2 Light curve analysis and the long-term oscillation

Table 1 lists the type-3 stars in the SMC and LMC where two photometric periods were found while Fig. 1 shows some examples of light curves. MACHO light curves were also inspected to confirm the long-term variations and search for colour variability. MACHO identifications are also provided along with the OGLE-II names in Table 1. In general, the long-term periodicity was evident from the first inspection of the light curve. The period P₂ in these cases was calculated using algorithms based on the F (variance ratio) statistics and fourier transform giving consistent results. The results shown in Table 1 reveal that the long-term variations have periods between 140-960 d. The period error was calculated as the half width at half maximum of the periodogram peak. Trial and error tests showed that these are upper limits for the error and the accuracy of P₂ in Table 1 is likely better than shown by a factor 2 at least. After finding P₂, we applied a non-linear least squares fit to the data of the type:

$$I = I_{0} - A_{I} {\rm {sin} (2 \pi (HJD-HJD_{0})}/P_{2})$$ (1)

where I₀, A_I and HJD₀ were parameters to be determined while the period P₂ remained fixed. The half-amplitudes A_I found were between 0.03 and 0.26 mag. The residuals to the best fit were examined with the same period searching routines employed for the analysis of the long-term oscillation. The surprising result was that in all cases listed in Table 1 we find a second period P₁, in the range of 2-16 days, which is listed along with their error in Table 1. For non-eclipsing stars this error was found in the same way that for P₂, but for 5 eclipsing stars the accuracy was usually much better and the error was estimated interactively.

We also studied the MACHO data for all the variables in our sample, constructing the b, r and b-r light curves, and searched for periodicities. We confirmed the periods found on the basis of the OGLE-II database, and found that for all cases the long-term oscillation is redder at maximum (Fig. 2). In general, the colour variations are of very low amplitude (Table 1).