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Astron. Astrophys. 325, 559-562 (1997)

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3. Oscillation frequencies and amplitudes

We use least squares Fourier decomposition to determine oscillation spectra, applying the technique of known constituents as described e.g. by Andreasen (1987). For each trial frequency, [FORMULA], a simultaneous least squares solution is calculated for the amplitude of the known constituents (with known frequencies) and amplitude and phase of a sine term with the trial frequency. The resulting reduction factor, S([FORMULA]), gives the fractional decrease in the least squares error (norm) obtained by including the trial frequency. A series of S-spectra is calculated. In each spectrum the frequency which gives maximum S is determined, and this frequency is included as a known constituent in the following spectra.

Preliminary analysis of the full time-series for BT Cnc showed the presence of several outlying observations with excessive residuals. Removing 22 outliers we accept a time-series of 682 observations. In the following we first discuss this series.

Fig. 1 shows the window function and four power spectra and Table 2 gives detailed information on the resulting four-frequency solution. The upper panel a) gives the window function centered on 10 cycles per day (c/d). Besides the 1 c/d alias's, considerably lower side-lobes separated by [FORMULA] c/d are seen. Panels b) and c) uniquely identify the primary and secondary oscillations with main properties given in Table 2. The main oscillation has frequency [FORMULA] c/d, amplitude 17.7 mmag and phase 2.61 radian with Epoch Feb. 14.8 1994 (HJD = 2449398.3054). Formal errors of both amplitude and phase are very small, 0.2 mmag and 0.01 radian, respectively. Due to the fact that our time-basis is only about 3 d, we cannot determine very accurate frequencies and are not able to separate e.g. close frequency pairs. Comparing results of many numerical experiments (see later) we estimate the uncertainty in our frequencies to be about [FORMULA] c/d.

[FIGURE] Fig. 1. Power spectra of BT Cnc showing the reduction factor S as function of trial frequency. S gives the fractional reduction in the lsq-norm obtained by including the trial frequency. The upper panel a shows the window function. Panels b and c identify primary and secondary oscillation, respectively; see text for details. Panels c -e show the spectrum obtained by using all frequencies already identified as known constituents. Panels d and e show mainly noise


Table 2. Main pulsation properties of BT Cnc according to the four-frequency solution illustrated in Fig. 1. Phases are given with Epoch HJD = 2449398.3054 in column 3. The rms-scatter given in column 4 is the scatter obtained by successive inclusion of the four frequencies

Panels c) - e) of Fig. 1 show S-spectra resulting from taking the already known frequencies as known constituents. From Panel c) it is seen that the S-value, S = 0.72, of the new frequency, [FORMULA] c/d, is even higher than the value found for the frequency of the primary oscillation, [FORMULA]. Closer inspection of Panel b) shows the presence of a top at about 7.0 c/d, which does not belong to the main pattern. Taking the primary oscillation as a known constituent, this top becomes the center of the [FORMULA] -pattern, which also closely resembles the window function.

While the primary and secondary oscillations are unambiguously identified, the following possible oscillations indicated in Table 2 have much smaller S-values and oscillation amplitudes, and no pattern in the S-spectrum resembling the window function can be identified. From Table 2 it is also seen that inclusion of the last two Fourier terms decrease the rms-scatter only marginally.

The difficult problem of deciding whether Fourier terms of low significance correspond to physical oscillations or are due to noise has been studied extensively in the literature, e.g. recently by Garrido & Rodríguez (1996) and Pardo & Poretti (1997). They emphasize that before searching for low-amplitude modes, both harmonics of the high-amplitude modes, i.e. 2 [FORMULA], 3 [FORMULA], ... , etc., and the interaction modes with frequencies [FORMULA], n and m integer, must be taken into account. In particular, for determination of the optimal set ([FORMULA], [FORMULA]), harmonics and interaction frequencies must be included. In the present case we find that these terms all have small amplitudes. Fitting ([FORMULA], [FORMULA]) taking into account also [FORMULA], [FORMULA], [FORMULA] and [FORMULA], we determine the optimal solution with maximum S given in Table 3. Here we have removed four more outliers, so that 678 observations are used. Table 3 also gives the results of searching for two further independent oscillations, and Fig. 2 shows the corresponding S-spectra and a comparison of the solution including all terms of Table 3 with the observations for each night. It is seen that several tops with S [FORMULA] 0.02 are present. However, we have not been able to identify a pattern that is robust to small changes in the calculation procedure.


Table 3. Main pulsation properties of BT Cnc according to the solution with harmonics and interaction modes (in upper part). In column 4 the mode identification is given. The last two lines give main properties of possible modes identified by spectra (Fig. 2) with all previously known frequencies as known constituents

[FIGURE] Fig. 2. Two S-spectra of BT Cnc identifying the last two modes given in Table 3 (upper panels) and the individual observations of each night shown together with the solution of Table 3 (lower panels). The rms-scatter of this solution is 3.0 mmag

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998