Analysis of time series of solar-like oscillations – Applications to the Sun and HD 52265 Dissertation zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultäten der Georg-August-Universität zu Göttingen vorgelegt von Thorsten Stahn aus Göttingen Göttingen 2010 Bibliografische Information der Deutschen Nationalbibliothek Die Deutsche Nationalbibliothek verzeichnet diese Publikation in der Deutschen Nationalbibliografie; detaillierte bibliografische Daten sind im Internet über http://dnb.d-nb.de abrufbar. D7 Referent: Prof. Dr. Stefan Dreizler Korreferent: apl. Prof. Dr. Laurent Gizon Tag der mündlichen Prüfung: 5.10. 2010 ISBN 978-3-942171-47-2 uni-edition GmbH 2011 http://www.uni-edition.de c Thorsten Stahn This work is distributed under a Creative Commons Attribution 3.0 License Printed in Germany Summary The study of stellar oscillations allows one to constrain the structure and dynamics of stel- lar interiors to a precision that cannot be achieved with other methods. This dissertation focusses on the measurements of the parameters (frequencies, amplitudes, linewidths, and rotational splitting) of the global modes of solar-like oscillations observed in the Sun and in the Sun-like star HD 52265. The thesis is organized in three main parts: (i) an im- plementation and validation of a global fit of stellar oscillation power spectra using solar observations, (ii) an application of this method to original 4-month-long CoRoT observa- tions of the solar-like star HD 52265, and (iii) an extension of the fitting method to time series with gaps. My main results concern HD 52265: the mode frequencies of the radial, dipole and quadrupole modes are measured with the highest precision achieved so far for a solar-type star and, for the first time, I measure unambiguously the effect of rotation on oscillations in a solar-type star. Analysis of 14 years of disk-integrated solar observations with SoHO/VIRGO: This data set spans more than a full sunspot cycle. Based on the maximum likelihood method I perform global fits of the solar oscillation power spectra using 4 months of observations at a time. The global parametric model prescribes smooth variations of the parameters with radial order. I determine the parameters of solar p modes with degrees ` 2 ranging ≤ over 14 consecutive radial orders. The model parameters include rotation and the incli- nation angle of the rotation axis to the line of sight. By comparison with published mode frequencies, I find that my measurements are essentially unbiased and have errors that are consistent with expectations, hence validating my global fit for application to Sun-like stars other than the Sun. Analysis of 4 months of CoRoT observations of HD 52265: This star is particularly interesting as it is a Sun-like star and it hosts a planet. At the time of writing, the observa- tions of HD 52265 provide the best oscillation power spectrum which is currently avail- able for a distant solar-like star. In particular, the radial, dipole, and quadrupole modes of oscillation are well resolved over a range of 10 consecutive radial orders. The derived large and small separations, ∆ν = 98.84 0.12 µHz and δν = 8.14 0.20 µHz, provide ± ± strong constraints on the density, mass, and age of the star. For example, the precision on the mean stellar density is 0.4% and the precision on the mass is about 2%. Even though the azimuthal components of the non-radial modes of oscillation are not resolved there is direct evidence for rotational broadening of the peaks in the power spectrum, which allows to measure the stellar angular velocity Ω and the inclination angle i of the rotation axis. Combined with estimates of rotation derived from the modulation of the light curve by the passage of starspots, I find 2.1 Ω/Ω 2.5 and 27◦ i 55◦. Assuming ≤ ≤ ≤ ≤ III Summary the stellar rotation axis and the normal to the orbital plane of the companion are aligned, i = ip, the radial velocity measurement of Mp sin ip can be turned into a constraint on the mass of the companion: 1.3 M /M 2.4. This strongly suggests that HD 52265b is ≤ p Jup ≤ a planet and not a brown dwarf, while it also illustrates the connections between astero- seismology and exoplanet science. Fitting method for the analysis of gapped time series: I implement a new MLE method to determine the mode parameters of solar-like oscillations with higher precision and less bias compared to a standard fitting method. In the case of a 2 week observation of one single mode of solar-like oscillation with a signal-to-noise ratio / = 6, and using a S N typical single-site observation window (duty cycle of 30%), the frequency estimate of the new method is by a factor of two more precise than the estimate obtained with the old method. The analysis of ground-based observations of stellar oscillations should benefit from the application of the new fitting method. IV Contents 1 Introduction 1 1.1 General purpose of helio- and asteroseismology . .1 1.2 Observations of solar-like oscillations . .2 1.3 Basic properties of solar-like oscillations . .5 1.3.1 Other types of stellar oscillations . .7 1.4 Interpretation of stellar oscillation frequencies . .7 1.5 Maximum likelihood estimation of stellar oscillation parameters . .9 2 Implementation and validation of a global fit of stellar oscillation power spec- tra using solar observations 11 2.1 Solar observations with SoHO/VIRGO . 11 2.2 Parameterization of the global fit . 14 2.2.1 Fit of the solar background noise . 14 2.2.2 Mode frequencies and rotational splitting . 17 2.2.3 Mode linewidths . 19 2.2.4 Mode amplitudes and mode visibility . 21 2.2.5 Asteroseismic constraints on Ω and i ................ 21 2.3 Global fit of the VIRGO data . 22 2.3.1 Mode frequencies . 25 2.3.2 Mode linewidths . 32 2.3.3 Oscillation amplitudes . 34 2.3.4 Solar rotation and the inclination of the rotation axis . 38 2.4 Discussion: Is the global fit good enough for asteroseismology? . 43 3 Asteroseismic analysis of the solar-like star HD 52265 47 3.1 The solar-like star HD 52265 . 47 3.1.1 The planet HD 52265b . 50 3.2 Observation of HD 52265 with CoRoT . 51 3.3 The power spectrum of HD 52265 . 51 3.3.1 Echelle spectrum . 55 3.3.2 Power at low frequencies: signature of stellar rotation . 56 3.4 Determination of the stellar background noise . 57 3.5 Modifications to the parameterization of the oscillation power spectrum . 58 3.6 Extraction of p-mode parameters and estimation of errors . 59 3.7 Global fit of the p-mode oscillation spectrum . 60 3.7.1 Mode frequencies . 60 V Contents 3.7.2 Mode linewidths . 70 3.7.3 Oscillation amplitudes . 76 3.7.4 Stellar rotation and the inclination of the rotation axis . 79 3.7.5 Fit A or Fit B: Which fit is better? . 88 4 Fourier analysis of gapped time series: maximum likelihood estimation 89 4.1 Introduction . 89 4.2 Statement of the problem . 90 4.2.1 The observed signal in Fourier space . 90 4.2.2 Statistics of the unconvolved signal . 92 4.3 Joint PDF of the complex Fourier spectrum . 93 4.4 Maximum likelihood estimation of stellar oscillation parameters . 95 4.4.1 Solar-like oscillations . 95 4.4.2 Deterministic oscillations plus white noise . 95 4.5 The old way: fitting the power spectrum and ignoring the correlations . 96 4.5.1 Solar-like oscillations . 97 4.5.2 Deterministic oscillations plus white noise . 97 4.6 Simulation of artificial time series . 97 4.6.1 Synthetic Window Functions . 98 4.6.2 Solar-like oscillations . 99 4.6.3 Deterministic sinusoidal oscillations plus white noise . 100 4.7 Testing and comparing the fitting methods for solar-like oscillations . 100 4.7.1 Window function with a 30% duty cycle . 100 4.7.2 Different window functions . 105 4.7.3 Cramér–Rao lower bounds . 108 4.7.4 Different signal-to-noise ratios . 109 4.7.5 Different mode lifetimes . 116 4.7.6 Impact of the initial guess . 122 4.8 Testing and comparing the methods for sinusoidal deterministic oscilla- tions plus white noise . 124 4.9 Conclusion . 127 5 Efficient maximization of the joint PDF: Cholesky decomposition 129 5.1 The Cholesky decomposition . 129 5.2 Testing the Cholesky decomposition . 130 6 Discussion 135 6.1 HD 52265: A remarkable data set for asteroseismology . 135 6.2 On the determination of global stellar parameters of HD 52265 . 135 6.3 Asteroseismic constraint on the mass of the exoplanet HD 52265b . 140 6.4 Other implications for the characterization of exoplanetary systems . 140 6.5 On ground-based observations for asteroseismology . 143 Bibliography 145 VI List of Figures 1.1 Disk-integrated observation of solar acoustic oscillations . .4 1.2 Asteroseismic Hertzsprung-Russell diagram . .8 2.1 Fourteen years of Sun-as-a-star data obtained with SoHO/VIRGO . 12 2.2 Oscillation power spectrum of a 120 day VIRGO time series . 13 2.3 Echelle spectrum of a 120 day VIRGO time series . 13 2.4 Fit of the background noise of a 120 day VIRGO time series . 15 2.5 Parameterization of solar p-mode frequencies (` = 0, 1) . 18 2.6 Parameterization of solar p-mode frequencies (` = 2) . 19 2.7 Parameterization of solar mode linewidths . 20 2.8 Global fit of the solar oscillation power spectrum I . 23 2.9 Global fit of the solar oscillation power spectrum II. 24 2.10 Distribution of the initial guesses and the fits of the frequency parameters 26 2.11 Solar-cycle variation of the mode frequencies .