3 Analysis of the light curve

We made use of a period-finding package consisting of single-frequency Fourier and multiple-frequency least-squares techniques (Breger 1990). First, an amplitude spectrum of the whole data set was computed out to the Nyquist frequency ($\approx 100$ cycles/day). No variations with time scales shorter than 2 hours were found and therefore the analysis was restricted to frequencies smaller than 12 c/d.

In Fig. 1 one can readily see that two different kinds of variability are present: long-term variations with a time scale of days and variations with a time scale of several hours; this is quite similar to the behaviour of HD 35914 (Paper I).

We first searched for possible periodicities in the long-term variations, since these appear to dominate the light curve. No periodic signals were found, again similar to HD 35914. Therefore, we turned to the short-term variability.

However, at this point caution is warranted: to examine the faster variations, one first needs to filter out the long-term variability, e.g. by adjusting the nightly zeropoints. Since we never observed a full cycle of the short-term variations during a single night, this can generate artifacts. In particular, signals with periods longer than an observing night will be suppressed. We refer to Paper I for a more detailed discussion of this problem (and how to take care of it).

To start a search for a typical time scale present in the short-term variations, we begin with an estimate of its possible range. As can be seen from Fig. 1, we never covered a full cycle during a single observing night. Therefore, the time scale must be longer than 8 hours.

Keeping the limitations explained above in mind, we set all nightly mean magnitudes to zero and calculated an amplitude spectrum of these data. This is shown in Fig. 2. A peak at 2.69 c/d with its alias structure dominates. To check how much the zeropoint adjustments affect this analysis, we computed an amplitude spectrum of the longest runs (T>4.5 hours) only. Such a plot closely resembles Fig. 2, suggesting that the zeropoint adjustments of the short runs have negligible influence on our results.

 

Figure 2: The amplitude spectrum of the M 2-54 data after adjusting the nightly zeropoints

What is the underlying time scale of the short-term variations? Three peaks in Fig. 2 are interesting, namely those at 1.68, 2.69 and 3.69 cycles/day. The latter peak must be an alias, since it corresponds to a period of 6.5 hours. Therefore, we should have seen a full cycle of such variations in five of our runs, which is not the case.

Calculating single-frequency fits to our data (and adjusting the zeropoints accordingly) yields somewhat lower rms residuals for the 2.69 c/d variation compared to the 1.68 c/d time scale (11.6 mmag vs. 12.1 mmag). However, we do not dare to suggest that this is the correct frequency, since these residuals are considerably higher than the scatter per single data point estimated in Sect. 2. This can be due to two reasons:

• The variations are not strictly periodic.
• The long-term variability could not be completely removed from the data. Therefore, remaining trends can affect the analysis. This may also result in different amplitudes for different cycles - which is apparent in Fig. 1.

The first hypothesis can be checked by calculating (O-C) diagrams. However, since we did not observe too many light maxima or minima, such an analysis is not of much use. The remaining possibility is to calculate phase diagrams relative to the two frequencies under consideration. We did this and show the result in Fig. 3.

 

Figure 3: Upper panel: a phase plot of the adjusted M 2-54 data relative to a frequency of 1.68 c/d. Lower panel: the same, but relative to the 2.69 c/d frequency

The most interesting feature in Fig. 3 clearly is that the phase diagram relative to the 2.69 c/d frequency is much smoother. This also suggests that this frequency is more likely to correspond to the correct time scale of the short-term light variations of M 2-54.

To summarize the results of our light-curve analysis, the variability of M 2-54 is quite similar to that of HD 35914 (Paper I). Two time scales are present: apparently non-periodic variations of the mean magnitude with a time scale of days plus quasiperiodic variability with a time scale of several hours. The latter time scale is most likely about 8.9 hours, but a 14.3-hour modulation cannot be ruled out.