4. Data analysis
4.1. Star numbers
It is interesting to compare the number of detected stars in Wh band with those in V and R bands. The total number of stars with at least one Wh measurement is 2927; however, since the field was not exactly the same during all the nights, the detected stars are located in a region of IC1613 which is actually slightly larger than 3:077x3:077. Moreover, the number of detections depend on the seeing conditions and sky background (see Table 1). We have considered therefore the Wh frame taken in the same night of V and R frames, with similar seeing conditions. The resulting number of detections is 1217 for Wh, 687 for V and 827 for R. Fig. 5 shows the corresponding distribution.
4.2. Variable stars
Different criteria for the detection of variability were adopted for comparing the capabilities of the various methods, particularly in the difficult cases given by the uncertainties due to crowding problems.
Firstly we used the variability index J (Stetson 1996). For each star the pairs of observations were considered, each with a weight , where k indicates the pair of observations . If the time separation between two subsequent observations was less than about three hours hour, they were considered as a pair. When the weight was =1.0, while when , . In this way, longer sequences of closely spaced observations had larger weight than sequences with similar number of observations but largely separated in time. The index J was redefined in order to take into account how many times a given star was measured, , where is the total weight a star would have if measured in all the images (see Kaluzny et al. 1998). As expected (Stetson 1996), most of the stars have values which are close to zero. The adopted threshold, allowed the selection of 136 candidate variable stars.
For another test for variability we considered 1491 light curves consisting of not less than 34 points, i.e. corresponding to stars which can be identified in the majority of the frames. They were checked for variability by means of two different methods, that is, their variances have been compared with two different noise estimates. At first, the noise component of a light curve was defined as the least variance found among the fainter objects than the examined one: a star was regarded as variable if its light variance, computed without taking the 10% most scattered measurements into account, exceeded 10 times this level. The white noise component of each time series was evaluated also from the root-mean-square difference between closely consecutive data, i.e., in our case, between measurements performed during the same night. When the light variance was inconsistent (i.e. larger than 3) with this noise definition, the object was classified as a variable star. These approaches have to be considered as complementary. The first one is based on no more than a rough estimate of the noise, which doesn't depend only on the measured magnitude but also, e.g., on the crowding in the image. On the other hand, rapid variations with time scales of some hours may escape detection with the second method. Combining both approaches, more than 250 candidate variable stars were singled out for a further detailed analysis.
Finally, some time was also spent for analyzing the data set on a star-by-star basis. A simple program was developed which computed the variance reduction for the time series, identified the maximum peak in the power spectrum, and showed the changes of peak and variance reduction when taking off progressively the most deviating points from the time series. A good indicator of variability was the stability of the power spectrum peak, even when the variance reduction was not very significant. In this way it was possible to detect variable stars with relatively small amplitude.
From the comparison of the three approaches we got the indication that, for an uneven data sampling as in the present case, the automatic methods should adopt very low threshold levels in order to detect variable stars which have low amplitude/noise ratio: on the one hand, this low threshold level yield also a large number of candidates which turn out to be nonvariable stars, and on the other hand some variable stars were found with the star-by-star analysis below such thresholds.
The uncertainties in the analysis for the detection of variable stars and the determination of their periods are mainly related to the number of bad points (characterized by a large DAOPHOT estimated error) in comparison with the number of good points, and to the significant aliases produced by the data sampling. The bad points are produced essentially by two causes: crowding, which implies a bad identification of the stars related to the variable seeing conditions, and occasional slight deformations of one stellar image which is interpreted by DAOPHOT as two close stars. The average number of bad points which must be discarded for obtaining a reasonable time series is about 2 per star in the case of Cepheids, and the maximum number of discarded points is 6. In general, stars with not less than 24 data points have been considered.
The data sampling is such that often there are significant aliases. Fig. 6 shows the spectral window. As we can see there is a strong alias with a complex structure at 1 c/d, so that if is the true period, we should expect strong aliases at the frequencies and . Usually there are not significant problems for intermediate periods. Often a visual comparison of the data phased with the different altenatives is sufficient to solve the possible ambiguities. On the other hand the aliasing makes it difficult to discriminate between long periods (say d) and periods very close to 1 d ( d), even if it appears more reasonable to expect, at least on a statistical basis, that most of these stars are long period objects. Another relevant ambiguity regards the possible very short period Cepheids ( d), for which, due the generally small amplitudes and therefore the low , it is difficult to judge if the best phasing is for c/d or .
© European Southern Observatory (ESO) 1999
Online publication: August 25, 1999