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Astron. Astrophys. 332, 149-154 (1998)

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Appendix A

The Fast Fourier Transform (FFT) spectrum of the data of Fig. 1 is shown in Fig. 5. Above [FORMULA] (the drawn line) a number of 8-10 features are present. However our sampling is rather irregular. Therefore we have complemented the Fourier analysis with another method of period determination: the PDM. The PDM method (Phase Dispersion Minimization, Stellingwerf 1978) is efficient to use on irregularly spaced data (as ours) and ideally suited to highly nonsinusoidal time variations. It is simply an automated version of the classical method of distinguishing between possible periods: a light curve for each trial period is obtained and the scatter of the observations computed. The most likely periods are those that produce the least observational scatter: in correspondence to these periods we will therefore observe minima in the spectrum, contrary to what happens with the FFT analysis, where the most likely periods appear as maxima. In the top panel of Fig. 5 the results of the two methods are plotted one above the other. The horizontal lines indicate the [FORMULA] level for each method: only those peaks simultaneously exceeding these lines (below it for the PDM and above it for the FFT analysis) are here considered.

[FIGURE] Fig. 5. Spectra by FFT and PDM for observations and artificial data

Among the 8-10 periods found by FFT only 5 of them agree, within their error bars ([FORMULA] days) with those determined by PDM, those at: 14.3 [FORMULA] days (14.4 [FORMULA] for PDM), 24.1 [FORMULA] (25.5 [FORMULA] for PDM), 34 [FORMULA] (33.2 [FORMULA] for PDM), 39 [FORMULA] 2 (36 [FORMULA] 2 for PDM) and at 71 [FORMULA] days (75 [FORMULA] 8 for PDM).

The dominant period for both methods is that at 25 days; the order of importance of the other periods is unclear. Moreover some periods could be subharmonics of dominant periods, like that at 71 days which seem to be a multiple of that at 14 days. In order to identify among the five periods the dominant ones we have applied the two methods, PDM and FFT, to artificial data: merely a sum of sinusoidal functions sampled at our observing points without noise added. Assuming only the minimum combination of two periods, one of the two that of 25 days, we have tried the 4 combinations of it with the other 4 periods. The sum of two sinusoidal functions at 25 and 14 days has revelead to be the successful combination; the only one, as shown in the lowest panel of Fig. 5, able to reproduce the results shown in the upper panel of Fig. 5. The peaks at 34, 39 and 70.9 days (33.2, 35.7 and 75.1 days by PDM) are determined as well in the spectra of the artificial data and therefore can be discarded as dominant periods. In particular we note that the peak at 34 days (at 33 days in the PDM) is only the result of a beat of the two main frequencies (i.e. 1/33=1/14.4- 1/25.5).

In conclusion, by using the PDM and artificial data we could establish that among the many features present in the FFT spectrum only the two at 25.5 and at 14.4 days are meaningful.

The period present in the PDM analysis at 158.7 days, although not present in the FFT analysis, is worthy of special attention because a long term periodicity clearly affects the data of Fig. 1. As shown in Sect. 2 of this paper, this periodic variation has a square wave trend rather than sinusoidal one. This explains why it is determined by the PDM, which is, as mentioned above a method "ideally suited for highly nonsinusoidal time variations"(Sterllingwerf, 1978).

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998