3. Discussion

3.1. Period determination techniques

Period determination techniques are based on different approaches. Among the principal ones the following can be considered: cycle counting methods, Fourier analysis (which is based on adjusting to periodic functions) and phase minimization and maximum entropy method. The analysis of the variable stars currently presented was carried out at three different places but in all the same basic mathematical principle was considered, i.e., Fourier analysis and least squares fitting methods. The period determination analysis was carried out originally utilizing the MUFRAN (MUlti FRequency ANalysis) package developed at the Konkoly Observatory by Kollath (1990). The MUFRAN is a collection of methods for period determination and sine fitting for observational data and graphic routines for the visualization of the results. It is based both on a Discrete Fourier Transform (Deeming 1975) and calculates the FFT of each track using equidistant spacing by "extirpolation'' (see Press & Rybicki 1989) in a faster way. Least square fitting provides the amplitudes and phases of fixed frequencies with error estimates which provide a rough idea of the quality of the fit. Whitening can be done and further analysis of the residuals can be repeated.

The second method utilized was the MFF that has been utilized extensively by the Mexican group (see, for example, Peniche et al. 1989). This method follows a series of steps, the first of which is to determine, by a classical Fourier transform, a first guess frequency. This is input in the MFF that is capable of adjusting to periodic functions up to ten frequencies and provide a refinement to each frequency considering all simultaneously. Goodness of fit is evaluated numerically by means of several statistical parameters of which the correlation coefficient and error are the most commonly employed. Once one frequency is determined, prewhitening is done and the residuals are analyzed as before in the Fourier transform, but the analysis in the MFF method is done from the original data considering the increasing number of frequencies being determined. Computing limitations restrict the resolution of the number of steps in which each frequency is swept but, in the end, accuracies in frequencies better than a few thousands c/d are attained.

The third and fourth methods utilized were those developed by the variable star group of the Instituto de Astrofisica de Andalucia, Spain: MINFRE and AnaFre (Analisis de Frecuencias). MINFRE, as the aforementioned methods, is based in a Fourier transform in order to obtain a first guess frequency which is then refined by sweeping frequencies by least squares fitting, in an interval which contains it, putting special emphasis on those cases in which aliasing becomes a problem. The optimization criteria being the minimization of the residuals. The best frequency is used to prewhiten the data and the procedure is repeated to obtain a second frequency. The whole procedure is repeated until the noise level is reached. Then, with all the frequencies obtained, the best set of frequencies is used to simultaneously solve, by least squares, for all the amplitudes and phases. Due to the high power of the computer used, a very small frequency step is considered reaching limits of frequencies restricted only by the accuracy given by the time span of the observations. AnaFre is a computing program designed to carry out the spectral decomposition of experimental data in a graphic and interactive environment. The algorithm that carries out the spectral analysis can be selected at any moment and so far it consists of Fourier Transform or least square fitting and is able to set the frequency interval and the accuracy with which the spectrum is calculated. Once the spectrum of the frequency spectrum has been obtained, the program suggests a frequency and by least squares calculates the corresponding amplitudes and phases. As soon as the frequency has been accepted the process is repeated iteratively as long as the adjustment is significant. This is done by the minimization of the residuals by the Levenberg-Marquardt method which requires that the estimation of the coefficients be adjusted (frequency, amplitude and phase for all the determined frequencies) by a simultaneous least squares fit for all the frequencies (General Linear Least Squares). Levenberg-Marquardt algorithm provides the optimum coefficients as well as the covariant matrix utilized in the estimation of the error of the fit. All the interface, both graphic and that used for the spectral decomposition are carried out utilizing National Instruments' LabVIEW program.

All the data previously mentioned were also analyzed by the method developed by Breger (PERIOD, 1991) which can fit and improve multiple frequencies simultaneously without prewhitening and gives as output the best frequencies, amplitudes, phases and zero-point. In principle, since all the aforementioned methods follow the same principle, they should give the same results. In practice, however, complications sometimes arise due, mainly, to i) the quality and quantity of the data which translates in complications such as aliasing; ii) the resolution given by the sweeping steps in the frequency analysis and iii) the problem of frequency shift, which is important when a close frequency is present.

3.2. Results

The results obtained for each star are presented in the following section.

KW 284. In a recent paper Belmonte et al. (1994) write on the nature of this star. KW 284 was first reported to be a possible variable by Rolland et al. (1991) and has been listed in Rodriguez et al. (1994) with a very small amplitude of pulsation of 0.004 mag and a relatively large period of 0.149 d. However, Belmonte et al. (1994), unaware of its variability, used it as a reference star in their STEPHI IV network, but the analysis of their data led them to conclude that the oscillation frequency detected by Rolland et al. (1991) is very probably the 69.5 Hz detected in the STEPHI IV data so, according to them, KW 284 could be a long period Delta Scuti star located in the upper part of the Main Sequence. They conclude that more data is needed to establish its frequency spectrum with high accuracy.

Aware of this possibility of variability of KW 284, special emphasis was made to assure its constancy during the reduction of the observation seasons. A comparison on a nightly basis was done for KW 284 versus KW 150 whenever possible. The three analyses, MUFRAN, MFF and MINFRE, coincide. No signs of variability were detected in this star either in 1982 or 1985. No traces of the peak at 6.05 c/d determined by Belmonte et al. (1994) were found. Hence it can be concluded that, at least in the observed seasons, this star remained constant. There is a possibility, of course, that a long term variation exists because, as has already been mentioned, night means were subtracted from the data inhibiting the possibility of detection of such periods. Nevertheless, when the frequency analysis was made on the data of the problems stars reduced with the mean of KW 150 and KW 284 without subtracting the mean nightly value, the periodograms showed some peaks at low frequencies in a consistent manner, which might imply a long term variability of one of the reference stars. For this reason the light curves were obtained each night and the mean value of the difference of KW 150 and KW 284 was subtracted. The mean value of each night was subtracted from the data.

The problem of the dual findings on this star remains. We have not found short amplitude variations of KW 284 in short time scales, although the variability reported by Rolland et al. (1991) is unquestionable. More, KW 284 has been found, by the speckle observations (Mason et al. 1993; McAlister et al. 1989), as an occultation binary system. However, this binarity cannot account for the variations found. Special care must be taken in the future when monitoring this star since it has been proven to vary for some time

KW 204. After the study of Breger (1972a,b) in which its discovery was noted, an amplitude of variation of
0.07 mag and a short period of pulsation of 0.11 d were assigned to this pulsator. However, these values were assigned with only few observed nights. KW 204 was later observed by Gupta & Bhatanagar (1974) for one night. This same star, was also observed by Guerrero et al. (1979). (This paper will be, hereinafter, referred to as GMS). They reported observations made in 1975 and 1976 in Johnson's UBV colors and the analysis of their data led them to derive three frequencies: 7.14417, 7.17378 and 9.77704 c/d; with these frequencies and their ratios they concluded that this star is pulsating with one radial mode and one non-radial mode. However, their data was obtained in a time span of one year and with only thirteen nights of observations. Another complexity in their data is constituted by the spacing between consecutive points since their data was obtained in Johnson's three UBV filters so the time span between consecutive observations in one filter increases.

In 1979 Breger reanalyzed the data of GMS and his previously unpublished 1967/68 data of the same star, KW 204. A multiple-frequency non-linear least squares analysis which searches for up to three frequencies simultaneously led him to derive a set of three frequencies different from those previously derived by GMS, namely, 9.777, 7.881 and 5.957 c/d. He identified these frequencies with the second radial overtone, the first overtone and the fundamental mode, respectively; the period ratios of these new frequencies indicated radial pulsation. The lack of agreement between the interpretation of the modes of pulsation between Breger (1979) and GMS, along with the goal of studying Delta Scuti star variables in clusters was one of the motivations for observing this star in the first previously mentioned 1982 campaign.

The application of MFF to this GMS set of data gave a maximum at 9.78 c/d. Prewhitening of this frequency left a small indication of other frequencies at 7.2394 and 8.2418 c/d which were, practically, impossible to determine and which did not significantly increase the correlation coefficient. When MUFRAN was employed with this GMS data set, the frequencies derived were those of GMS, namely 9.7770, 7.1489 and 7.1788 c/d. However, if the three shortest nights were not included in the analysis, the result gave only the first one. Similar analysis was carried out for the 1982 season obtained jointly at the SPM and the Ege University observatories. This set is constituted of six nights, of which five are almost consecutive and one 18 days apart, at two observatories located at different longitudes which changes completely the one cycle per day pattern attained with observations obtained from only one site.

The application of the MFF program to this 1982 data set gave the following results: The frequency obtained from the Fourier transform, 9.78 c/d was refined to 9.780 c/d and subtracted from the data. Application of a second run of Fourier yielded a peak at another maximum at 8.75 c/d. Both were swept with the MFF method and converged to 9.779 and 8.755 c/d with a correlation coefficient of 0.93. The same 1982 data set was further analyzed without considering the data of the night of 2445077, which is too separated from the rest of the observed days and complicates the window pattern. The analysis carried out gave the same basic frequency of 9.780 c/d which was prewhitened. A second peak at 9.88 c/d was derived. Both frequencies converged to 9.781 and 9.887 c/d with a correlation coefficient of 0.942. The residuals were at the noise level with a small peak at 6.54 c/d.

The analysis of the 1985 data utilizing both the MFF and the MINFRE methods gave the frequencies of 9.7901, 12.9244 and 8.2475 c/d with a discrepancy in the middle one which was of 11.935 with the MFF method. The correlation coefficient gave R² of 0.765 and 0.748, respectively. Similarly, the frequency data sets of GMS and Breger (1980) and MFF (1982) with the 1985 data set gave correlation coefficients of 0.686, 0.699, and 0.672. The new codes of AnaFre and PERIOD gave, with data of the 1985 season correlation coefficients, R², of 0.785. In this sense one could conclude that the behavior of the star in 1985 is best described by these last frequency sets. Emphasis should be made that the number of frequencies in each case is not always the same.

Since several frequency sets have been obtained from different data sets, compiled in Table 7 (click here), what would be desirable now is to test all frequencies in all data sets. Since the correlation coefficient gives a definite mathematical parameter of goodness of fit and on the explanation of the behavior of the data with respect to the frequencies assumed, several runs in the MFF with the different data sets were done. In this sense, with the different data sets the different frequencies proposed gave the correlation coefficients, R², listed at the bottom of Table 7 (click here). An explanation of Table 7 (click here) makes it easier to understand. The heading of the columns lists either the author or the computing code. On the other hand, the correlation coefficients at the bottom provide the goodness of the fit for the several frequency sets involved; in parenthesis are the data sets considered. From this statistical indicator it can be concluded, for example, that the frequency set that best describes the behavior of the GMS data set is that of GMS; that the best frequency set in 1982 is that of Breger (GMS) whereas in 1985 is that of AnaFre (1985) or PERIOD. The set with the highest correlation coefficient has been written in bold characters. For these, the amplitude in thousands of magnitude for each frequency is given in parenthesis.

 

frequency GMS Breger MFF MFF AnaFre PERIOD MUFRAN(ampl)

(c/d) (1979) (GMS) (1982) (1985) (1985) (1985) MFF(ALL)

f₁ 7.14417 9.777 9.781 9.790 9.765 9.765 9.780

f₂ 7.17378 7.881 9.887 11.935 11.909 11.909

f₃ 9.77704 5.957 8.248 6.980 6.980

R²(1982) 0.090 0.929 0.9268 0.813 0.681 0.681 0.895

R²(GMS) 0.873 0.816 0.5094 0.253 0.451 0.451 0.747

R²(1985) 0.686 0.699 0.672. 0.761 0.785 0.785 0.637

R²(82-85) 0.617 0.724 0.740 0.218 0.590 0.588 0.762

R²(ALL) 0.427 0.545 0.706 0.202 0.448 0.464 0.780

 

From the analysis summarized in Table 7 (click here) and the previous discussion it becomes immediately clear that our understanding of this star, despite all the observations, is still weak. One frequency around 9.8 c/d maintains its presence in all the analysis and all the observed seasons. However, the presence of all the other frequencies is, apparently, either a function of the data analysis or rests in the true nature of the star. It should be kept in mind that none of the data has a high temporal sampling: in GMS because they observed it simultaneously in several filters and in the 1985 season (with only two nights) because a large number of stars were observed. The best coverage, in this sense, is that of SPM-Ege in 1982 and the results indicate that 9.78 c/d frequency is more plausible for describing the nature of the star.

It is interesting to note that although GMS and Breger (1980) interpret the pulsational nature of KW 204 in two radically different ways, both sets of frequencies fit the observations obtained in 1982. It is also remarkable that with only one frequency both seasons, GMS and 1982, can be fairly well explained. To test this last assumption a data set constituted of both 1982 and 1985 was built and analyzed with MUFRAN, AnaFre and PERIOD packages. All of them produced the single frequency 9.77910 c/d with residuals 0.007 mag. Prewhitening this frequency indicated the presence of another at 9.508 c/d but of a much lower amplitude. It is remarkable that only one frequency can explain such separate data strings. Encouraged by this result, the data set of GMS was added, for which it was necessary to establish a zero level of this set to homogenize the entire set, which was then analyzed with the aforementioned computing packages. All gave just one frequency of 9.78007 c/d, with a correlation coefficient of 0.7976, which explains the behavior of this star in a time span of 12 years. In fact, the different frequency sets of this whole data set gave R² much lower than that obtained for just one frequency. With the new extension, that of 1997, an exceedingly large time basis has been acquired, 22 yr (78622 cycles). The analysis of the data in this time interval in the MUFRAN package yielded a frequency of 9.7800672. This same frequency was obtained in the MFF method which gave a correlation coefficient of 0.7803, which although low, is obtained considering the whole data set. The predictions compared to the observations show a remarkable correct phasing in an exceedingly large time span, see Fig. 1 (click here).

  
Figure 1: Light curves of KW204. Dots, observed points; continuous line is the prediction with the frequency derived in the present paper. The time span is of 22 yrs

Up to now there have been ambiguities in the determination of the pulsational nature of KW 204. Since the interpretation of the modes of pulsation depends strongly on the period ratios of the frequencies found, it is exceedingly important to accurately determine such frequencies. It is a well- known fact (Petersen 1975) that if the period ratios are near P₀ /P₁ = 0.76 and P₁ /P₂ = 0.81, etc. one can deduce the presence of radial pulsation. If the ratios found do not fit this scheme, they are generally interpreted as non-radial modes. It is interesting to note that although GMS and Breger (1980) interpret the pulsational nature of KW 204 in two radically different ways, both sets of frequencies fit the observations obtained in 1982. It is also remarkable that with only one frequency all seasons can be fairly well explained.

Henry et al. (1977) have found that KW 204 is anomalous among the stars within the Praesepe cluster since it has an abnormally strong K line and slightly weak metal lines. Breger (1980) has also found that the amplitude of KW 204 appears variable on a time scale of years. According to Breger (1980) the 1975/6 fit shows no change in amplitudes during the year, but the amplitudes during 1967/8 were definitely smaller; following Bregers assertion, we have calculated the mean amplitudes for the new observed seasons. Then, the amplitudes have changed from 0.076 mag in 1967 to 0.083 mag in a time span of one year. When GMS observed it the amplitude was 0.047 mag; then the new values of the amplitude were of 0.070 mag in 1982 and of 0.050 mag in 1985. However, we have found that the amplitude changes drastically within a season if it is long enough as in the case of 1982.

KW 207. Before the last campaigns of the Delta Scuti Network (Breger et al. 1993) and the STEPHI IV (Belmonte et al. 1994; Pérez-Hernández et al. 1995), the previous available data of this star was scarce. It was first observed by Breger (1970) for two nights in 1968 and 1969. Later, Bossi et al. in (1977) (BGM) observed this star for four nights separated 35 days; furthermore, only two of these nights are longer than the assumed period, therefore we worked mainly with their longest nights, HJD 2443173 and 203. A periodogram analysis of the first three nights led them to derive a period of 0.0534d; they reported that the last and the longest night did not fit this frequency but they found more and different frequencies from the analysis of this night: a main peak at 0.059d, and another much less marked peak at 0.068d; however, the 0.0534d component seemed to have disappeared. Since they didn't report their observations, gross values of their photometry were obtained directly from their reported light curves. The obtained data of this star in 1982 consisted of two consecutive nights, separated from that of Bossi et al. (1977) by five years: therefore, a separate analysis of each season was carried out, and no attempt to phase lock was considered.

A periodogram of the two longest nights of the Bossi et al. (1977) data was carried out with MUFRAN. The data was obtained directly from their plots and the frequencies derived from this analysis yield 17.4456 and 13.3707 c/d. An analysis with PERIOD of the same data result in the following frequency set: 17.3116 and 13.5041 c/d which obviously present an aliasing problem. In order to discriminate which frequency set better fits the data, the correlation coefficient in MFF was determined. For MUFRAN it was of 0.851 whereas for PERIOD was 0.8497 implying numerically that the frequency set derived from MUFRAN adjusted better but by no means this result is conclusive since the data set is constituted of only two nights quite separated in time. Later, each frequency was tested in the MFF method with the 1982 data sweeping in a frequency interval of 17.0 to 19.0 c/d, i.e., an interval that covers all the frequencies found by the periodograms. The maximum peak was at 18.781 c/d.

More recently, the frequencies of this star were accurately determined in a three continent campaign with the participation of five observatories organized by Breger et al. (1993). The frequencies they determined were 19.76, 17.36, 16.69, 18.62 and 19.87 c/d above the restrictive signal/noise ratio < 4.0 and another, 23.91 c/d slightly below such criterion. However, attempts to fit the set of six-frequency or even only the four-frequency solutions to the older data were disappointing. It is in this sense that the data presented here, and obtained long ago, are important to verify the pulsational nature of the star. A slightly different set of frequencies was obtained by Perez-Hernandez et al. (1995) from the STEPHI IV campaign on this and on KW 323 (BN Cnc) during a three week, three-continental run. A set of six frequencies of 16.63, 16.865, 17.366, 18.636, 19.777 and 19.86 c/d was obtained, a set very close to that found by Breger et al. (1993).

The analysis carried out on the 1985 data gave, with the MUFRAN, MFF and the MINFRE methods, basically the same results. The frequencies determined were 19.817 and 18.616 c/d. Further analysis in the MINFRE gave a small peak at 21.046 c/d. However, the correlation coefficient R² was low, of 0.53 that could be due to the noise in the data or to the presence of the other reported frequencies that could not be obtained from the available data. As with KW 204, goodness for each frequency set was tested for all the available data and, again, the correlation coefficient R² was utilized for deciding which frequency set better describes the behavior of the star. The only available data sets in this case were those of 1982 with only one observatory, SPM, and the data obtained in the coordinated campaign between the Pisèstetö observing station and SPM. In each one the different frequency sets gave R² for 1982 and for 1985 listed in Table 8 (click here). In this sense, the set of frequencies that best describes the available data is that of PERIOD for the 1985, see Fig. 2 (click here) for the prewhitening procedure, or STEPHI IV for the 1982 data but none of them adequately describe the behavior of the star.

  
Figure 2: Periodograms of KW207 showing the effect of the prewhitening. Y axis is at the same scale and prewhitening steps increase from top to bottom. The last spectrum has reached the noise level

 

frequency BGM Breger STEPHI MFF MUFRAN MFF AnaFre (ampl) PERIOD

(c/d) (1977) (1993) IV (1982) (1985) (1985) (1985) (1985)

f₁ 18.72 19.76 19.777 18.786 19.815 19.816 19.798(3) 19.798

f₂ 16.95 17.36 17.366 9.784 9.790 9.788(3) 9.788

f₃ 14.70 16.69 16.630 16.892 16.893 16.881(2) 16.881

f₄ 19.60 18.62 18.636 18.637(2) 18.637

f₅ 19.87 19.860

f₆ 23.91 16.865

R²(1982) 0.797 0.852 0.867 0.786 0.789 0.805 0.805

R²(1985) 0.18 0.343 0.376 0.400 0.400 0.455 0.455

 

KW 323. The Delta Scuti star KW 323, like KW 207, was observed in the STEPHI IV campaign. The analysis of the data led Perez-Hernandez et al. (1995) to determine the set of frequencies of 263.6, 266.7, 297.8, 300.4 and 327.2 (Hz (corresponding, in c/d, to 22.75, 23.043, 25.730, 25.920 and 28.270 c/d). It was previously observed by Kovacs (1981) who derived a set of frequencies listed in Table 9 (click here) and by Breger (1970) who first reported it as variable. However, Kovacs (1981) utilized stars KW 385 and KW 284 as a references, the former was later found to be variable (Paparo & Kollath 1990). A similar analysis as with the previous stars, produced the results presented in Table 9 (click here).

 

frequency Kovacs STEPHI MFF AnaFre (ampl) MUFRAN PERIOD

(c/d) (1981) IV (1985) (1985) (1985) (1985)

f₁ 25.76213 22.750 25.762 25.760(3) 25.760 25.760

f₂ 22.87924 23.043 6.905 6.906(2) 6.909 6.906

f₃ 15.29977 25.730 24.971 24.971(1) 24.971 24.971

f₄ 17.88238 25.920 12.872(1) 12.872

f₅ 12.64201 28.270

f₆ 30.42958

R²(1985) 0.316 0.290 0.370 0.402 0.370 0.402

 

The correlation coefficient was again used to test the goodness of each frequency set; the numerical values of R² obtained are presented in Table 9 (click here). This implies that the set of frequencies of either AnaFre or PERIOD gives the best adjustment, but the fit is still rather poor.

KW 45. The coincidence of the results derived from the MFF and the MINFRE methods is remarkable. The first two frequencies derived are basically the same, of 24.99 and of 11.01 c/d. The correlation coefficient is still low, 0.54. Further sweeping with the MINFRE yields frequencies at 4.66, 16.67 and 9.44 c/d but of much lower amplitudes. The residual obtained considering the first two of the aforementioned frequencies are of 0.0035 mag, certainly noise level. These same results were obtained with the remaining of the frequency determination methods. The results obtained are presented in Table 10 (click here). From the correlation coefficient presented in the last line of Table 10 (click here), it can be seen that all the frequency sets fit the data equally poorly, but slightly better for the frequencies refined by PERIOD.

 

frequency MUFRAN MFF MINFRE AnaFre (ampl) PERIOD (c/d)

f₁ 24.963 24.967 24.99 24.957(9) 24.957

f₂ 27.903 26.909 11.01 26.902(5) 26.902

f₃ 9.101 9.103 4.66 9.102(3) 9.102

f₄ 16.67

f₅ 9.44

R²(1985) 0.554 0.560 0.411 0.566 0.563

 

KW 154. The analysis clearly shows the existence of two frequencies the first of which is determined to be around 17.024 c/d. The analysis made with the period searching methods gave the results presented in Table 11 (click here). In the frequency sets determined, two frequencies consistently appear, one around 17.03 and another at 16.30 c/d which have a low correlation coefficient of 0.37. The additional frequencies shown correspond to peaks of much lower amplitudes in the periodogram.

 

frequency MUFRAN MFF (ampl) AnaFre PERIOD

(c/d) (1985)

f₁ 17.038 17.024(4) 17.040 17.040

f₂ 16.3041 16.305(2) 16.300 16.300

f₃ 4.051(2) 2.025 2.025

f₄ 18.682(2)

R²(1985) 0.362 0.467 0.438 0.438

 

KW 445. A similar analysis with the above mentioned methods was carried out on the data of this star. This data consisted of the 1985 observations made at SPM and Piszkéstetö. Although the window function improved, the results are by no means conclusive. The frequencies obtained are shown in Table 12 (click here) for the sake of completeness, but in all cases the correlation coefficient was very poor. A new observation season is being planned to obtain better data which will allow the period determination of this star.

 

frequency MUFRAN MFF AnaFre (ampl) PERIOD

(c/d) (85)

f₁ 10.9914 20.488 20.488(2) 20.488

f₂ 25.8767 11.325 15.119(2) 15.119

f₃ 4.2912 15.124 26.863(2) 26.863

f₄ 11.326(2) 11.326

R²(1985) 0.077 0.183 0.330 0.330