Optimizing Weights for the Detection of Stellar Oscillations: Application to alpha Centauri A and B, and beta Hydri

These “raw” weights can then be adjusted to minimize the noise level in the final power spectrum by identifying and revising those uncertainties that are too optimistic, and at the same time rescaling the uncertainties to be in agreement with the actual noise levels in the data. We have previously described the application of this process to velocity observations of solar-like oscillations (Butler et al. 2004;Bedding et al. 2007;Leccia et al. 2007;Arentoft et al. 2008).

These noise-optimized weights can be further adjusted to minimize the sidelobes in the spectral window that arise from daily gaps. This was done on a night-by-night basis for our two-site observations of α Cen A (Bedding et al. 2004), α Cen B (Kjeldsen et al. 2005) and β Hyi (Bedding et al. 2007). However, that procedure was not automatic and is therefore impractical for our recent multi-site campaign on Procyon (Arentoft et al. 2008), which involved observations with 11 spectrographs spread over nearly four weeks. We have therefore developed a new method for obtaining the sidelobe-optimized weights (Kjeldsen et al., in prep.).

The new method operates with two timescales. All data segments of a certain length (2 hr, for example) are required to have the same total weight Figure 1.

The time series of weights for α Cen A for the three schemes. Black symbols show data from UCLES and grey symbols show UVES. Upper panel: noise-optimized weights (Butler et al. 2004); middle panel: night-bynight sidelobe-optimized weights (Bedding et al. 2004); lower panel: sidelobeoptimized weights using the new method.

throughout the time series, with the relaxing condition that variations on long time scales (>12 hr, for example) are allowed. The method produces the cleanest possible spectral window in terms of suppressing the sidelobes. In order to test the method, we have applied it to the published data on α Cen A and B, and β Hyi, allowing a comparison for the three stars of the resulting power spectra and spectral window to those coming from using the original noise-and sidelobeoptimized weights.

The data discussed below originate from three spectrographs: UVES on the ESO VLT at Paranal, Chile; UCLES on the AAT at Siding Spring Observatory in Australia; and HARPS on the 3.6-m ESO telescope at La Silla, Chile.

The α Cen A data considered here are those for which the sidelobe-optimization procedure was first developed. The data consist of dual-site observations with UVES and UCLES. Figure 1 shows the time series of weights for three optimization schemes discussed above. The upper panel shows the noise-optimized weights (Butler et al. 2004), the middle panel shows the sidelobe-optimized weights obtained by adjusting on a night-by-night basis (Bedding et al. 2004) while the lower panel shows the sidelobe-optimized weights obtained with the new method. The UCLES data (black symbols) can be identified in the upper panel as the 5 nights with relatively low weight, as compared to the three nights of UVES data (grey symbols).

As can be seen by comparing the upper two panels in Fig. 1, the sidelobeoptimized spectral window in Bedding et al. (2004) was obtained by reducing the weights of the first two UCLES nights and increasing those of the UCLES data in the period with observations from both telescopes. This resulted in a significant decrease of the sidelobes but it also increased the noise level in the power spectrum because higher weight was given to lower-precision data. There was also a decrease in frequency resolution due to the suppression of the first two nights of observations, effectively shortening the time base of the observations.

Figure 2 shows the spectral windows for the three weighting schemes and Fig. 3 shows the corresponding power spectra. In Table 1 we give the noise levels, effective observing times and frequency resolutions, where the latter are given as the FWHM of the spectral window in power.

Table 1 shows that the two sidelobe-optimization methods result in similar noise levels and both, as expected, have higher noise than that in the noiseoptimized power spectrum. However, the new method results in a longer effective observing time and better frequency resolution than the night-by-night sidelobe-optimized weights from Bedding et al. (2004). The improvement can be understood by looking at Fig. 1, where we see that the new sidelobe-optimized weights are more homogeneous and give higher weight to

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