3 Period analysis

 

 

Table 2: The result of the period analysis. f₄ and f₆ seems to be caused by an unresolved frequency pair, thus discarded from further analysis.

No.	freq.	ampl.	S/N
	(d-1)	(mmag)
f1	9.1565	24.3	27.6
f2	9.4649	8.4	9.5
f3	15.4036	5.5	8.3
(f4)	9.8800	4.9	5.6
f5	15.9013	3.5	5.3
(f6)	9.7777	3.3	4.1

The period analysis was performed by means of standard Fourier-analysis with subsequent prewhitening steps. For this we have used Period98 of Sperl (1998) which also includes multifrequency least squares fitting of the parameters.

Figure 3: Amplitude spectra of the whole V light curves ( left column) and close-up views of the main peaks ( right column). Six frequencies can be identified with S/N larger than 4.

First we decided to analyse the merged transformed data. As has been mentioned above, the use of the same comparison star suggested a common analysis of the standard V-band data. The variable has very similar B-V and b-y indices as the comparison, thus the colour-dependent terms in the standard transformation equations are only small corrections. Either the mean brightness (6 $.\!\!^{\rm m}$65) or the extrema of the light curve are considered, the standard V data originated from Johnson-BV and Strömgren-by measurements agree very well (the difference does not exceed 0 $.\!\!^{\rm m}$005). However, some spurious low-frequency components yielded by the period analysis enforced us to reject this homogeneity assumption. During the prewhitening steps two low-frequency components (at 0.0007 d^-1 and 0.5 d^-1) appeared suggesting: 1. a possible shift by a few millimagnitudes of the mean brightness from Johnson and Strömgren photometry and 2. possible slow and low-amplitude variation of the comparison star mimicking changes of the mean brightness on a daily basis. A close look at the comparison minus check magnitudes revealed indeed some slight changes of the daily averages with no systematic short-term variations. Therefore, we have adjusted the individual light curves (21 together) by subtracting nightly mean values (the differences are of order of a few millimagnitudes). Although the low-frequency components have been removed, we have to admit that the mean values on some nights are fairly uncertain because of shorter time-spans than mean periods. That is the reason why the most critical subset obtained on JD 2 451 785 was excluded from the frequency analysis.

The calculated amplitude spectra are presented in Fig. 3, where we show the individual frequency spectra after consecutive prewhitenings. In order to illustrate the difficulty of the whole analysis we also present close-up views of the main peaks. Besides the strong 1/day alias structure the 1/year is also strong. A certain amount of ambiguity cannot be excluded due to the complex spectral window and finite spectral resolution.

The primary peak at f₁=9.15650 d^-1 is in very good agreement with the Hipparcos result (9.15642 d^-1). In every step of the prewhitening procedure we allowed all of the parameters to vary to get the "best'' Fourier-fit of the light curve. In order to check the reality of the components, we have determined the signal to noise ratios (S/N) following the suggestions of Breger et al. (1993). The calculated S/N for the sixth components is 4.1 so it satisfies the proposed criterion of Breger et al. (S/N(real) > 4) for accepting ambiguous peaks in the frequency spectrum. Therefore, the final set consists of six frequencies ranging from 9.15 c/d to 15.9 c/d and it is summarized in Table 2.

The period analysis has been checked by making use of the Hipparcos Epoch Photometry data consisting of 117 points. They were analysed separately and the calculated frequency spectrum (middle panel in Fig. 4) yields essentially to the same dominant period, that has been used to phase the data (top panel in Fig. 4). After prewhitening with this frequency, no further periodicity can be inferred from these data (bottom panel in Fig. 4). A further consistency check is a comparison of Hipparcos and our dominant frequencies. It is shown in Fig. 5, where a close-up to the main peaks is presented. As Hipparcos has good spectral window (in sense of having only weak and asymmetric sidelobes), the very good agreement of the main peaks supports our frequency analysis (see Jerzykiewicz & Pamyatnykh 2000 for a recent discussion on the use of Hipparcos photometric data).

Since the bulk of the data was obtained in 2001, a separate analysis of the 2001 data was carried out to yield insights into the reality of the six-frequency solution. Furthermore, the noise in the Strömgren vdata is significantly reduced compared to transformed Johnson V magnitudes (i.e. the full amplitude is larger and the photometric accuracy is better through the v filter). The prewhitening and simultaneous nonlinear parameter fitting resulted in largely similar frequencies at several 1/year aliases of the finally adopted set (for instance, even the dominant component at 9.15 d^-1 is approximately a 1/year alias of the true frequency). That is why we favoured frequencies resulted from the whole two-years long data.

Figure 4: The phase diagram of the Hipparcos data taken from the Hipparcos Epoch Photometry database calculated with a period of 0.1092130 d (top panel) and the frequency spectrum (middle panel). No secondary period can be inferred from these data (bottom panel).

Taking the six frequencies in Table 2, strong interaction between f₄ and f₆ is obvious. A visual inspection of the corresponding close-up views in Fig. 3 reveals that the peaks are broader than expected from the length of the data (the horizontal axes in the right column have the same frequency scales). It is suggested that some very close components may exist which are beyond our spectral resolution (approximately $1.5/\Delta T$, Loumos & Deeming 1978). For our data $\Delta T=$ 729 d, which corresponds to a  $\Delta f\approx0.002$ d^-1. Most recently, Breger & Bischof (2002) studied close frequency pairs ( $\Delta f<0.06$ d^-1) in $\delta$ Scuti stars. A detailed discussion of the behaviour of BI CMi led these authors to conclude that very close frequency pairs do indeed exist and their presence should be taken into account when planning photometric observations with the best available spectral resolution. We conclude that similar close frequencies might also exist in V784 Cas making difficult to interpret the presently available data. As a result, we kept only four frequencies for further analysis. The observed individual V light curves are compared with the four-component harmonic fit in Fig. 6. We turn back to the adopted frequencies in Sect. 5.

Figure 5: A close-up to the main peaks in the frequency spectra calculated from our data (1999-2001) and Hipparcos Epoch Photometry.

Figure 6: The observed individual V light curves (gray dots) with the four-component harmonic fit.