Free Access
Volume 537, January 2012
Article Number A30
Number of page(s) 15
Section Stellar structure and evolution
Published online 23 December 2011

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Appendix A: Normalization of the power density spectra

Table A.1

Characteristics of the different methods.

Different methods have been used and compared for the analysis of the power excess (Huber et al. 2009; Mosser & Appourchaux 2009; Hekker et al. 2010; Kallinger et al. 2010; Mathur et al. 2010). A comprehensive comparison of the properties and characteristics of the methods was given by Hekker et al. (2011a), but only dealt with the large separation Δν and the frequency νmax of maximum oscillation signal. A comparison of complementary analysis methods applied to the Kepler short-cadence data by Verner et al. (2011) presented results on the maximum mode amplitude, but not for red giants. For the current work on red giants, parameters characterizing the oscillation power excess have been compared.

To ensure a correct comparison of results obtained with different methods, it was first necessary to normalize the outputs. The computation of power density spectra performed with Lomb-Scargle periodograms has taken into account the correction for the duty cycle of each target, in order to obtain spectra with the frequency resolution corresponding to the total observation time and frequency up to the Nyquist frequency (νNyquist = 283.2  μHz) related to the mean time sampling δtLC of the Kepler long-cadence data. We chose to compute power density spectra (PDS) with the following normalization: a white noise signal recorded at the Kepler long-cadence sampling δtLC with a noise level σt (in ppm) gives a PDS with a spectral density σν, such that where N is the number of points in the time series. The characteristics of the methods used for this work are briefly described in Table A.1.

Appendix B: Smoothing

Most of the methods use a smoothed spectrum to obtain the global parameters of the Gaussian envelope (Sect. 3). Since the width of the Gaussian envelope of red giant oscillation (Eq. (1)) is narrow (Mosser et al. 2010), this step must be performed carefully. An example for a typical red-clump star is given in Fig. B.1. The parameters Hνmax and Bνmax have been calculated for the local description of the background smoothed with different filter sizes and are summarized in Table B.1. We have used a Gaussian window for performing the smoothing and have tested different values of the FWHM. The mean value of the large separation acts as a natural characteristic frequency, so we expressed the FWHM in units of Δν and have tested values varying from 0.75   Δν to 2.0   Δν. When this value increases, the measured value of νmax decreases, with variations as large as 4%, depending on the method. In parallel, Hνmax decreases and Bνmax increases when the filter width increases, which results in variations of the height-to-background ratio (HBR) larger than a factor of two. As expected, δνenv also increases with . This variation can be easily understood by recognizing that increasing the smoothing will spread the power of the oscillations relative to the background. We consider an acceptable compromise to be a Gaussian filter with a FWHM equal to the mean large separation Δν. Such a width is large enough for smoothing the influence of individual contributions of different degrees, and narrow enough to avoid the dilution of the global parameters, as shown by the example given in Table B.1: when the parameter gets larger than 1, all terms show significant variations.

thumbnail Fig. B.1

Influence of the width of the smoothing parameter for a typical red-clump giant (KIC 1161618). The different colors indicate the different values of the smoothing (in units of Δν).

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Table B.1

Influence of the smoothing on the seismic global parameters of KIC 1161618.

This study shows the importance of the smoothing parameter. A systematic variation induced by the smoothing as large as 4% in νmax is important because this parameter plays a key role in scaling relations for seismic parameters (Mosser et al. 2010; Huber et al. 2010). Since νmax is used for deriving the seismic mass and radius (Mosser et al. 2010; Kallinger et al. 2010; Hekker et al. 2011b), a bias should be avoided in order to avoid subsequent biases in the stellar parameters. A systematic relative error σνmax on νmax translates into a bias in the determination of the derived stellar parameters, of the order of σνmax and 3   σνmax for, respectively, the relative precision of the radius and mass.

A variation of a factor of two of the HBR depending on the smoothing means that studying the partition of energy between the oscillation and the background requires a careful description. According to the solar example, longer time-series are certainly required to reach the necessary precision to draw firm conclusions.

Appendix C: Mode identification

The determination of the mode visibility (Sects. 4 and 7) requires the complete identification of the p-mode spectrum. This is automatically given by the red giant universal oscillation pattern (Mosser et al. 2011b), with the parametrization of the

dimensionless factor ε of the asymptotic development (Tassoul 1980). The function εν) enables the identification of the radial modes (C.1)with Δν the large separation averaged in the frequency range  [νmax − δνenv,νmax + δνenv] . For simplicity, we introduce the reduced frequency, dimensionless and corrected for the ε term of the Tassoul equation: (C.2)According to Huber et al. (2010) and Mosser et al. (2011b), ε is mainly a function of the large separation. For radial modes, the reduced frequency is very close to the radial order, except for possible very small secondary-order terms in εν) that do not depend directly on Δν and do not hamper the mode identification.

The determination of the evolutionary status of the giants (Sect. 4.2), namely the measurement of the g-mode spacing of  = 1 mixed modes, is then achieved with the automated method described by Mosser et al. (2011a). This could be done for about 674 targets out of 1043 with high signal-to-noise ratio time series. This method measures the g-mode spacing of the mixed modes from the Fourier spectrum of the oscillation spectrum windowed with narrow filters centered on the expected locations of each pure  = 1 pressure mode.

© ESO, 2012

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