry. In the preceding sections of thischapter we showed that we were able to match models that have a meandeviation between the theoretically calculated modes and observations of lessthan a few  Hz. Matching models to a fractional accuracy of about 0.05%is an amazing accomplishment considering the complexity of the models andthe sensitivity of the pulsation modes to the magnetic perturbations. (Thehigh frequency oscillations in the Sun are matched to a fractional accuracyof about 0.3%). Those models are; however, a poor statistical match becausethe MOST frequencies are determined to such a high precision. The currentstate of the observations of roAp star oscillations is so highly precise thatthe model calculations are being stretched to the limits of their applicability.How well can we distinguish between reasonable models of HR 1217 giventhe small match probabilities with our current grid?The short answer is that matching modelled frequencies to (ultra-)preciseobservations (like those from MOST) is an extremely sensitive tool to dis-criminate between models. In Fig. 5.20 we plot the minimum frequencydeviation required to match 50 and 75% (left and right panels of that  gure,2025.3. On discriminating between modelsrespectively) of the observed frequencies of Tab. 5.3 as a function of magnetic eld strength. (The model mass and composition Z/X are also identi ed inthat  gure.) Immediately we can see that there are only 142 of 51,795 modelsthat match 50% of the observed frequencies below a 1  Hz threshold. Wecan 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 matchingthreshold to between  2{3  Hz. In that frequency range, 75% of the mostobservations are matched by 69 of the 51,795 models. Put another way, 136models within our grid match between 6 and 8 of the MOST observations( 50{66%) to an accuracy that is less than 1  Hz. Figure 5.21 shows thesame type of plot but we require matching 100% of the MOST observationslisted in Tab. 5.3. There are 58 models that can match 100% of the observedperiodicities with frequency deviations that are not greater than 6  Hz.2035.3. On discriminating between modelsFigure 5.20: Minimum frequency deviations required to match 50 and 75% of the MOSTobservations as a function of magnetic  eld strength. The left panel plots the minimumfrequency deviation required to match 50% of the MOST observations. The colour baron the bottom identi es the mass of the model. Symbols shown in the legend identifythe composition (Z/X) ranges ( sub-solar Z/X,  solar Z/X,  super-solar Z/X) of themodels. 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  Hz and 69 models (= 0.1% of the models in the grid)match 75% of the MOST periodicities with deviations between 2{3  Hz.2045.3. On discriminating between modelsFigure 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 frequencydeviations between 4.7{6  Hz.The model (labelled LPC) with the smallest frequency deviation required2055.3. On discriminating between modelsto match 100% of the MOST observations (Tab. 5.3) has properties thatare listed in Tab. 5.9. The frequency deviations for model LPC are shownin 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 frequenciesat about 2600 and 2800  Hz are ‘ = 0 and 2 modes, respectively. The modesalternate in a pattern that would suggest that the closely (small or secondorder) spaced modes are in ‘ = 1, 3 and ‘ = 0, 4 pairs. Of course, thematch in model LPC gives a small spacing that is between 2{3 times whatis observed (. 3  Hz). The magnetic  eld of model LPC is 2.4 kG and isconsistent with the lower end of the L uftinger et al. (2008) measurement (Bbetween 2{4.4 kG).2065.3.OndiscriminatingbetweenmodelsT able 5.9: Some prop e rt ie s of a selected mo del of HR 1217. This mo del has the smallest fr e qu e n c y deviation (= 4.759  Hz)required to matc h all of the MOST frequencies. The column lab elled as probabilit y lab el refers to the order of the probabili t yfrom eac h of the Gaussian or Laplace pr obabilities. F or example, 1 G is the most probable Gauss i an probab ilit y and 10 Lw ould b e the 10th most probable Laplace probabilit y . The probabilit y ratio is de ned as the Gaussian probabilit y of th e mo deldivided b y the Gaussian probabilit y of the most probab le mo del (se e T ab. 5.4). The mo dels in this table w ere matc hed to allfrequencies listed in T ab. 5.3 using 50  3  errors.Mo del ID M/M  X Z Age (Gyr)  M conv :cor e /M ?B (kG) M conv :env : /M ? Log 10 (T eff ) Log 10 (L/L  ) Log 10 (R/R  )   (  Hz)cut-o (  Hz) median deviation (  Hz) mean deviation (  Hz)  deviation (  Hz) probabilit y lab el probabilit y ratioLPC 1.75 0.740 0.022 0.65 1. 8 0.17462.4 0.00 3.866 0.9222 0.2535 70.501966.12 2. 55 2.43 1.60 3460 G, 1740 L 7.476  10  352075.3. On discriminating between modelsFigure 5.22: Deviations between the frequencies of model LPC (Tab. 5.9) and theMOST frequencies. The model LPC has the smallest frequency deviation ( 4.7  Hz) ofall the models that is needed to match 100% of the MOST observations. This model isstill 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 MOST2085.3. On discriminating between modelsobservations, we next see if our observed variations are matched more often toa 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 theangular degrees (‘ = 0{4). A match occurs if the best  t model frequenciesfall within the 3 uncertainties for the periodicities listed in Tab. 5.3. Thehistograms showing the matched ‘ values for each of the observed frequenciesare 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 nostatistically preferred ‘ value for any of the observed frequencies. This is alsotrue for magnetic models with B = 2 and 4 kG (Fig. 5.24) and B = 6 and 8kG (Fig. 5.25). The modelled frequency matches tend to alternate betweeneven and odd modes. If we assume that for a particular frequency an oddor even mode match is equally likely, we could immediately eliminate half ofthe models in our grid by applying an external constraint on the mode parityfor that match.2095.3. On discriminating between modelsFigure 5.23: Histograms of matched model frequencies. A match occurs if the best  tmodel frequencies fall within the 3 uncertainties for the periodicities listed in Tab. 5.3.The left panels present histograms of the matched frequencies in all of the magnetic andnon-magnetic models. The right panels show matches in cases where B = 0 kG. Eachcolumn shows modes with di erent angular degrees, increasing from ‘ = 0 at the top to‘ = 4 at the bottom. There are no preferred matches for ‘ values. Matches near 2650  Hzseem to be matched less often for all ‘ values when compared to the other frequencies.2105.3. On discriminating between modelsFigure 5.24: Histograms of matched model frequencies. The format is the same as that inFig. 5.23. The left panels now show only models with B = 2 kG, and the right panels onlythose models with B = 4 kG. Model frequencies near 2650  Hz are again under-matched,especially for the ‘ = 0, B = 4 kG case.2115.3. On discriminating between modelsFigure 5.25: Histograms of matched model frequencies. The format is the same as thatin Fig. 5.23. The left panels now show only models with B = 6 kG, and the right panelsonly those models with B = 8 kG. The ‘ = 0 and 2 modes tend to be matched more oftennear 2800  Hz in the 8 kG models. As in the cases with di erent magnetic  elds, modelsmatched to the observed frequencies near 2650  Hz are less common.2125.3. On discriminating between modelsWe have used the largest grid of magneto-acoustic modes to date to searchfor a match to the most precise photometric data ever measured on the roApstar HR 1217. We are able to match models to within (on average) a few Hz. This represents an impressive accomplishment and gives hope that fu-ture advancements in theoretical modelling of roAp stars will yield accurateand meaningful constraint on the physics of these unique stellar structures.The matches explored are not perfect and local variations in the probabilityspace of our modelled parameters occur for each of the matching scenariosdiscussed in this chapter. The magnetic  eld perturbations are cyclic in na-ture, and as such, leads to parameter degeneracies when matching models toobservations. It is important to note that by using a small grid of models,one could, conceivably,  nd a reasonable match (with say a mean deviationof 1{2  Hz) to the observations. Such a match may not be (and is most likelynot) the best global match to the data. Likewise, the large span of Z usedin our model grid introduces degeneracies that allow a large range in modelmass and evolutionary state to enter the luminosity-e ective temperature er-ror box into which HR 1217 falls (x4.1.1). In terms of probabilities, Figs. 5.3,5.4, 5.10, and 5.17 show probabilities for the magnetic  eld strengths thatare broadly distributed in groups ranging from 1{5 kG and 8{10 kG. Thereare no preferred models in our grid that have a magnetic  eld strength of B= 0 kG. The probability space around the mass parameter favours modelswith masses less than about 1.5 M and the composition (Z/X) seems toloosely favour values less than 0.02. If the mean is used as a discriminantbetween models, there are no preferred parameter combinations. The meanor median frequency deviation; when used as the only matching statistic,2135.3. On discriminating between modelsdoes not provide enough of a constraint on the models to draw even broadconclusions. The ranges in parameters we give based on our probabilitydistributions should be used as guidelines for future modelling e orts andshould not be taken as strict statistical limits on the physical parametersof HR 1217. We do not place con dence limits on any of the physical pa-rameters of HR 1217 from our modelling e orts. The lack of a statisticallymeaningful match, and complications arising from the degeneracies in someof the model parameters, makes claims of precise parameter limits dubious atbest. A model independent measure of the parity of the observed oscillationswould greatly constrain the matched model properties. We will discuss thequality of the matches and suggest alternate modelling avenues that shouldbe explored in the (following) concluding chapter.214Chapter 6Summary, conclusions, &future workThe roAp star HR 1217 exhibits a rich eigenspectrum that obeys the regularasymptotic (x 1.1.1) spacing between its oscillation modes. The observedfrequencies are spaced by an alternating pattern of  33.5 and 34.5  Hz |values that are interpreted as half of the large spacing   | with an im-portant deviation. A global, multisite, photometric campaign (Kurtz et al.,1989) revealed an anomalous frequency spacing of  50  Hz that could notbe explained by asymptotic theory. Much later, Cunha (2001) theorized thatthe strong magnetic  eld ( 2{5 kG) of HR 1217 could signi cantly perturbits normal oscillation modes. She predicted the existence of a \missing mode"that would explain the anomalous increase in the regular spacings. At nearlythe same time as Cunha’s prediction there was another (global) photometriccampaign 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 numberof new periodicities that agreed well with the predictions of Cunha. A non-asymptotic, potentially magnetically perturbed, frequency spacing of  15 Hz was found at about 2785  Hz along with a frequency group that showeda spacing of 2.5  Hz. This was cautiously interpreted as a potential second215Chapter 6. Summary, conclusions, & future workorder separation, or small spacing (x 1.1.1), that could possibly constrainthe age of HR 1217. It was shown for the  rst time that the magnetic  eldplays a direct role in the pulsation spacings of a roAp star. The photomet-ric variations identi ed in these earlier ground-based campaigns are shownschematically in Fig. 1.7.Motivated by these earlier results, MOST observed HR 1217 near con-tinuously as a Fabry target (x 2.1) for about 30 days in late 2004. Thesatellite collected more than 68,000 photometric data points with a duty cy-cle of 95%. These data represent the ultimate photometric measurements onHR 1217. We used CAPER (Cameron et al., 2006, 2008); a tool developedspeci cally for time-series analysis of MOST data (x 2.2), to determine theoscillation frequencies of HR 1217. The Fourier parameters extracted fromthe MOST light curve are summarized in Tab. 3.1. The uncertainties inour  tted parameters were determined using 100,000 bootstrap samples. Intotal there were 29 frequencies identi ed, each having a S/N  3.5 (& 2.5 detection) to within uncertainties. (The reduction is described in full detailin chapter 3.) The most precise amplitude determined from the MOST datais 6  mag, 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(x 1.4) below:1. It has been known since the early 1980s that HR 1217 is a multiperiodicvariable star with near equally spaced modes, alternately spaced by 33.5 and 34.5  Hz. The most recent con rmation of this was theanalysis of the Whole Earth Telescope data by Kurtz et al. (2005), thatachieved a duty cycle of 36% and reached an unprecedented precision216Chapter 6. Summary, conclusions, & future workof 14  mag for a ground-based photometric study. We will be ableto unambiguously identify the spacings in the MOST data because ofits near continuous coverage and ultra-high precision. Are the alreadyidenti ed periodicities constant over time for all observations on HR1217, or do they vary over time, pointing to a selective mode dampingmechanism 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 untilthe recent results of Kurtz et al. (2005, 2002). Those authors identi eda previously unidenti ed frequency near 2790  Hz that could only beexplained as a magnetic perturbation Cunha (2001). Explaining thatfrequency (or lack of that frequency) has driven the theory of roAp staroscillations for more than two decades. Is this frequency identi ed inthe MOST data on HR 1217 and is it stable since the last observationsof Kurtz et al. (2005)?The average di erence between the frequencies extracted from the data ofKurtz et al. (2005) and those from the MOST data (x3.3) is 0.08 0:09  Hz(3 uncertainty), indicating that all frequencies identi ed 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-basedcollaboration of Kurtz et al. (1989), even though there are some di erencesin the amplitudes of the dominant peaks (e.g., x 3.3). Frequencies near the\missing mode" at  2790  Hz (present in the WET data of Kurtz et al.2005, but were not found in the Kurtz et al. 1989 data) are identi ed in theMOST photometry. This suggests that those frequencies are stable over a217Chapter 6. Summary, conclusions, & future workbaseline of at least 4 years. It should be noted that these frequencies showthe largest deviations ( 0:2  Hz) from those determined by Kurtz et al.(2005) because they are located near a harmonic of the orbital period of theMOST satellite (16  165  Hz = 2640  Hz and 17  165  Hz = 2805  Hz)and are potentially in uenced by the stray light (x 2.1). That being said,the deviations between the MOST frequencies near 2790  Hz and the corre-sponding Kurtz et al. (2005) frequencies are near their 3 uncertainties. Anydeviations between the MOST and Kurtz et al. (2005) frequencies are wellbelow the resolution of 1/T = 0.4  Hz. Because the MOST and WET datashow common frequencies, the alternating values of  33.5 and  34.5  Hzremain the same. Those values are consistent with acoustic modes that arealternating 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 x 1.3).3. We expect to observe a number of new periodicities in the data becauseof the high level of photometric precision and the near contiguous datasampling of the MOST satellite. Can new periodicities be identi edand, if so, can they be used to constrain the physics of this magneticpulsator?4. A recent spectroscopic study on HR 1217 by Mkrtichian & Hatzes(2005) identi ed new frequencies at  2553 and 2585  Hz. These fre-quencies are interesting because they approximately match the alter-nating  33.5 and 34.5  Hz pattern previously observed in HR 1217,but were not identi ed in the photometry of Kurtz et al. (2005). Doesthe MOST data uncover those potentially new periodicities?218Chapter 6. Summary, conclusions, & future workThere were 7 new frequencies identi ed in the MOST data (x3.3). Recur-ring frequency spacings of 1.5 and 2.5 Hz are evident in the MOST data.Main sequence models of A-type stars (e.g., x 4.2) have small spacings (x1.1.1) that are about 3  Hz and higher. This small spacing can potentiallyconstrain the age of the star because its value is most sensitive to changesin 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, thesmall spacing (e.g., Dziembowski & Goode 1996). This was also illustratedin x 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 thatit is not immediately obvious whether or not the frequencies are rotationalsplittings or \small spacings". A repeating pattern of  1.5 and  2.5  Hzis evident if  9 is chosen as the main (not a rotational splitting) frequency inthe grouping. However, the uncertainties suggest that the di erence between 10 and  11 is not a rotational splitting. One would expect the frequency set 10,  11, and  12 to be a rotational triplet based on the alternating    =2.It is possible that magnetic e ects are causing a non-repeating pattern of 1.5 and  2.5  Hz, or that the closely spaced modes are convolved insuch a way that a stable least squares frequency solution exists where  10and  11 are spaced close to, but not at (within the 3 errors) the rotationfrequency of the star. Because a spacing of 2.5  Hz had been observed byKurtz et al. (2005), and there is more than one example of this spacing inthe MOST data, we believe that these spacings are not related to unresolvedrotational splittings. It should be noted here that the rotation frequencyobserved in the MOST data (0.924  0.047  Hz, see x 3.2 and 3.3) is de-219Chapter 6. Summary, conclusions, & future worktermined 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 frequencymeasurements (x 3.2).A new MOST frequency at  1 = 2603:832  Hz is inconsistent with thenew frequencies at  2553 and 2585  Hz 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 thatare not present in photometric data. This was illustrated for the case of theroAp star HD 134214 (Cameron et al. 2006,x2.3.1) where MOST photometryidenti ed a single periodicity during a short observing run while Kurtz et al.(2006) identi ed up to 6 periodicities in their radial velocity data (not ob-served synchronously with MOST). It is possible that there is some unknownselection e ect in the atmospheres of roAp stars that re ects certain modesat di erent depths, causing results from spectroscopy to sometimes be dis-crepant with photometric measurements.5. In order to determine the reliability and uniqueness of the periodicitiesin a multiperiodic variable, we need to develop tools that can test theprecision and resolution of our data set. Techniques we have developedfor other MOST targets will be applied to the HR 1217 data. The res-olution of our data set and the assertion that closely spaced frequenciesare spaced by the rotation rate of the star will be tested.The bootstrap technique described in x 2.2 provides a means to test ifwe are extracting resolved (or unresolved) frequencies in the data. In x 3.3220Chapter 6. Summary, conclusions, & future workwe 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  t to become unstable and the new frequencies (Tab. 3.3) clearlyinteract with the previously identi ed, resolved, parameters of Tab. 3.1 (seeFig. 3.8). Because of this, we are con dent that we have extracted onlyresolved frequencies.The ultra-precise photometric measurements of the MOST satellite reveala number of new periodicities that are spaced close to what is expected foran asymptotic small spacing (.3  Hz) and one new frequency that is spacedby an amount ( 15  Hz) that is reminiscent of the magnetically perturbedmodes identi ed by Kurtz et al. (2005). We next summarize the modellinge orts we used to describe the observed oscillations in HR 1217.In recent years there have been a number of studies that have attemptedto model speci c roAp stars. Cunha et al. (2003) examined a small numberof stellar models for HR 1217 in hopes of determining its evolutionary status.They did not include the e ects of a magnetic  eld 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  Equ and 10Aql using  10 A-star models (including magnetic perturbations using themethod of Saio & Gautschy 2004) and data collected by MOST. HD101065;a.k.a. Przybylski’s star, was studied by Mkrtichian et al. (2008), who alsouse only a few stellar models (again using the method of Saio & Gautschy2004) to explore the physics of that star. Their results were discussed in moredetail in chapter 5. Brandao et al. (2008) also use only a few (non-magnetic)models to study the spectrum of  Cir that was obtained from photometry221Chapter 6. Summary, conclusions, & future workcollected with the WIRE (NASA’s Wide Field Infrared Explorer) satellite(Bruntt et al., 2009; Buzasi, 2002). Each of the above studies places somelimits on the properties of the star they were studying, even without a thor-ough parameter space search. Interestingly, the modelling e orts referencedabove that included magnetic e ects arrived at \best matched" magnetic eld strengths that were about a factor of 2 larger than what was measuredfrom spectropolarimetric observations.Our study explores the magnetic e ects on the pulsation frequencies formodels of HR 1217 for the  rst time and signi cantly extends the numberof models and the breadth of the parameter space explored when comparedto earlier asteroseismic studies on HR 1217, or any other roAp star. Thehigh quality MOST data and this extensive grid of state-of-the-art models ofmagneto-acoustic modes for HR 1217 is a  rst, and is unlikely to be surpassedin the near future. Our grid covers the observable characteristics of HR1217 (Te = 7400+100 200 K, L = 7.8  0.7 L , B  2{5 kG), and includesmagnetic perturbations to the calculated normal modes of the models. Thesemodels are outlined in detail in chapter 4. This grid extensively coveredthe parameter space (x 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 = 0.700to 0.740 in steps of 0.020) and frequency range (‘ = 0{4 over the range1900{3100  Hz.) appropriate for HR 1217. Mixing length parameters of = 1:4; 1:6; and 1.8 are adopted and the models in the grid have a largespacing (   (M/R3)1=2) that ranges from  57{81  Hz. The pulsationmode calculation was tested using a new method developed by Kobayashi(2007,x4.2.1) for terrestrial seismology. The modes calculated using the JIG222Chapter 6. Summary, conclusions, & future workcode were con rmed by comparison to modes calculated using Kobayashi’smethod. (Showing the consistency between the modes calculated using thetwo di erent pulsation codes was the last of the modelling goals outlined inx 1.3) Our model grid, composed of 569 A-star models that fall in the HR1217 luminosity{e ective temperature error bounds, is the most extensivegrid of models used to study the magnetic e ects on the pulsation modesof roAp star to date. The magnetic perturbations to the eigenfrequencies ofthe models were estimated using the variational method of Cunha & Gough(2000); see x 4.3, assuming a dipolar magnetic  eld, with polar strengthsranging from 1 to 10 kG in steps of 0.1 kG. This brings the total number ofmagneto-acoutic pulsation models to 51,795.The modelling goals outlined in x 1.3 have a common thread: Namely,how closely can we match the frequencies (and or spacings) observed byMOST, and what constraints can we place on the physical parameters of thestar?In chapter 5 we showed that we are able to match models to within (onaverage) a few  Hz. A mean match of a few  Hz represents a fractional accu-racy of about ( 1.5  Hz / 2700  Hz =) 0.05%. For comparison, the high fre-quency oscillations in the Sun are matched to a fractional accuracy of about0.3%. This is an extraordinary accomplishment considering the complexity ofour models and the large perturbations introduced by the inclusion of a mag-netic  eld. The matches obtained are not perfect; but, matching modelledfrequencies to the MOST frequencies has proven to be an extremely sensitivetool to discriminate between models. For example, 136 models within ourgrid match between 6 and 8 ( 50{66%) of the MOST observations to an223Chapter 6. Summary, conclusions, & future workaccuracy that is less than 1  Hz. That represents only (100  136/51,795=) 0.3% of the models in our grid. We were able to show that there are58 models that can match 100% of the observed periodicities with frequencydeviations that are not greater than 6  Hz (x 5.3). There are about (100  17579/51796 =) 33% of the models in our grid that match the observed largespacing of HR 1217 ( 68  Hz) to within about 1.25  Hz. (To arrive at avalue for the large spacing we calculated an average over all n and ‘ for fre-quencies in the range of 2618.600 to 2792.249  Hz.) Using the large spacingas a discriminant between models is less e ective than matching individualfrequencies. The matching of the observed frequencies to our modelled modeseliminates a large percentage of parameter space that is allowed within theluminosity-e ective temperature error bounds for HR 1217. We cannot  nda model that matches both the magnetically perturbed modes (spaced by 15  Hz) and the oscillations spaced by the small spacing (.3  Hz) simulta-neously. This presents a signi cant challenge to the modelling of roAp starsand 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 verylow. While we achieve average deviations of a few  Hz between our measuredperiodicities and the modelled eigenmodes, the measured precision on theMOST frequencies is on the order of 0.01  Hz. The magnetic  eld perturba-tions are cyclic in nature, and as such, there are parameter degeneracies whenmatching models to observations. The large span of Z used in our model gridintroduces an indeterminacy that allow a large range in model mass and evo-lutionary state to enter the luminosity-e ective temperature error box into224Chapter 6. Summary, conclusions, & future workwhich HR 1217 falls (x4.1.1). The magnetic  eld of HR 1217 was estimatedby Bagnulo et al. (1995) to be 3.9 kG. More recently, L uftinger et al. (2008)measured a  eld variation over the rotation period of HR 1217 of 2.2 to 4.4kG. We have illustrated in chapter 5 that probabilities of a model match forthe magnetic  eld strengths broadly distributed in groups ranging from 1{5kG and 8{10 kG. In previous studies on roAp stars (e.g., Gruberbauer et al.2008, and the others mentioned above), the modelled magnetic  eld strengthswere about a factor of 2 larger than the spectroscopically measured values.The range in probability for a match to the magnetic  eld strength of HR1217 in our models are consistent with both the spectroscopically determinedvalues and twice those values. We conclude that there are probably magnetic eld matches for those other roAp stars that are consistent with external mea-sures, provided the model parameter space is explored in more detail. Thereare no preferred models in our grid that have a magnetic  eld strength of B =0 kG. The probability space around the mass parameter favours models withmasses less than about 1.5 M and the composition (Z/X) seems to looselyfavour values less than 0.02. There is no meaningful constraint on the mixinglength parameter  . If the mean is used as a discriminant between models,there are no preferred parameter combinations because there is a wide rangeof models that have low mean deviation matches. The ranges in parameterswe give based on our probability distributions should be used as guidelinesfor future modelling e orts and should not be taken a strict statistical limitson the physical parameters of HR 1217. We do not place con dence limits onany of the physical parameters of HR 1217 from our models. The lack of astatistically meaningful match, and complications arising from the degenera-225Chapter 6. Summary, conclusions, & future workcies in some of the model parameters, makes claims of precise limits dubiousat best. The current state of the observations of roAp star oscillations is sohighly precise that the model calculations are being stretched to the limitsof their applicability.In stellar seismology we usually look for the closest matched model andthen explore di erent physics to see how the models can be improved. Stellarmodels do not generally correspond to simple analytical models where oneparameter can be tweaked to match an observation without having to changeother parameters as well. That is simply why we have explored a largecoverage in the parameter space relevant for models of HR 1217. As anaside, most of the modes observed in the Sun could not be matched in the80s and early 90s. The best helioseismic agreement at high frequencies stillshow a discrepancy that is many standard deviations from the observations.Our grid of magnetically perturbed oscillation models represents a unique rst step to try to constrain the properties of a roAp star. The modelswe calculate use standard physics and focus on the addition of a magneticperturbation to the oscillation modes as an important ( rst) non-standardaddition. Calculating our models came at a signi cant computational costso we focused on as wide of a parameter space as we possibly could. Thiscomes at the expense of  ne grid resolution. We found that our current gridresolution in age and mass is not su cient to be able to properly constrainthe properties of HR 1217. Take, for example, model E of Tab. 4.1. Over itscorresponding evolutionary sequence the large spacing changes from 80 to 68 Hz. There were 11 models with detailed seismic information output. Thechange in   is not equally spaced along an evolutionary track and ranges226Chapter 6. Summary, conclusions, & future workby about 0.2  Hz near 68  Hz to about 1  Hz near 80  Hz. We expecta percentage change in   to roughly correspond to the same percentagechange for a given mode along an evolving set of models. That means thefrequencies between the models in our grid (that are adjacent in age) changefrom about 5 to 40  Hz near 2700  Hz. A model that matches frequencies towithin about 5  Hz can potentially be changed in age slightly to get a bettermatch | at least in the non-magnetic case. The behaviour of the modesin the magnetic models is more di cult to predict because of the non-linearcoupling between frequencies, magnetic  eld strength and the structure ofthe outer layers of the model.So where should future modelling e orts be focused? Recent, non-standard,models were used to explore the excitation physics of the roAp stars (Gautschyet al. 1998,Balmforth et al. 2001,Saio 2005, and Th eado et al. 2005,2009). Al-though some of those models excited oscillations in a very speci c frequencyrange, or over speci c temperature and luminosity ranges, none were success-ful in calculating a globally applicable excitation model. This leaves us witha number of unsolved problems in our understanding of the roAp star pulsa-tions that should also be addressed. These include: 1) the understanding ofwhat excites the observed periodicities in these stars, 2) how the pulsationmodes interact with the chemical gradients in the upper stellar atmosphere,3) why some roAp stars show frequencies above the theoretical acoustic cut-o frequency, and 4) what the e ects of alternate magnetic geometries areon the pulsation modes. We point to recent reviews by Kurtz & Martinez(2000), Cunha (2007), Saio (2008) and Shibahashi (2008) to highlight thecurrent e orts within the asteroseismic community to address these issues.227Chapter 6. Summary, conclusions, & future workBecause previous studies using non-standard model physics focused onthe excitation of the calculated modes, they did not highlight what a ecttheir model alterations had on the individually calculated model frequen-cies and spacings. This is necessary to de ne which non-standard additionsshould be added to future model grids to improve the matches to observa-tions. A  ducial model sequence should be de ned so that modellers withinthe roAp community can easily compare and contrast the addition of newphysics to a standard set of models. To increase the e ciency of grid cal-culations, we suggest that a study be undertaken that de nes the minimumspacing in the basic stellar parameters (M, Z/X,   , B) that is needed tointerpolate modelled frequencies to within a sub- Hz accuracy. Interpolationwithin a small grid of models having a relatively large spread in parametersis generally untrustworthy. We suspect that the sharp jumps in frequencycaused by the introduction of a magnetic  eld could signi cantly complicatethe interpolation of modelled frequencies. Finally, we found that there is noapparent constraint on ‘ from our model grid. We need a model indepen-dent constraint on ‘ (or at least the parity of ‘) to get  rm asterosesimicconstraints.The relatively small deviations between our theoretical models and theMOST observations give us hope that future advancements in theoreticalmodelling of roAp stars will yield accurate and meaningful constraint on thephysics of the unique Ap stellar structures. There is the potential to thor-oughly test the role a magnetic  eld plays on the structure of these starsand to infer the properties of that  eld in the hidden stellar interior. WhenMOST observed HR 1217 in late 2004, it set the standard for photometric228Chapter 6. Summary, conclusions, & future workobservations of roAp stars. The oscillation spectrum of HR 1217 extractedfrom the MOST data is the most precise and complete spectrum ever ob-served for this star, and other roAp stars. 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Monograph serieson nonthermal phenomena in stellar atmospheres244Appendix AHD 127756 and HD 217543parameter tablesTables of  t parameters identi ed in the slowly pulsating Be (SPBe) starsHD 127756 and HD 217543 by the MOST team (Cameron et al., 2008).245AppendixA.HD127756andHD217543parametertablesT able A.1: F requencies i de n ti ed in the star HD 127756 b y the MOST team (Came r on et al., 2008)#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   1 0.0335 7.2 3.60 8.68  0.0009 0.0023 0.2 0.6 0.11 0.302 0.0739 2.2 0.59 6.97  0.0020 0.0055 0.1 0.4 0.22 0.583 0.1300 8.8 3.65 9.50  0.0007 0.0017 0.1 0.4 0.07 0.194 0.1664 7.3 1.27 8.31  0.0004 0.0010 0.2 0.4 0.06 0.165 0.1957 5.0 2.64 8.71  0.0007 0.0018 0.2 0.5 0.10 0.246 0.2530 1.8 4.11 5.12  0.0031 0.0078 0.2 0.4 0.32 0.817 0.2827 1.4 4.27 3.89  0.0032 0.0079 0.2 0.4 0.34 0.888 0.7108 2.3 3.31 5.74  0.0015 0.0038 0.2 0.6 0.15 0.379 0.7393 1.7 5.22 4.80  0.0016 0.0040 0.2 0.5 0.19 0.5210 0.7740 5.1 4.80 9.04  0.0005 0.0013 0.2 0.5 0.06 0.1711 0.8035 6.7 4.24 8.16  0.0005 0.0013 0.3 0.7 0.03 0.0812 0.8280 7.2 4.13 8.22  0.0005 0.0015 0.3 0.7 0.03 0.0913 0.8702 2.4 2.57 5.17  0.0015 0.0048 0.2 0.4 0.15 0.4514 0.9149 10.2 4.18 11.12  0.0004 0.0011 0.1 0.4 0.04 0.1015 0.9595 6.5 2.67 9.80 + 0.0006 0.0015 0.1 0.4 0.06 0.17c ontinue d on next p age246AppendixA.HD127756andHD217543parametertablesT able A.1: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   : : : : : : 0.9595 6.5 2.67 9. 80  0.0006 0.0015 0.1 0.4 0.06 0.1616 1.0236 4.2 0.31 8.74  0.0007 0.0021 0.1 0.4 0.08 0.2217 1.0595 2.3 0.06 5.00  0.0015 0.0042 0.1 0.4 0.15 0.4118 1.1297 1.9 4.55 4.00  0.0024 0.0064 0.2 0.5 0.28 0.7219 1.1418 6.2 1.65 8.24  0.0007 0.0016 0.2 0.5 0.08 0.2120 1.5301 1.2 4.16 4.04  0.0021 0.0051 0.1 0.3 0.22 0.5621 1.7761 1.8 5.56 5.65  0.0017 0.0047 0.1 0.4 0.19 0.4622 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 0.3023 1.8441 5.1 4.67 9.11  0.0010 0.0027 0.2 0.7 0.09 0.2424 1.8685 2.1 4.81 5.08  0.0019 0.0054 0.2 0.5 0.18 0.4925 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 0.1926 2.0171 1.1 3.44 3.87  0.0028 0.0076 0.1 0.4 0.32 0.7927 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 0.19c ontinue d on next p age247AppendixA.HD127756andHD217543parametertablesT able A.1: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   28 2.1160 1.8 0.11 5.10  0.0014 0.0036 0.1 0.4 0.15 0.4029 2.7814 1.5 2.77 5.22  0.0018 0.0049 0.1 0.4 0.21 0.5430 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 1.01248AppendixA.HD127756andHD217543parametertablesT able A.2: F requencies i de n ti ed in the star HD 217543 b y the MOST team (Came r on et al., 2008)#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   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 0.302 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 0.433 0.1201 13.3 1.76 8.28  0.0012 0.0043 0.2 0.7 0.10 0.364 0.1806 5.7 3.03 4.59  0.0016 0.0071 0.3 0.8 0.12 0.575 0.2454 4.6 3.97 5.10  0.0017 0.0074 0.3 0.9 0.12 0.626 0.2904 1.9 2.03 3.15  0.0042 0.0121 0.2 1.7 0.27 1.017 0.3133 3.1 4.67 4.40  0.0024 0.0099 0.3 2.2 0.19 0.758 0.3987 2.0 5.78 3.97  0.0022 0.0053 0.2 0.6 0.19 0.599 0.5147 1.6 2.80 3.32  0.0034 0.0091 0.2 0.6 0.31 0.8210 0.5668 1.7 4.37 3.17  0.0034 0.0091 0.2 0.6 0.32 0.7911 0.6154 2.0 5.61 3.09  0.0026 0.0062 0.2 0.5 0.23 0.5612 0.7772 1.3 1.87 3.11  0.0026 0.0068 0.2 0.5 0.23 0.6113 1.0006 2.1 2.68 3.67  0.0019 0.0051 0.2 0.5 0.16 0.42c ontinue d on next p age249AppendixA.HD127756andHD217543parametertablesT able A.2: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   14 1.5826 2.4 0.72 4.36  0.0016 0.0048 0.2 0.5 0.16 0.4415 1.7164 2.2 1.85 4.21  0.0020 0.0053 0.2 0.5 0.19 0.5216 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 0.4617 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 0.4418 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 0.1819 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 1.2820 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 0.4121 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 0.0222 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 0.28c ontinue d on next p age250AppendixA.HD127756andHD217543parametertablesT able A.2: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   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 0.5224 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 0.5325 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 0.3926 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 0.4227 2.3674 3.0 1.59 4.27  0.0016 0.0046 0.2 0.6 0.15 0.4028 3.7755 2.0 4.00 4.00  0.0019 0.0049 0.2 0.5 0.18 0.5129 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 0.4230 4.0017 3.4 2.13 4.52  0.0014 0.0037 0.2 0.5 0.13 0.3531 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 0.3732 4.1079 2.6 1.03 4.08  0.0025 0.0068 0.2 0.6 0.20 0.52c ontinue d on next p age251AppendixA.HD127756andHD217543parametertablesT able A.2: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   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 0.2434 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 0.3535 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 0.7336 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 0.3637 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 0.4138 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 0.5239 4.6210 1.6 5.40 3.38  0.0029 0.0072 0.2 0.5 0.27 0.7340 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 0.46252Appendix BFrequency lists in cycles day 1It is more common to use units of  Hz when presenting both theoreticaland observational results on roAp stars. However, there are still cases whenunits of cycles day 1 are used. The latter units are particularly useful whendescribing the  ne (rotational) spacings observed in roAp stars. In this ap-pendix we reproduce the tables in the main text with frequencies in units ofcycles day 1.253AppendixB.Frequencylistsincyclesday  1T able B.1: T able 3.1 with frequencies in units of cycles da y  1#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   1 224.9711 39 2.22 3.8  0.0025 0.0069 6 15 0.34 0.932 226.2470 50 3.29 4.2  0.0021 0.0057 6 16 0.27 0.723 226.3244 125 5.70 9.2  0.0013 0.0034 6 16 0.11 0.294 226.4061 68 1.68 6.3  0.0024 0.0064 6 16 0.28 0.745 229.1326 96 3.36 7.5  0.0015 0.0039 6 16 0.14 0.356 229.2127 365 5.56 24.2  0.0004 0.0010 6 16 0.04 0.107 229.2955 92 2.07 7.8  0.0011 0.0028 6 16 0.10 0.278 231.9695 43 2.02 3.7  0.0047 0.0125 6 16 0.38 1.109 232.0409 31 2.83 3.1  0.0078 0.0209 9 26 0.55 2.1810 232.1187 123 1.26 8.9  0.0010 0.0025 7 17 0.10 0.2611 232.1931 467 4.46 29.7  0.0003 0.0008 6 16 0.03 0.0812 232.2747 139 1.13 10.9  0.0008 0.0020 6 16 0.07 0.1813 234.9202 51 4.50 5.8  0.0044 0.0118 6 16 0.25 0.6814 235.0043 319 6.04 23.3  0.0005 0.0012 6 16 0.05 0.1315 235.0870 764 1.83 41.6  0.0002 0.0005 6 16 0.02 0.05c ontinue d on next p age254AppendixB.Frequencylistsincyclesday  1T able B.1: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   16 235.1654 177 4.59 12.4  0.0009 0.0023 6 16 0.10 0.2617 235.3077 41 2.08 3.1  0.0029 0.0084 6 16 0.39 1.1018 237.9862 110 1.81 8.5  0.0014 0.0037 6 16 0.17 0.4519 238.0660 118 3.45 8.9  0.0013 0.0034 6 16 0.17 0.4520 238.1214 67 2.75 6.2  0.0033 0.0093 6 16 0.36 1.0521 238.1983 39 5.14 3.1  0.0039 0.0138 6 17 0.53 1.6722 240.8659 24 4.72 2.7  0.0110 0.0274 6 16 0.90 2.5923 240.9585 37 0.24 3.6  0.0030 0.0083 6 16 0.29 0.8024 241.0277 33 5.56 3.3  0.0041 0.0136 7 16 0.38 1.1625 241.0911 37 2.85 4.4  0.0063 0.0169 7 18 0.45 1.2426 241.1755 82 5.00 6.8  0.0028 0.0073 7 17 0.29 0.7427 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 1.0628 242.4082 84 1.10 6.9  0.0013 0.0033 6 15 0.15 0.3729 242.4896 81 2.09 7.1  0.0016 0.0041 6 16 0.16 0.44255AppendixB.Frequencylistsincyclesday  1T able B.2: T able 3.2 with frequencies in units of cycles da y  1 . Columns with headers  i -  j use i and j to denote the frequencyn um b er from T able B.1. The a v erage rotation s eparat ion (spacings 9 : : : 18 & 27 : : : 35) is 0.0798  0.0041  Hz (3  = 0.0112 Hz).#  i -  j  y [  Hz]   y 3   y #  i -  j  y [  Hz]   y 3   y1 6 - 3 2.8883 0.0013 0.0035 19 11 - 8 0.2236 0.0047 0.01252 11 - 6 2.9804 0.0005 0.0012 20 11 - 9 0.1522 0.0079 0.02093 15 - 11 2.8939 0.0003 0.0009 21 17 - 15 0.2208 0.0029 0.00844 19 - 15 2.9790 0.0013 0.0035 22 21 - 19 0.1324 0.0041 0.01425 23 - 19 2.8926 0.0032 0.0090 23 26 - 23 0.2170 0.0041 0.01116 3 - 1 1.3533 0.0028 0.0077 24 28 - 26 1.2326 0.0031 0.00807 17 - 8 3.3382 0.0055 0.0150 25 17 - 9 3.2669 0.0084 0.02258 21 - 17 2.8906 0.0048 0.0162 26 26 - 21 2.9773 0.0048 0.01569 3 - 2 0.0773 0.0026 0.0069 27 16 - 15 0.0785 0.0009 0.002610 4 - 3 0.0816 0.0026 0.0069 28 19 - 18 0.0797 0.0017 0.005211 6 - 5 0.0800 0.0017 0.0043 29 20 - 19 0.0555 0.0035 0.009512 7 - 6 0.0828 0.0009 0.0026 30 21 - 20 0.0769 0.0052 0.016413 9 - 8 0.0715 0.0095 0.0242 31 23 - 22 0.0926 0.0112 0.0285c ontinue d on next p age256AppendixB.Frequencylistsincyclesday  1T able B.2: c ontinue d#  i -  j  y [  Hz]   y 3   y #  i -  j  y [  Hz]   y 3   y14 10 - 9 0.0778 0.0078 0.0207 32 24 - 23 0.0693 0.0052 0.015615 11 - 10 0.0743 0.0009 0.0026 33 26 - 25 0.0845 0.0069 0.018116 12 - 11 0.0816 0.0009 0.0017 34 27 - 26 0.0748 0.0052 0.017317 14 - 13 0.0841 0.0043 0.0121 35 29 - 28 0.0814 0.0017 0.005218 15 - 14 0.0827 0.0009 0.0017257AppendixB.Frequencylistsincyclesday  1T able B.3: T able 3.3 with frequencies in units of cycles da y  1 . F requencies n um b ered 30 and higher are the unresolv edcomp onen ts.#  [cycles da y  1 ] A [  mag]  [rad]    A   3   3  A 3   1 224.9710 38 2.22  0.0023 5 0.26 0.0067 16 0.112 226.2480 45 3.25  0.0020 5 0.21 0.0060 17 0.503 226.3196 106 5.96  0.0021 12 0.20 0.0047 15 0.754 226.4012 64 2.07  0.0037 5 0.34 0.0072 24 1.425 229.1339 95 3.20  0.0012 5 0.16 0.0042 17 0.366 229.2125 364 5.59  0.0003 5 0.04 0.0011 16 0.087 229.2938 81 2.13  0.0017 9 0.14 0.0052 17 0.318 231.9733 43 1.67  0.0038 5 0.35 0.0073 30 0.959 232.0429 40 3.02  0.0042 10 0.44 0.0133 16 0.1110 232.1167 111 1.37  0.0018 10 0.10 0.0037 36 0.1211 232.1927 475 4.50  0.0003 9 0.05 0.0007 16 0.2312 232.2726 131 1.30  0.0015 8 0.17 0.0031 16 0.8813 234.9166 54 5.18  0.0036 8 0.52 0.0099 16 0.5014 235.0042 320 6.11  0.0003 5 0.06 0.0010 28 0.5815 235.0875 761 1.81 + 0.0004 8 0.02 0.0012 36 0.12c ontinue d on next p age258AppendixB.Frequencylistsincyclesday  1T able B.3: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]    A   3   3  A 3   : : : : : : 235.0875 761 1.81  0.0004 8 0.02 0.0012 36 0.1216 235.1680 184 4.45  0.0012 6 0.13 0.0026 16 0.2317 235.3059 31 2.33  0.0024 9 0.27 0.0073 16 0.8818 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 0.4519 238.0632 112 4.02  0.0024 6 0.22 0.0057 28 0.5820 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 1.2921 238.1960 33 5.35  0.0027 6 0.29 0.0099 16 1.2422 240.8660 24 4.73  0.0041 5 0.44 0.0123 15 1.3723 240.9585 36 0.19  0.0026 5 0.28 0.0077 15 0.8424 241.0280 34 5.48  0.0031 5 0.31 0.0093 15 0.9525 241.0901 35 3.02  0.0029 5 0.30 0.0086 16 0.8726 241.1750 82 5.04  0.0013 5 0.14 0.0039 16 0.4127 241.2498 58 1.50  0.0022 5 0.22 0.0067 16 0.6928 242.4082 84 1.10  0.0011 5 0.12 0.0032 15 0.36c ontinue d on next p age259AppendixB.Frequencylistsincyclesday  1T able B.3: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]    A   3   3  A 3   29 242.4896 82 2.09  0.0011 5 0.13 0.0034 15 0.3730 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 4.3831 226.4391 31 5.19  0.0085 6 0.54 0.0164 17 1.5032 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 2.0833 229.2515 51 3.47  0.0029 5 0.31 0.0098 17 0.9934 232.0707 57 2.84  0.0029 8 0.27 0.0120 24 1.2035 232.2356 55 2.10  0.0037 7 0.47 0.0086 18 0.9236 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 2.7337 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 1.6838 235.1226 77 3.06  0.0030 10 0.20 0.0073 21 0.7639 238.0214 48 3.48  0.0051 6 0.49 0.0119 18 1.2840 238.0867 55 6.07 + 0.0074 17 0.49 0.0251 42 1.76c ontinue d on next p age260AppendixB.Frequencylistsincyclesday  1T able B.3: c ontinue d#  [cycles da y  1 ] A [  mag]  [rad]    A   3   3  A 3   : : : : : : 238.0867 55 6.07  0.0074 17 0.49 0.0251 42 2.5741 241.2817 28 0.19  0.0042 5 0.46 0.0128 15 1.39261AppendixB.Frequencylistsincyclesday  1T able B.4: T ab le 5.3 with frequencies in units of cycles da y  1 . F requency uncertain ties mark ed b y  are a v erage uncertain tiesbased on the separation from the mark ed frequency to the nearest (  rotationally split) frequency plus its uncertain t y .#  [cycles da y  1 ] A [  mag]  [rad]  S/N   3    A 3  A   3   1 224.9711 39 2.22 3.8  0.0024 0.0069 6 15 0.34 0.933 226.3244 125 5.7 9.2  0.0013 0.0034 6 16 0.11 0.296 229.2126 365 5.56 24.2  0.0003 0.0010 6 16 0.04 0.19 232.0409 31 2.83 3.1  0.0775  0.0822  9 26 0.55 2.1811 232.1931 467 4.46 29.7  0.0003 0.0008 6 16 0.03 0.0815 235.0870 764 1.83 41.6  0.0002 0.0004 6 16 0.02 0.0517 235.3077 4 1 2.08 3.1  0.0029 0.0084 6 16 0.39 1.119 238.0660 118 3.45 8.9  0.0013 0.0034 6 16 0.17 0.4521 238.1983 3 9 5.14 3.1  0.0421  0.0499  6 17 0.53 1.6723 240.9585 3 7 0.24 3.6  0.0029 0.0083 6 16 0.29 0.826 241.1756 8 2 5 6.8  0.0029 0.0073 7 17 0.29 0.7428 242.4082 8 4 1.1 6.9  0.0422  0.0444  6 15 0.15 0.37262Appendix CBootstrap distributionsAll bootstrap distributions for the  t parameters given in Tab. 3.1 are pre-sented below. In each  gure the top panels show the phase ( ) in radians,the middle panels are the amplitudes (A) in mmag, and lower panels are fre-quencies ( ) in  Hz. Increasing parameter numbers from Tab. 3.1 are readleft to right for each  gure. Thus, distributions for parameter set ( 1, A1, 1) are shown in the left panels while distributions for the set ( 2, A2,  2)are shown on the right. In each plot the green and red lines show the 1 and 2 uncertainties calculated by counting 68% and 95% of the realizations, cen-tred on the best  t parameter, respectively. The pink lines show the best  tparameter with the standard deviation calculated from the analytic formulaand the blue dots are the mean of the distribution with the standard erroron that mean. (Note: The phases are calculated by referencing to the timeof the  rst MOST observation, HJD 2451545.00 + 1769.42 days.)263Appendix C. Bootstrap distributionsFigure C.1: Bootstrap distributions for  t parameters 1 and 2 in Tab. 3.1.264Appendix C. Bootstrap distributionsFigure C.2: Bootstrap distributions for  t parameters 3 and 4 in Tab. 3.1.265Appendix C. Bootstrap distributionsFigure C.3: Bootstrap distributions for  t parameters 5 and 6 in Tab. 3.1.266Appendix C. Bootstrap distributionsFigure C.4: Bootstrap distributions for  t parameters 7 and 8 in Tab. 3.1.267Appendix C. Bootstrap distributionsFigure C.5: Bootstrap distributions for  t parameters 9 and 10 in Tab. 3.1.268Appendix C. Bootstrap distributionsFigure C.6: Bootstrap distributions for  t parameters 11 and 12 in Tab. 3.1.269Appendix C. Bootstrap distributionsFigure C.7: Bootstrap distributions for  t parameters 13 and 14 in Tab. 3.1.270Appendix C. Bootstrap distributionsFigure C.8: Bootstrap distributions for  t parameters 15 and 16 in Tab. 3.1.271Appendix C. Bootstrap distributionsFigure C.9: Bootstrap distributions for  t parameters 17 and 18 in Tab. 3.1.272Appendix C. Bootstrap distributionsFigure C.10: Bootstrap distributions for  t parameters 19 and 20 in Tab. 3.1.273Appendix C. Bootstrap distributionsFigure C.11: Bootstrap distributions for  t parameters 21 and 22 in Tab. 3.1.274Appendix C. Bootstrap distributionsFigure C.12: Bootstrap distributions for  t parameters 23 and 24 in Tab. 3.1.275Appendix C. Bootstrap distributionsFigure C.13: Bootstrap distributions for  t parameters 25 and 26 in Tab. 3.1.276Appendix C. Bootstrap distributionsFigure C.14: Bootstrap distributions for  t parameters 27 and 28 in Tab. 3.1.277Appendix C. Bootstrap distributionsFigure C.15: Bootstrap distributions for  t parameters 29 in Tab. 3.1.278Appendix DMagnetic perturbation (Saio)plotsPlots of magnetically perturbed pulsation modes are presented below. Theplots give perturbed frequencies as a function of magnetic  eld (Saio plots)for degrees of ‘ ranging from 0 to 4. (See the legend at the top of each plotfor degree identi cations.) The caption below each plot identi es the modelusing the same labelling system used in Tables 5.1, 5.2, 5.4, 5.5, 5.9, 5.6,and 5.8. Each plot identi es (with a solid vertical line) the closest matchedmodel listed in the aforementioned tables found in chapter 5. The observedfrequencies from the HR 1217 data (Tab. 3.1) are shown as horizontal blackdots in each plot.279Appendix D. Magnetic perturbation (Saio) plotsFigure D.1: Saio plots showing the best  t models M1 (top) and M2 (bottom) in Tab. 5.1.280Appendix D. Magnetic perturbation (Saio) plotsFigure D.2: Saio plots showing the best  t models M3 (top) and M4 (bottom) in Tab. 5.1.281Appendix D. Magnetic perturbation (Saio) plotsFigure D.3: Saio plots showing the best  t models M5 (top) and M6 (bottom) in Tab. 5.1.282Appendix D. Magnetic perturbation (Saio) plotsFigure D.4: Saio plots showing the best  t models M7 (top) and M8 (bottom) in Tab. 5.1.283Appendix D. Magnetic perturbation (Saio) plotsFigure D.5: Saio plots showing the best  t models M9 (top) and M10 (bottom) in Tab. 5.1.284Appendix D. Magnetic perturbation (Saio) plotsFigure D.6: Saio plots showing the best  t models M9 (top) and MM1 (bottom) in Tables5.1 and 5.2.285Appendix D. Magnetic perturbation (Saio) plotsFigure D.7: Saio plots showing the best  t models U1 (top) and U2 (bottom) in Tab. 5.4.286Appendix D. Magnetic perturbation (Saio) plotsFigure D.8: Saio plots showing the best  t models U3 (top) and U4 (bottom) in Tab. 5.4.287Appendix D. Magnetic perturbation (Saio) plotsFigure D.9: Saio plots showing the best  t models U5 (top) and U6 (bottom) in Tab. 5.4.288Appendix D. Magnetic perturbation (Saio) plotsFigure D.10: Saio plots showing the best  t models U7 (top) and U8 (bottom) in Tab. 5.4.289Appendix D. Magnetic perturbation (Saio) plotsFigure D.11: Saio plots showing the best  t models U9 (top) and U10 (bottom) in Tab. 5.4.290Appendix D. Magnetic perturbation (Saio) plotsFigure D.12: Saio plots showing the best  t models UU1 (top) and UU2 (bottom) inTab. 5.5.291Appendix D. Magnetic perturbation (Saio) plotsFigure D.13: Saio plots showing the best  t models GS (top) and MS (bottom) in Tab. 5.6.292Appendix D. Magnetic perturbation (Saio) plotsFigure D.14: Saio plots showing the best  t models PGS (top) and LPC (bottom). Modelproperties are given in Tabs. 5.8 and Tab. 5.9, respectively.293


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