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Astron. Astrophys. 350, 513-516 (1999)

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3. Time series analysis

We performed the time series analysis of V 368 Cep photometry with the TSPA method from Jetsu & Pelt (1999: Paper I), where four detailed application examples are also given. A prolonged time series of V1794 Cyg photometry was recently analysed with this method in Jetsu et al. (1999: Paper II). The parameters of our second order ([FORMULA]) light curve model


were the mean (M), the amplitudes ([FORMULA], [FORMULA]), and the frequency (f, i.e. the period [FORMULA]). The free parameters were [FORMULA] or [FORMULA] for fixed f. Excluding the two aforementioned probable flares, the normalized magnitudes y within every subset (Paper I: Eq. 17) were derived in those UBVRI passbands that contained measurements during at least seven nights (nts). Only subsets with [FORMULA] were modelled. The frequency f was a free parameter in the nonlinear modelling of y with the TSPA method. This gave the periods, and the primary and secondary minimum epochs (Table 3: P, [FORMULA] and [FORMULA]). Linear modelling with these subset ephemerides [FORMULA] gave the mean and total amplitude of the UBVRI light curves (Table 4: [FORMULA] and [FORMULA]). All error estimates were determined with bootstrap (Paper I: Sect. 4).


Table 3. The estimates for the photometric rotation period (P), and the epochs of the primary and secondary minima in HJD-2440000 ([FORMULA]) for [FORMULA] subsets. Note that some estimates are rejected with the rules [FORMULA], [FORMULA] and [FORMULA] (see the end of Sect. 3)


Table 4. The light curve mean (M) and total amplitude (A) in UBVRI for [FORMULA] subsets. Note that the M estimates in SET=4 and 5 are magnitude differences between V 368 Cep and HD219522 in U and [FORMULA]. The errors in parenthesis are [mmag]

The observing window in photometry is usually one sidereal day (i.e. [FORMULA]). If the real period is P, then spurious periods are induced at


where [FORMULA] and [FORMULA] (Tanner 1948). Combining [FORMULA] with [FORMULA] predicts [FORMULA], [FORMULA], [FORMULA] and [FORMULA]. These five alternatives explain the whole period finding history of V 368 Cep in our Table 1. The most recent study presented "unambiguous indications" for [FORMULA], and also mentioned some spurious periodicities (Bossi & La Franceschina 1995). Unfortunately, Eq. 1 connecting the real and spurious periodicities does not identify the correct one, i.e. the probability of mistaking a spurious periodicity for a real one should never be underestimated, as reminded by Table 1 in Paper I. Thus we decided to perform the TSPA method analysis between [FORMULA] and [FORMULA], i.e. the above five alternative periodicities were tested against each other. The window periods [FORMULA] within each subset were determined with the Deeming (1975) method.

The [FORMULA] for y with the [FORMULA] and [FORMULA] periodicities were the smallest in 4 and 3 subsets out of eight, respectively. The [FORMULA], [FORMULA] and [FORMULA] periodicities could be rejected with this [FORMULA]-criterion. Nevertheless, this criterion could not separate [FORMULA] from [FORMULA]. A unique solution was obtained from the linear correlation coefficients r between the phase residuals [FORMULA] and [FORMULA] (Paper I: Sect. 5). The r for [FORMULA] reached a significance higher than 0.001 in two subsets. No such cases occurred for [FORMULA]. Since the [FORMULA] and [FORMULA] for spurious periods correlate, and those for real periodicity do not, we conclude that the real periodicity in V 368 Cep is [FORMULA]. The normalized magnitudes are displayed in Fig. 1 with such periodicities. Criteria for accepting or rejecting the TSPA method modelling results were also presented in Papers I & II. The rejection rules were

  • [FORMULA]: If   the distribution of the model residuals or that of the M, P, A, [FORMULA] or [FORMULA] bootstrap estimates is not gaussian, then the P, [FORMULA] and [FORMULA] estimates are rejected.

  • [FORMULA]: P, [FORMULA] and [FORMULA] of [FORMULA] subsets are rejected.

  • [FORMULA]: Those [FORMULA] not present in 95% of the bootstrap models are "unreal".

Comparison between Tables 3 and 4 confirmed that the [FORMULA] rejections occur for low amplitude light curves (SET=2, 5 and 6). The [FORMULA] rejection in SET=8 only implies that this secondary minimum may be "unreal", but the model itself is reliable. In conclusion, reliable periodicity detection in V 368 Cep succeeded for [FORMULA] with a subset length of about one month, and [FORMULA].

[FIGURE] Fig. 1. The normalized magnitudes (y) with the ephemerides of Table 3

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© European Southern Observatory (ESO) 1999

Online publication: October 4, 1999