4 Analysis

A comprehensive analysis of a pulsating variable star is based on the assumption that the star's pulsation behaviour is determined by the inner structure of the stars, which most of the time does not change dramatically over a few years or a decade or so. Evolutionary period changes or instability in the amplitudes should be expected according to the evolutionary and pulsation theory but the stationarity of the pulsation as an initial assumption could be accepted.

If we accept the idea mentioned above, we could see our comprehensive analysis as a set of data where the same physical process is sampled at different times and sites (which means different longitude in our case) with different data distribution involving two runs of coordinated observation. The distinct subsets of data could be regarded as a physical process convolved with different window pattern, which produce different distribution of side lobes around the real frequencies. Although none of the subsets are long enough to get a well-established solution for the real physical process going on in the star, the peaks which appear in each subset suggest a high probability of a peak being a real frequency.

However, we should emphasize that in a comprehensive analysis some of the frequencies and specially their amplitudes can be regarded to be on a level of a guess not a fact. In the case of data sets like this a search for amplitude variability is hopeless. No phases and errors are given in our paper neither for the frequency solution for the subsets nor for the finally accepted frequency solution because it would suggest that the presented mathematical description of the light variation of 57 Tau is ready for theoretical modelling.

The multi-frequency analysis of 57 Tau was performed with the MUFRAN programme (Kolláth [1990]). MUFRAN (MUlti FRrequency ANalysis) is a collection of methods for period determination, sine fitting for observational data and graphics display routines. At each step, all previous frequencies were computed and successive prewhitening was avoided.

   
4.1 Subsets

As a first step, the different data sets were analysed separately. Of course the shorter time period covered by these independent data sets does not allow the precise value of the frequencies to be determined or closely spaced frequencies to be resolved. We are not going to discuss the frequency analyses of subsets in detail, but the frequencies detected from the independent subsets or different combinations of data sets, with their amplitudes, are summarized in Table 4. Those frequencies, which are identified also in the unprewhitened spectrum of the subsets and maybe more free of bias than the others, are marked by asterisk.

We should include some general remarks. First, it is very encouraging that for different data sets, and spectral window patterns, several similar frequencies are found. As a consequence of the shortness of the independent data sets, the frequencies are not properly resolved around 18.2 and 20.4 c/d. Depending on the data set one or two dominant peaks are seen in this frequency range but the dominant ones are not always the same. Several frequencies are common to each sample, although in some cases, it is hard to decide between values that differ by $\pm (1-2)$ c/d. This is a typical result for the discontinuous data sets. However, some guidelines exist that can help to distinguish between the aliases. It is obvious that not all of the peaks situated around 18 c/d are real frequencies, because there are amplitude variations in the observed light curve from cycle to cycle. Such variations could be better described by beat of frequencies further away from each other.

   
4.2 The whole data set

As a last step of the analysis, all the available data from 1981-1995 were analysed together. We should emphasize that the solutions are obtained from properly sampled spectra. Regarding the longest time base, the spectra were calculated for each 4 c/d wide range separately, with 90001 points. Figure 5 displays the spectral window, the original and residual amplitude spectra of 57 Tau.

For easier graphical representation we show compressed spectra, i.e. only the maximum values are plotted in each 0.0022 d^-1 wide bins of the spectra. Contrary to undersampled spectra the frequencies and amplitudes of the peaks are undisturbed after compression.

 

 

Table 5: Finally accepted frequency solution for 57 Tau. ^* Marks the frequencies that can be localized in the unprewhitened spectrum

Frequencies	Ampl. in V	Ampl. in B
c/d	mmag	mmag
f1 0.16208	1.27	1.27
f2 0.65745*	1.26	1.20
f3 0.80250*	1.80	2.33
f4 1.11933	1.07	0.83
f5 7.22323	1.00	1.23
f6 14.16131	0.95	1.08
f7 16.73835	1.06	1.07
f8 17.25689*	2.07	2.24
f9 18.21986*	1.41	1.32
f10 20.21810	1.35	1.27
f11 20.44054*	1.24	1.51
f12 24.55519*	1.41	1.62

The best mathematical representation of the pulsation spectrum of 57 Tau for the entire (1981-95) time-base, supposed a static pulsational arrangement, is given in Table 5. Amplitudes are given for both y & b colours. The residual spectrum in Fig. 5. is obtained after prewhitening with the finally accepted frequency solution.

   
4.2.1 Low frequency domain

In principle, low frequency terms can be an artifact introduced by small misalignements between different subsets (instrumental effects). Such kind of low frequency terms can be hardly cleaned by limited number of frequencies. For 57 Tau four frequencies were accepted in the low frequency domain. The two frequencies at 0.8025 and 0.6575 c/d are the most certain. Although the difference between the two frequencies is near to a value of an alias peak in the spectral window, the two frequencies can be independently found in the spectrum. After prewhitening with these two, another frequency at 0.805 c/d appeared but a prewhitening with a frequency at 0.162 c/d resulted a better residual spectrum. This frequency is not involved in Table 4 since it has not appeared in most of the subsets. The frequency at 1.119 c/d, similar to that obtained from the Chinese data set (see Table 4), appeared only in the last stage of the analysis. The steps of prewhitening can be found in Paparó et al. ([1999]). The low frequency part of the residual spectrum in Fig. 5 seems to be completely clean. After prewhitening with all frequencies a peak at 3.02141 c/d with 0.69 mmag amplitude appeared. The fact that the same periodicity appeared in the analyses of the comparison stars, as a result of small errors in the determination of atmospheric extinction, confirms that the previous prewhitening process removed real power.

   
4.2.2 p-mode region

Two or three frequencies seem to be excited around 17.2 or 18.2 and one or two around 20.4, although in most subsets only two of these frequencies can be localized in the unprewhitened spectrum. The frequenices accepted in the final solution are the result of many trials but should not be regarded as definite, as solutions in the case of the best-studied $\delta$ Scuti stars (even the integer part can be ambiguous). The frequencies at 7.2 and 24.5 c/d do not appear to be disturbed by other frequencies but the proper cycle counting (+1 c/d alias) is questionable. The frequencies at 14.1 and 16.7 c/d may be influenced by the previous steps of prewhitening.

Although the amplitudes of frequencies are extremely low, all of them past the significance test discussed earlier. The residual spectrum in the p-mode region is not completely cleaned. The spectrum shown in Fig. 5 (bottom panel) has a significant peak at f₁₃ = 29.8386 c/d, however an inspection of the fit indicates large amplitude fluctuations of this mode. Because of the small signal to noise ratio, it is impossible to obtain time-dependent amplitudes for this oscillation. From amplitude fitting we could not even determine whether this mode exists permanently in the data. However the phase of the oscillation can give valuable information.

Figure 6: The Fourier-phase diagram for the frequency at 29.83 c/d value. The well-defined phase value over the entire dataset represent the coherent behaviour of the periodicity as mentioned in the text

To check whether this oscillation is present coherently in the whole data set, we calculated the Fourier phase diagram (Paparó et al. [1998]) for this frequency (Fig. 6). To avoid errors due to other periodicities we first prewhitened the data with all 12 other frequencies. The Fourier phases are definitely between 0 and $\pi$, i.e., the oscillations should be coherent over all observational time spans (there is only less then half a period shift during 150000 periods).

If the 29.838 c/d frequency, with stationary amplitude, is added to the accepted set of solutions, a poorer overall fit to the data is obtained, and asymmetric cycles are created. This frequency was therefore omitted from the finally accepted solution.

Figure 7: Fit of the light curves for 57 Tau with the finally accepted frequencies from Table 5. The date are given as HJD 2400000+. The overall fit of the observed light curve is satisfactory in those parts where the stationary pulsational behaviour of 57 Tau dominates

A natural test of the quality of frequency solution is given by the fitting of the observed light curves. In Fig. 7 the fitted light curves are shown using the finally accepted frequency solution. Before we comment on the fit, we find it noteworthy that not only are there very low amplitudes of each excited mode (the amplitude ratio of the dominant modes for $\theta$ Tucanae and 57 Tau is 7.5 - Paparó et al. [1996]) but there are complex light variations too. Specifically, these features are remarkable where only one cycle has "extremely'' large amplitude both in minimum and maximum compared to the neighbouring cycles. Such dramatic changes in amplitude from one cycle to the other could hardly be explained by a missing frequency of a static pulsational arrangement.

In our conclusion the overall fit of the observed light curve is satisfactory in those parts where the stationary pulsational behaviour of 57 Tau dominates. The accepted frequency solution can yield the phase of minima and maxima in those parts. In some cases, the amplitudes of the synthetic light curves do not fit the observed one, where both lower or higher amplitudes occur. These cases suggest that more frequencies very close to the values used in the fit might be also excited. These frequencies could cause discrepancies only in the amplitudes not in phase.

   
4.3 Single unusual cycles

The light variation of 57 Tau tends to resemble a slowly varying sinusoidal when S/N is large enough. It is hard to say anything about nights when the amplitude of light variation does not exceed a S/N of 1. On some nights, however, single unusual cycles either with larger amplitude than the surrounding cycles or with asymmetric shape occur.

We focus the readers attention to the most remarkable cases, although similar events could be happening more often. In Fig. 8 those cycles are shown where the S/N ratios are high and variations are large in amplitude or asymmetrical in shape. On night HJD 2444918 and 2446764 the single cycles have larger amplitude than the previous and consecutive cycles. On the other two nights, at HJD 2445339 and 2450001, the cycles look like a typical cycle of an RR Lyrae star, the ascending branch is very steep and the descending branch is less steep.

Figure 8: Single unusual cycles of 57 Tau. The panels are marked as HJD 240000+. A physical process, gaining large enough observable size, from time to time, seems to be localized in the single unusual cycles

We should emphasize the following. As Fig. 8 shows the cited single unusual cycles were obtained by different (well-experienced) observers with different photometers on different sites. It is therefore extremely unlikely to be an instrumental effect. Each case was observed near to meridian, hence an explanation pertaining to the potential large-airmass differential observations is also ruled out. Although the amplitude of the general fit is exceptionally wrong for such cycles, they are still in phase with the fit. The previous and following cycles are well-fitted for these nights, and only the single cycles behave in an unusual manner. A physical process, gaining large enough observable size, from time to time, seems to be localized in the single unusual cycles.

A check for a possible connection between the frequency at 29.83 c/d and the unusual behaviour of the single cycles was carried out. Although the inclusion of this frequency marginally improves the fit for the single unusual cycles, it is hard to establish any definite connection because its effect is tiny.

It is well-known that a comprehensive study of a pulsating variable star, especially if the frequencies are not properly resolved, is not as convincing as, for example, a multi-site WET campaign. To support our finding of single unusual cycles in 57 Tau, the paper on CD $-24^\circ$ 7599 (=XX Pyx) observed in a WET campaign, published by Handler et al. ([1996]) is noteworthy. The data are excellent quality, the general fit is extremely good. However, the cycle at 9.85 in their Fig. 1. (see in Fig. 9 on larger scale) seems to be similar to what we see in 57 Tau. Although it is explained in the paper mentioned above as a consequence of the large airmass, both before and after(!) the cycle, the amplitude is small and the fit is the same quality as it is near to the meridian for except the unusual single cycle. Of course, such a tiny effect can be localized much better if it occurs more often in a star.

Figure 9: Unusual cycle of XX Pyx observed in a WET campaign. Both before and after(!) the cycle, the amplitude is small and the fit is the same quality as it is near to the meridian for except the unusual single cycle