2. Photometry

Three available sets of photometric data turned out useful for our work.

  
Figure 1: The light curve resulting from our observations of 1992: the magnitude difference 14 Lac - 2 And (top) is compared, in the same scale, to the difference between the comparison magnitudes HD 217101-2 And (bottom)

• The 516 values (in the instrumental UBV system) obtained by us at the Observatorio Astronómico Nacional (San Pedro Martir, BC, Mexico) during the above quoted campaign (September 1992). These differential measurements were performed using the 0.84 m telescope equipped with a photon counting photometer; the light detector was a dry ice cooled GaAs phototube and the adopted V filter was made up of two elements (2 mm GG595 + 2 mm BG18). The data have not been transformed to Johnson's system since, in view of the complex variability pattern expected in o And, we preferred to increase our time coverage rather than to spend time in observing standard stars. As comparison stars (in common with o And) we used 2 And, HD 217101 and 10 Lac; all the magnitude differences are given with respect to the reference star 2 And. Our large set of data allowed us, in order to do it, to shift all our comparison objects to the same ficticious magnitude with great accuracy. The light stability of these stars has been checked also removing them by turns from the data files: such tests resulted all in increasing the variance of the light curves of o And and 14 Lac. Considering the size of our observational field (10 Lac is about 8 to the SW of 2 And), the data have been processed using the technique developed by Poretti & Zerbi (1993) in order to deal with the variations in extintion coefficient during each night. The resulting light curve is shown in Fig. 1 (click here), interested people can get it in electronic form.
• The light curves in the DAO photometric system published in Hill et al. (1976), defined by 72 differential measurements in each of the filters (35), (44) and (55), comparable to the and V standard ones respectively.
• The UBV photometry performed from August to November 1969 at the Astrophysical Station of Serra La Nave (CT, Italy) of the Osservatorio Astronomico di Catania by M. Rodonò, processed by L. Mantegazza and mentioned in Mantegazza (1980). These observations (Mantegazza 1995) consist of 210 , 188 and 192 values.

The basic information about these light curves is presented in Table 1 (click here), where the white noise contained in each time series has been evaluated from the root-mean-square difference between closely consecutive data. The corresponding signal-to-noise ratio, obtained assuming this white noise value as representative of the whole noise, is given in decibels: fans of the classical astronomical units are reminded that .

 

1969 1971-1973 1992

No of measurements 210 188 192 72  516 

No of nights 32 57  16 

Total useful obs. time (hours) 86 76 70 48  102 

Baseline (days) 120 120 103 805  16 

Standard Deviation (mmag) 29.0 27.3 25.6 35.7  18.0  21.8  24.3 

White Noise (mmag) 14.6 15.9 7.4 9.1  5.2  4.6  4.7 

S/N ratio (dB) 4.7 2.8 10.4 11.6  10.4  13.2  14.1 

 

  
Figure 2: Non-periodic (top) and periodic (bottom) component of the ultraviolet light curve published by Hill et al. (1976) resolved by the LIN algorithm (Bossi & La Franceschina 1995). The periodic term is phased with a frequency of 0.09916 d ^-1

  
Figure 3: Frequencies of the non-sinusoidal periodic component observed in the light variations at different epochs. The length of the rectangles represents the duration of the observational seasons, the height the error bar on the frequency values

Working on the assumption of simple periodic variations, first of all we have to determine their frequencies in the different epochs. So all the light curves, except the (35) series published by Hill et al. (1976), have been analized with this object using the PDM method (Stellingwerf 1978): this procedure is our most effective tool for analysing highly non-sinusoidal periodicities. Hill's ultraviolet curve, which presents a combination of a strongly non-sinusoidal periodic component with a long term trend, has been subjected to a MPDM analysis (Bossi & La Franceschina 1995). As shown in Fig. 2 (click here), the LIN algorithm, presented in the quoted paper, has been able to resolve this time series into a non-periodic component and a periodic one with a frequency of 0.09916 d ^-1 . The results are summarized in Table 2 (click here) and shown in Fig. 3 (click here). The errors have been determined by means of the standard statistical approach assuming each time series to consist of a periodic part (with the addition of a long term trend in the case of (35)), described by a derived mean curve, and white noise. It is easy to verify the consistency of the frequencies derived from simultaneous observations at different wavelengths, while this frequency may or may not be constant at different times: the difference between the value obtained from the yellow curve of Hill et al. (1976) and the one derived from our V curve corresponds to more than 3. The significance of this gap depends on the just quoted assumption of simple periodic signals.

 

Epoch U or DAO(35) Freq. (d ^-1 ) B or DAO(44) Freq. (d ^-1 ) V or DAO(55) Freq. (d ^-1 ) 1969 1971 - 1973 1992

 

A stronger evidence supports the presence of changes in amplitude and shape of the light patterns with timescales of years.

We can describe effectively some features of this evolution by means of a Fourier decomposition of the examined light curves, assumed, in each of the three observational epochs, to be periodic with the previously determined frequency. In all seven time series, the basic frequency and its first harmonic appear enough to represent the periodic component of the signal. In Table 3 (click here) we show the behaviour of three meaningful parameters: their combined standard deviation , their amplitude ratio a₂/a₁ and their phase difference (for the last definition see e.g. Simon & Lee 1981).

 

Band 1969 1971 - 1973 1992 V or DAO(55) B or DAO(44) U or DAO(35)

a₂/a₁ V or DAO(55) B or DAO(44) U or DAO(35) V or DAO(55) B or DAO(44) U or DAO(35)

 

The growth of resulting from the first line of the table cannot be considered as an evidence of amplitude changes: the observed amplitude of the light variations is a wavelength-sensitive parameter (we can observe a greater variability in the yellow and ultraviolet measurements than in the blue ones) and the examined sets of data were produced in different epochs by means of different equipments. Nevertheless, it is interesting to compare this pattern with the observations performed in 1980 and 1981 by Garrido et al. (1983), which found no significant variation of the magnitude of this star.

  
Figure 4: Values of the amplitude ratio and of the phase difference between the basic frequency and its first harmonic resulting from the Fourier decomposition of the V and DAO(550 light curves represented in polar co-ordinates ()

The changes in the shape of the yellow (V or DAO(55)) light curve, described by a₂/a₁ and , are shown in Fig. 4 (click here), where the amplitude ratio and the phase difference are represented in polar co-ordinates with the respective errors. As we can verify observing Table 3 (click here), this behaviour, unlike the amplitude of the light curve, shows no significant dependence on the wavelength.

Further information is given by a frequency analysis of the magnitude changes performed dropping the monoperiodic restriction. The results, obtained using Vanicek's (1971) method and adjusting the outcomes by means of a simultaneous nonlinear least squares fit, are syntesized in Table 4 (click here). The values relative to the DAO(35) curve result from simultaneous fits, optional in Vanicek's algorithm, with a cubic polynomial, describing the above quoted long term trend, in addition to the sinusoids.

 

Epoch   Band

1969 U B V

1971-1973 DAO(35) DAO(44) DAO(55)

1992 V

 

The scanned frequency range allows us to exclude the presence, in all the available sets of photometric data, of detectable rapid changes besides the well-known light variations on a time scale of days: just this time scale, rather than its quite common variability pattern (see e.g. Balona et al. 1987, or van Vuuren et al. 1988), characterizes this interesting Be star.

Moreover, a cross-check of the frequency spectrum of our mexican data against the one, obtained through the same procedure, of the light variations simultaneously detected in o And reassured us once again about the constancy of our comparison stars: no common periodicity has been detected in both time series.

Finally, the data obtained in 1969 hold in particular our attention. Their interest, besides the appearence of a frequency instead of the basic one, lies in the ratio of this frequency to : and respectively in the U, B and V bands. The differences between these values and 3/2, corresponding to 12, more than 11 and 9 respectively, are meaningful without doubt. Therefore, we cannot consider the resulting light curve as strictly periodic. This fact, which might be interpreted as an indication of multiperiodic variations, describes at least a transient stage of the variability pattern of this star occurred during the first observational season. It may represent therefore a further indication of non-stationary behaviour, this time on a scale of months.