Curvelet analysis of asteroseismic data: The Procyon noise-reduced MOST power spectrum

1. Curvelet analysis of asteroseismic data: • The Procyon noise-reduced MOST power spectrum • R. A. García (1,2) , P. Lambert (1,2), J. Ballot (2,3) , S. Pires (4,1) , • P.A.P. Nghiem(1,2), S. Turck-Chièze(1,2) & J.M. Matthews(5) • (1)  DSM/DAPNIA/Service d’Astrophysique, CEA/Saclay, 91191 Gif sur Yvette, France • (2)  AIM - Unité Mixte de Recherche CEA - CNRS - Université Paris 7 - UMR n°7158, CEA/Saclay, 91191 Gif sur Yvette, France • (3)  Max-Planck-Institut für Astrophysick, Karl-Schwarzschild-Str.1, Postfach 1317, 85741 Garching, Germany • (4)  DSM/DAPNIA/EDI, CEA/Saclay, 91191 Gif sur Yvette, France • (5)  Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver V6T IZI, Canada COROT Week 9, 5-9 December 2005, ESTEC, Netherlands

2. INTRODUCTION: • Recent photometric observations of Procyon A by MOST have shown no clear p-mode signal while power spectra of ground-based velocity measurements show power excess and peaks interpreted as evidence for the presence of p-modes. • The statistical distribution of the highest peaks in MOST is not consistent with simple noise and has been interpreted by some as possible p-mode signal at roughly the expected level. • We aim to enhance the S/N ratio of the MOST observations to a level high enough to see if those peaks are part of a bum in power caused by the presence of acoustic mode power. • A curvelet filtering is applied to the echelle diagram in order to enhance curved structures. • After filtering a clear bump appears in the power spectrum around 1 mHz as already observed from ground.

3. CONTEXT: • Many independent reports of p-mode properties have been published based on very high-resolution spectrometric techniques from ground-based observations: Martic et al. 1999, 2004; Egenberger et al. 2004. • The first spaced-based asteroseismic photometric mission MOST could not uncover the presence of p-mode signal in the amplitude spectrum (Matthews et al. 2004). • Bedding et al. (2005) argued that the non-detection of oscillations in Procyon A by MOST could be explained considering that p-modes could be severely diluted by intrinsec convective noise which was expected to be stronger in light than in velocity. They conclude that the distribution of the highest peaks was not consistent with simple noise and could be the signature of p-mode power. • Robinson et al. (2005) performed hydrodynamic simulations of convection in a Procyon model. They suggest that the convection characteristics in Procyon are different from those of the Sun. Their theoretical results are consistent with both the light and velocity measurements.

4. RAW POWER SPECTRUM: • MOST data: January 8 to February 9 2004 • 32 days with 99% duty cycle • First 20 days with a mean sampling of 15s; the last 12 days with a mean sampling of 7.5s  rebin the latter to obtain a mean interval of 15s. • Low-frequency trend removed • Computing the power spectrum: iterative least-squares fitting of sine waves (SWF) • Normalisation of the spectrum by the standard deviation of the noise at high frequency (4-5 mHz) • The Earthshine scattering modulated by the orbital period (101.413min): power is removed from the spectrum in bands of 2 Hz wide centred on the orbital frequency and its harmonics

5. RAW POWER SPECTRUM: Smoothed spectrum with a boxcar filter 270 Hz 2-order polynomial fit (in log-log parameter space) to the background noise

6. METHOD: THE CURVELET ANALYSIS • We apply a new multiscale image processing technique, the curvelet transform, to the MOST data. This transform was developed to deal with images containing highly anisotropic patterns (Candès & Donoho 1999). • The curvelet transform can be used to reduce the noise in the asteroseismic echelle diagrams (Lambert et al. 2005) and consequently allows us to obtain a power spectrum with an enhanced S/N. • To proceed a noise reduction using this transform, we follow the implementation of Starck, Candès & Donoho 2002.

7. FILTERED POWER SPECTRUM: • How do we proceed ? • Estimation of the large spacing 0 (FFT or autocorrelation of the raw spectrum) to have a folding frequency • Building of the raw echelle diagram • Filtering with the curvelet transform • Unfolding the filtered echelle diagram • Repeating this operation with different values of 0 slightly different (±1 bin) and averaging to reduce the dependence to the exact folding frequency

8. FILTERED POWER SPECTRUM: Smoothed spectrum with a boxcar filter 270 Hz 2-order polynomial fit (in log-log parameter space) to the background noise Removed Orbital harmonics

9. Rank 1 : 449.5 Hz (1  ) Rank 2 : 394.9 Hz (3  ) Rank 3 : 727.2 Hz (1  ) Rank 4 : 759.6 Hz (3  ) Rank 5 : 691.8 Hz (3  ) Rank 6 : 887.6 Hz (2  ) Rank 7 : 746.5 Hz (2  ) Rank 8 : 1212.6 Hz (1  ) Rank 9 : 801.4 Hz (1  ) Rank 10 : 1049.7 Hz (1  ) Rank 11 : 340.0 Hz (>4  ) Rank 12 : 856.3 Hz (2  ) Rank 13 : 1041.2 Hz (>4  ) Rank 14 : 942.4 Hz (1  ) Rank 15 : 766.5 Hz (>4  ) • FILTERED POWER SPECTRUM: • Power-based rank test: taking the 15 highest peaks ranked by their power. • Comparison between the frequencies of the 15 ranked peaks and the p-modes reported by • Martic et al. 2004: 6, 9, 12 peaks at 1-, 2- and 3- of the central frequencies given in table 2. • Eggenberger et al. 2004: 4 peaks compatible inside 3- • Probability of having these results caused by pure noise: • Monte Carlo simulation • Probabilities having, at least, 6, 9, 12 peaks compatible with 1-, 2- and 3- error are 7.6%, 12.4% and 13.3%

10. Martic et al. 2004 Eggenberger et al. 2004 • FILTERED POWER SPECTRUM: Vertical lines: the 15 ranked peaks 1 2 3 from frequencies reported by Martic et al. 2004 The bar width represents the observational error  = 2Hz

11. CONCLUSION : Uncovering the presence of a clear bump in the MOST power spectrum for the same position of the one already measured from ground-based observations.A power rank based test of the 15 highest peaks in this region gives coincidences with previous identified p-mode frequencies from Martic et al. 2004