Issue 
A&A
Volume 537, January 2012



Article Number  A30  
Number of page(s)  15  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201117352  
Published online  23 December 2011 
Online material
Appendix A: Normalization of the power density spectra
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 ν_{max} of maximum oscillation signal. A comparison of complementary analysis methods applied to the Kepler shortcadence 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 LombScargle 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 (ν_{Nyquist} = 283.2 μHz) related to the mean time sampling δt_{LC} of the Kepler longcadence data. We chose to compute power density spectra (PDS) with the following normalization: a white noise signal recorded at the Kepler longcadence sampling δt_{LC} with a noise level σ_{t} (in ppm) gives a PDS with a spectral density σ_{ν}, 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 redclump star is given in Fig. B.1. The parameters H_{νmax} and B_{νmax} 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 ν_{max} decreases, with variations as large as 4%, depending on the method. In parallel, H_{νmax} decreases and B_{νmax} increases when the filter width increases, which results in variations of the heighttobackground ratio (HBR) larger than a factor of two. As expected, δν_{env} 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.
Fig. B.1
Influence of the width of the smoothing parameter for a typical redclump giant (KIC 1161618). The different colors indicate the different values of the smoothing (in units of Δν). 

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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 ν_{max} is important because this parameter plays a key role in scaling relations for seismic parameters (Mosser et al. 2010; Huber et al. 2010). Since ν_{max} 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 σ_{νmax} on ν_{max} translates into a bias in the determination of the derived stellar parameters, of the order of σ_{νmax} and 3 σ_{νmax} 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 timeseries 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 pmode 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 ε(Δν) enables the identification of the radial modes (C.1)with Δν the large separation averaged in the frequency range [ν_{max} − δν_{env},ν_{max} + δν_{env}] . 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 secondaryorder terms in ε(Δν) 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 gmode spacing of ℓ = 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 signaltonoise ratio time series. This method measures the gmode 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 ℓ = 1 pressure mode.
© ESO, 2012
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