Modelling Β Pictoris' Pulsations to Weigh Its Giant Planet
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Modelling β Pictoris' pulsations to weigh its giant planet M.Sc. Research in Astronomy - Cosmology: Minor Research Project Author: Martijn S. S. L. Oei B.Sc. Supervisor: Prof. dr. Ignas A. G. Snellen Last revision: July 31, 2016 Title page background: Visualisation of asteroseismological pulsations in a star orbited by a hot, rocky planet. Credits: Gabriel P´erez D´ıaz,Instituto de Astrof´ısica de Canarias (Servicio MultiMedia) Author contact details: Martijn Simon Soen Liong Oei B.Sc. Leiden Observatory, Leiden University, The Netherlands E-mail: [email protected] [email protected] Website: home.strw.leidenuniv.nl/~oei 1 Contents 1 Introduction 11 1.1 The exoplanet revolution . 11 1.2 Planets around massive stars . 11 1.3 Asteroseismology . 12 1.4 Aims of this work . 12 1.5 Case study: Beta Pictoris . 13 2 Pulsational noise on stellar radial velocities 14 2.1 Data set . 14 2.2 From spectra to radial velocities . 15 2.3 Reducing the measured spectra . 15 2.4 Reducing the reference spectrum . 16 2.5 The spectral cross correlation function . 17 2.5.1 Defining cross correlation . 17 2.5.2 Defining autocorrelation . 17 2.5.3 Types of spectral CCFs . 18 2.6 Creating measured spectral CCFs . 18 2.7 Determining radial velocities . 20 2.7.1 Approximating the no-pulsation MCCF . 21 2.7.2 Fitting the no-pulsation MCCF to other MCCFs . 21 2.8 High frequency variations . 21 3 Building a stellar model 23 3.1 Model idealisations . 23 3.2 Coordinate system conventions . 24 3.3 From observer plane to stellar surface . 25 3.4 The stellar coordinate axes . 26 3.5 From cartesian ΣW coordinates to spherical ΣS coordinates . 28 3.6 Pulsational velocity . 29 3.7 Rotational velocity . 30 3.8 Radial velocity . 31 2 3.9 Limb darkening . 31 3.10 Rest frame spectrum . 32 3.11 Observable spectrum . 35 3.12 Spectral cross correlation function . 36 3.13 Spectral cross correlation function residuals . 37 3.14 Physical parameter overview . 38 3.14.1 Basic physical parameters . 38 3.14.2 Pulsational and starspot-related physical parameters . 38 4 Pulsational effects on the spectral CCF 40 4.1 Gaussian blobs . 41 4.2 An equatorial blob moving along ^r .......................... 42 4.3 An equatorial blob moving along φ^ .......................... 44 4.4 An equatorial blob moving along θ^ .......................... 46 4.5 Sine waves . 46 4.6 A sine wave moving along ^r .............................. 48 4.7 A sine wave moving along φ^ .............................. 51 5 Attenuating pulsational noise 53 5.1 Determining stellar parameters . 53 5.1.1 Apparent equatorial rotational velocity . 53 5.1.2 Fitting the no-pulsation MCCF . 55 5.2 Creating SCCRs . 57 5.2.1 Superposition of 2 waves . 57 5.2.2 Superposition of 3 or more waves . 58 5.2.3 Linearity of SCCRs . 60 5.3 Fitting SCCRs to MCCRs . 61 5.3.1 Parameter estimation . 62 5.3.2 Fitting procedure . 63 5.3.3 Fitting results . 63 5.4 Improving on RV measurements . 65 5.4.1 Creating CMCCF . 65 5.4.2 Recalculating RVs . 65 6 Discussion 69 6.1 Stellar model inconsistencies . 69 6.2 High frequency variations . 70 7 Summary and conclusion 71 8 Acknowledgements 73 9 Bibliography 75 3 A Additional figures 79 A.1 Chapter 4 . 79 A.2 Chapter 5 . 81 4 List of Figures 2.1 Upper graph: More than 1000 HARPS β Pictoris spectral cross correlation func- tions (MCCFs) plotted on top of eachother show RV-dependent variability. Lower graph: This variability between the MCCFs is quantified at each radial velocity via the MCCF standard deviation. 19 2.2 Upper graph: Hundreds of HARPS β Pictoris radial velocities spread over several years determined by fitting a no-pulsation spectral CCF to each of the MCCFs. Lower graph: Radial velocities seem to fluctuate during a single night (MJD 54542) with a frequency ∼ 101 min. Similar patterns are observed in other nights. 22 3.1 Stellar latitude maps for various inclination angles made with our model. Each band covers a 10° range of latitudes. As far as we know, the fourth image from left represents the way that β Pictoris is viewed from the Solar System. 25 3.2 Longitude maps for a rotating star at an inclination angle of 45° for various times as made with our model. The sharp red-blue boundary indicates the direction of x^S(t), which of course rotates a full 2π rad during a stellar day. For β Pictoris, td =16h......................................... 28 3.3 Radial velocity maps for a star rotating at a constant angular velocity seen under the same inclination angles as used in Figure 3.1, again made with our model. No pulsations are introduced. Note that the only difference between the maps is the direction from which the star is observed. The colours indicate the Dopper shift of a green emission line - red signals redshift while blue means blueshift. 31 3.4 Relative intensity maps for stars with various limb darkening coefficients, again made with our model. No starspots are introduced. The inclination angle is irrelevant, as are the pulsation and rotation parameters. The limb darkening co- efficients a0; a1; a2; ::: are given directly underneath each map's description. From left to right: Lambertian radiator, exempli gratia a blackbody; star with constant emissivity throughout and vanishing absorption coefficient (for some wavelength); our estimate of β Pictoris limb darkening profile from fitting the σ-clipped mean MCCF (see Chapter 5); Solar limb darkening profile at 550 nm (Pierce, 2002); Solar limb darkening profile at 2 µm (Pierce, 2002). 33 5 3.5 Comparison between normalised spectra with a single absorption line. Upper graph: a Gaussian profile is indicated with a thick green line, with various Voigt profiles overplotted. Lower graph: differences between the Gaussian profile and each Voigt profile. The lower γ is, the closer the Voigt profile resembles the Gaussian. 35 3.6 Two simultaneously plotted simulated single-lined stellar spectra. In green: spec- trum with single Gaussian absorption line that would be measured when looking at stellar surface patches at rest with respect to the observer. In blue: the rota- tionally broadened spectrum, which an observer infinitely far from the rotating star would observe. No pulsations were added to the star. 36 4.1 Basic lay-out of our pulsation visualiser, which provides a graphical overview of the effect of pulsations and star spots on stellar spectra and spectral CCF-derived quantities. Upper row, from left to right: the first three maps indicate the pul- sational velocity of each point of the stellar surface in the local radial, azimuthal and polar directions, respectively. The fourth map graphically represents the ra- dial velocity due to pulsations and, most importantly for stars like β Pic, due to rotation. In order to raise the association with redshift and blueshift, positive radial velocity is denoted by red and negative radial velocity is denoted by blue. The fifth map shows the limb darkening profile or relative intensity map of the stellar disk. Lower row, from left to right: the rest frame and observable spectrum overplotted. The big central graph shows the raw spectral CCF. The above of the rightmost graphs gives the time-evolving part that is due to pulsations, and the other shows the partial derivative with respect to time thereof. 41 4.2 Pulsation velocity maps for a star with 3 blobs of moving material at various times at an inclination angle of 90°, which is considered suitable for β Pictoris. Over the course of half a stellar rotation period, the static pulsation pattern moves in and out of view, causing a time-dependent perturbation to the observable spectrum. Exactly what changes are made depends on whether the pulsation velocity displayed is pointing in the local radial, azimuthal or polar directions. 42 4.3 Three pulsation visualiser snapshots of the process in which a single equatorial Gaussian blob turning in and out of view by stellar rotation distorts the observable spectrum and CCF. In this case, the blob moves in the radial direction. For clarity, −1 vA = 50 km s was used. 43 4.4 Three pulsation visualiser snapshots of the process in which a single equatorial Gaussian blob turning in and out of view by stellar rotation distorts the observable spectrum and CCF. In this case, the blob moves in the azimuthal direction. Just −1 like for Figure 4.3, vA = 50 km s was used. 45 6 4.5 Sinusoidal pulsation velocity maps for various wave numbers N for a star seen under an inclination angle of 60°. Upper row: pulsation patterns on the visible part of the stellar surface. Lower row: plate car´eeprojection world maps of the pulsation patterns that were `wrapped around' the stellar spheres in the upper row. These maps allow one to see the whole pulsation pattern imprinted on the star at once - the azimuthal angle coordinate goes from 0 to 2π rad from left to right in these maps, the polar angle from 0 to π rad. 47 4.6 A pulsation visualiser snapshot of the process in which a single, static, sinusoidal wave pattern seemingly changing due to stellar rotation distorts the observable spectrum and CCF. In this case, the material in the wave moves in the radial direction. 48 4.7 Simulated spectral cross correlation function residuals for a rotating star to which sinusoidal pulsations in the radial direction are added, shown simultaneously for various times. The positive and negative SCCR amplitudes are also shown for each radial velocity via the dashed lines.