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6 Variable stars analysis: The minimum entropy method

Because of poor weather conditions and unexpected instrumental problems the 1998 campaign was finished slightly earlier than planned in August 1998. The final number of observations including both campaigns was less than originally expected: a total of 76 good quality nights with an average of 35 observations per window. Even if the data is good enough for the discovery of variables, this has limited our analysis of the variables found (see below).

When we finished all the analysis with the Class32 program, we had finally obtained data from 124976 objects in the 12 databases. The AlarmFlag of each object was checked and the stars that presented a significant variation (sufficient number of observations, reasonable photometric errors, large magnitude variations, etc.) selected. The total number of objects per window and number of variables found are in Table 5.

Table 5: The final number of objects and variable stars found in each window

Window& $n^\circ$\space of stars& $n^\circ$\s...
 ...454& 23\\ LU& 8630& 20\\ LV& 9680& 26\\ Total& 124976& 479\\ \hline\end{tabular}

Among the variables we have found (see the complete listings in Appendix B), a major subset shows hints of periodic variations. However, if we do not have clear evidence to say that the star is periodic, its classification is highly compromised. As stressed before, the main problem is that we do not have as yet chromatic/spectral information about the objects. Since the variable classes are defined mainly by the spectral characteristics (especially in the case of aperiodic variable stars), the study of the individual stars given in the Appendix B is a natural and necessary next step. It should be remarked that the temporal covering of our data is not (generally speaking) broad enough to affirm whether a given star that does not show periodic variations is actually aperiodic.

In the case of the stars with hints of periodic light curves it is important to estimate the period (or periods) for a tentative classification. Several algorithms to perform period calculation exist, of which the Fourier method is the most popular. However, as is well-known, the Fourier method works with equally-spaced points from a temporal series, which is never the case for astronomical measures. Many authors (see, for example, Cuypers 1987) have developed a modified Fourier analysis for non-equally-spaced points. These works have generally resulted in computational time-consuming algorithms. In our analysis we have adopted a new method (the minimum entrophy method), developed by Cincotta et al. (1995).

This method is based on the minimization of the information entropy for the true period(s). The rigorous mathematical formulation can be found in Cincotta et al. (1996), for example. We will briefly describe the method.

\includegraphics [width=8.8cm,clip]{ds8752f5.eps}\end{figure} Figure 5: S vs. pj diagram for the star LI471, a known W Virginis
\includegraphics [width=8.8cm,clip]{ds8752f6.eps}\end{figure} Figure 6: Light curve of the star LI471

Consider a temporal series (u(ti)) and the set of periods to test pj (j = 1,...,n). According to Cincotta et al. (1995) we have calculated the phase of the temporal series and created a unitary (normalized) plane $(\phi,u)$,where $\phi$ is the phase. For each trial period pj, the light curve has been constructed and distributed in $(\phi,u)$ plane, that is divided in an arbitrary number (N) of partitions. The next step was to calculate the probability ($\mu_{i}$) for a point to fall in one partition, dividing the number of points in each partition by the total number of points in the temporal series. Finally we have computed the entropy as
S = -\sum_{i=1}^N\mu_{i}\ln(\mu_{i})\end{displaymath} (8)
If the trial period is not the real one, the light curve points will be uniformly distributed in the plane $(\phi,u)$ and the entropy will be maximum. On the other hand, if the trial period matches is the actual period, the points will be limited to a small region of the plane and we will have a minimum of the entropy. Therefore, we expect that minima in the S vs. pj diagram correspond to the actual period(s).

The method works very well when the light curve is sufficiently sampled (empirically, $\geq 50$ points for the period determination to be strictly independent of the partition number (N)). This is not quite the case of our databases, although the actual number of points is not too low in some cases. Thus, the application of the method remains feasible, but noisier S vs. pj diagrams are obtained. On the other hand, the computing processing time is less time-consuming than the modified Fourier analysis and, in principle, we believed suitable for our extensive databases (see discussion below). Another well-known problem is the presence of the harmonics, which can cause some confusion when we have few points.

Simulations and comparisons between minimum entropy and modified Fourier analysis were performed by Cincotta et al. (1995). Their tests showed that the minimum entropy method is more efficient for the resolution of multiple periods than the Fourier analysis. However, the accuracy of the estimated periods are not known yet and we intend to collaborate with the development of the method with its massive application to our databases. A comparison between several methods to determine periods present in our data will be the subject of a future study.

For simplicity, we consider that the stars have only one period. We shall present some examples from our databases to see how this works and evaluate the results. As we say before, the method is independent of the partition number when the series is well sampled, this isn't our case and, for each star, the calculations were done for several configurations of parameters, varying the number of partition (N) and trial period interval. We considered a period as found when the minimum persisted for all tried configuration (the differences between minima in each configuration are in the less significative digits).

The star LI471 is a W Virginis previously identified, and belonging to GCVS (General Catalogue of Variable Stars, Khopolov et al. 1988), with an attributed 11.49 days period. The light curve of the star (with our data) can be seen in Fig. 6. Figure 5 shows the S vs. pj diagram for that star which is noisy as expected. Nevertheless, the minimum we have obtained by applying the minimum entropy method is 11.50 days, consistent with the catalogued value. This example shows that, in spite of the noise, accurate determinations are possible in an efficient form.

\includegraphics [width=8.5cm,clip]{ds8752f7.eps}
\quad \end{figure} Figure 7: S vs. pj diagram for the star LA1552, classified as
a Mira. The estimated period is 323.2 days

\includegraphics [width=8.5cm,clip]{ds8752f9.eps}\end{figure} Figure 8: S vs. pj diagram for the star BJ878, a RR Lyrae. The estimated period was 0.33 days

\includegraphics [width=8.5cm,clip]{ds8752f8.eps}\end{figure} Figure 9: Light curve of the star LA1552

\includegraphics [width=8.5cm,clip]{ds8752f10.eps}\end{figure} Figure 10: Light curve of the star BJ878

Besides the problem of the scarcity of the points, our photometric errors are quite large, which in turn provokes more noise in the diagrams. As a result of the analysis we have checked that the determination of small periods is easier and more reliable than the determination of large periods (like Miras). This fact is closely related to the definition of the phase (see Cincotta et al. 1995) which is inversely proportional to the period. To give an example, we discuss the star LA1552, not known before and preliminary classified as a Mira, which reflects this difficulty very well (the light curve can be seen is Fig. 9). Figure 7 shows the S vs. pj diagram. We estimate the period in 323.2 days, but in this case the method is more dependent on the number the partitions.

As a further example of the capability of the method to find periods smaller than 1 day, we show the case of BJ878, a RR Lyrae star already known and belonging to GCVS (Fig. 10). The catalogued period is 0.37 days and our best estimation was 0.33 days. Figure 8 shows the S vs. pj diagram, with a well resolved minimum. In fact, it is not impossible that the catalogued period has to be corrected if further studies confirm the present value.

Even in cases where periodicity is suggested by the data, we have not been able to determine period for all those candidate stars. Some of them have so few points that it was impossible to draw a reliable conclusion. These stars are indicated with "NC" (not calculated) in the nineth column in the catalogue presented in Appendix B. The stars that we believe to be periodic, but for which a period could not be reliably found, are indicated with "NF" (not found) in the same column.

\includegraphics [width=15.6cm,clip]{ds8752f11.eps}\end{figure} Figure 11: Light curves of the stars BG1159, LD610, BE1802 and LB1034

\includegraphics [width=15.5cm,clip]{ds8752f12.eps}\end{figure} Figure 12: Light curves of the stars LD315, BE1152, BG1275 and LV95

\includegraphics [width=15.5cm,clip]{ds8752f13.eps}\end{figure} Figure 13: Light curves of the stars LA2402, LA1925, BG801 and LR101

To perform a preliminary variable star classification, we have based our judgement on:

All stars positions were compared with the ones of the variable objects belonging to the GCVS and NSV (New Suspected Variables, Kukarkin et al. 1982) catalogues and to those of the SIMBAD database using WWW interface. Among the 479 stars that showed significant light variations found in our database, only 16 were already present in other catalogues (including the IRAS catalog (Beichman et al. 1988), the work of Hazen (1996) on the variables in NGC 6558 (in the center of BE window), the CCDM (Catalog of Double and Multiple stars) by Dommange (1983), besides the quoted GCVS and NSV catalogues). Therefore, 96.7% of the variables in the new databases were unknown until now.

Considering our limitations, we restricted the classification of our variables to four broad classes defined as:

The result of this analysis can be appreciated in the tenth column of the tables in Appendix B, which constitutes our very preliminary classification of the variable stars and serves as a starting point for more comprehensive studies of these objects.

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